Fundamentals of Accelerator Physics and Technology %5Bpresentation Slides%5D %282006%29 WW

Fundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

Simulation and Measurement LabSimulation and Measurement Lab

Overview of simulation software, Overview of simulation software, measurement equipment measurement equipment

and assignmentsand assignments

SorenSoren PrestemonPrestemon


Program layoutProgram layoutSimulation and Measurements

Soren Prestemon

Three simulation and three measurement assignmentsThree simulation and three measurement assignments

Final project: Design an accelerator Final project: Design an accelerator (for high energy physics or for synchrotron radiation)(for high energy physics or for synchrotron radiation)

…2 Days!…2 Days!

Simulations:1. Dipole/quadrupole fields

Code Magnet/Poisson2. Cavity modeling

Code Superfish3. Synchrotron radiation

Code SynRad

Measurements:1. Dipole or quadrupole fields

Hall probe2. Microwave cavity

Network analyzer3. Insertion device

Hall probe

+

+

+

2 Days

2 Days

2 Days

Note: the simulation and measurement lab is 30% of your grade


WorkgroupsWorkgroupsSimulation and Measurements

Soren Prestemon

Eight teams Eight teams –– labworklabwork is a group effort!is a group effort!• Labwork is designed as 2-day exercise (lab+simulation)

Each exercise will require a written reportLab reports for the first week (2 reports) are due Monday, Feb. 23;Lab reports for the second week are due before the final exam

The last 2-day exercise will consist of the design of a storage ring

1 1a-M2 1a-C3 1b-M4 1b-C5 2-M6 2-C7 3-M8 3-C

Key

Day\Group # 1 2 3 4 5 6 7 81/16/20051/17/2005 1 2 3 4 5 6 7 81/18/2006 2 1 4 3 6 5 8 71/19/2006 5 6 7 8 1 2 3 41/20/2006 6 5 8 7 2 1 4 31/23/2006* 7 8 5 6 7 8 5 61/24/2006 8 7 6 5 8 7 6 51/25/20061/26/20061/27/2006*

* previous lab assignements due

Design of a Storage Ring

Lab and Calculations Overview

1a1b234

Dipole Field measurementsQuadrupole field measurementsMicrowave cavity measurements

Dipole/Quadrupole calculationsDipole/Quadrupole calculationsCavity simulation

Design of a storage ringInsertion device measurments Insertion device radiation properties

Computer labMeasurement lab


Lab groupsLab groupsCourseIntroduction

Jeffrey Ball, ORNL, ae Molly Scannell, BNL, bs Khalimov Mirkomal, Arifov Institue of Electronics, gs Anthony Bavuso, Jefferson Lab, bsee

Hannes Bartosik, Tech. University of Vienna, gs Jonah Weber, LBNL, bsee Danielle Sanzone, SLAC, bap

Ekaterina Danilova, ORNL, ms Nicola Pozzobon, University of Pisa , gs Heritier Makamizile Mbo, Congo-Fr. Electric Company, gs

William Chase, Brookhaven National Lab, bsp Lukas Jagerhofer, Tech. University of Vienna, gs Lynn Garren, Fermilab, phd

Paul Cummings, Embry-Riddle Aeronautical U., ug Muhammad Jamil, Konkuk University, gs Everette R. Martin, Fermilab, bsee

Laurie Elizabeth (Elisa) Dowell, Naval Research Lab, msrs Kristine Ferrone, Brookhaven National Laboratory, bsaAndres Gomez Alonso, U. Politecnica de Catalunya & CERN, gs

Sammie Garvin, George Mason University, bs Robert Hensley, Embry-Riddle Aeronautical University, ug Artur Paytyan, Fermilab, ms

Mohammad Adil Khan, Kyungpook National University, gs Valentina Previtali, Genova University & CERN, gs Eric Tse, SLAC, bap

3

2

84

7

1

6

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Introduction to lab exercises:Introduction to lab exercises:Dipole/Dipole/quadrupolequadrupole exerciseexercise

Simulation and Measurements

Soren Prestemon

Learn about design and purpose of magnetic elementsLearn about design and purpose of magnetic elements

The magnetic elements form the backbone of a storage ring:The magnetic elements form the backbone of a storage ring:•• Dipoles steer the beam, bending it on a circular path;Dipoles steer the beam, bending it on a circular path;•• QuadrupolesQuadrupoles focus the beam, essential to keep the electrons focus the beam, essential to keep the electrons from divergingfrom diverging

Simulations:Dipole and quadrupole magnetic

system designGoal: understand excitation curves,

saturation effects, purpose of poles and yokes, field harmonics, beam steering and focusing using the code Magnet

Measurements:Dipole or quadrupole field measurements Goal: Learn to assemble a model magnet

and a Hall probe; measure spatial field profiles, excitation curves; understand current and current density, integrated fields, and beam steering or focusing as a function of current


Introduction to lab exercises:Introduction to lab exercises:Accelerating cavity exerciseAccelerating cavity exercise


Soren Prestemon

Learn about RF cavity design and measurementsLearn about RF cavity design and measurements

Accelerating cavities serve:Accelerating cavities serve:•• in booster rings, to bring electrons up to nominal in booster rings, to bring electrons up to nominal operating energy, operating energy,

•• in the main ring to compensate for radiation energy in the main ring to compensate for radiation energy loss (e.g. synchrotron radiation)loss (e.g. synchrotron radiation)

Simulations:RF cavity simulationGoal: understand electric and

magnetic fields that can exist in a cavity; find resonant modes and frequencies, distinguish modes from harmonics

Measurements:Pillbox cavity measurements Goal: Estimate base resonant frequency

from geometry of a cavity; understand fields excited by two antennas installed in the device; measure resonant curve and impact of geometry on cavity tune


Introduction to lab exercises:Introduction to lab exercises:Accelerating cavity exerciseAccelerating cavity exercise


Soren Prestemon

Learn about insertion devices for synchrotron radiationLearn about insertion devices for synchrotron radiation

Insertion devices characterize third generation SR sources:Insertion devices characterize third generation SR sources:•• Provide intense SR tailored to specific scientific experimentsProvide intense SR tailored to specific scientific experimentsThey also serve in damping rings to reduce They also serve in damping rings to reduce emittance emittance

Simulations:Synchrotron radiation generationGoal: understand the radiation

characteristics associated with bending magnets, wigglers, and undulators, including spectral range and photon beam polarization

Measurements:Field measurements and characterization

of a wiggler magnet Goal: Using a Hall probe, understand

field characteristics and resulting radiation properties of a wiggler; evaluate impact of a wiggler on the electron beam (e.g. steering and focusing)


Introduction to lab exercises:Introduction to lab exercises:Storage ring designStorage ring design


Soren Prestemon

Use knowledge learned during the course to design a ringUse knowledge learned during the course to design a ring

There are a large number of design choices in the design of a There are a large number of design choices in the design of a storage ring; the scientific purpose serves to dictate the storage ring; the scientific purpose serves to dictate the optimal parameter choicesoptimal parameter choices

Design process:Choose to design either a SR storage ring or a HEP machineChoose basic design point: electron or proton energy, ring diameterChoose realistic ring elements (dipole field strengths, quadrupole fields, etc)Incorporate elements discussed during the course (particle sources,

accelerating structures, chromatic aberration correction, vacuumcomponents, etc)

Simulate performance using program BeamOptics: plot betatron functions, perform particle tracking, modify design to yield reasonable lifetime


Simulation and Measurement LabSimulation and Measurement LabGoalsGoals

• The goal of the lab is to provide a hands-on appreciation for the design, fabrication, and measurements associated with some key accelerator components– The lab questions should be addressed as well as

possible, but above all we want you to understand the issues involved

– You are free to do further measurements, play with design modifications, etc

– An additional computer exercise (extra credit!) on quadrupole focusing can be addressed in your spare time – you would find it useful for the final project

… and ask questions!

1


Lecture No. 10Lecture No. 10

Injection & ExtractionInjection & Extraction

Fernando Fernando SannibaleSannibale

2


IntroductionIntroductionInjection & ExtractionF. Sannibale

•• Every accelerator complex Every accelerator complex that includes a storage ring that includes a storage ring

requires an injection system. requires an injection system.

•• With increasing complexity, With increasing complexity, when more than one ring is when more than one ring is present, extraction systems present, extraction systems make their appearance and make their appearance and more injection systems are more injection systems are

added. added.

CERNCERN

•• Systems used for injection Systems used for injection can be used as well for extraction can be used as well for extraction

by simple “ mirror reflection” .by simple “ mirror reflection” .

•• In fixed target experiments, In fixed target experiments, where the beam is extracted and where the beam is extracted and

sent to a target, the extraction sent to a target, the extraction systems can assume different systems can assume different

characteristics.characteristics.

3


The Typical Injection SectionThe Typical Injection SectionInjection & ExtractionF. Sannibale

matching elementsmatching elements

Septa magnetsSepta magnets

Fast kickerFast kicker

Storage RingStorage Ring

TransferlineTransferline

•• MatchingMatching : for an efficient injection, the optical functions in the trans: for an efficient injection, the optical functions in the transferfer--line line and in the storage ring must be the same at the injection point.and in the storage ring must be the same at the injection point.

•• Septum MagnetSeptum Magnet : Special magnet with a “ thin” wall that allows to place the : Special magnet with a “ thin” wall that allows to place the magnet close to the storage ring orbit. Can operate in DC or in magnet close to the storage ring orbit. Can operate in DC or in pulsed mode.pulsed mode.

•• Fast KickerFast Kicker : It is the pulsed element that gives the final kick that puts t: It is the pulsed element that gives the final kick that puts the he injected beam on the storage ring orbit. Its pulse must last forinjected beam on the storage ring orbit. Its pulse must last for less than a less than a

ring revolution period for avoiding kick the beam again. ring revolution period for avoiding kick the beam again. •• In some injection schemes, a slow orbit bump localized in the sIn some injection schemes, a slow orbit bump localized in the septum eptum

region, brings the beam closer to the septum wall.region, brings the beam closer to the septum wall.

4


Injection MatchingInjection MatchingInjection & ExtractionF. Sannibale

Matching is fundamental for protons and ionsMatching is fundamental for protons and ions, because of the , because of the absence of damping.absence of damping.

Any injection error is immediately exported to the storage ring Any injection error is immediately exported to the storage ring generating generating emittanceemittance growth.growth.

Matching is less critical for electrons and positrons because daMatching is less critical for electrons and positrons because damping mping washeswashes--out any injection error within the acceptance of the storage rinout any injection error within the acceptance of the storage ringg

By matching at the injection point the optical and the dispersioBy matching at the injection point the optical and the dispersion n functions between the functions between the transferlinetransferline and the storage ring, one ensures a and the storage ring, one ensures a “ smooth” transition for the beam from the injector to the storag“ smooth” transition for the beam from the injector to the storage ring.e ring.

Anyway, for some specific application a good matching is importaAnyway, for some specific application a good matching is important also nt also for electrons and positrons. For example, for electrons and positrons. For example, toptop--off operationoff operation (quasi(quasi--

continuous injection) in synchrotron light sources and lepton continuous injection) in synchrotron light sources and lepton colliderscollidersrequires a good matching in order to minimize the perturbation (requires a good matching in order to minimize the perturbation (noise) noise)

that the injection transient can generate during users’ data takthat the injection transient can generate during users’ data taking.ing.

5


Septum MagnetSeptum MagnetInjection & ExtractionF. Sannibale

DAΦΦΦΦNE Frascati

•• Special care in the design must be Special care in the design must be used for avoiding field leakage that used for avoiding field leakage that

will affect the beam orbitwill affect the beam orbit

•• Septa can be in or out of Septa can be in or out of vacuum, DC or pulsed.vacuum, DC or pulsed.

6

DAΦNE

Krasnykh SLAC


Fast KickersFast KickersInjection & ExtractionF. Sannibale

Normally, the kicker pulse duration must be shorter than a revolNormally, the kicker pulse duration must be shorter than a revolution ution period. Typical kickers pulses have ~ 100 ns duration. New machiperiod. Typical kickers pulses have ~ 100 ns duration. New machines nes

such as the ILC and other special applications are asking for such as the ILC and other special applications are asking for challenging kickers with few ns pulse duration.challenging kickers with few ns pulse duration.

LNLS-Campinas

SPEAR 3

or discrete elements PulseForming Network (PFN)

DistributedPFN

7


Single Turn InjectionSingle Turn InjectionInjection & ExtractionF. Sannibale

••The transfer matrix from the septum to the kicker position is giThe transfer matrix from the septum to the kicker position is given by:ven by:

KickerKicker

Septum

Septum

θ

µµ= π/2= π/2= π/2= π/2= π/2= π/2= π/2= π/2Optimum phase advanceOptimum phase advance

( )

( ) ( ) ( )

′

−−++

−

+=

′ S

S

KK

S

SK

SKSK

SKSS

K

K

K

x

x

a

a

x

x

µµββ

ββµααµαα

µββµµββ

sincoscossin1

sinsincos

By imposing thatBy imposing thatat the kicker at the kicker xxKK = 0= 0

ϕβ

αµtan

cot=

+−=′ S

S

SS xx

ϕ

xxSS is defined by the required stay clear at the septum position. is defined by the required stay clear at the septum position.

Now using these values at the septum, one Now using these values at the septum, one can calculate the angle at the kicker position:can calculate the angle at the kicker position:

θµββ

tansin

1 =−=′ S

KS

K

xx

In order to “ place” the beam on the ring reference orbit, the anIn order to “ place” the beam on the ring reference orbit, the angular kick gular kick must be equal to must be equal to ––θθθθθθθθ. . Note that the minimum Note that the minimum θ θ θ θ θ θ θ θ is obtained when is obtained when µµµµµµµµ = = ππππππππ/2/2

xxSS

8


Stacking BeamStacking BeamInjection & ExtractionF. Sannibale

•• If multiple injection in the same bucket is requiredIf multiple injection in the same bucket is required ((stackingstacking), the ), the previous scheme can modified as follows:previous scheme can modified as follows:

θ θ

ππππππππ betatronbetatron phase advancephase advance

Kicker 1Kicker 1Kicker 2Kicker 2

π/2π/2π/2π/2π/2π/2π/2π/2 phase advancephase advance

•• The new kicker “ preThe new kicker “ pre--kicks” the kicks” the storedstored beam so that when it will pass beam so that when it will pass through Kicker 1 (simultaneously with the injected bunch) it wilthrough Kicker 1 (simultaneously with the injected bunch) it wil l be placed l be placed

back on the nominal closed orbit.back on the nominal closed orbit.

9


OffOff--EnergyEnergyInjection SchemeInjection Scheme

Injection & ExtractionF. Sannibale

•• By using a lattice with nonzero horizontal dispersion in the seBy using a lattice with nonzero horizontal dispersion in the septum area a ptum area a special injection scheme can be used.special injection scheme can be used.

No fast kicker is required. Because of the damping requirement, No fast kicker is required. Because of the damping requirement, such a such a scheme can be used only for electrons and positrons. scheme can be used only for electrons and positrons.

•• The injected beam will move on dispersion orbits for few dampinThe injected beam will move on dispersion orbits for few damping times g times until the radiation damping will bring it at the nominal energy until the radiation damping will bring it at the nominal energy merging with merging with

the stored beam.the stored beam.

SeptumSeptum

SxOnOn--energy beam trajectoryenergy beam trajectory

ηInjInj x

p

p=

∆

0

0≠η

A particular offA particular off--energy beam trajectoryenergy beam trajectory

Injx

SInj xx ≈

Several rings use or successfully tested the offSeveral rings use or successfully tested the off--energy injection schemes. energy injection schemes. For example APS, HERA, CESR, …For example APS, HERA, CESR, …

10


Electrons, ProtonsElectrons, Protonsand Heavy Ionsand Heavy Ions


•• There is a fundamental difference between the two cases of lighThere is a fundamental difference between the two cases of light particles t particles (electron and positrons) and of heavier ones (proton, ions, ...)(electron and positrons) and of heavier ones (proton, ions, ...)..

Stored Beam

Injected Beam

•• Electrons and positronsElectrons and positrons benefit from benefit from synchrotron radiation dampingsynchrotron radiation damping. The . The injected beam within few damping times oscillates down to the stinjected beam within few damping times oscillates down to the stored beam ored beam

merging with it. merging with it. LiouvilleLiouville theorem is not violated because synchrotron theorem is not violated because synchrotron radiation is a nonradiation is a non--Hamiltonian phenomenon. The equilibrium distribution of Hamiltonian phenomenon. The equilibrium distribution of

the stored beam is usually the stored beam is usually gaussiangaussian..

•• For For protons and ionsprotons and ions LiouvilleLiouville theorem does not allow the “ merging” of the theorem does not allow the “ merging” of the beams and one needs to use special schemes that allow to fill thbeams and one needs to use special schemes that allow to fill the whole e whole

available transverse acceptance. The final distribution is “ irreavailable transverse acceptance. The final distribution is “ irregular” .gular” .

11


MultiturnMultiturn InjectionInjectionfor Protons and Ionsfor Protons and Ions


•• The typical scheme uses a The typical scheme uses a fast orbit bumpfast orbit bump in the septum area in in the septum area in combination with a lattice tuned on a combination with a lattice tuned on a resonant resonant betatronbetatron tunetune::

•• New beam is injected in the same bucket every revolution periodNew beam is injected in the same bucket every revolution period. .

•• The bump is properly shifted between the injected bunches. The bump is properly shifted between the injected bunches.

•• While the resonant tune generates a constant rotation in the While the resonant tune generates a constant rotation in the transverse phase space.transverse phase space.

•• By selecting the proper combination of tune and “ speed” of the By selecting the proper combination of tune and “ speed” of the bump, it bump, it is possible to fill the phase space in an almost uniform way (beis possible to fill the phase space in an almost uniform way (beam am

“ painting” ).“ painting” ).•• For example, for a fractional For example, for a fractional

horizontal tune of 0.25:horizontal tune of 0.25:•• The first of such schemes was The first of such schemes was

used in 1953 at the used in 1953 at the COSMOTRON Cyclotron in COSMOTRON Cyclotron in

BrookhavenBrookhaven

x′

x

x′

x1

x′

x12

x′

x

13

2

x′

x1

4

3

2

x′

x15

4

32

x′

x12 6

5

4

3

x′

x1

23

4

5

6

•• Longitudinal painting is also possible by properly changing theLongitudinal painting is also possible by properly changing the energy energy of the injected beam every injection cycle.of the injected beam every injection cycle.

12


Charge Exchange InjectionCharge Exchange InjectionInjection & ExtractionF. Sannibale

Stored protonsStored protons

Stripping foilStripping foilStripping foilStripping foil


HH--


stripped protonsstripped protons

Stripping foilStripping foil

Bump MagnetBump Magnet

On 1963, G.I. On 1963, G.I. BudkerBudker and G.I. and G.I. DimovDimov at at NovosibirskNovosibirsk conceived a new conceived a new multiple injection scheme involving Hmultiple injection scheme involving H-- ions:ions:

•• The stripping foil must be very efficient in removing the electThe stripping foil must be very efficient in removing the electrons from rons from the negative ions (~99%) and at the same time the perturbation tthe negative ions (~99%) and at the same time the perturbation the foil he foil

induces on the stored proton beam should be as small as possibleinduces on the stored proton beam should be as small as possible..•• Carbon and aluminum ~ 10 to 100 Carbon and aluminum ~ 10 to 100 µµµµµµµµm thickness are typically used.m thickness are typically used.

•• Stripping is aStripping is a--non Hamiltonian phenomenon so non Hamiltonian phenomenon so LiouvilleLiouville theorem does theorem does not apply and multiple injections on the same phase space area anot apply and multiple injections on the same phase space area are re

allowed.allowed.

13


ExploitingExploitingBetatronBetatron ResonancesResonances


•• By setting the lattice on tune By setting the lattice on tune resonancesresonances, one can generate , one can generate “ islands” in the phase space. “ islands” in the phase space.

The example shows a 4The example shows a 4thth order order resonanceresonance

Simulation by M. Giovannozzi

•• In fact, In fact, resonancesresonances and and phase space islands have phase space islands have

been efficiently exploited for been efficiently exploited for both injecting or extracting both injecting or extracting

the beamthe beam

•• ResonancesResonances are usually are usually dangerous and carefully dangerous and carefully avoided in designing and avoided in designing and operating a storage ring.operating a storage ring.

•• Anyway, there are exceptions. Anyway, there are exceptions.

14


Resonant SchemesResonant SchemesInjection & ExtractionF. Sannibale

Simulation by M. Giovannozzi

•• The 4The 4thth resonance shown in the resonance shown in the previous viewgraph can be used for a previous viewgraph can be used for a

resonant extraction scheme for resonant extraction scheme for example.example.

•• The simulation shows a slow bump The simulation shows a slow bump bringing the beam close to the septum bringing the beam close to the septum

wall and a fast kick extracting the wall and a fast kick extracting the beam from the storage ring to other beam from the storage ring to other

side of the septum wall into the side of the septum wall into the transferlinetransferline..

•• Schemes like this are studied for Schemes like this are studied for multimulti--turn extractionturn extraction as required from as required from some fixed target applications. In the example, because of the 0some fixed target applications. In the example, because of the 0.25 tune, .25 tune, the islands rotate 90 deg on every turn and are extracted in fouthe islands rotate 90 deg on every turn and are extracted in four turns. r turns.

The central core is then extracted by the fast kicker. The central core is then extracted by the fast kicker.

•• Injection schemes exploiting Injection schemes exploiting betatronbetatron resonancesresonances, have been , have been successfully tested in few storage rings. In the AURORA ring frosuccessfully tested in few storage rings. In the AURORA ring from m Sumitomo for example, they use a halfSumitomo for example, they use a half--integer resonance scheme. integer resonance scheme.

15


ReferencesReferencesInjection & ExtractionF. Sannibale

•• G.H. ReesG.H. Rees , Injection, CAS , Injection, CAS -- 5th General accelerator physics course,5th General accelerator physics course, CERN 94CERN 94--0101

•• RendeRende SteerenbergSteerenberg , talk at AXEL , talk at AXEL –– 2005, March 17, 2005.2005, March 17, 2005.

16


Possible Possible HomeworksHomeworksInjection & ExtractionF. Sannibale

•• In designing a single turn injection system, we have the choiceIn designing a single turn injection system, we have the choice of of locating our kicker at two different positions, say A and B. At locating our kicker at two different positions, say A and B. At A the A the

betatronbetatron phase advance from the septum position is phase advance from the septum position is ππππππππ/2 and the /2 and the horizontal beta function is 3m. AT B the phase advance is 5/8 horizontal beta function is 3m. AT B the phase advance is 5/8 ππππππππ but the but the beta function is 7 m. Where would you place the kicker? If the rbeta function is 7 m. Where would you place the kicker? If the required equired

stay clear at septum imposes a distance between the injected beastay clear at septum imposes a distance between the injected beam m trajectory and the closed orbit of 2 cm, what is the required antrajectory and the closed orbit of 2 cm, what is the required angular kicker gular kicker

for storing the beam? The beta function at the septum is 5 m.for storing the beam? The beta function at the septum is 5 m.

•• If we want to upgrade the above system for stacking the beam byIf we want to upgrade the above system for stacking the beam byplacing a second kicker in the mirror symmetry point with respecplacing a second kicker in the mirror symmetry point with respect of the t of the

septum magnet. What would it be the required angular kicker in tseptum magnet. What would it be the required angular kicker in the he second kicker?second kicker?

•• If the RF frequency in our ring is 500 MHz and the harmonic numIf the RF frequency in our ring is 500 MHz and the harmonic number is ber is 400, what will be the maximum pulse duration that the kicker pul400, what will be the maximum pulse duration that the kicker pulse can se can

have?have?

1



Lifetime in Storage RingsLifetime in Storage Rings


2


IntroductionIntroductionLifetimeF. Sannibale

•• Particles are lost in accelerators because of the accelerator fParticles are lost in accelerators because of the accelerator f inite aperture.inite aperture.

•• The limiting aperture in accelerators can be The limiting aperture in accelerators can be physicalphysical or or dynamicdynamic ..The vacuum chamber defines the physical aperture, while transverThe vacuum chamber defines the physical aperture, while transverse and se and

momentum acceptances of the accelerator define the dynamical onemomentum acceptances of the accelerator define the dynamical one. .

•• Processes important for the lifetime include: Processes important for the lifetime include: elastic and inelastic residual elastic and inelastic residual gas scatteringgas scattering , , scattering with the other particles in the beamscattering with the other particles in the beam , , quantum quantum

lifetime for electrons and positronslifetime for electrons and positrons , , tune tune resonancesresonances , …, …

•• Many processes can excite particles on orbits larger than the nMany processes can excite particles on orbits larger than the nominal. ominal. If the displacement in the new orbit is larger than the apertureIf the displacement in the new orbit is larger than the aperture, the particle is , the particle is

obviously lost.obviously lost.

•• For most applications beam needs to be stored for as long as poFor most applications beam needs to be stored for as long as possible, so it ssible, so it is very important to contain the above effects within acceptableis very important to contain the above effects within acceptable values.values.

•• Such a requirement has important consequences on the design conSuch a requirement has important consequences on the design constraints. straints. For example, limiting the effects of the residual gas scatteringFor example, limiting the effects of the residual gas scattering pushes pushes

towards ultra high vacuum technologies.towards ultra high vacuum technologies.

•• Damping plays a major role in the electron/positron case. For pDamping plays a major role in the electron/positron case. For protons and rotons and heavy ions, lifetime is usually much longer but any perturbationheavy ions, lifetime is usually much longer but any perturbation will will

progressively buildprogressively build--up and generate losses.up and generate losses.

3


The Concept of LifetimeThe Concept of LifetimeLifetimeF. Sannibale

By defining the By defining the lifetime lifetime ττττττττ as:as:

•• From the last equation, one can see that the lifetime is defineFrom the last equation, one can see that the lifetime is defined as the time d as the time required for the beam to reduce its number of particles to required for the beam to reduce its number of particles to 1/1/ee of the initial of the initial

value.value.

•• Lifetime due to the individual effects (gas, Lifetime due to the individual effects (gas, TouschekTouschek, …) can be similarly , …) can be similarly defined. The total lifetime will be then obtained by summing thedefined. The total lifetime will be then obtained by summing the individual individual

contributions: contributions:

....1111

321

+++=ττττ

•• With this definition, the problem of calculating the lifetime iWith this definition, the problem of calculating the lifetime is reduced to the s reduced to the evaluation of the single lifetime components.evaluation of the single lifetime components.

( ) constantwithdttNdN ≡−= αα

τteNN −= 0ατ 1=

•• In a loss process, the number of particles lost at the time In a loss process, the number of particles lost at the time t t is proportional to is proportional to the number of particles present in the beam at the time the number of particles present in the beam at the time tt::

4


Is the Constant LifetimeIs the Constant LifetimeModel Accurate?Model Accurate?

LifetimeF. Sannibale

•• In fact, in most of the electron storage rings the lifetime actIn fact, in most of the electron storage rings the lifetime actually depends on ually depends on current.current.

•• Additionally, synchrotron radiation by hitting the vacuum chambAdditionally, synchrotron radiation by hitting the vacuum chamber transfers er transfers the energy required to the molecules trapped in the vacuum chambthe energy required to the molecules trapped in the vacuum chamber wall to er wall to

be released (be released (gas gas desorptiondesorption). ).

•• Because of this, for higher stored currents, the synchrotron raBecause of this, for higher stored currents, the synchrotron radiation diation intensity increases generating more intensity increases generating more desorptiondesorption and increasing the pressure and increasing the pressure in the vacuum chamber (in the vacuum chamber (dynamic pressuredynamic pressure). This will increase the scattering ). This will increase the scattering of the beam with the residual gas, with a consequent reduction oof the beam with the residual gas, with a consequent reduction of the beam f the beam

lifetime.lifetime.

•• The previous model, where the lifetime was assumed constant, isThe previous model, where the lifetime was assumed constant, is often too often too simple for describing the case of real accelerators.simple for describing the case of real accelerators.

•• Anyway, for reasonably small variations of the current, the conAnyway, for reasonably small variations of the current, the constant lifetime stant lifetime assumption is locally valid and it is widely used.assumption is locally valid and it is widely used.

•• In fact, the In fact, the TouschekTouschek effecteffect (discussed later), whose contribution dominates (discussed later), whose contribution dominates the losses in many of the present electron accelerators, dependsthe losses in many of the present electron accelerators, depends on current. on current.

When the stored current decreases with time, the losses due to When the stored current decreases with time, the losses due to TouschekTouschekdecrease as well and the lifetime increases.decrease as well and the lifetime increases.

5


Example of Lifetime in RealExample of Lifetime in RealAcceleratorsAccelerators


ALSALS

DADAΦΦΦΦΦΦΦΦNENE

Electrons Positrons

HERAHERA

PhotonPhoton--FactoryFactory

6


Dynamic Aperture andDynamic Aperture andMomentum AcceptanceMomentum Acceptance


•• Quite often in existing storage rings, the aperture is not limiQuite often in existing storage rings, the aperture is not limited by the ted by the vacuum chamber size.vacuum chamber size.

CornellILC-DR

•• In fact, In fact, nonlinearitiesnonlinearities in the fields of the magnets create resonance in the fields of the magnets create resonance “ islands” in the phase space that can capture particles with lar“ islands” in the phase space that can capture particles with large amplitude ge amplitude

orbits and bring them in collision with the vacuum chamber.orbits and bring them in collision with the vacuum chamber.

•• This effect creates a “ virtual” aperture for This effect creates a “ virtual” aperture for the machine which is usually referred as the the machine which is usually referred as the

dynamic aperturedynamic aperture

•• Due to their strong nonlinear nature, Due to their strong nonlinear nature, dynamic apertures can be calculated only dynamic apertures can be calculated only

numerically.numerically.

•• In the longitudinal plane, the In the longitudinal plane, the momentum acceptancemomentum acceptance is limited by is limited by the the size of the RF bucketsize of the RF bucket or by the or by the dynamic aperture for the offdynamic aperture for the off--momentum particlesmomentum particles. In fact, off. In fact, off--energy particles in energy particles in dispersivedispersiveregions can hit the dynamic aperture of the ring even if their regions can hit the dynamic aperture of the ring even if their

momentum difference is still within the limits of the RF acceptamomentum difference is still within the limits of the RF acceptance.nce.

7


Cross SectionCross Sectionof a Scattering Eventof a Scattering Event


•• In scattering processes it is useful to define as In scattering processes it is useful to define as cross section cross section σσσσσσσσ the the event rate per unit incident flux and per target particleevent rate per unit incident flux and per target particle..

•• For an interaction with cross section For an interaction with cross section σσσσσσσσ, the number of events per second , the number of events per second (event rate), in the rest frame of the particle group 2 for exam(event rate), in the rest frame of the particle group 2 for example, is given ple, is given

by:by:

( )( ) σσσφ IRIR VvnnVnvnNdt

dNN 12121121 ====

where where nn11 and and nn22 are the densities of the two groups of particles are the densities of the two groups of particles vv11 is the is the velocity of the particle group 1 and velocity of the particle group 1 and VVIRIR is the volume of the region where is the volume of the region where

the two particles interact.the two particles interact.

•• The equation above applies for uniform densities The equation above applies for uniform densities nn11, , nn2 2 and constant and constant σσσσσσσσ. . For the more general case where these quantitiesFor the more general case where these quantities depend on position, the depend on position, the

above expression must be replaced with:above expression must be replaced with:

=IRV

dzdydxnnvN 211 σ

•• Let us consider two groups of particles. Particles in the same Let us consider two groups of particles. Particles in the same group have group have same momentum and are distributed in uniform spatial distributiosame momentum and are distributed in uniform spatial distributions.ns.

8

Incident positiveparticles


Gas Lifetime:Gas Lifetime:Elastic ScatteringElastic Scattering


•• When a charged particle passes close to a residual gas moleculeWhen a charged particle passes close to a residual gas molecule, it is , it is deflected by the electric field of the molecule nucleus.deflected by the electric field of the molecule nucleus.

•• This phenomenon is a particular case of This phenomenon is a particular case of Coulomb scatteringCoulomb scattering and it is usually referred as and it is usually referred as

Rutherford scatteringRutherford scattering, after the name of the , after the name of the English scientist that first discovered it in 1911.English scientist that first discovered it in 1911.

•• Rutherford experiments were quite important Rutherford experiments were quite important because proved that the atom mass is not because proved that the atom mass is not uniformly distributed (Thomson model) but uniformly distributed (Thomson model) but

instead concentrated in a very small positively instead concentrated in a very small positively charged part of the atom, the nucleus. charged part of the atom, the nucleus.

•• The equation of motion for the problem can be solved showing thThe equation of motion for the problem can be solved showing that the at the trajectories of the scattered particles are hyperbolae (trajectories of the scattered particles are hyperbolae (KeplerKepler problem).problem).

•• In the process, the incident particle does not loose energy, soIn the process, the incident particle does not loose energy, so this kind of this kind of scattering is referred as scattering is referred as elasticelastic ..

•• In a storage ring, when a beam particle scatters with a residuaIn a storage ring, when a beam particle scatters with a residual gas molecule l gas molecule it undergoes to it undergoes to betatronbetatron oscillations. If the oscillation amplitude is larger than oscillations. If the oscillation amplitude is larger than

the ring acceptance the particle is lost.the ring acceptance the particle is lost.

9


Rutherford ScatteringRutherford ScatteringCross SectionCross Section


•• Rutherford calculated the differential cross section for the elRutherford calculated the differential cross section for the elastic scattering astic scattering of a charged particle with a nucleus:of a charged particle with a nucleus:

( ) ( ) ][2sin

1

24

14

22

20

MKSpc

ZeZ

d

d IncR

θβπεσ

=

Ω

where where ZZIncInc e e is the charge of the incident particle, is the charge of the incident particle, ZeZe is the charge of the is the charge of the nucleus, nucleus, ββββββββcc and and pp are the velocity and the momentum of the incident particles are the velocity and the momentum of the incident particles

and and θθθθθθθθ is the scattering angle. is the scattering angle. ΩΩΩΩΩΩΩΩ is the solid angle.is the solid angle.

•• In deriving the previous equation, screening effects of the atoIn deriving the previous equation, screening effects of the atom electrons m electrons and direct inelastic scattering with the atoms electrons were neand direct inelastic scattering with the atoms electrons were neglected glected

because small. Nucleus recoil has been neglected as well.because small. Nucleus recoil has been neglected as well.

θr

ϕ x

z

y

•• For small angles, the screening from the molecule electrons musFor small angles, the screening from the molecule electrons must be taken t be taken into account and for large scattering the nucleus finite size muinto account and for large scattering the nucleus finite size must be st be

considered. considered.

10




•• In the case of a beam of In the case of a beam of NN particles scattering on a residual gas molecule in particles scattering on a residual gas molecule in a storage ring, the accelerator aperture will limit the scatterea storage ring, the accelerator aperture will limit the scattered angle to some d angle to some

value value θθθθθθθθMAXMAX. For scattered angles larger than . For scattered angles larger than θθθθθθθθMAXMAX the particle will be lost.the particle will be lost.

•• If If nn is the gas molecule density, is the gas molecule density, AATT the beam transverse size, the beam transverse size, LL the ring the ring length, length, TT the revolution period and the revolution period and ββββββββcc the beam velocity, then:the beam velocity, then:

RmoleculesparticlesbeamGas

Ndt

dN σφ−=

•• By using the definition of cross section, the rate of losses isBy using the definition of cross section, the rate of losses is given by: given by:

L

c

A

N

TA

N

TTparticlesbeam

βφ == LnAN Tmolecules = θθσϕσσπ

θ

π

dd

ddd

d

d

MAX

R

Lost

RR sin

2

0 Ω

=ΩΩ

=

( )

−=

π

θ θθθ

βεπβπ

MAX

d

pc

ZeZcNn

dt

dN Inc

Gas 2sin

sin

42 4

22

20

( ) ( )][

2tan

1

4 2

22

20

MKSscatteringelasticgasforrateLoss

pc

ZeZcNn

dt

dN

MAX

Inc

Gas θβεπβπ

−=

11




760][

0TorrP

nMn =

•• The number of molecules for cubic centimeter, for a gas at 0 The number of molecules for cubic centimeter, for a gas at 0 ooCC and at 760 and at 760 TorrTorr, is given by the , is given by the Loschmidt'sLoschmidt's constantconstant nn00 = 2.68675 = 2.68675 xx 10102525 mm--33

( ) A

TIncTorr

Gas pc

ZeZPNcMn

dt

dN

εβ

βεπβπ 4

7604

22][

20

0

−≅

•• If we assume that our gas is composed by If we assume that our gas is composed by MM--atomic molecules and that its pressure is atomic molecules and that its pressure is

PP, then the density of the gas is:, then the density of the gas is:

•• For a ring with acceptance For a ring with acceptance εεεεεεεεAA and for small and for small θθθθθθθθ, , the maximum scattering angle at the scattering the maximum scattering angle at the scattering

point is:point is:( ) ( )ss

T

AMAX β

εθ =

•• For an For an estimateestimate, we can replace , we can replace ββββββββTT with its with its average value along the ring:average value along the ring:

T

AMAX β

εθ =

( )GastNN τ−= exp0

][4760

2

20

20

][

MKSZeZ

pc

nMcP T

A

IncTorrGas β

εββ

πετ

≅

•• Which used with the previous results gives:Which used with the previous results gives:

That integrated:That integrated:

with:with:

12


Gas Lifetime:Gas Lifetime:Inelastic ScatteringInelastic Scattering


•• In the inelastic scattering, the incident particles In the inelastic scattering, the incident particles lose energylose energy during the during the process.process.

•• We can distinguish two main phenomena:We can distinguish two main phenomena:

•• Gas Gas bremsstrahlungbremsstrahlung : the incident particle is : the incident particle is deflected by the molecule nucleus and because of deflected by the molecule nucleus and because of

the transverse acceleration, radiates a photon. the transverse acceleration, radiates a photon. This effect is important for relativistic particlesThis effect is important for relativistic particles

•• Atom excitationAtom excitation : the interaction brings the atom : the interaction brings the atom to ionization or into an excited state. to ionization or into an excited state.

The effect is important for non relativistic particlesThe effect is important for non relativistic particles

•• For both the processes, if the amount of lost energy is beyond For both the processes, if the amount of lost energy is beyond the the momentum acceptance of the ring the particle is lost. momentum acceptance of the ring the particle is lost.

•• The lifetime contribution due to inelastic scattering is calculThe lifetime contribution due to inelastic scattering is calculated following ated following the same steps used for the elastic case, replacing the cross sethe same steps used for the elastic case, replacing the cross section for the ction for the

elastic scattering with the sum of the two crosselastic scattering with the sum of the two cross--section terms for the inelastic section terms for the inelastic case. case.

13


Gas Gas BremsstrahlungBremsstrahlungLifetimeF. Sannibale

•• BremsstrahlungBremsstrahlung in accelerators is important for in accelerators is important for relativistic electrons and positrons.relativistic electrons and positrons.

( ) ( ) ( )[ ]

−

−+

−

+

−= 1112

011

2

00

20

2

13

2ln

3

41

3

4 κϕκϕκκϕκκκ

ακ

σE

ZEE

rZ

d

d B

( )κκκ

−=

00

20

311 100EE

cm

Z

functions screening

energy particleE

particlethebylostenergyenergy photon

constantstructurefinehc

e

radiuselectronclassicalmr

≡≡

≡≡

≡==

≡×= −

21

0

0

2

150

,

137

1

4

10818.2

ϕϕ

κπε

α

•• The differential crossThe differential cross--section was first section was first calculated by calculated by BetheBethe and and HeitlerHeitler::

Where:Where:

with:with:

•• For high relativistic electrons, the screening is maximum and tFor high relativistic electrons, the screening is maximum and the crosshe cross--section becomes (section becomes (complete screening casecomplete screening case):):

−+

−

+

−=

0

2

00

20

2

19

1ln

3

1209.51

3

44

EZ

EE

rZ

d

d B κκκκ

ακ

σ

nucleus

photon

14


Gas Gas BremsstrahlungBremsstrahlungLifetimeF. Sannibale

•• After some manipulations, a simple expression for the lifetime After some manipulations, a simple expression for the lifetime contribution contribution due to gas due to gas bremsstrahlungbremsstrahlung can be obtained:can be obtained:

∆−=

0

ln3

41

E

E

L

c A

RBremτ

•• Here, Here, ∆∆∆∆∆∆∆∆EEAA is the is the energy acceptanceenergy acceptance of the storage ring.of the storage ring.LLRR is is radiation lengthradiation length of the gas and is defined as the length required to of the gas and is defined as the length required to the particle to lose (1 the particle to lose (1 -- ee--11) of its energy when traveling trough the gas.) of its energy when traveling trough the gas.

+≅31

220

183ln4

9

21

ZnZr

LR

α

•• In a real accelerator, the residual gas is a combination of difIn a real accelerator, the residual gas is a combination of dif ferent ferent molecular species. Anyway, it turns out that the average molecular species. Anyway, it turns out that the average <<ZZ22>> over the over the different species is ~ 50 which is approximately the value for ndifferent species is ~ 50 which is approximately the value for nitrogen.itrogen.

••This allow to write with good approximation:This allow to write with good approximation:

( ) ][0][

1

ln

14.153

nTorrAhoursBrem PEE∆

−≅τ

15


Atomic ExcitationAtomic ExcitationLifetimeF. Sannibale

•• The differential crossThe differential cross--section for the atomic excitation contribution is very section for the atomic excitation contribution is very similar to the one for the similar to the one for the bremsstrahlungbremsstrahlung::

( ) ( ) ( )[ ]

−

−+

−

+

−= 2122

021

2

00

20. 1

3

2ln

3

81

3

4 κψκψκκψκκκ

ακ

σE

ZEE

rZ

d

d Exc

( )κκκ

−=

00

20

322 100EE

cm

Zfunctions screening≡21,ψψwith different:with different: and:and:

•• For the electron accelerator case, these differences make the cFor the electron accelerator case, these differences make the crossross--section section for atomic excitation much smaller than the for atomic excitation much smaller than the bremsstrahlungbremsstrahlung one.one.

2ZofinsteadZ 48 ofinstead

•• For extremely relativistic particles, the complete screen case For extremely relativistic particles, the complete screen case gives:gives:

−+

−

+

−=

0

2

00

20 1

9

1ln

3

2085.71

3

44

EZ

EE

Zr

d

d Exc κκκκ

ακ

σ

16


Gas Lifetime: Example ofGas Lifetime: Example ofVacuum RequirementsVacuum Requirements


•• We already saw that for We already saw that for electronselectrons in the approximation of in the approximation of <<ZZ22> ~ 50,> ~ 50, we we have for the gas have for the gas bremsstrahlungbremsstrahlung lifetime:lifetime:

•• In the same approximation, the inelastic gas scattering lifetimIn the same approximation, the inelastic gas scattering lifetime becomes:e becomes:

( ) ][0][

1

ln

14.153

nTorrAhoursBrem PEE∆

−≅τ

[ ]][

][

][

][2025.10

mT

mA

nTorr

GeVhoursGas P

E

βε

τ µ≅

•• Evaluating these expressions for the typical electron ring caseEvaluating these expressions for the typical electron ring case, one finds , one finds that the requirement on vacuum is for that the requirement on vacuum is for dinamicdinamic pressures of the order of the pressures of the order of the

nTorrnTorr..

17


TouschekTouschek EffectEffectLifetimeF. Sannibale

•• The The TouschekTouschek is the dominant effect limiting the lifetime in many of is the dominant effect limiting the lifetime in many of the modern electrons storage rings.the modern electrons storage rings.

•• Particles in the bunch are subjected to Particles in the bunch are subjected to betatronbetatron oscillations.oscillations.Coulomb scattering between the particles can transfer transverseCoulomb scattering between the particles can transfer transverse

momentum to the longitudinal plane.momentum to the longitudinal plane.•• If this extra momentum brings the two scattered particles beyonIf this extra momentum brings the two scattered particles beyond the d the

momentum acceptance of the ring, then the particles are lost. momentum acceptance of the ring, then the particles are lost.

•• This process is usually referred as the This process is usually referred as the TouschekTouschekeffecteffect after the Austrian scientist that discovered it.after the Austrian scientist that discovered it.

•• The first observation was done in The first observation was done in the early 60’s in the early 60’s in FrascatiFrascati at ADA, at ADA, the electronthe electron--positron accelerator positron accelerator conceived by conceived by TouschekTouschek and the and the

first ever built. first ever built.

19211921--19781978

ADAADA

18



•• The effect can be properly investigated in the center of mass The effect can be properly investigated in the center of mass system (CMS),system (CMS),where the particles are nonwhere the particles are non--relativistic.relativistic.

−=Ω θθβ

σ242

20.

sin

3

sin

44r

d

d Tousc

•• In this frame, the Coulomb scattering between two particles of In this frame, the Coulomb scattering between two particles of the same the same specie and with equal but opposite momentum specie and with equal but opposite momentum pp, is regulated by the , is regulated by the MöllerMöller

differential crossdifferential cross--sections:sections:

velocityCMSc ≡β

ψcosppS ≡•• In the CMS, the longitudinal component of the In the CMS, the longitudinal component of the momentum due to the scattering is:momentum due to the scattering is:

•• Which in the laboratory system becomes:Which in the laboratory system becomes: ψγγβγ cos~ ppEc

pp SSS =

−=′

where the ~ sign is a good approximation because the particles awhere the ~ sign is a good approximation because the particles are nonre non--relativistic in the CMS.relativistic in the CMS.

•• The last equation shows how the momentum transfer in the laboraThe last equation shows how the momentum transfer in the laboratory tory system is system is amplified by a factor amplified by a factor γγγγγγγγ..

ϕθ

ψp

s

yx

ϕψθ cossincos =ϕψψ ddd sin=Ω

19



•• If If γ γ γ γ γ γ γ γ ppss is larger than the momentum acceptance is larger than the momentum acceptance ∆∆∆∆∆∆∆∆ppAA, both the scattered , both the scattered particles are lost. And the condition for losing a particle becoparticles are lost. And the condition for losing a particle becomes:mes:

••The The MöllerMöller crosscross--section can now be section can now be integrated within this limits obtaining:integrated within this limits obtaining:

µγ

ψ =∆>p

pAcos

•• After some additional algebra and assuming After some additional algebra and assuming gaussiangaussian distributions, we distributions, we finally obtain the finally obtain the TouschekTouschek lifetime for a lifetime for a flat beamflat beam::

( ) ( )( )T

AXSYXTousch

Cpp

Ncr ζσσσσπγ

πτ 2

0233

20

.

1

4

1

∆′=

+−= µ

µβπσ ln1

1824

20

.

rTousch

( ) 5.178.1

1ln~1ln

2

112

−

−

−= −

∞

T

u

TTTT due

uu

uC

Tζζζ

ζζε

2

0

′∆

=X

AT p

p

σγζwithwith

andand

•• A similar equation can be obtained for the case of round beams.A similar equation can be obtained for the case of round beams.

and where the approximate expression can be used for and where the approximate expression can be used for ζζζζζζζζTT < 1.< 1.

20


Minimizing Minimizing TouschekTouschekLossesLosses


•• In many electron (and positron) storage rings, high current andIn many electron (and positron) storage rings, high current and small small emittancesemittances are usually required. This makes of the are usually required. This makes of the TouschekTouschek effect the major effect the major

responsible for particle losses in such rings.responsible for particle losses in such rings.

( ) ( )( )T

AXSYXTousch

Cpp

Ncr ζσσσσπγ

πτ 2

0233

20

.

1

4

1

∆′=

•• In most In most colliderscolliders, the energy is usually fixed, larger , the energy is usually fixed, larger emittancesemittances (and thus (and thus larger beam sizes) are welcome while the current should be as hilarger beam sizes) are welcome while the current should be as high as gh as

possible. Short bunches are preferred (hourglass effect).possible. Short bunches are preferred (hourglass effect).

Acceptanceˆ20

2

RFVp

pRF

A ∝∆

•• Depending on the application, a tradeoff between the different Depending on the application, a tradeoff between the different requirements requirements must be defined. must be defined.

•• In synchrotron light sources, higher beam energy and longer bunIn synchrotron light sources, higher beam energy and longer bunches ches (harmonic cavities) can be used, while (harmonic cavities) can be used, while emittancesemittances (and beam sizes) must be (and beam sizes) must be

small and the current must be high.small and the current must be high.

•• For all applications, the momentum For all applications, the momentum acceptance must be maximizedacceptance must be maximized

•• Dynamic apertureDynamic aperture

21


Quantum LifetimeQuantum LifetimeLifetimeF. Sannibale

At a fixed observation point along a storage ring, the transversAt a fixed observation point along a storage ring, the transverse motion of a e motion of a particle is sampled as a pure sinusoidal oscillation: particle is sampled as a pure sinusoidal oscillation:

( ) yxTtaxTTT ,sin =+= ϕωβ β

Ta β±

Usually, tunes are chosen in order to avoid Usually, tunes are chosen in order to avoid resonancesresonances. In such a situation at . In such a situation at a fixed a fixed azimuthalazimuthal position, a particle turn after turn sweeps all possible position, a particle turn after turn sweeps all possible

positions between the envelope:positions between the envelope:

In the presence of synchrotron radiation, photon emission randomIn the presence of synchrotron radiation, photon emission randomly changes ly changes the “ invariant” the “ invariant” aa and consequently changes the trajectory envelope as well.and consequently changes the trajectory envelope as well.

Cumulative photon emission can bring the particle envelope beyonCumulative photon emission can bring the particle envelope beyond the ring d the ring acceptance in some acceptance in some azimuthalazimuthal point and the particle is lost.point and the particle is lost.

The explained loss mechanism is responsible for the soThe explained loss mechanism is responsible for the so--called called transversetransversequantum lifetime.quantum lifetime.

Similar arguments apply also for the longitudinal plane and the Similar arguments apply also for the longitudinal plane and the longitudinal longitudinal quantum lifetimequantum lifetime can be defined as well.can be defined as well.

22


Quantum LifetimeQuantum LifetimeLifetimeF. Sannibale

•• Quantum lifetime was first estimated by Quantum lifetime was first estimated by BruckBruck and Sands:and Sands:

( )lifetimequantumTransverse

yxTAA TT

T

TDQ TT

,2exp 222

2

=≅ σσττ

( )lifetimequantumalLongitudin

E EADQ LL

22 2exp σττ ∆≅

eVE

U

Ve

Eh

EJ

U

VeF

Eh

EJE RF

C

LRF

C

L

E

A

81

01

0

01

02

2

1008.1

ˆ2

ˆ

2

×≅

−≈

=

∆ παασ

yxTE

where ETTTT ,

2

0

2 =

+= σηεβσ

For an For an isoiso--magnetic ring:magnetic ring:

•• Transverse quantum lifetime sets the minimum requirement for thTransverse quantum lifetime sets the minimum requirement for the e transverse aperture, while the longitudinal one defines the minitransverse aperture, while the longitudinal one defines the minimum mum

momentum acceptance necessary from the lifetime point of view.momentum acceptance necessary from the lifetime point of view.

•• Quantum lifetime very strongly Quantum lifetime very strongly depends on the ratio between depends on the ratio between

acceptance and acceptance and rmsrms size.size.Values for this ratio of 6 or little larger Values for this ratio of 6 or little larger

are usually required.are usually required.

23


Tune Tune ResonancesResonancesLifetimeF. Sannibale

•• Tune Tune resonancesresonances are carefully avoided in designing storage rings.are carefully avoided in designing storage rings.

•• In fact, particles trapped in a resonance can be quickly lost.In fact, particles trapped in a resonance can be quickly lost.Lower order Lower order resonancesresonances are usually more dangerous. are usually more dangerous.

(<~ 12(<~ 12thth for protons and <~ 4for protons and <~ 4th th for electrons)for electrons)

ntegersiimlandorderresonancemliml yx ,,≡+=⋅+⋅ νν

ALSALS

•• Anyway, imperfections, nonlinear effects and Anyway, imperfections, nonlinear effects and phenomena associated with momentum phenomena associated with momentum

diffusion can bring the particle on a resonance.diffusion can bring the particle on a resonance.

•• Examples of common tune shift effects in Examples of common tune shift effects in storage rings: storage rings: non linear non linear multipolemultipole terms in terms in magnetic fieldsmagnetic fields (tune shift on amplitude), (tune shift on amplitude),

beambeam--beam effects during collisionbeam effects during collision , , ““ wakefieldswakefields” (tune shift on current), ” (tune shift on current), inelastic inelastic

gas scattering combined with nonzero gas scattering combined with nonzero chromaticitychromaticity , …, …

•• The working point in the tune plane must be carefully selected The working point in the tune plane must be carefully selected in order to in order to minimize the impact of all such effects.minimize the impact of all such effects.

24


Remarks on the Lifetime forRemarks on the Lifetime forProtons and Heavy IonsProtons and Heavy Ions


•• In protons and heavy ions storage rings no damping is present. In protons and heavy ions storage rings no damping is present. As a As a consequence any perturbation on the particle trajectories buildsconsequence any perturbation on the particle trajectories builds--up and can up and can

eventually lead to the loss of the particle.eventually lead to the loss of the particle.

•• Other effects such as Other effects such as elastic gas scatteringelastic gas scattering , , molecule excitationmolecule excitation , , fluctuations in the fluctuations in the

magnetic and RF fieldsmagnetic and RF fields , , Coulomb scattering Coulomb scattering (intra(intra--beam scattering)beam scattering), …,, …,

add up to generate a lifetime of the order of add up to generate a lifetime of the order of hundreds of hours typically.hundreds of hours typically.

•• On the other hand, important loss mechanisms On the other hand, important loss mechanisms for electrons, become negligible for protons. for electrons, become negligible for protons.

These include for example, These include for example, TouschekTouschek and gas and gas bremsstrahlungbremsstrahlung scattering.scattering.

•• Quite often in Quite often in colliderscolliders, the interaction between the colliding beams, the so, the interaction between the colliding beams, the so--called called beambeam--beam effectbeam effect , becomes the main mechanism of losses., becomes the main mechanism of losses.

25


ReferencesReferencesLifetimeF. Sannibale

•• A. A. WrulichWrulich , Single Beam Lifetime, , Single Beam Lifetime, CAS CAS -- 5th General accelerator physics course, CERN 945th General accelerator physics course, CERN 94--0101

•• M. SandsM. Sands , The Physics of Electron Storage Rings. An Introduction, , The Physics of Electron Storage Rings. An Introduction, SLAC Report 121 UCSLAC Report 121 UC--28 (ACC) (1970)28 (ACC) (1970)

26


Possible HomeworkPossible HomeworkLifetimeF. Sannibale

•• Show how different lifetime contributions add up together for tShow how different lifetime contributions add up together for the total he total lifetime. Explain the physical meaning of the constant lifetime. Explain the physical meaning of the constant αααααααα (the inverse of the (the inverse of the

lifetime).lifetime).•• Estimate the lifetime for the DAEstimate the lifetime for the DAΦΦΦΦΦΦΦΦNE electron beam. Use the line in the NE electron beam. Use the line in the

lifetime plot in the examples viewgraph.lifetime plot in the examples viewgraph.•• Calculate the number of molecules per cmCalculate the number of molecules per cm3 3 for a gas of Nfor a gas of N22 at the pressure of at the pressure of

1 1 nTorrnTorr..

•• Estimate the lifetime due to elastic gas scattering for a 1.9 Estimate the lifetime due to elastic gas scattering for a 1.9 GeVGeV electron electron beam at a pressure of 1 beam at a pressure of 1 nTorrnTorr. Assume that the gas is mainly N. Assume that the gas is mainly N2 2 (Z=7), that (Z=7), that

the average ring beta function is 1.5 m and that the ring acceptthe average ring beta function is 1.5 m and that the ring acceptance is 10ance is 10--66 m. m. Remember that Remember that εεεεεεεε00=8.8543=8.8543 x x 1010--1212 F mF m--11..

•• For the same ring of the previous problem, calculate the lifetiFor the same ring of the previous problem, calculate the lifetime due to gas me due to gas bremsstrahlungbremsstrahlung for the case of 1% relative momentum acceptance.for the case of 1% relative momentum acceptance.

•• For the same ring, calculate also the For the same ring, calculate also the TouschekTouschek lifetime for a bunch current lifetime for a bunch current of 10 of 10 mAmA and average and average rmsrms beam sizes of 100 beam sizes of 100 µµµµµµµµm, 10 m, 10 µµµµµµµµm and 1 cm, for x, y and m and 1 cm, for x, y and

s respectively. Assume an average s respectively. Assume an average rmsrms value for x’ of 60 value for x’ of 60 µµµµµµµµradrad..

•• Finally, estimate for the same ring the longitudinal quantum liFinally, estimate for the same ring the longitudinal quantum li fetime when fetime when the longitudinal damping time is 10 ms.the longitudinal damping time is 10 ms.

27


Spherical CoordinatesSpherical CoordinatesAnd Solid AngleAnd Solid Angle


θr

ϕ x

z

y

θϕθϕθ

cos

sinsin

cossin

rz

ry

rx

===

θϕθ dddrrdzdydx sin2=

θϕθ ddd sin=Ω

πθθϕππ

4sin0

2

0

==Ω ddd

1



Collective effects.Collective effects.Single and Single and MultibunchMultibunch InstabilitiesInstabilities


2


IntroductionIntroductionCollective EffectsF. Sannibale

•• Charged particles in a beam mutually interact and interact withCharged particles in a beam mutually interact and interact with the the conductive walls of the vacuum chamber.conductive walls of the vacuum chamber.

•• These effects are usually referred as These effects are usually referred as collective effectscollective effects..

•• For example, we already saw that the For example, we already saw that the TouschekTouschek effect can generate losses effect can generate losses in electron storage rings. Other examples include space charge lin electron storage rings. Other examples include space charge l imiting the imiting the

minimum minimum emittanceemittance and the maximum current in proton storage rings, beamand the maximum current in proton storage rings, beam--beam effects reducing the luminosity performance in beam effects reducing the luminosity performance in colliderscolliders, …, …

•• Collective effects play a major role, quite often limiting the Collective effects play a major role, quite often limiting the final final performance of an accelerator.performance of an accelerator.

•• In designing high performance accelerators, collective effects In designing high performance accelerators, collective effects need to be need to be carefully taken into account and solutions for minimizing these carefully taken into account and solutions for minimizing these effects need effects need

to be adopted.to be adopted.

•• Additionally, collective effects make particles within the buncAdditionally, collective effects make particles within the bunch and h and between bunches “ communicate” , allowing for single bunch and between bunches “ communicate” , allowing for single bunch and multibunchmultibunch

instabilities. instabilities.

•• Solutions can be Solutions can be passivepassive, when in the design phase the parameters are , when in the design phase the parameters are chosen in order to contain collective effects, or chosen in order to contain collective effects, or activeactive where the accelerator where the accelerator

operates above instability threshold but operates above instability threshold but feedbackfeedback systems damp the systems damp the instabilities down.instabilities down.

3


SpaceSpace--ChargeChargeCollective EffectsF. Sannibale

We already saw how Coulomb scattering between two particles in tWe already saw how Coulomb scattering between two particles in the he beam can generate particle losses by the beam can generate particle losses by the TouschekTouschek Effect.Effect.

But Coulomb interaction is also responsible for the soBut Coulomb interaction is also responsible for the so--called called space space chargecharge effect. In this case, the generic particle in the bunch experieeffect. In this case, the generic particle in the bunch experiences nces the the collective collective Coulomb force due to the field generated by the charge of Coulomb force due to the field generated by the charge of

all the other particles in the bunch.all the other particles in the bunch.

Such fields, referred also as Such fields, referred also as selfself--fieldsfields, are quite nonlinear and their , are quite nonlinear and their evaluation usually requires numerical techniques.evaluation usually requires numerical techniques.

Anyway, by using the proper approximation, it is possible to obtAnyway, by using the proper approximation, it is possible to obtain ain analytical solutions that gives us some useful insights on the eanalytical solutions that gives us some useful insights on the effect.ffect.

4


TengTeng SolutionSolutionand the and the GaussianGaussian casecase

Collective EffectsF. Sannibale

( ) ( ) ,1

,1

00

ybab

Exbaa

E yx +=

+= λ

πελ

πε

•• By assuming a By assuming a continuouscontinuous (non(non--bunched) beam with bunched) beam with constant linear charge constant linear charge density density λλλλλλλλ and with a and with a stationary uniform ellipticalstationary uniform elliptical distribution in the transverse distribution in the transverse

plane, plane, TengTeng in 1960 found the following expression for the fields in 1960 found the following expression for the fields insideinside the the beam:beam:

( ) ( ) xbaa

cBy

bab

cB yx +

=+

−= λβπµλβ

πµ 00 ,

•• For a more realistic For a more realistic gaussiangaussian distribution in the transverse plane distribution in the transverse plane and for and for xx << << σσσσσσσσxx and and yy << << σσσσσσσσyy::

( ) ( ) ,2

1,

2

1

00

yExEyxy

yyxx

x σσσλ

πεσσσλ

πε +=

+= ( ) ( ) x

cBy

cB

yxxy

yxyx σσσ

λβπ

µσσσ

λβπ

µ+

=+

−=2

,2

00

•• For both cases the fields scale linearly with For both cases the fields scale linearly with x x and and yy, and:, and:

,, xyyx Ec

BEc

Bββ =−=

with with aa and and bb the ellipse halfthe ellipse half--axes, and the beam moving along axes, and the beam moving along zz with velocity with velocity ββββββββcc..

5


Effects of the SpaceEffects of the Space--ChargeChargeCollective EffectsF. Sannibale

( )BvEqF

×+=

•• Such space charge fields exert forces on the beam particles, whSuch space charge fields exert forces on the beam particles, whose ose intensities are given by the intensities are given by the LorentzLorentz Force:Force:

( ) ( ) ( )xqEcBEqF xyxx22 11 βλββ −∝−=−=

•• The last equations show that the The last equations show that the space charge forces become negligible for space charge forces become negligible for relativistic beamsrelativistic beams. .

•• Such a situation generates a Such a situation generates a betatronbetatron tune shift with currenttune shift with current for the particles for the particles in the core of the beam.in the core of the beam.

And using theAnd using theprevious relations:previous relations: ( ) ( ) ( )yqEcBEqF yxyy

22 11 βλββ −∝−=+=

•• They also show that for the nonThey also show that for the non--relativistic beam, the forces are repulsive relativistic beam, the forces are repulsive and proportional to the distance from the beam center. and proportional to the distance from the beam center.

•• This is equivalent to a defocusing This is equivalent to a defocusing quadrupolequadrupole in both planes with strength in both planes with strength proportional to the current in the beam. proportional to the current in the beam.

•• For the nonFor the non--core particles the linear dependence of the force breaks and core particles the linear dependence of the force breaks and numerical calculations are required for the evaluation of the spnumerical calculations are required for the evaluation of the space charge ace charge

effects.effects.

6


Vacuum Chamber Effects:Vacuum Chamber Effects:Image ChargeImage Charge


•• Particle beams requires ultra high vacuum pressures, that can bParticle beams requires ultra high vacuum pressures, that can be achieved e achieved inside special metallic vessels called inside special metallic vessels called vacuum chambersvacuum chambers..

•• For the Maxwell equations, the electric field associated with tFor the Maxwell equations, the electric field associated with the particle he particle beam, must terminate perpendicularly on the chamber beam, must terminate perpendicularly on the chamber equipotentialequipotential

conductive walls.conductive walls.

•• In the lab system the beam electromagnetic field of a relativisIn the lab system the beam electromagnetic field of a relativistic particle is tic particle is transversely confined within an angle of ~transversely confined within an angle of ~ 1/1/γγγγγγγγ (where (where γ γ γ γ γ γ γ γ is the particle energy is the particle energy

in rest mass units).in rest mass units).

•• This boundary conditions requires that the same amount of chargThis boundary conditions requires that the same amount of charge but with e but with opposite sign, travels on the vacuum chamber together with the opposite sign, travels on the vacuum chamber together with the beam. Such beam. Such

charge is referred as the charge is referred as the image chargeimage charge..

NegativeNegativeCharged BeamCharged Beam

γ1



7


Vacuum Chamber Wake FieldsVacuum Chamber Wake FieldsCollective EffectsF. Sannibale

•• The beam and its electromagnetic field travel inside the vacuumThe beam and its electromagnetic field travel inside the vacuum chamber chamber while the image charge travels while the image charge travels onon the chamber itself. the chamber itself.

•• Any variation on the chamber profile, on the chamber material, Any variation on the chamber profile, on the chamber material, or on the or on the material properties breaks this configuration.material properties breaks this configuration.

•• The result is that the beam loses a (usually small) part of it The result is that the beam loses a (usually small) part of it is energy that is energy that feeds the electromagnetic fields that remain after the passage ofeeds the electromagnetic fields that remain after the passage of the beam. f the beam.

Such fields are referred as Such fields are referred as wake fieldswake fields..

•• Vacuum chamber wake fields generated by beam particles, mainly Vacuum chamber wake fields generated by beam particles, mainly affect affect trailing particles and in the case of ultratrailing particles and in the case of ultra--relativistic beams can relativistic beams can onlyonly affect affect

trailing particles.trailing particles.

8


Wake Fields and InstabilitiesWake Fields and InstabilitiesCollective EffectsF. Sannibale

•• Wake fields are transient effects, they are generated during thWake fields are transient effects, they are generated during the beam e beam passage and then last for a finite amount of time that depends opassage and then last for a finite amount of time that depends on the n the

particular wake and on the geometry of the vacuum chamber.particular wake and on the geometry of the vacuum chamber.•• If the wake field lasts for the duration of a bunch (hundreds oIf the wake field lasts for the duration of a bunch (hundreds of f psps typically), typically), particles in the bunch tail can interact with the wakes due to tparticles in the bunch tail can interact with the wakes due to the particles in he particles in

the head and the head and single bunch instabilitiessingle bunch instabilities can be triggered (distortion of the can be triggered (distortion of the longitudinal distribution, bunch lengthening, …).longitudinal distribution, bunch lengthening, …).

•• If the wake field lasts longer, for example for the distance inIf the wake field lasts longer, for example for the distance in time between time between bunches (several ns typically), wakes from leading bunches can ibunches (several ns typically), wakes from leading bunches can interact with nteract with

following bunches and potentially generate following bunches and potentially generate multimulti --bunch or coupled bunch bunch or coupled bunch instabilitiesinstabilities..

9


Wake PotentialsWake PotentialsCollective EffectsF. Sannibale

•• For this reason, in investigating longitudinal wake fields we cFor this reason, in investigating longitudinal wake fields we consideronsider only the only the electric component of the wake fieldselectric component of the wake fields. .

•• In practical cases, wake potentials are integrated over some fiIn practical cases, wake potentials are integrated over some finite length. The nite length. The length of the integration path must be longer than the wake fiellength of the integration path must be longer than the wake field maximum d maximum

extension.extension.

( ) ( ) ,,,,,, ∞

∞−

⋅−=− sdttrrsEttrrV leadtrailtrailleadWleadtrailtrailleadW

•• It is often convenient to deal with wake It is often convenient to deal with wake potentials instead of wake fields. The potentials instead of wake fields. The wake wake potentialpotential is defined as the energy variation is defined as the energy variation

induced by the wake field of the leading induced by the wake field of the leading particle on the unit charge trailing particle.particle on the unit charge trailing particle.

•• Wake fields effects can be divided into longitudinal and transvWake fields effects can be divided into longitudinal and transversal. In the ersal. In the longitudinal case the wakes affect the energy of the particles, longitudinal case the wakes affect the energy of the particles, while in the while in the

transverse case is their transverse momentum to be affected.transverse case is their transverse momentum to be affected.

xy

sleadr

trailr

leadq

leadlead tvs =

trailtrail tvs =trailq

•• In our results, In our results, ss = = vtvt with with vv constant, this is a very good approximation for constant, this is a very good approximation for relativistic particles but it is also a reasonable assumption forelativistic particles but it is also a reasonable assumption for the cases where r the cases where

the wake induced energy variation is small respect to the particthe wake induced energy variation is small respect to the partic le energy .le energy .

10


Wake FunctionsWake FunctionsCollective EffectsF. Sannibale

•• The The wake functionwake function is instead defined as the energy variation induced by the is instead defined as the energy variation induced by the wake field of a unit charge leading particle on the unit charge wake field of a unit charge leading particle on the unit charge trailing particle.trailing particle.

( ) ( )lead

leadtrailtrailleadWleadtrailtraillead q

ttrrVttrrW

−=− ,,,,

•• The The total wake potentialtotal wake potential for a bunch with charge distribution for a bunch with charge distribution ii with:with:

( ) Nqtdrdtri =

,

is given by:is given by: ( ) ( ) ( ) −= tdrdtrittrrWtrV trailtrailtrailtrail

,,, ,

•• The total wake potential gives the energy variation that the trThe total wake potential gives the energy variation that the trailing particle ailing particle experiences due to the wakes of the whole bunch.experiences due to the wakes of the whole bunch.

•• Very often in real accelerators, we deal with distributions thaVery often in real accelerators, we deal with distributions that “ live” in t “ live” in the neighbor of the bunch center. In this case, it is sufficientthe neighbor of the bunch center. In this case, it is sufficient to use the to use the

wakes on axis that can be obtained by setting wakes on axis that can be obtained by setting rr and and rrtrailtrail = 0= 0 in the in the previous expressions (previous expressions (monopole wakemonopole wake approximation).approximation).

11


Coupling ImpedanceCoupling ImpedanceCollective EffectsF. Sannibale

•• The wake function represents the interaction of the beam with tThe wake function represents the interaction of the beam with the external he external environment in the environment in the time domaintime domain..

( ) ( ) ttwithderrWrrZ trailj

trailtrail −== −∞

∞− τττω ωτ,,,,

•• As for other phenomena, the equivalent As for other phenomena, the equivalent frequency domainfrequency domain analysis can be analysis can be very useful giving a different insight and additional interpretavery useful giving a different insight and additional interpretations.tions.

•• The frequency domain “ alter ego” of the wake function is the The frequency domain “ alter ego” of the wake function is the coupling coupling impedanceimpedance, measured in Ohm and defined as the , measured in Ohm and defined as the Fourier transform of the Fourier transform of the

wake functionwake function::

•• If If II is the Fourier transform of the charge distribution, the Fourieis the Fourier transform of the charge distribution, the Fourier transform r transform of the total induced voltage is simply given by:of the total induced voltage is simply given by:

( ) ( ) ( )ωωω ,,,,,~

rIrrZrrV trailtrail

=•• And the time domain expression can be obtained by the inverse FAnd the time domain expression can be obtained by the inverse Fourier ourier

transform:transform:

( ) ( )∞

∞−

= ωωπ

τ ωτ derrVrrV jtrailtrail ,,

~2

1,,

12


Interpretation of theInterpretation of theCoupling ImpedanceCoupling Impedance


•• The coupling impedance is a complex quantity with real and imagThe coupling impedance is a complex quantity with real and imaginary parts:inary parts:

( ) ( ) ( )ωωω ,,,,,, trailjtrailRtrail rrZjrrZrrZ +=

•• There is a strong analogy between wake field and electronic cirThere is a strong analogy between wake field and electronic circuit theories. cuit theories. This can be exploited and wakes can be represented by equivalentThis can be exploited and wakes can be represented by equivalent circuits.circuits.

iB CLR

•• Analogously to the circuit case, the resistive part of the coupAnalogously to the circuit case, the resistive part of the coupling impedance ling impedance is responsible for the beam losses, while the imaginary part defis responsible for the beam losses, while the imaginary part def ines the ines the

phase relation between the beam excitation and the wake potentiaphase relation between the beam excitation and the wake potential.l.

( )L

CRQ

LCjQ

RZ R

R

R

==

−+

= ,1

,

1

ω

ωω

ωω

ω

•• For example, the impedance of a parallel RLC circuit is often aFor example, the impedance of a parallel RLC circuit is often associated to ssociated to the impedance of the sothe impedance of the so--called called high order modeshigh order modes (HOM), single resonance (HOM), single resonance

wakes in the vacuum chamber. wakes in the vacuum chamber.

( ) ( ) ( )

>

−

−−−

<

= −

014

411sin411cos

00

2

22

2

ττω

τω

τ

τ τω

Q

QQ

C

eW RR

QR

13


NarrowNarrow--band and Broadband and Broad--bandbandCoupling ImpedancesCoupling Impedances


•• Using this RLC model the Using this RLC model the HOMsHOMs can be classified in two main categories. can be classified in two main categories. •• NarrowNarrow--band impedancesband impedances. These modes are characterized by relatively high . These modes are characterized by relatively high QQand their spectrum is narrow. The associated wake last for a reland their spectrum is narrow. The associated wake last for a relatively long time atively long time

making this modes important for making this modes important for multibunchmultibunch instabilities.instabilities.

•• BroadBroad--band impedancesband impedances. These modes are characterized by a low . These modes are characterized by a low QQ and their and their spectrum is broader. The associated wake last for a relatively sspectrum is broader. The associated wake last for a relatively short time making hort time making

this modes important only for single bunch instabilities.this modes important only for single bunch instabilities.

14


Transverse Wake Fields CaseTransverse Wake Fields CaseCollective EffectsF. Sannibale

•• A similar approach and definitions can be used for the transverA similar approach and definitions can be used for the transverse wake se wake case.case.

•• The The transverse wake functiontransverse wake function defines the defines the transverse momentum kick per transverse momentum kick per unit leading charge and unit trailing charge unit leading charge and unit trailing charge due to the wake fields.due to the wake fields.

•• Transverse wake fields are excited when the beam passes out of cTransverse wake fields are excited when the beam passes out of centerenter. If . If the displacement is small enough only the the displacement is small enough only the dipoledipole term proportional to the term proportional to the displacement is important. In such a situation, the displacement is important. In such a situation, the transverse dipole wake transverse dipole wake

function,function, defined as the transverse wake function for unit displacement, defined as the transverse wake function for unit displacement, can can be used.be used.

•• Longitudinal and transverse wakes are representation of the samLongitudinal and transverse wakes are representation of the same 3D wake e 3D wake field and are linked each other by the Maxwell equations. field and are linked each other by the Maxwell equations.

The soThe so--called called PanofskyPanofsky--Wenzel relationsWenzel relations allow to calculate one wake allow to calculate one wake component when the other is known.component when the other is known.

•• The The transverse coupling impedancetransverse coupling impedance is defined as the Fourier transform of is defined as the Fourier transform of the transverse wake function times the transverse wake function times j .j .

15


Impedance of AcceleratorsImpedance of AcceleratorsCollective EffectsF. Sannibale

•• In a real accelerators, the vacuum chamber has a very complex sIn a real accelerators, the vacuum chamber has a very complex shape and hape and includes many components that can potentially have “ trapped” HOMincludes many components that can potentially have “ trapped” HOM..

•• In summary, when the beam transits along the vacuum chamber it In summary, when the beam transits along the vacuum chamber it excites excites wake fields. These can be classified in three main categories:wake fields. These can be classified in three main categories:

•• Anyway, not all the wakes excited by the beam can be trapped inAnyway, not all the wakes excited by the beam can be trapped in the the vacuum chamber. In fact, for a given vacuum chamber geometry, itvacuum chamber. In fact, for a given vacuum chamber geometry, it exist a exist a cutoff frequencycutoff frequency such that modes with frequency above cutoff propagates such that modes with frequency above cutoff propagates

along the chamber:along the chamber:

-- wake fields that travels with the beam (such as the space chargwake fields that travels with the beam (such as the space charge);e);-- wake fields that are localized in some resonant structure in thwake fields that are localized in some resonant structure in the e

vacuum chamber (narrow and broad band HOM);vacuum chamber (narrow and broad band HOM);-- high frequency wakes, above the vacuum chamber cutoff, that high frequency wakes, above the vacuum chamber cutoff, that propagates along the vacuum chamber. This last category does propagates along the vacuum chamber. This last category does not generate any net interaction with the beam unless they are not generate any net interaction with the beam unless they are

synchronous with the beam itself.synchronous with the beam itself.

sizetransversechamberaa

cfCutoff ≡≈

16


Impedance of AcceleratorsImpedance of AcceleratorsCollective EffectsF. Sannibale

•• Narrow band impedances, such as the ones due to the RF cavitiesNarrow band impedances, such as the ones due to the RF cavities and and potentially to other components such as kickers and similar strupotentially to other components such as kickers and similar structures, are ctures, are

well localized and are usually treated independently.well localized and are usually treated independently.

•• In the case of broad band impedances, the contributions of the In the case of broad band impedances, the contributions of the single single components (bellows, vacuum ports, transitions, diagnostics, …) components (bellows, vacuum ports, transitions, diagnostics, …) are are

individually calculated (mainly numerically) and/or measured andindividually calculated (mainly numerically) and/or measured and then added then added together.together.

•• Most of the circular accelerators with relatively long bunches Most of the circular accelerators with relatively long bunches (greater than (greater than few tens of few tens of psps rmsrms) present a total coupling impedance which is mainly ) present a total coupling impedance which is mainly

inductive. Scaled models, such as the inductive. Scaled models, such as the broadband resonatorbroadband resonator (parallel RLC (parallel RLC with with Q Q ~ 1~ 1 and and 22πωπωπωπωπωπωπωπωR R ~ ~ ffCutoffCutoff) or such as the empirical “) or such as the empirical “ SPEAR scalingSPEAR scaling” model, ” model,

are often used to represent with some success the total broadbanare often used to represent with some success the total broadband d impedance of such accelerators.impedance of such accelerators.

•• Concerning the wake fields that propagates with the beam, we alConcerning the wake fields that propagates with the beam, we already ready glanced on the space charge “ wake” . Another example of such a kiglanced on the space charge “ wake” . Another example of such a kind of nd of

wake is the one represented by the wake is the one represented by the resistive wall impedanceresistive wall impedance, where the finite , where the finite resistivityresistivity of the vacuum chamber walls generates a wake that can be often of the vacuum chamber walls generates a wake that can be often

expressed as the impedance of a highexpressed as the impedance of a high--frequency broadfrequency broad--band resonator.band resonator.

17


The Synchrotron RadiationThe Synchrotron Radiation“ Wake” Field“ Wake” Field


•• The wake field due to synchrotron radiation, belongs to the catThe wake field due to synchrotron radiation, belongs to the category of the egory of the wakes that propagates with the beam.wakes that propagates with the beam.

•• Such a wake is important only for the relativistic particle casSuch a wake is important only for the relativistic particle case.e.

•• Relativistic particles on a curved trajectory emit synchrotron Relativistic particles on a curved trajectory emit synchrotron radiation (SR). radiation (SR). The SR fields propagates in a cone of emission centered on the tThe SR fields propagates in a cone of emission centered on the tangent to angent to

the beam trajectory at the emission point and with ~the beam trajectory at the emission point and with ~ 1/1/γγγγγγγγ aperture.aperture.

•• The fields propagate at the speed of light, while the particlesThe fields propagate at the speed of light, while the particles move on the move on the curved trajectory. For this reason, even if the particles are recurved trajectory. For this reason, even if the particles are relativistic the lativistic the

projection of their speed on the tangent direction is smaller thprojection of their speed on the tangent direction is smaller than an cc..

•• In other words, the SR wake field due to a particle in the tailIn other words, the SR wake field due to a particle in the tail of the bunch of the bunch can reach and interact with a particle in the head!can reach and interact with a particle in the head!

This is exact the opposite of what happens with vacuum chamber wThis is exact the opposite of what happens with vacuum chamber wakes.akes.

18


Single Bunch EffectsSingle Bunch EffectsCollective EffectsF. Sannibale

Broadband impedances have important effects on accelerators.Broadband impedances have important effects on accelerators.

In electron storage rings in the presence of radiation damping, In electron storage rings in the presence of radiation damping, the the equilibrium distributions at low current are usually equilibrium distributions at low current are usually gaussiangaussian. By increasing . By increasing the current per bunch, the wakes become stronger and can generatthe current per bunch, the wakes become stronger and can generate non e non

gaussiangaussian equilibrium distributions.equilibrium distributions.

In In linacslinacs and in heavy particle accelerators, broad band impedances can and in heavy particle accelerators, broad band impedances can generate generate emittanceemittance and energy spread growth.and energy spread growth.

In all accelerators, if the current per bunch is increased furthIn all accelerators, if the current per bunch is increased further, the wakes er, the wakes can become strong enough to generate single bunch instabilities can become strong enough to generate single bunch instabilities that can that can severely change the characteristics of the bunch and/or generateseverely change the characteristics of the bunch and/or generate particle particle

losses.losses.

In what follows, some examples (not a complete review!) of such In what follows, some examples (not a complete review!) of such cases will cases will be given.be given.

19


Potential Well Distortion:Potential Well Distortion:The SR Wake CaseThe SR Wake Case


•• The example concerns the case of the synchrotron radiation (SR)The example concerns the case of the synchrotron radiation (SR) wake in wake in electron storage rings.electron storage rings.

•• The SR wake due to the particles in the tail of The SR wake due to the particles in the tail of the bunch interacts with particles in the head by the bunch interacts with particles in the head by

changing their energy.changing their energy.•• This generates a distortion of the normally parabolic RF potentThis generates a distortion of the normally parabolic RF potent ial (ial (potential potential

well distortionwell distortion). In this situation, the bunch is forced to a new equilibrium w). In this situation, the bunch is forced to a new equilibrium with ith a nona non--gaussiangaussian longitudinal distribution. longitudinal distribution.

•• When a storage ring is tuned for short bunches When a storage ring is tuned for short bunches (~ few (~ few psps rmsrms), the SR becomes the dominant ), the SR becomes the dominant

wake.wake.HeadTail

nominal bunchdistribution

SR wake accelerates bunch head

Total potential wake(gaussian case)

20


Potential Well Distortion:Potential Well Distortion:The General CaseThe General Case


•• The potential well distortion mechanism, shown for the case of The potential well distortion mechanism, shown for the case of the SR wake, the SR wake, is actually quite general and common to all kind of wakes in eleis actually quite general and common to all kind of wakes in electron rings.ctron rings.

•• Remembering that wakes can be represented by the real and imagiRemembering that wakes can be represented by the real and imaginary part nary part of the coupling impedance, some common “ rules” can be derived.of the coupling impedance, some common “ rules” can be derived.

•• The The real (resistive) partreal (resistive) part of the coupling of the coupling impedance generates impedance generates asymmetric asymmetric

distortionsdistortions and and lengtheninglengthening of the bunch of the bunch distribution. The distribution. The bunch center of mass bunch center of mass moves towards a different RF phasemoves towards a different RF phase to to

compensate for the wake induced compensate for the wake induced energy losses. energy losses.

•• The The imaginary (reactive) partimaginary (reactive) part of the of the coupling impedance generates coupling impedance generates

symmetric distortionssymmetric distortions of the bunch of the bunch distribution. The bunch distribution. The bunch center of mass center of mass does not movedoes not move (no energy losses). It (no energy losses). It

generates generates bunch lengthening or bunch lengthening or shorteningshortening..

L1

L3

L2

L3 > L2 > L1

Z = j ω LαC > 0

R3 > R2 > R1

R1 R2R3

Z = R

21


Single Bunch Instabilities:Single Bunch Instabilities:The SR Wake CaseThe SR Wake Case


Simulated instability showing Simulated instability showing the the microbunchingmicrobunching. .

Courtesy of Marco Courtesy of Marco VenturiniVenturini

•• In an electron storage ring, if the current per In an electron storage ring, if the current per bunch is above a specific threshold, the SR bunch is above a specific threshold, the SR

wake can drive a wake can drive a microbunchingmicrobunching instabilityinstability in in the electron bunch.the electron bunch.

10.5mA

28.8mA

100ms806040200

Time (msec)

40.0mA

10 10 mAmA

29 29 mAmA

40 40 mAmA

Time (Time (msecmsec))

Bo

lom

eter

sig

nal

(V

)B

olo

met

er s

ign

al (

V)

ALS DataALS Data

( ) ( ) ][2 0312332121

0 UnitsMKSrCcemA q π=

( ) 32

29

23611

23

210

21 cos

1

λγ

ρ

α

ϕ s

C

sRF

bJVfh

AI

>

•• The SR wake becomes strong enough to create The SR wake becomes strong enough to create temporary microtemporary micro--structures in the bunch that structures in the bunch that

radiates strong “ bursts” of coherent radiates strong “ bursts” of coherent synchrotron radiation in the farsynchrotron radiation in the far--infrared.infrared.

22


Single Bunch Instabilities:Single Bunch Instabilities:Beam Break UpBeam Break Up


•• When a bunch enters offWhen a bunch enters off--axis in a axis in a linaclinac structure it excites transverse structure it excites transverse wakes.wakes.

•• If the impedance associated with the wake is broadIf the impedance associated with the wake is broad--band, the head of the band, the head of the bunch can excite the wakes that will deflect the tail of the bunbunch can excite the wakes that will deflect the tail of the bunch.ch.

•• The effect was first observed in 1966 at SLAC in the 2 miles loThe effect was first observed in 1966 at SLAC in the 2 miles long ng linaclinac of the of the SLC (Stanford Linear SLC (Stanford Linear ColliderCollider) and was responsible for luminosity limitation.) and was responsible for luminosity limitation.

•• In long high current/bunch In long high current/bunch linacslinacs the effect can build up and the bunch can the effect can build up and the bunch can be distorted into a “ banana” like shape. This effect is known asbe distorted into a “ banana” like shape. This effect is known as singlesingle--bunch bunch

beam break upbeam break up (SBBU).(SBBU).

t0 t1 t2t3 t4

t5

t6

t0 t1 t2t3 t4

t5

t6

23

Horizontal size ina dispersive region


Single Bunch Instabilities:Single Bunch Instabilities:Microwave InstabilityMicrowave Instability


•• The total broad band impedance of a storage ring is also responThe total broad band impedance of a storage ring is also responsible of another sible of another longitudinal single bunch instability known as the longitudinal single bunch instability known as the microwave instabilitymicrowave instability..

•• When the current per bunch is larger than the instability thresWhen the current per bunch is larger than the instability threshold:hold:

•• The net effect on the bunch is an The net effect on the bunch is an increase of the energy spreadincrease of the energy spread above threshold above threshold with a consequent with a consequent increase of the bunch lengthincrease of the bunch length and of the beam and of the beam transverse size transverse size

in in dispersivedispersive regions.regions.

( )nZe

EEI EC

peak//

2002 σπα

>

the single particles get excited by the wakes on exponentially gthe single particles get excited by the wakes on exponentially growing rowing longitudinal oscillations. Because nonlongitudinal oscillations. Because non--linearitieslinearities, the oscillation frequency , the oscillation frequency

changes with amplitude limiting the maximum amplitude and in moschanges with amplitude limiting the maximum amplitude and in most of the cases t of the cases no particle loss happens.no particle loss happens.

24


Multi Bunch InstabilitiesMulti Bunch InstabilitiesCollective EffectsF. Sannibale

•• In the case of narrowIn the case of narrow--band impedances the wake generated by one bunch band impedances the wake generated by one bunch can last long enough to interfere with other bunches or with thecan last long enough to interfere with other bunches or with the bunch itself bunch itself

in subsequent turns. In this situation multiin subsequent turns. In this situation multi --bunch instabilities can be excited.bunch instabilities can be excited.

•• High current accelerators are carefully designed in order to miHigh current accelerators are carefully designed in order to minimize broad nimize broad band and narrow band impedances. Anyway, even in the best conceiband and narrow band impedances. Anyway, even in the best conceived ved

accelerator, the impedance accelerator, the impedance cannot vanishcannot vanish and there will be always a current and there will be always a current threshold above which the beam will become unstable. threshold above which the beam will become unstable.

If the accelerator is required to operate above the instability If the accelerator is required to operate above the instability threshold, threshold, active active feedback systemsfeedback systems are necessary for damping down the instabilities.are necessary for damping down the instabilities.

•• Despite these difficulties, properly designed accelerators withDespite these difficulties, properly designed accelerators with low overall low overall broadbroad--band impedance, carefully damped band impedance, carefully damped HOMsHOMs and active longitudinal and and active longitudinal and

transverse bunch by bunch feedbacks achieved very remarkable restransverse bunch by bunch feedbacks achieved very remarkable results.ults.Currents of few Amps have been stored in electron and positron mCurrents of few Amps have been stored in electron and positron machines achines

(PEP 2, KEK(PEP 2, KEK--B, DAB, DAΦΦΦΦΦΦΦΦNE, …) and of many tens of NE, …) and of many tens of mAmA in proton machines (SPS, in proton machines (SPS, TEVATRON, HERA, …).TEVATRON, HERA, …).

25


Coupled Bunches ModesCoupled Bunches ModesCollective EffectsF. Sannibale

n = 1n = 1

n = 2n = 2

n = 3n = 3

n = 4n = 4

From Dan Russell'sMultiple DOF Systems

26


MultiMulti --Bunch InstabilitiesBunch InstabilitiesMechanismMechanism


( ) [ ]( ) [ ]( ) dampingradiationDnnt

nn tet Dn ≡+= +−αϕωϕϕ αω

0Im Resinˆ

•• By using the model of coupled harmonic oscillators, every mode By using the model of coupled harmonic oscillators, every mode can be can be characterized by a complex frequency characterized by a complex frequency ωωωωωωωω and by the equation of a damped and by the equation of a damped

oscillator:oscillator:

•• The oscillation becomes unstable (antiThe oscillation becomes unstable (anti--damping) when:damping) when:

( )alwaysDD 00]Im[ ><+ ααω

•• Wakes fields produce a shift of the imaginary part of the frequWakes fields produce a shift of the imaginary part of the frequency:ency:

( )nS

CBn Z

E

eI ω

ναω ≈∆ ]Im[

•• Depending on the signs of the momentum compaction and of the Depending on the signs of the momentum compaction and of the impedance, some modes can become unstable when the current per bimpedance, some modes can become unstable when the current per bunch is unch is

increased.increased.

•• Feedback systems increase Feedback systems increase ααααααααDD so that to increase the threshold for the so that to increase the threshold for the instabilities.instabilities.

27


““ Good” Wake FieldsGood” Wake FieldsCollective EffectsF. Sannibale

•• WakefieldWakefield--based acceleration schemes.based acceleration schemes.Strong R&D and very promising results.Strong R&D and very promising results.

•• Wake fields are commonly exploited in Wake fields are commonly exploited in diagnostic systems used for the diagnostic systems used for the

characterization of the beam properties.characterization of the beam properties.

•• Bunches in electron storage rings with longitudinal Bunches in electron storage rings with longitudinal distribution asymmetrically distorted by wakedistribution asymmetrically distorted by wake--fields emit fields emit

coherent synchrotron radiation at much higher coherent synchrotron radiation at much higher frequencies than bunches with nominal frequencies than bunches with nominal gaussiangaussian

distribution. This can be exploited for designing fardistribution. This can be exploited for designing far--infrared synchrotron light sources with revolutionary infrared synchrotron light sources with revolutionary

performances.performances.

Bad Wake God Wakes

•• Not all of these “ evanescent and ghostly” wakes are bad in acceNot all of these “ evanescent and ghostly” wakes are bad in accelerator lerator applications. In fact, there are few examples were wakes play a applications. In fact, there are few examples were wakes play a positive role:positive role:

Laser

Gas

Gas jet nozzle

e- bunchPlasmachannel

THzRadiation

L’OASISL’OASIS

28


ReferencesReferencesCollective EffectsF. Sannibale

L. Palumbo, V. G. L. Palumbo, V. G. VaccaroVaccaro, M. , M. ZobovZobov, “ Wake fields and impedances” , CERN, “ Wake fields and impedances” , CERN--9595--0606

A. A. ChaoChao, “ Physics of Collective Beam Instabilities in High Energy , “ Physics of Collective Beam Instabilities in High Energy Accelerators” , WileyAccelerators” , Wiley--InterscienceInterscience Pub. (1993).Pub. (1993).

A. A. ChaoChao, M. , M. TignerTigner, “ Handbook of Accelerator Physics and Engineering” , , “ Handbook of Accelerator Physics and Engineering” , Word Scientific Pub. (1998).Word Scientific Pub. (1998).

29


Possible HomeworkPossible HomeworkCollective EffectsF. Sannibale

•• Calculate, the repulsive force that a proton with a 100 Calculate, the repulsive force that a proton with a 100 µµµµµµµµm displacement from m displacement from the beam center will experience due to the space charge from thethe beam center will experience due to the space charge from the other other

protons in the beam. The beam has a circular profile with protons in the beam. The beam has a circular profile with rmsrms size of 2 mm size of 2 mm and an energy of 2 and an energy of 2 GeVGeV. The linear charge density is of 0.7 . The linear charge density is of 0.7 nC/mnC/m. Estimate if . Estimate if

the effect on the particle integrated over one turn is significathe effect on the particle integrated over one turn is significant or not (the nt or not (the ring length is 100 m). Compare with the case of an electron beamring length is 100 m). Compare with the case of an electron beam with the with the

same characteristics.same characteristics.

•• In principle, a particle accelerator built in the space (orbitiIn principle, a particle accelerator built in the space (orbiting around the ng around the earth for example) could be built without a vacuum chamber. Willearth for example) could be built without a vacuum chamber. Will the the

particles in such an accelerator be subjected to any wake field?particles in such an accelerator be subjected to any wake field? Please Please explain your answer.explain your answer.

1



Real Accelerators.Real Accelerators.Errors and Diagnostics.Errors and Diagnostics.


2


IntroductionIntroductionReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• In designing and constructing an accelerator, physicists and enIn designing and constructing an accelerator, physicists and engineers do gineers do their best in making a perfect job and in foreseeing any possibltheir best in making a perfect job and in foreseeing any possible operation e operation

mode for their device.mode for their device.

•• In this lecture, we will briefly introduce the more typical (anIn this lecture, we will briefly introduce the more typical (and predictable) d predictable) errors affecting real accelerators. We will also discuss with soerrors affecting real accelerators. We will also discuss with some more me more details examples (not a complete list) of diagnostic systems anddetails examples (not a complete list) of diagnostic systems and beam beam

measurements used for correcting for those errors.measurements used for correcting for those errors.

•• In most of the cases, the In most of the cases, the ideal machineideal machine remains just a concept and one has remains just a concept and one has to deal with more real objects where construction tolerances andto deal with more real objects where construction tolerances and unpredicted unpredicted

phenomena generate effects that need to be measured and correctephenomena generate effects that need to be measured and corrected.d.

3


Magnet Misalignment EffectsMagnet Misalignment EffectsReal AcceleratorsErrors & Diagnostics

F. Sannibale

A A multipolemultipole of order of order nn, with a , with a tilt errortilt error will will present present a “ skew” component of order a “ skew” component of order nn with with

amplitude proportional to the tilt angle.amplitude proportional to the tilt angle.

A A multipolemultipole of order of order nn, with a , with a displacement displacement errorerror will present will present all the all the multipolarmultipolar

components with order components with order i = 1, 2, …, n i = 1, 2, …, n -- 11..x∆

y∆

x∆

θ

4


Dipole ErrorDipole ErrorComponent EffectComponent Effect

Real AcceleratorsErrors & Diagnostics

F. Sannibale

•• If we have dipolar errors in If we have dipolar errors in NN different ring locations, they generate different ring locations, they generate NN kicks kicks θθθθθθθθii and a total orbit distortion given by: and a total orbit distortion given by:

( ) ( )( ) ( ) ( ) ( )[ ] yxwsss

ssw

N

iiwiiw

w

w ,cossin2 1

=+−= =

πϕϕνθβπν

β

•• Because of radiation damping, positrons and electrons converge Because of radiation damping, positrons and electrons converge into the into the distorted orbit in roughly a damping time. Protons and heavier pdistorted orbit in roughly a damping time. Protons and heavier particles articles oscillates around the distorted closed orbit without converging oscillates around the distorted closed orbit without converging into it.into it.

•• In the case of a single kick at the position In the case of a single kick at the position ss, the displacement induced the , the displacement induced the kick at the same point kick at the same point ss is given by:is given by:

( ) ( ) yxwssw ww ,cot2

1 == πνθβ

•• If a If a corrector or steering magnetcorrector or steering magnet (small dipole magnet capable of generating (small dipole magnet capable of generating a kick a kick θθθθθθθθ) has a ) has a beam position monitorbeam position monitor (BPM) nearby, by kicking the beam and (BPM) nearby, by kicking the beam and using the previous relation, the beta function at that point canusing the previous relation, the beta function at that point can be measured.be measured.

•• We saw that a displacement error in a magnet generates a We saw that a displacement error in a magnet generates a dipole dipole componentcomponent in its center. This term induces a beam in its center. This term induces a beam orbit distortion.orbit distortion.

Betatron phases

•• Note that for integer tunes no closed orbit exists. Note that for integer tunes no closed orbit exists.

5


Orbit Correction BasicsOrbit Correction BasicsReal AcceleratorsErrors & Diagnostics

F. Sannibale

Or in matrix representation, when:Or in matrix representation, when:

( ) ( ) ( )[ ] Njssss

uM

iijii

j

j ...,,2,1cos)(sin2

)(

1

=+−=− =

πϕϕνθβπν

β

By using By using MM correctors magnets, we can find a set of kicks that cancels thecorrectors magnets, we can find a set of kicks that cancels thedisplacement of the beam at the BPM positions. This is obtained displacement of the beam at the BPM positions. This is obtained when: when:

NN uuu ...,,, 21=u

The kicks that need to be applied to the steering magnets for coThe kicks that need to be applied to the steering magnets for correcting the rrecting the closed orbit distortion, can be obtained by inverting the previoclosed orbit distortion, can be obtained by inverting the previous equation:us equation:

By measuring the orbit distortion in By measuring the orbit distortion in N N BPMsBPMs along the ring, we find the set of along the ring, we find the set of displacements:displacements:

( ) ( ) ( )[ ]πϕϕνπνββ

+−==− ij

ij

ijMN ssss

Mwith cossin2

)()(Mu

NM uM 1−−=The elements of the The elements of the response matrixresponse matrix MM, can be calculated from the machine , can be calculated from the machine

model, or measured by individually exciting each of the correctomodel, or measured by individually exciting each of the correctors and rs and measuring the induced displacement in each of the measuring the induced displacement in each of the BPMsBPMs..

6


QuadrupoleQuadrupole ErrorErrorComponent EffectComponent Effect


F. Sannibale

QuadrupoleQuadrupole error components (gradient errors) can be due to misalignment error components (gradient errors) can be due to misalignment of higher order of higher order multipolarmultipolar magnet (magnet (sextupolessextupoles, , octupolesoctupoles, …) or due to error , …) or due to error

in the currentin the current--strength calibration of strength calibration of quadrupolequadrupole magnets.magnets.

strengthquadrupolep

GqkwhereLk QQ ==≅∆

02πβν

Gradient errors generate a Gradient errors generate a betatronbetatron tune shifttune shift equal to:equal to:

LL is the is the quadrupolequadrupole magnetic length, magnetic length, GG is its gradient and is its gradient and qq and and pp00 are the are the particle charge and momentum respectively. The previous equationparticle charge and momentum respectively. The previous equation can be can be used for measuring the beta function at the used for measuring the beta function at the quadrupolequadrupole position, when the position, when the

tune shift for small change of the magnet strength is measured.tune shift for small change of the magnet strength is measured.

It can be shown that It can be shown that quadrupolequadrupole gradient errors makes the gradient errors makes the halfhalf--integer integer resonance unstableresonance unstable..

Tilt errors in Tilt errors in quadrupolequadrupole magnets generate magnets generate couplingcoupling between the vertical and between the vertical and the horizontal planes. On the other hand, on purpose tilted the horizontal planes. On the other hand, on purpose tilted quadrupolesquadrupoles

((skew skew quadrupolesquadrupoles) can be used for compensating the coupling due to lattice ) can be used for compensating the coupling due to lattice nonnon--linearitieslinearities..

7

ALSALS


NonNon--ideal Magnets:ideal Magnets:MultipolarMultipolar TermsTerms


F. Sannibale

•• Good design and construction can minimize but not Good design and construction can minimize but not cancel the cancel the multipolarmultipolar field presence. Additionally, in most field presence. Additionally, in most

of storage rings of storage rings sextupolesextupole (and sometimes (and sometimes octupoleoctupole) ) magnets are added on purpose for the compensation of magnets are added on purpose for the compensation of

chromatic effects and for improving the dynamic aperture.chromatic effects and for improving the dynamic aperture.

•• MultipolarMultipolar field components field components introduce nonintroduce non--linearitieslinearities that that

generate a shift in the generate a shift in the betatronbetatronfrequency for large amplitude frequency for large amplitude

oscillations (oscillations (tune shift on amplitudetune shift on amplitude).).

•• These tune shifts can bring particles on tune These tune shifts can bring particles on tune resonancesresonancesgenerating particle losses (generating particle losses (dynamic aperturedynamic aperture).).

•• Simplified geometries, imperfections and mechanical tolerances Simplified geometries, imperfections and mechanical tolerances in the in the design and construction of accelerator magnets, populates the acdesign and construction of accelerator magnets, populates the accelerators celerators

with a plethora of with a plethora of higher order higher order multipolarmultipolar termsterms..

•• On the other hand, these frequency shifts generate On the other hand, these frequency shifts generate dede--coherencecoherence in the in the oscillations with a damping effect on instabilities (oscillations with a damping effect on instabilities (Landau dampingLandau damping).).

8

SOLEILSOLEIL


Power Supply FluctuationsPower Supply FluctuationsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• Fluctuations in the power supply currentFluctuations in the power supply current of the accelerator magnets can of the accelerator magnets can limit the performance of an accelerator. limit the performance of an accelerator.

•• Jitter in the dipole magnet power supplies generates fluctuatioJitter in the dipole magnet power supplies generates fluctuations in the ns in the beam energy inducing jitter in the tunes and orbit fluctuations beam energy inducing jitter in the tunes and orbit fluctuations as well.as well.•• Jitter on Jitter on quadrupolequadrupole magnet power supplies generate magnet power supplies generate betatronbetatron tune tune

fluctuations that can bring particles on tune fluctuations that can bring particles on tune resonancesresonances and generate particle and generate particle losses.losses.

•• Any power supply fluctuation will be transferred to beam (ampliAny power supply fluctuation will be transferred to beam (ampli fied in the fied in the case of strong focusing machines) affecting the ultimate performcase of strong focusing machines) affecting the ultimate performance of the ance of the

accelerators. accelerators.

•• Power supply stability requirements Power supply stability requirements strongly depend on which part of the strongly depend on which part of the

accelerator the magnet is located. accelerator the magnet is located. Typical relative stability requirements range Typical relative stability requirements range from few units of 10from few units of 10--33 for beam transferfor beam transfer--lines lines power supplies to about 10power supplies to about 10--55 for the case of for the case of

storage ring power supplies.storage ring power supplies.

9


Other PerturbationsOther PerturbationsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• Several other perturbations can affect the proper operation of Several other perturbations can affect the proper operation of an an accelerator.accelerator.

•• Last but not least, accelerators are designed for specific applLast but not least, accelerators are designed for specific appl ications that ications that often require detectors using high magnetic fields.often require detectors using high magnetic fields.

This is the typical case for high energy physics experiments in This is the typical case for high energy physics experiments in colliderscolliders, or of , or of insertion devices for radiation production in light sources. Theinsertion devices for radiation production in light sources. These fields if not se fields if not

compensated can have a strong impact on the accelerator performacompensated can have a strong impact on the accelerator performance. nce.

•• Very large accelerators are sensitive to the earth magnetic fieVery large accelerators are sensitive to the earth magnetic field, to the moon ld, to the moon phases, to neighbor railway stations, …phases, to neighbor railway stations, …

•• All accelerators are sensitive to environmental fields and variAll accelerators are sensitive to environmental fields and variables: stray ables: stray magnetic fields due to equipment or to high power electric cablemagnetic fields due to equipment or to high power electric cables, presence s, presence

of other accelerators, temperature variations, fluctuations of tof other accelerators, temperature variations, fluctuations of the main AC he main AC power, ground motion, vibrations, …power, ground motion, vibrations, …

•• In order to minimize and compensate for the effects due to all In order to minimize and compensate for the effects due to all these these perturbations and errors, an efficient beam diagnostics system nperturbations and errors, an efficient beam diagnostics system need to be eed to be

used.used.

10


ElectromagneticElectromagneticBeam Position MonitorsBeam Position Monitors


F. Sannibale

11

ay∆

AV

BV

α

β

w

w


ElectromagneticElectromagneticBeam Position MonitorsBeam Position Monitors


F. Sannibale

In electromagnetic In electromagnetic BPMsBPMs, the image charge in an , the image charge in an electrode is proportional to the beam current and to electrode is proportional to the beam current and to

the angle included between the beam and the the angle included between the beam and the electrode extremes:electrode extremes:

βα

IGV

IGV

B

A

==

)()( yayawBut ∆+=∆−= βα

ya

wIGV

ya

wIGV

B

A

∆+=

∆−=

yIya

wGVV BA ∆

∆−=−

22

2

a

y

VV

VV

BA

BA ∆=+−

BA

BA

VV

VVay

+−

≅∆2

aIya

wGVV BA 22

2

∆−=+

In addition to this geometric In addition to this geometric effect, the field lines tend to effect, the field lines tend to

cluster closely in the region of cluster closely in the region of the nearest electrode (the the nearest electrode (the EE field field

must be perpendicular to the must be perpendicular to the walls). For this geometry, this walls). For this geometry, this gives an additional factor two:gives an additional factor two:

12


““ Button” Type Button” Type BPMsBPMsReal AcceleratorsErrors & Diagnostics

F. Sannibale

!bohba =+

SORSOR--RingRing

DCBA

DCBA

DCBA

DBCA

VVVV

VVVVKy

VVVV

VVVVKx

++++−+

=∆++++−+

=∆)()(

,)()(

APSAPS

AVBV

CVDV

•• Capacitive type (derivative response), low coupling impedance, Capacitive type (derivative response), low coupling impedance, relatively relatively low sensitivity, best for storage rings.low sensitivity, best for storage rings.

PEP IIPEP II

DELTADELTA

•• Typical Typical geometry used geometry used in the presence in the presence of synchrotron of synchrotron

radiation.radiation.

13


StriplineStripline ElectrodeElectrodeReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• Transmission line type, relatively high beam impedance, high seTransmission line type, relatively high beam impedance, high sensitivity, nsitivity, directionality capability, best for directionality capability, best for linacslinacs and and transferlinestransferlines..

FNALFNALInjectorInjector

cL2

0201 0 ZorRandZRUsually LL ==

1LR 2LR

0Z

OutV

L

1LR 2LR

0Z

OutV

L

cLt 2=

1LR 2LR

0Z

OutV

L

cLt =

WallI

Blue color: inverted polarity pulseBlue color: inverted polarity pulse

No signal on No signal on RRLL22!!

1LR 2LR

0Z

WallI

OutV

L

0=tVacuum Chamber

StriplineElectrode

14


StriplineStripline BPMBPMReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• StriplineStripline structures are also widely used as the “ kicker” in transverse astructures are also widely used as the “ kicker” in transverse and nd longitudinal feedback systems.longitudinal feedback systems.

SLACSLACLCLSLCLS

OutV

OutV

FNAL InjectorFNAL Injector

HERA HERA StriplineStripline BPMBPM

SPRING 8 KickerSPRING 8 Kicker

15


Other Other BPMsBPMsReal AcceleratorsErrors & Diagnostics

F. Sannibale

Photon Photon -- BPMBPM

∆∆∆∆y

e-

e-

SR

Beam

“ Blades”

FMBFMBBESSY II,BESSY II,ALS,ALS,SLS,SLS,LNLSLNLS

The intensity of the modes in the resonant The intensity of the modes in the resonant structure is proportional to the beam offsetstructure is proportional to the beam offset

In In resonant resonant BPMsBPMs the the beam excites modes in beam excites modes in

resonant structuresresonant structures

TTF BPMTTF BPM

16

SNSSNS


Beam Profile Monitors:Beam Profile Monitors:Wire ScannersWire Scanners


F. Sannibale

•• A moveable wire scans the beam transversally.A moveable wire scans the beam transversally.

OxfordOxford--DanfisikDanfisik

KEKKEK

•• The interaction between the beam The interaction between the beam and the wire generates (by ionization, and the wire generates (by ionization, bremsstrahlungbremsstrahlung, atomic excitation, …) , atomic excitation, …)

a “ shower” of secondary emission a “ shower” of secondary emission particles proportional to the number particles proportional to the number

of beam particles hitting the wire.of beam particles hitting the wire.

•• The secondary particles (mainly electrons The secondary particles (mainly electrons and photons) are detected and the beam and photons) are detected and the beam transverse profile can be reconstructed.transverse profile can be reconstructed.

••The wire material can be a metal, carbon, or …The wire material can be a metal, carbon, or … a laser beam (Compton scattering,a laser beam (Compton scattering,neutralization)neutralization)

35

30

25

20

15

10

5

0

Wire

sig

nal (

mV

)

-20 -10 0 10 20

Actuator position (mm)

600

500

400

300

200

100

0

Laser notch (microV

)

BNLBNL--SNSSNSHH-- beambeamprofileprofile

17


Virtual PhotonsVirtual PhotonsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• In the “ Particle Sources” lecture, we already saw that accordinIn the “ Particle Sources” lecture, we already saw that according to quantum g to quantum field theory, a photon with a large enough energy can “ oscillatefield theory, a photon with a large enough energy can “ oscillate” between the ” between the

states of virtual electronstates of virtual electron--positron pair and of real photon. positron pair and of real photon. γ

γ

−e•• The opposite is also true. An electron moving in The opposite is also true. An electron moving in

the free space can be considered as “ surrounded” the free space can be considered as “ surrounded” by a cloud of by a cloud of virtual photonsvirtual photons that appear and that appear and disappear and that indissolubly travel with it.disappear and that indissolubly travel with it.

•• Nevertheless, in particular situations, the electron can receivNevertheless, in particular situations, the electron can receive a “ kick” that e a “ kick” that separates it from the photons that become real.separates it from the photons that become real.

–– when the electron moves on a curved trajectory, the transverse when the electron moves on a curved trajectory, the transverse acceleration induces the separation. This is the case of acceleration induces the separation. This is the case of synchrotron radiationsynchrotron radiation..

–– when a relativistic electron moves inside a media and the speedwhen a relativistic electron moves inside a media and the speedof light in the media is smaller than the particle velocity, theof light in the media is smaller than the particle velocity, then the n the separation can happen. This is the case of the separation can happen. This is the case of the CerenkovCerenkov radiation.radiation.

–– when a relativistic electron moves inside a nonwhen a relativistic electron moves inside a non--homogeneous homogeneous media, then the separation can happen. This is the case of the media, then the separation can happen. This is the case of the transition (diffraction) radiationtransition (diffraction) radiation. .

18


Photon BasedPhoton BasedBeam Profile MonitorsBeam Profile Monitors


F. Sannibale

•• Photon diagnostics exploiting all the described emission mechanPhoton diagnostics exploiting all the described emission mechanisms are isms are widely used for measuring the transverse and longitudinal profilwidely used for measuring the transverse and longitudinal profiles of es of

relativistic beams.relativistic beams.

A. LumpkinA. Lumpkin

•• In fact, the spatial distribution of the photons reproduces exaIn fact, the spatial distribution of the photons reproduces exactly the ctly the particle distribution of the beam and can be conveniently used fparticle distribution of the beam and can be conveniently used for the or the

characterization of the beam.characterization of the beam.•• Monitors exploiting transition and Monitors exploiting transition and CerenkovCerenkov radiation are relatively invasive radiation are relatively invasive

and are mainly used in single pass or fewand are mainly used in single pass or few--turns accelerators. turns accelerators.

•• The angular distribution of the photons depends on several beamThe angular distribution of the photons depends on several beamparameters. This fact can be exploited for the measurements of qparameters. This fact can be exploited for the measurements of quantities uantities

other than the beam distribution as well.other than the beam distribution as well.

19


Synchrotron RadiationSynchrotron RadiationMonitorsMonitors


F. Sannibale

•• Synchrotron radiation, very abundant in electron and positron aSynchrotron radiation, very abundant in electron and positron accelerators and ccelerators and present in very high energy proton storage rings, is widely usedpresent in very high energy proton storage rings, is widely used for transverse for transverse

and longitudinal beam profile measurements. and longitudinal beam profile measurements.

•• One of the appealing features of such One of the appealing features of such diagnostic systems is that they are nondiagnostic systems is that they are non--

invasive.invasive.

ALSALS

•• The resolution of these monitors are The resolution of these monitors are limited by the geometry of the system and limited by the geometry of the system and

by the radiation diffraction.by the radiation diffraction.•• The geometric limitation requires The geometric limitation requires small aperture systems while the small aperture systems while the

diffraction term requires large diffraction term requires large apertures and shorter photon apertures and shorter photon

wavelengths. Tradeoff solutions must wavelengths. Tradeoff solutions must be adopted.be adopted.

•• Typical resolutions in electron Typical resolutions in electron storage rings using hard xstorage rings using hard x--ray ray

photons range between few and photons range between few and tens of microns.tens of microns.

20


Other Beam Profile MonitorsOther Beam Profile MonitorsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• The simplest beam profile monitor is probably the one using The simplest beam profile monitor is probably the one using fluorescent fluorescent screensscreens intercepting the beam.intercepting the beam.

•• The beam particles hitting the screen material excite the atomsThe beam particles hitting the screen material excite the atoms that that subsequently radiate a photon in the visible range when decayingsubsequently radiate a photon in the visible range when decaying back to the back to the

ground state. ground state. •• The resulting image of the beam on the screen is then viewed byThe resulting image of the beam on the screen is then viewed by a a ccdccdcamera and eventually digitized by a frame grabber for further acamera and eventually digitized by a frame grabber for further analysis.nalysis.

•• Such monitors are destructive and typically are used only in beSuch monitors are destructive and typically are used only in beam am transferlinestransferlines..

••Another category of beam profile monitors are the Another category of beam profile monitors are the ionization chambersionization chambers..

•• In this monitor, a gas in a dedicated portion of the vacuum chaIn this monitor, a gas in a dedicated portion of the vacuum chamber is mber is ionized by the passage of the beam.ionized by the passage of the beam.

Depending on the scheme used, either the electrons or the ionizeDepending on the scheme used, either the electrons or the ionized atoms can d atoms can be detected for the beam profile reconstruction. Time of flight be detected for the beam profile reconstruction. Time of flight analysis of the analysis of the

ionized particles are usually necessary.ionized particles are usually necessary.

•• Because of their Because of their perturbativeperturbative nature, these monitors are mainly used in nature, these monitors are mainly used in single pass accelerators.single pass accelerators.

21

Image fromMax Planck Institute web site

Photocathode

STREAK CAMERA SchemeSTREAK CAMERA Scheme


PhotonPhoton--BasedBasedLongitudinal Profile MonitorsLongitudinal Profile Monitors


F. Sannibale

•• In photonIn photon--based longitudinal beam profile monitors, detectors such as based longitudinal beam profile monitors, detectors such as streak camerasstreak cameras, , fast photodiodes and fast photodiodes and photomultipliersphotomultipliers are used.are used.

In the streak camera case, time resolution of several hundreds oIn the streak camera case, time resolution of several hundreds of f fsfs can be can be achieved.achieved.

APSAPS

LEPLEPESRFESRF

•• Streak cameras with an additional couple of Streak cameras with an additional couple of sweeping electrodes (orthogonal to the other sweeping electrodes (orthogonal to the other

one) have single bunchone) have single bunch--single turn capabilities single turn capabilities and can be used for the characterization of and can be used for the characterization of

single and single and multibunchmultibunch intabililitiesintabililities..

22


More on LongitudinalMore on LongitudinalProfile MonitorsProfile Monitors


F. Sannibale

•• FemtosecondsFemtoseconds resolution resolution (or even smaller) can be (or even smaller) can be

achieved by achieved by interferometricinterferometrictechniques involving techniques involving

coherent light in the Farcoherent light in the Far--IR IR (coherent synchrotron (coherent synchrotron

radiation, coherent transition radiation, coherent transition radiation, …) or by electroradiation, …) or by electro--

optic techniques using nonoptic techniques using non--linear crystals and laser linear crystals and laser

probing.probing.

•• For relatively long bunches ~ 100 For relatively long bunches ~ 100 psps or longer or longer electromagnetic pickups can be efficiently electromagnetic pickups can be efficiently

used.used.DADAΦΦΦΦΦΦΦΦNENE

AccumulatorAccumulator

•• In this example, the beam inside the DAIn this example, the beam inside the DAΦΦΦΦΦΦΦΦNE NE Accumulator (~ 150 Accumulator (~ 150 psps rmsrms) is measured by ) is measured by

using the signal from a using the signal from a striplinestripline..

L’OASIS L’OASIS -- LBLLBL

L’OASIS L’OASIS -- LBLLBL

23


Current Monitors: Faraday CupCurrent Monitors: Faraday Cupand Current Transformersand Current Transformers

Real Accelerators Errors & Diagnostics

F. Sannibale

•• Current transformersCurrent transformers are used for measuring the AC component of the are used for measuring the AC component of the beam current:beam current:

L

RforI

N

RI

RLi

RLi

N

RV beambeamOUT >>≅

+= ω

ωω

1

beamI

N

OUTVR

BERGOZBERGOZ

•• Conceptually, the Conceptually, the Faraday cupFaraday cup is the simplest among the current monitors.is the simplest among the current monitors.

•• Electrostatic fields with the proper sign can be Electrostatic fields with the proper sign can be added in order to avoid that primary and added in order to avoid that primary and

secondary (emission) charged particles can secondary (emission) charged particles can leave the cup affecting the measurement.leave the cup affecting the measurement.

•• For short bunches, if the shape of the bunch needs to be measurFor short bunches, if the shape of the bunch needs to be measured as well, ed as well, the FC has to be designed as a transmission line in order to prethe FC has to be designed as a transmission line in order to present a good sent a good

high frequency response.high frequency response.

beamI

OUTVcore

typermeabiliHigh

OutV“Cup” electrode

24


Beam Current Monitors:Beam Current Monitors:The DC Current TransformerThe DC Current Transformer


F. Sannibale

•• For measuring the average beam current (DC component), the For measuring the average beam current (DC component), the parametric parametric current transformercurrent transformer or or DC current transformerDC current transformer (DCCT) is used:(DCCT) is used:

•• The DCCT uses two high The DCCT uses two high permeability cores driven to permeability cores driven to

saturation by a low frequency saturation by a low frequency current modulation.current modulation.

Demodulator

beamI

OutV.2 harnd

Hz200~

•• The signals from two secondary coils The signals from two secondary coils of the cores are mutually subtracted.of the cores are mutually subtracted.

•• Because of the nonBecause of the non--linear linear magnetization curve of the core magnetization curve of the core

material, this difference signal is zero material, this difference signal is zero only when the beam current is zero.only when the beam current is zero.•• In the presence of beam current this difference signal is nonIn the presence of beam current this difference signal is non--zero and in zero and in

particular shows a second harmonic component.particular shows a second harmonic component.•• A current proportional to the amplitude of this component is feA current proportional to the amplitude of this component is fed back into a d back into a

third coil in order to compensate for the beam current and to mathird coil in order to compensate for the beam current and to make the ke the difference signal zero.difference signal zero.

•• At equilibrium, the current flowing in this third coil is equalAt equilibrium, the current flowing in this third coil is equal in amplitude to in amplitude to the beam current but opposite in sign.the beam current but opposite in sign.

25

DADAΦΦΦΦΦΦΦΦNENE--LINACLINAC

ZZ = = RR


Beam Current Monitors:Beam Current Monitors:Wall Current Monitors.Wall Current Monitors.


F. Sannibale

•• The bandThe band--width of such monitors is limited to width of such monitors is limited to few GHz.few GHz.

•• Additionally, in the described configuration Additionally, in the described configuration they can radiate and/or pickthey can radiate and/or pick--up high frequency up high frequency

electromagnetic noise.electromagnetic noise.

OUTV

beamI

ZChamberVacuum

beamOUT IZV =FNALFNAL

ElectronsElectrons

PositronsPositrons

•• For limiting such a noise, a metallic For limiting such a noise, a metallic shield loaded with ferrites (inductive shield loaded with ferrites (inductive

loading) can be used.loading) can be used.

FerritesShieldMetallic

26


Beam SpectrumBeam SpectrumReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• In the presence of In the presence of betatronbetatron and/or synchrotron oscillations, sidebands and/or synchrotron oscillations, sidebands around each of the revolution harmonics will appear at the frequaround each of the revolution harmonics will appear at the frequencies:encies:

•• By detecting the passage of a By detecting the passage of a particle at a fixed particle at a fixed azimuthalazimuthal

position the following time domain position the following time domain signal can be observed.signal can be observed.

•• By Fourier transforming or by By Fourier transforming or by using a spectrum analyzer, the using a spectrum analyzer, the same signal in the frequency same signal in the frequency

domain will appear as:domain will appear as:

03T 04T0 0T 02T t

I

……

L

v

Tf ==

00

1Revolution harmonicsRevolution harmonics

03 f 04 f0 0f 02 f f

df

dP

……

syxwQofpartfractionalnff wS ,,0 =±=

•• In the case of a multiIn the case of a multi --particle beam, because of the nonparticle beam, because of the non--zero momentum zero momentum spread and machine nonspread and machine non--linearitieslinearities, the particles have slightly different , the particles have slightly different

oscillation frequencies. As a consequence the spectral lines wiloscillation frequencies. As a consequence the spectral lines wil l show a finite l show a finite thickness.thickness.

InstabilityInstabilitysidebandssidebands

27


SchottkySchottky Noise MonitorsNoise MonitorsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• SchottkySchottky noise cannot be used in noise cannot be used in electron and positron machines electron and positron machines

because in those accelerators, the because in those accelerators, the noise due to synchrotron radiation noise due to synchrotron radiation quantum fluctuations is strong and quantum fluctuations is strong and

covers the covers the SchottkySchottky noise. noise.

FNALFNAL

•• By using resonant electromagnetic pickBy using resonant electromagnetic pick--ups ups (cavity or (cavity or waveguidewaveguide) the signal resulting from ) the signal resulting from the motion of all the particles can be detected. the motion of all the particles can be detected.

•• In fact, In fact, SchottkySchottky noise monitors are actually the main nonnoise monitors are actually the main non--invasive invasive diagnostic tool used in heavy particle storage rings. Quantitiesdiagnostic tool used in heavy particle storage rings. Quantities that can be that can be

measured include longitudinal and transverse tunes, momentum sprmeasured include longitudinal and transverse tunes, momentum spread and ead and beam current.beam current.

•• Because the motion of the particles is essentially Because the motion of the particles is essentially independent, such a signal appears as a noise and independent, such a signal appears as a noise and

it is usually referred as the it is usually referred as the SchottkySchottky noise noise (SN). (SN). SN find applications in beam diagnosticsSN find applications in beam diagnostics

FNALFNAL

Synchrotronsidebands

Revolutionharmonic

28


Tune MeasurementTune MeasurementReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• In electron and positron machines, in order to measure the In electron and positron machines, in order to measure the betatronbetatron tunes tunes in the absence of instabilities, coherent beam oscillations needin the absence of instabilities, coherent beam oscillations need to be excited.to be excited.

•• Synchrotron tune can be measured by modulating the RF phase or Synchrotron tune can be measured by modulating the RF phase or amplitude and by measuring the induced sidebands using the sum samplitude and by measuring the induced sidebands using the sum signal ignal from a pickfrom a pick--up.Theup.The same detection part of the same detection part of the betatronbetatron tune measurement tune measurement

system can be used.system can be used.


HorizontalVertical

0π

KICKER

BAND-PASSFILTER

0

0 0

π

BEAMPOSITIONMONITOR

AMPLITUDE

A

PHASE

NETWORKANALYZERHP 4195 A

RATIO B/A

RF OUT B

100 WCLASS A

100 WCLASS A

29


Beam Characterization:Beam Characterization:Chromaticity MeasurementChromaticity Measurement


F. Sannibale

Chromaticity is measured in storage rings by changing the momentChromaticity is measured in storage rings by changing the momentum um (energy) of the beam and by recording the induced tune variation(energy) of the beam and by recording the induced tune variations.s.

The beam momentum is usually changed by varying the RF frequencyThe beam momentum is usually changed by varying the RF frequency..In this way, the revolution period is modified and the particlesIn this way, the revolution period is modified and the particles are forced into are forced into trajectories with different curvature in the dipole magnets. Thitrajectories with different curvature in the dipole magnets. This can happen s can happen

only if the particles change their momentum.only if the particles change their momentum.

RF

RF

CCCC f

f

f

f

T

T

L

L

p

p ∆=

∆−=∆=∆=∆

ηηηη1111

0

0

000

yxwpp

Qww ,

0

=∆∆

=ξ

RHICRHIC

Fits

The experimental The experimental data are fitted by a data are fitted by a

polynomial function.polynomial function.The fitting function The fitting function

calculated at the calculated at the nominal momentum nominal momentum

gives the linear gives the linear chromaticity.chromaticity.

30


Orbit MeasurementsOrbit MeasurementsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• In most of acceleration applications, the beam orbit needs to bIn most of acceleration applications, the beam orbit needs to be very stable.e very stable.In In colliderscolliders, counter, counter--rotating beams with transverse size in the nanometer scale rotating beams with transverse size in the nanometer scale

need to overlap for collision, while in light sources orbit stabneed to overlap for collision, while in light sources orbit stability requirements are ility requirements are often on the order of a micron. often on the order of a micron.

•• We saw how accelerator imperfections can generate orbit distortWe saw how accelerator imperfections can generate orbit distort ions.ions.•• Such orbits need to be carefully Such orbits need to be carefully measuredmeasured and and correctedcorrected. .

In addition, orbit feedbacks are often used to ensure the requirIn addition, orbit feedbacks are often used to ensure the required stability.ed stability.•• In circular machines, the transverse beam trajectory can be appIn circular machines, the transverse beam trajectory can be approximated by a roximated by a

sinusoid oscillating at the sinusoid oscillating at the betatronbetatron frequency. frequency. NyquistNyquist theorem states that we need to sample the orbit in a number of theorem states that we need to sample the orbit in a number of positions positions

at least twice the at least twice the betatronbetatron tune number. With some contingency, at least four tune number. With some contingency, at least four BPMsBPMs per 2per 2ππππππππ betatronbetatron phase advance are used in circular and linear accelerators.phase advance are used in circular and linear accelerators.•• Absolute orbit measurements suffer of Absolute orbit measurements suffer of accuracy limitations. In fact, the actual accuracy limitations. In fact, the actual

center of magnets and center of magnets and BPMsBPMs is not exactly is not exactly known. known.

Measured closed orbits are often referred Measured closed orbits are often referred to a “ golden orbit” , which is usually to a “ golden orbit” , which is usually

obtained by the obtained by the beambeam--based based alignment of alignment of the beam to the center of the the beam to the center of the quadrupolesquadrupoles..

NSRL NSRL -- HefeiHefei

31


Momentum and MomentumMomentum and MomentumSpread MeasurementSpread Measurement


F. Sannibale

•• In In linacslinacs and and transferlinestransferlines the momentum and momentum the momentum and momentum spread are mainly measured by spectrometer systems.spread are mainly measured by spectrometer systems.

•• The beam enters in the field of a dipole magnet The beam enters in the field of a dipole magnet where particles with different where particles with different momentamomenta follows follows

different trajectories.different trajectories.

0ppBLUE <

0ppRED >

Detector

Bend•• The particle position is then measured on a detector The particle position is then measured on a detector

downstream the magnet.downstream the magnet.

•• The spectrometer resolution is limited by the intrinsic beam siThe spectrometer resolution is limited by the intrinsic beam size at the ze at the detector plane and by field nondetector plane and by field non--linearitieslinearities..


Secondary emissionhodoscope

Spectrometercontrol window

DADAΦΦΦΦΦΦΦΦNENE--LinacLinac

32


Momentum Spread & Momentum Spread & EmittanceEmittanceMeasurement in RingsMeasurement in Rings


F. Sannibale

•• In electron and positron storage rings the equilibrium beam In electron and positron storage rings the equilibrium beam emittanceemittance and and the momentum spread can be measured by the combined measurement the momentum spread can be measured by the combined measurement of at of at

least two transverse beam profiles at two different ring locatioleast two transverse beam profiles at two different ring locations.ns.

2,11

12/12

=≡

+

+= indexsystemi

px

pxixiirms

ση

κεβ

212

221

21

22

22

21

1 xxxx

xrmsxrmsx

xx

ηβηβηη

κεε

−−

=+

=

212

221

22

112

2

2

xxxx

xrmsxrmsp xx

p ηβηβββσ

−−

=

ε: emittance

p: momentum

β: beta function

η: dispersion

κ: emittance ratio

xrms: rms beam size

•• The beam size at a particular The beam size at a particular azhimutalazhimutal position is given by:position is given by:

•• If the beam size is measured in two different points of the rinIf the beam size is measured in two different points of the ring and the g and the optical functions at such points are known, then:optical functions at such points are known, then:

33


EmittanceEmittance MeasurementMeasurementIn In LinacsLinacs and and TranferlinesTranferlines


F. Sannibale

•• A “ popular” technique for measuring the A “ popular” technique for measuring the emittanceemittance in in linacslinacs or or transferlinestransferlinesuses the souses the so--called “called “ three gradient methodthree gradient method” . ” .

•• The gradient (focusingThe gradient (focusing--defocusing strength) defocusing strength) of a of a quadrupolequadrupole is changed and the related is changed and the related transverse beam profiles are recorded by a transverse beam profiles are recorded by a

detector downstream the detector downstream the quadrupolequadrupole. .

Quadrupole

Beam profilemonitor

•• The measurement requires a minimum of 3 different The measurement requires a minimum of 3 different quadrupolequadrupole gradients but gradients but the accuracy can be improved if more points are taken. the accuracy can be improved if more points are taken.

•• The beam size at the detector is defined by the The beam size at the detector is defined by the beam beam emittanceemittance and by the local beta function. and by the local beta function.

The The emittanceemittance is an invariant while the beta is an invariant while the beta changes with the changing changes with the changing quadrupolequadrupole gradient. gradient.

•• An analytical expression linking the transverse An analytical expression linking the transverse profiles with the beam profiles with the beam emittanceemittance can be derived can be derived

and used for fitting the experimental data. and used for fitting the experimental data.

•• From the fit, the values for the From the fit, the values for the emittanceemittance and and for the optical functions at the for the optical functions at the quadrupolequadrupole

position can be finally extracted.position can be finally extracted.

DADAΦΦΦΦΦΦΦΦNENELINACLINAC

34


More MeasurementsMore MeasurementsReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• Lifetime measurementsLifetime measurements: by current monitors, by beam loss monitors, by : by current monitors, by beam loss monitors, by modifying beam parameters in order to discriminate among differemodifying beam parameters in order to discriminate among different nt

contributions, …contributions, …

•• Coupling impedance measurementsCoupling impedance measurements: by characterizing the tune shift on : by characterizing the tune shift on current, by measuring instability thresholds, by energy spread current, by measuring instability thresholds, by energy spread

measurements, …measurements, …

•• Optical function measurementsOptical function measurements: by single : by single quadrupolequadrupole gradient perturbation, gradient perturbation, by phase advance between position monitors, by response matrix, by phase advance between position monitors, by response matrix, by energy by energy

momentum variation for dispersion function measurement, …momentum variation for dispersion function measurement, …

•• Momentum acceptance measurementsMomentum acceptance measurements: by changing the particle momentum : by changing the particle momentum in combination with lifetime measurements, by modifying accelerain combination with lifetime measurements, by modifying accelerator tor

parameters for discriminating among different contributions, …parameters for discriminating among different contributions, …

•• NonNon--linearitieslinearities and dynamic aperture measurementsand dynamic aperture measurements: by kicking the beam : by kicking the beam transversely and characterizing the tune shift on amplitude, by transversely and characterizing the tune shift on amplitude, by frequency frequency

map analysis, by demap analysis, by de--coherence measurements, …coherence measurements, …

•• Transverse coupling measurementsTransverse coupling measurements: by transverse beam profile monitors, : by transverse beam profile monitors, by response matrix, by closest tune approach, …by response matrix, by closest tune approach, …

•• ……

35


ReferencesReferencesReal AcceleratorsErrors & Diagnostics

F. Sannibale

M. M. SerioSerio, “, “ DiagnosticaDiagnostica e e misuremisure” , Seminars on DA” , Seminars on DAΦΦΦΦΦΦΦΦNE, February 2000.NE, February 2000.

M. M. ZolotorevZolotorev, “ Radiation and Acceleration” , tutorial, LBL 2005., “ Radiation and Acceleration” , tutorial, LBL 2005.

R. R. LittauerLittauer, “ Beam Instrumentation” , Proc. Physics of High Energy Particle , “ Beam Instrumentation” , Proc. Physics of High Energy Particle Accelerators (Stanford, CA, 1982); AIP Conf. Accelerators (Stanford, CA, 1982); AIP Conf. PorcPorc. 105 (1982) 869.. 105 (1982) 869.

M. M. MintyMinty, “ Diagnostics” , CAS Synchrotron Radiation and Free, “ Diagnostics” , CAS Synchrotron Radiation and Free--Electron Electron Lasers, Lasers, BrunnenBrunnen, Switzerland, July 2003., Switzerland, July 2003.

36


Possible HomeworkPossible HomeworkReal AcceleratorsErrors & Diagnostics

F. Sannibale

•• A misaligned A misaligned quadrupolequadrupole in an electron storage rings with 5.2 horizontal in an electron storage rings with 5.2 horizontal tune generates a horizontal closed orbit distortion of 2 mm at itune generates a horizontal closed orbit distortion of 2 mm at i ts own ts own

position. Calculate the kick that a corrector magnet inside the position. Calculate the kick that a corrector magnet inside the quadrupolequadrupoleneeds to apply for correcting the orbit. The horizontal beta at needs to apply for correcting the orbit. The horizontal beta at the the quadrupolequadrupole

is 3 m. is 3 m.

•• Describe the shape of the pulse from a matched Describe the shape of the pulse from a matched striplinestripline of 5 cm length of 5 cm length detecting a uniform distributed beam with 2 ns total length.detecting a uniform distributed beam with 2 ns total length.

•• Define the electronic circuit equivalent to a resistive wall cuDefine the electronic circuit equivalent to a resistive wall current monitor rrent monitor with a ferrite loaded shield. Calculate the frequency response owith a ferrite loaded shield. Calculate the frequency response of such a f such a

monitor.monitor.

•• Calculate the length of the detector of the FNAL Injector Calculate the length of the detector of the FNAL Injector striplinestripline in the in the figure on the “figure on the “ striplinestripline electrode” viewgraph.electrode” viewgraph.

1



Light Sources.Light Sources.Brightness and Insertion DevicesBrightness and Insertion Devices


Thanks to Herman Thanks to Herman WinickWinick and David Robinand David Robin

2


IntroductionIntroductionLight Sources Brightness & ID F. Sannibale

•• However other researchers soon realized that synchrotron However other researchers soon realized that synchrotron radiation was the brightest source of infrared, ultraviolet, andradiation was the brightest source of infrared, ultraviolet, and xx--

rays, and that could be very useful for studying matter on the rays, and that could be very useful for studying matter on the scale ofscale of atoms and moleculesatoms and molecules..

•• Electron accelerators were initially developed to probe Electron accelerators were initially developed to probe elementary elementary ((subnuclearsubnuclear) particles ) particles for the study of the fundamental nature of matter, for the study of the fundamental nature of matter,

space, time, and energy.space, time, and energy.

In the earlier times, synchrotron In the earlier times, synchrotron radiation was just considered as a radiation was just considered as a

waste product waste product limiting the limiting the performance achievable with lepton performance achievable with lepton

machines.machines.

••The first time synchrotron radiation was observed in an acceleraThe first time synchrotron radiation was observed in an accelerator was in tor was in 1947 from the 70 1947 from the 70 MeVMeV electron beam at the General Electric Synchrotron in electron beam at the General Electric Synchrotron in

New York State.New York State.

3


Interaction ofInteraction ofPhoton’s with MatterPhoton’s with Matter

Light SourcesBrightness & IDF. Sannibale

RadiographyRadiography

DiffractionDiffraction

Photoelectric EffectPhotoelectric Effect Compton ScatteringCompton Scattering

4


What is Synchrotron Radiation?What is Synchrotron Radiation?Light SourcesBrightness & IDF. Sannibale

•• We already showed that We already showed that synchrotron radiation is synchrotron radiation is

electromagnetic radiation emitted electromagnetic radiation emitted when charged particles are when charged particles are radiallyradially

acceleratedaccelerated (move on a curved (move on a curved path).path).

Electrons Electrons acceleratingaccelerating by by running up and down in a running up and down in a radio antenna emit radio radio antenna emit radio waves (long wavelength waves (long wavelength electromagnetic waves)electromagnetic waves)

Both cases are manifestation of the same physical phenomenon:Both cases are manifestation of the same physical phenomenon:Charged particles radiate when acceleratedCharged particles radiate when accelerated..

5


Why Do Particles RadiateWhy Do Particles Radiateunder Acceleration?under Acceleration?


F. Sannibale

•• We already saw that according to quantum field We already saw that according to quantum field theory, a particle moving in the free space can be theory, a particle moving in the free space can be considered as “ surrounded” by a cloud of considered as “ surrounded” by a cloud of virtual virtual

photonsphotons that appear and disappear and that that appear and disappear and that indissolubly travel with it.indissolubly travel with it.

γ

γ

−e

•• When When acceleratedaccelerated, the particle receives a “ kick” that can separate it from the , the particle receives a “ kick” that can separate it from the photons that become real and independently observable.photons that become real and independently observable.

In the field of the magnets in a In the field of the magnets in a synchrotron, charged particles synchrotron, charged particles

moves on a curved trajectory. The moves on a curved trajectory. The transverse acceleration, if strong transverse acceleration, if strong enough, allows for the separation enough, allows for the separation

and and synchrotron radiationsynchrotron radiation is is generated.generated.

••Lighter particles are “ easier” to accelerate and radiate photonsLighter particles are “ easier” to accelerate and radiate photons more more efficiently than heavier particles.efficiently than heavier particles.

6


The Classical PictureThe Classical PictureLight SourcesBrightness & IDF. Sannibale

•• Historically, the whole theory was developed well before quantuHistorically, the whole theory was developed well before quantum m mechanics was even conceived:mechanics was even conceived:

•• The description of synchrotron radiation presented in the previThe description of synchrotron radiation presented in the previous ous viewgraph made use of quantum field theory.viewgraph made use of quantum field theory.

PowerLarmor

ac

qP 2

30

2

6πε=

-- in in 1897 Joseph 1897 Joseph LarmorLarmor derived the derived the expression for the instantaneous total expression for the instantaneous total

power radiated by an accelerated charged power radiated by an accelerated charged particle.particle.

-- and in and in 1898 Alfred 1898 Alfred LienardLienard(before the relativity theory!) (before the relativity theory!) extended extended Larmor’sLarmor’s result to result to

the case of a relativistic the case of a relativistic particle undergoing particle undergoing

centripetal acceleration in a centripetal acceleration in a circular trajectorycircular trajectory

18571857--19421942

18691869--19581958

7


Longitudinal vs.Longitudinal vs.Transverse AccelerationTransverse Acceleration


22

3200

2

6

= ⊥⊥ dt

d

cm

qP

pγπε

2

||

3200

2

|| 6

=

dt

d

cm

qP

p

πεnegligible!

( )radiuscurvatureq

cP ==⊥ ρ

ρβγ

πε 2

42

06

•• Radiated power for transverse acceleration Radiated power for transverse acceleration increases dramatically with increases dramatically with energyenergy. This sets a practical limit for the maximum energy obtainable . This sets a practical limit for the maximum energy obtainable with a with a

storage ring, but makes the construction of synchrotron light sostorage ring, but makes the construction of synchrotron light sources urces extremely appealing!extremely appealing!

8


The BrightnessThe Brightnessof a Light Sourceof a Light Source


FluxFlux = # of photons in given = # of photons in given ∆λ∆λ∆λ∆λ∆λ∆λ∆λ∆λ//λλλλλλλλsecsec

BrightnessBrightness = # of photons in given = # of photons in given ∆λ∆λ∆λ∆λ∆λ∆λ∆λ∆λ//λλλλλλλλsec, mrad sec, mrad θθθθθθθθ, mrad , mrad ϕϕϕϕϕϕϕϕ, mm, mm22

Ω== dSdBrightnessd

NdFlux

λ

•• In one of the previous lectures, we already dealt with the concIn one of the previous lectures, we already dealt with the concept of ept of brightnessbrightnessand showed how this quantity is the one of the main parameters fand showed how this quantity is the one of the main parameters for the or the

characterization of a particle source.characterization of a particle source.•• We remind that We remind that brightness is definedbrightness is defined as the density of particle on the 6as the density of particle on the 6--D phase D phase

spacespace..•• The same definition applies to the photon caseThe same definition applies to the photon case, just taking into account that , just taking into account that

photons are bosons and that the photons are bosons and that the PauliPauli exclusion principle does not apply.exclusion principle does not apply.

•• This is an important advantage because, at least from the pointThis is an important advantage because, at least from the point of view of of view of quantum mechanics, no limitation to achievable photon brightnessquantum mechanics, no limitation to achievable photon brightness exists.exists.

•• From the above definitions, one can see that for a given flux, From the above definitions, one can see that for a given flux, sources with a sources with a smaller smaller emittanceemittance will have a larger brightness.will have a larger brightness.

9


How Bright is aHow Bright is aSynchrotron Light Source?Synchrotron Light Source?


10


Synchrotron RadiationSynchrotron RadiationAngular DistributionAngular Distribution


•• Radiation becomes more focused at higher energies. Radiation becomes more focused at higher energies.

Cone apertureCone aperture~ 1/~ 1/γγγγγγγγ

11


Time CompressionTime CompressionLight SourcesBrightness & IDF. Sannibale

12


Typical BandTypical Band--WidthWidthOf Synchrotron LightOf Synchrotron Light


t∆=∆ 1ω

•• Example for an electron ring with 1.9 Example for an electron ring with 1.9 GeVGeV and with a bending radius of 5 m: and with a bending radius of 5 m:

11819 101.3102.37.2 −− ×≅∆×≅∆≅ sstmml ω

nmf

cHzf

MAXMINMAX 61.0109.4

217 ≅=⇔×≅∆≈ λ

πω

θ

θ

γθ 1

2×≈

γρθρ 2≈=l

Very broad band!Very broad band!

13


Synchrotron RadiationSynchrotron RadiationElectromagnetic SpectrumElectromagnetic Spectrum


Coherent Synchrotron RadiationCoherent Synchrotron RadiationTHzTHz Synchrotron Light SourcesSynchrotron Light Sources

Synchrotron Light SourcesSynchrotron Light Sources

14


PolarizationPolarizationLight SourcesBrightness & IDF. Sannibale

15


What Properties Make SynchrotronWhat Properties Make SynchrotronRadiation so Useful?Radiation so Useful?

Light Sources Brightness & ID F. Sannibale

•• High brightness and fluxHigh brightness and flux

SR offers many characteristics of visible lasers SR offers many characteristics of visible lasers but into the xbut into the x--ray regime!ray regime!

•• Wide energy spectrumWide energy spectrum

•• Highly polarized and short pulsesHighly polarized and short pulses

Recapitulating the main properties of synchrotron radiation:Recapitulating the main properties of synchrotron radiation:

•• Partial coherencePartial coherence

•• High StabilityHigh Stability

16


How to ExploitHow to ExploitSynchrotron RadiationSynchrotron Radiation


Modern synchrotron light sources are accelerators optimized for Modern synchrotron light sources are accelerators optimized for the the production of synchrotron radiation.production of synchrotron radiation.

ALS LINACALS LINAC

ALSALS

ALS BoosterALS Booster

CIRCECIRCE

ESRFESRF-- FranceFrance

SPRING 8 SPRING 8 JapanJapan

APS APS -- USAUSA

17


ApplicationsApplicationsLight SourcesBrightness & IDF. Sannibale

•• MedicineMedicine•• BiologyBiology•• ChemistryChemistry•• Material ScienceMaterial Science•• Environmental ScienceEnvironmental Science•• and much moreand much more

18


Materials ScienceMaterials ScienceLight SourcesBrightness & IDF. Sannibale

Using SR to learn Using SR to learn how high how high

temperature temperature superconductors superconductors

workwork

Using SR to make Using SR to make miniature mechanical and miniature mechanical and electromechanical deviceselectromechanical devices

Visualizing magnetic Visualizing magnetic bits on a computer bits on a computer

hard drivehard drive

Understanding Understanding how debris causes how debris causes damage to aircraft damage to aircraft

turbinesturbines

19


Next generationNext generationof semiconductorsof semiconductors


EUV LithographyEUV Lithography

ALSALS

20


Chemistry and BiologyChemistry and BiologyLight SourcesBrightness & IDF. Sannibale

Studying Studying Anthrax Toxin Anthrax Toxin components to components to

develop develop treatment in the treatment in the

advanced stages advanced stages of infection.of infection.

Measuring very low levels of mercury in Measuring very low levels of mercury in fish and determining its chemical form.fish and determining its chemical form.

Cholera toxin attacking a gut cellCholera toxin attacking a gut cell

21


Protein CrystallographyProtein CrystallographyLight SourcesBrightness & IDF. Sannibale

LeukemiaLeukemia

Drug DesignDrug DesignGLEEVECGLEEVEC Understanding how protein’s are madeUnderstanding how protein’s are made

RibosomesRibosomes make the stuff of life. They are make the stuff of life. They are the protein factories in every living the protein factories in every living

creature, and they churn out all proteins creature, and they churn out all proteins ranging from bacterial toxins to human ranging from bacterial toxins to human

digestive enzymes digestive enzymes

22


Cellular ImagingCellular ImagingLight SourcesBrightness & IDF. Sannibale

This is an image taken with the xThis is an image taken with the x--ray microscope of a malariaray microscope of a malaria--

infected blood cell. Researchers at infected blood cell. Researchers at Berkeley Lab use pictures like this Berkeley Lab use pictures like this to analyze what makes the malariato analyze what makes the malaria--

infected blood cells stick to the infected blood cells stick to the blood vessels. blood vessels.

23


BiomedicineBiomedicineLight SourcesBrightness & IDF. Sannibale

Image of a human coronary artery Image of a human coronary artery taken with synchrotron radiation at taken with synchrotron radiation at

SSRLSSRL

before estrogen loss after estrogen loss

Studies of osteoporosis at SSRLStudies of osteoporosis at SSRL

These studies make use of the These studies make use of the penetrating power of Xpenetrating power of X--rays, rays,

rather than their short rather than their short wavelengthwavelength

24


Art & ArchaeologyArt & ArchaeologyLight SourcesBrightness & IDF. Sannibale

Sulfuric acid causing the Sulfuric acid causing the decay of the decay of the VasaVasa, the , the

Swedish warship which sank Swedish warship which sank in Stockholm harbor in 1628in Stockholm harbor in 1628

Virgin, Child, and Saint JohnVirgin, Child, and Saint John A A renaissance panel painting by Jacopo renaissance panel painting by Jacopo

SellaioSellaio or or FilippinoFilippino LippiLippi being restored at being restored at the Cantor Art Centerthe Cantor Art Center

25


XX--rays have enabledrays have enabledseminal scientific discoveriesseminal scientific discoveries


18 Nobel Prizes Based on X-rayWork

Chemistry1936: Peter Debye1962: Max Purutz and Sir John

Kendrew1976 William Lipscomb1985 Herbert Hauptman and Jerome

Karle1988 Johann Deisenhofer, Robert

Huber and Hartmut Michel1997 Paul D. Boyer and John E.

Walker2003 Peter Agre and Roderick

Mackinnon

Physics1901 Wilhem Rontgen1914 Max von Laue1915 Sir William Bragg and son1917 Charles Barkla1924 Karl Siegbahm1927 Arthur Compton1981 Kai Siegbahn

Medicine1946 Hermann Muller1962 Frances Crick, James Watson

and Maurice Wilkins1979 Alan Cormack and Godrey

Hounsfield

26


20 000 Users World20 000 Users World--WideWideLight SourcesBrightness & IDF. Sannibale

•• 54 in operation in 19 countries used by more than 20,000 scient54 in operation in 19 countries used by more than 20,000 scient istsists•• 8 in construction 8 in construction

•• 11 in design/planning 11 in design/planning

For a list of SR facilities around the world seehttp://ssrl.slac.stanford.edu/SR_SOURCES.HTML

www.sesame.org.jo

27


Steep Growth in BrightnessSteep Growth in BrightnessLight SourcesBrightness & IDF. Sannibale

Wilhelm Conrad Wilhelm Conrad RöntgenRöntgen (1845(1845--1923) 1923)

18451845

28


Growth in XGrowth in X--ray Brightness Comparedray Brightness Comparedto Growth in Computing Speedto Growth in Computing Speed


One millionOne billion

A million more

Computing speedX-rays

1015

1014

1013

1012

1011

1010

109

108

107

106

105

104

103

102

101

100

10-1

1023

1022

1021

1020

1019

1018

1017

1016

1015

1014

1013

1012

1011

1010

109

108

10-7

1960 1970 1980 1990 2000

Computing speed

X-ray Brightness

X-ray Brightness

Computing speed

29


How to Optimize a How to Optimize a Synchrotron Light SourceSynchrotron Light Source


•• The ultimate performance parameter of a synchrotron light sourcThe ultimate performance parameter of a synchrotron light source is the e is the brightness. brightness.

•• The battle for the brightness maximization is fought in two froThe battle for the brightness maximization is fought in two fronts:nts:

–– In the storage ring, by increasing the current and designing neIn the storage ring, by increasing the current and designing new w lattices capable of smaller lattices capable of smaller emittancesemittances. Current of hundreds of . Current of hundreds of mAmA and and lattices with ~ 1 nm lattices with ~ 1 nm emittanceemittance are presently used.are presently used.

–– In the ring elements where the synchrotron radiation is actuallIn the ring elements where the synchrotron radiation is actually y generated: dipole magnets and insertion devices. And this is whegenerated: dipole magnets and insertion devices. And this is where re spectacular improvements have been achieved!spectacular improvements have been achieved!

•• Light sources are usually classified for increasing brightness Light sources are usually classified for increasing brightness as:as:

–– 11stst generationgeneration: x: x--ray tubes.ray tubes.

–– 22ndnd generationgeneration: “ parasitic” synchrotron radiation sources from : “ parasitic” synchrotron radiation sources from dipoles in dipoles in colliderscolliders..

–– 33rdrd generationgeneration: dedicated storage rings with insertion devices: dedicated storage rings with insertion devices

–– 44thth generationgeneration: free electron lasers: free electron lasers

30


How Synchrotron RadiationHow Synchrotron Radiationis Generated in Storage Ringsis Generated in Storage Rings


Continuous spectrum characterized Continuous spectrum characterized by by εεεεεεεεcc = critical energy= critical energy

εεεεεεεεcc(keV(keV) = 0.665 B(T)E) = 0.665 B(T)E22(GeV)(GeV)

For exampleFor example: :

for B = 1.35 T E = 2 for B = 1.35 T E = 2 GeVGeV εεεεεεεεcc = 3.6keV= 3.6keV

+ harmonics at higher energy+ harmonics at higher energy

QuasiQuasi--monochromatic spectrummonochromatic spectrum with with peaks at lower energy than a wigglerpeaks at lower energy than a wiggler

K = γϕ where ϕ is the angle in each is the angle in each polepole

λ1 = λu

2γ2(1 + ) ~ (fundamental)(fundamental)K2

2 γ2

λU

ε1 (keV) = 0.95 E2 (GeV)

K2λu(cm) (1 + )

2

bending magnet

t1

t2

Bending Magnet — A “Sweeping Searchlight”

Wiggler — Incoherent Superposition

Dipoles

t3 t4

(10-100) γ –1

t5

γ –1

Undulator — Coherent Interference

(γ N)–1

t1

t2



Dipoles

t3 t4

(10-100) γ –1

t5

γ –1


(γ N)–1

wiggler - incoherent superposition

t1

t2



Dipoles

t3 t4

(10-100) γ –1

t5

γ –1


(γ N)–1

undulator - coherent interference

31


Bend MagnetBend MagnetLight SourcesBrightness & IDF. Sannibale

““ C” shaped for C” shaped for allowing to the allowing to the radiation to exitradiation to exit

NormalNormal--ConductiveConductive~ 1.5 T Max~ 1.5 T Max

32


Bend Magnet SynchrotronBend Magnet SynchrotronRadiation SpectrumRadiation Spectrum


Universal functionUniversal functionSpectrum:Spectrum:

Critical frequencyCritical frequency

33


Dipoles for Hard XDipoles for Hard X--raysraysLight SourcesBrightness & IDF. Sannibale

At the Advanced Light Source three of the existing thirty six 1.At the Advanced Light Source three of the existing thirty six 1.3 T 3 T dipoles were replaced by threedipoles were replaced by three 5 T superconducting dipoles 5 T superconducting dipoles

(“(“ superbendssuperbends” ).” ).

SuperbendSuperbendwithout cryostatwithout cryostat

SuperbendSuperbendwith cryostatwith cryostat

300.0

B (

T)

-180.0 -60.0 60.0 180.0

0.51.01.52.02.53.03.54.04.55.05.56.0

distance along beam (mm)-300.0

0.0

SuperbendSuperbendmagnetic fieldmagnetic field

34

Bending MagnetBending Magnet


Spectrum Energy DependencySpectrum Energy DependencyLight SourcesBrightness & IDF. Sannibale

Remark: The Remark: The distribution for longer distribution for longer wavelengths does not wavelengths does not

depend on energy.depend on energy.

35


Planar Planar UndulatorsUndulatorsLight SourcesBrightness & IDF. Sannibale

ParticleParticletrajectorytrajectory

Permanent MagnetsPermanent Magnets

Invented byInvented byKlaus Klaus HalbachHalbach

19241924--20002000

36


UndulatorUndulator RadiationRadiationLight SourcesBrightness & IDF. Sannibale

Photons emitted by different poles interfere transforming the Photons emitted by different poles interfere transforming the continuous dipolecontinuous dipole--like spectrum into a discrete spectrumlike spectrum into a discrete spectrum

The interference condition requires that, while traveling along The interference condition requires that, while traveling along one period of one period of the the undulatorundulator, the electrons slip by one radiation wavelength with respect to, the electrons slip by one radiation wavelength with respect to

the (faster) photon.the (faster) photon.

37


From From UndulatorUndulator RadiationRadiationto Wiggler Radiationto Wiggler Radiation


The spectrum of the The spectrum of the undulatorundulator radiation:radiation:

poleeachinanglebendingtheiswhereK U

ρλϕγϕ2

≈=

harmonicK stU 12

12

2

21

+=

γλλ

depends strongly on the depends strongly on the strength parameter strength parameter KK::

eB

cm0γβρ =Remembering that:Remembering that: One can see that One can see that KK is is proportional to the field proportional to the field BB::

In a permanent magnet In a permanent magnet undulatorundulator, , BB and consequently and consequently KKcan be modified by changing the gap height. The larger the can be modified by changing the gap height. The larger the

gap the lower the field.gap the lower the field.

When When B B is increased, both is increased, both KK and the and the “ wiggling” inside the “ wiggling” inside the undulatorundulator increase as increase as wellwell . With the larger wiggling, the overlap . With the larger wiggling, the overlap

between the radiated field (between the radiated field (1/1/γγγγγγγγ cone) cone) decreases and the interference is reduced. decreases and the interference is reduced. For For KK >> 1>> 1 no interference is present and no interference is present and

the the undulatorundulator presents the continuum presents the continuum spectrum typical of the spectrum typical of the wigglerwiggler..

GapGap

βλ B

cm

eK U

02≈

38


Elliptically PolarizingElliptically PolarizingUndulatorsUndulators


ALS EPU50 (1998)ALS EPU50 (1998)Pure permanent Pure permanent

magnet technology, magnet technology, Elliptically polarizing Elliptically polarizing

capability.capability.

Such a device allows Such a device allows for the complete control for the complete control of the polarization from of the polarization from

linear in to elliptical.linear in to elliptical.

The arrays of The arrays of permanent magnets permanent magnets can be mechanically can be mechanically

shifted modifying shifted modifying the polarization of the polarization of the radiated light.the radiated light.

39


Future of Synchrotron RadiationFuture of Synchrotron RadiationLight SourcesBrightness & IDF. Sannibale

•• Shorter Pulse LengthsShorter Pulse Lengths-- FemtoFemto (10(10--1212) and ) and AttosecondAttosecond (10(10--1515))

•• Higher BrightnessHigher Brightness-- Free Electron LasersFree Electron Lasers

•• Terahertz (TTerahertz (T--rays) rays) -- Coherent Synchrotron RadiationCoherent Synchrotron Radiation

…… (?)(?)

40


Free Electron Laser BasicsFree Electron Laser BasicsLight SourcesBrightness & IDF. Sannibale

•• In free electron lasers (FEL), a relativistic electron beam andIn free electron lasers (FEL), a relativistic electron beam and a laser overlap and a laser overlap and travel simultaneously inside an travel simultaneously inside an undulatorundulator..

•• Oscillating through the Oscillating through the undulatorundulator, the electron bunch interacts with the laser and , the electron bunch interacts with the laser and in a minor way with its own electromagnetic field created via spin a minor way with its own electromagnetic field created via spontaneous ontaneous emission. Depending on the relative phase between radiation and emission. Depending on the relative phase between radiation and electron electron

oscillation, electrons experience either a deceleration or acceloscillation, electrons experience either a deceleration or acceleration. eration.

•• The laser is tuned at the frequency of one The laser is tuned at the frequency of one of the of the undulatorundulator harmonics. The whole harmonics. The whole

undulatorundulator is included inside an optical cavity is included inside an optical cavity composed by two reflecting mirrors located composed by two reflecting mirrors located

at the two at the two undulatorundulator extremes. extremes.

••In such a schemes the laser beam In such a schemes the laser beam bounces many times back and forward bounces many times back and forward

inside the cavity and has multiple inside the cavity and has multiple interactions with the electron beam.interactions with the electron beam.

•• Through this interaction a longitudinal fine structure, the so Through this interaction a longitudinal fine structure, the so called called micromicro--bunchingbunching, is established which amplifies the electromagnetic field at th, is established which amplifies the electromagnetic field at the laser e laser

frequency.frequency.

41


The SASE FEL SchemeThe SASE FEL SchemeLight SourcesBrightness & IDF. Sannibale

•• In the selfIn the self--amplified spontaneous emission (SASE) FEL, there is no laser andamplified spontaneous emission (SASE) FEL, there is no laser and the the electron beam interacts only with its own spontaneous emission. electron beam interacts only with its own spontaneous emission.

•• In order to obtain a large gain in the SASE In order to obtain a large gain in the SASE scheme, a long and expensive scheme, a long and expensive undulatorundulator is is

required. In a “ conventional” FEL the required. In a “ conventional” FEL the undulatorundulator is much shorter because the is much shorter because the laser beam is relaser beam is re--circulated many times circulated many times

inside the cavity. Unfortunately, the highest inside the cavity. Unfortunately, the highest frequency achievable with such a frequency achievable with such a

configuration is limited to the nearconfiguration is limited to the near--UV UV because of the absence of efficient large because of the absence of efficient large

incidence angle mirrors for shorter incidence angle mirrors for shorter wavelengths.wavelengths.

SaturationSaturation

•• For such a scheme to work, one has to guarantee a good electronFor such a scheme to work, one has to guarantee a good electron beam quality beam quality and a sufficient overlap between radiation pulse and electron buand a sufficient overlap between radiation pulse and electron bunch along the nch along the

undulatorundulator. To achieve that, one needs a low . To achieve that, one needs a low emittanceemittance, low energy spread electron , low energy spread electron beam with an extremely high charge density in conjunction with abeam with an extremely high charge density in conjunction with a very precise very precise

magnetic field and accurate beam steering through the magnetic field and accurate beam steering through the undulatorundulator. .

42

XX--FELFEL


(Some) FEL Projects(Some) FEL ProjectsLight SourcesBrightness & IDF. Sannibale

43


ReferencesReferencesLight SourcesBrightness & IDF. Sannibale

•• Technical Design Report (TDR) for TESLA, Technical Design Report (TDR) for TESLA, Part V Part V The XThe X--ray free electron laserray free electron laser

44


Possible ProblemsPossible ProblemsLight SourcesBrightness & IDF. Sannibale

•• Calculate the critical energy in Calculate the critical energy in eVeV for the ALS for the ALS superbendssuperbends knowing that the knowing that the electron beam energy is 1.9 electron beam energy is 1.9 GeVGeV, the field is 5 T and the total deflection angle , the field is 5 T and the total deflection angle

for the magnet is 10 deg. Remember that the photon energy is givfor the magnet is 10 deg. Remember that the photon energy is given by en by hfhf(with (with hh the Planck constant, 6.626068 × 10the Planck constant, 6.626068 × 10--3434 mm22 kg / s, and kg / s, and ff the photon the photon

frequency)frequency)

•• Always for the ALS case, calculate the critical energy for the Always for the ALS case, calculate the critical energy for the normal bends normal bends knowing that the bending radius is 4.957 m and the total deflectknowing that the bending radius is 4.957 m and the total deflect ion angle for ion angle for

the magnet is 10 deg.the magnet is 10 deg.

•• Using the universal spectrum for the bending magnet radiation, Using the universal spectrum for the bending magnet radiation, calculate for calculate for both the above cases, the maximum radiated power in 0.1% bandwidboth the above cases, the maximum radiated power in 0.1% bandwidth when th when

400 400 mAmA electrons are stored ( the ring length is 197 m). Indicate at welectrons are stored ( the ring length is 197 m). Indicate at which hich photon energy is the maximum located.photon energy is the maximum located.

1



CollidersColliders and Luminosityand Luminosity


2


IntroductionIntroductionColliders &Luminosity

F. Sannibale

CollidersColliders for high energy particle physics experiments are surely one of for high energy particle physics experiments are surely one of the the most important application of particle accelerators.most important application of particle accelerators.

Actually, the developments in particle accelerators and of elemeActually, the developments in particle accelerators and of elementary particle ntary particle physics probably represent one of the best examples of synergy bphysics probably represent one of the best examples of synergy between etween

different physics disciplines.different physics disciplines.

CollidersColliders can be characterized by the different nature of the colliding can be characterized by the different nature of the colliding particles (leptons or hadrons) and by the different accelerationparticles (leptons or hadrons) and by the different acceleration scheme used scheme used

(linear or circular)(linear or circular)

In existing lepton In existing lepton colliderscolliders, electrons collide with positrons and a significant , electrons collide with positrons and a significant R&D is undergoing for the definition of a possible scheme for a R&D is undergoing for the definition of a possible scheme for a muonmuon

collidercollider..HadronHadron colliderscolliders include, protons colliding with protons or antiinclude, protons colliding with protons or anti--protons and protons and

heavy ion heavy ion colliderscolliders. .

Higher collision energies can be achieved with Higher collision energies can be achieved with hadronhadron colliderscolliders but “ cleaner” but “ cleaner” measurements can be done with lepton measurements can be done with lepton colliderscolliders..

3


IntroductionIntroductionColliders &Luminosity

F. Sannibale

In electronIn electron--positron collisions the particles annihilate and all the energy positron collisions the particles annihilate and all the energy at at the center of mass system is available for the generation of elethe center of mass system is available for the generation of elementary mentary

particles. particles.

Such particle generation can happen only if it exists a particleSuch particle generation can happen only if it exists a particle with rest mass with rest mass energy equal to the collision energy at the center of mass systeenergy equal to the collision energy at the center of mass system.m.

The energy of the colliding beams can be tuned on the rest mass The energy of the colliding beams can be tuned on the rest mass energy of a energy of a known particle for studying its properties, or can be scanned foknown particle for studying its properties, or can be scanned for the research r the research

of unknown particles.of unknown particles.

In In hadronhadron colliderscolliders, the quarks in the hadrons interact during the collision , the quarks in the hadrons interact during the collision and generate other particles. Because each and generate other particles. Because each hadronhadron is a combination of three is a combination of three

quarks, quarks, simultanoeussimultanoeus generation of different particles is possible.generation of different particles is possible.

Most of the particles generated during a collisions usually haveMost of the particles generated during a collisions usually have a short a short lifetime and decay in other particles. Particles detectors are dlifetime and decay in other particles. Particles detectors are designed in order esigned in order

to measure the particle itself when possible or to measure the pto measure the particle itself when possible or to measure the particles articles generated during the decay.generated during the decay.

4


High Energy Physics DetectorsHigh Energy Physics DetectorsColliders &Luminosity

F. Sannibale

CDFCDF

CDFCDF

KLOEKLOE

KLOEKLOE

5


Energy at theEnergy at theCenter of Mass SystemCenter of Mass System

Colliders &Luminosity

F. Sannibale

• Two particles have equal rest mass m0.Center of Mass Frame (CMF): Velocities are equal and opposite, total energy is Ecm.

• The quantity is invariant.

• In the CMF, we have

• While in the LF:

• And after some algebra we can obtain for relativistic particles:

212 EEEcm ≅

( ) 2122

0212

12

1

2

21

~~22

~~2

~~~~PPcmPPPPPP +=++=+

( ) 222

21 cEPP CM=+( )2

21 PP +

( )pcEP CM ,21 = ( )pcEP CM −= ,22

Laboratory frame (LF):

( )111 ,2~

pcEP = ( )222 ,2~

pcEP =

6


Interaction RegionInteraction RegionReference FrameReference Frame


F. Sannibale

x, y, z ≡≡≡≡ Lab. Reference Frame

P1P2

s+s-

y ≡ ≡ ≡ ≡ vertical axis

x ≡ ≡ ≡ ≡ horizontal axis

IP z ≡≡≡≡ longitud. axisαααα ≡ ≡ ≡ ≡ crossing angle

7


Basic DefinitionsBasic DefinitionsColliders &Luminosity

F. Sannibale

Luminosity:Luminosity:Event RateEvent Rate

for afor aUnit Cross Section EventUnit Cross Section Event

Event RateEvent Rateper Unit Incident Fluxper Unit Incident Flux

per Target Particleper Target Particle

Cross Section:Cross Section:

8


Counter RotatingCounter RotatingBeams CaseBeams Case


F. Sannibale

L ==== 2v f R dx dydz dt n++++ x, y, z, t(((( )))) n−−−− x, y,z, t(((( ))))

n±±±± x, y, z, t(((( ))))

dxdy dz n±±±± x, y, z, t(((( )))) ==== N±±±±

HeadHead--on Collisionon CollisionCounterCounter--rotating Beams rotating Beams with Longitudinal Speed with Longitudinal Speed vv

----

Single BunchSingle Bunch--

Revolution Frequency Revolution Frequency ffRR--

9


GaussianGaussian BeamBeamSingle Bunch LuminositySingle Bunch Luminosity


F. Sannibale

n−−−− x, y, z, t(((( )))) ==== N−−−−e

−−−−x2

2σσσσx−−−−2 −−−−

y2

2σσσσ y−−−−2 −−−−

z−−−−vt(((( ))))2

2σσσσ z−−−−2

2ππππ(((( ))))32 σσσσx−−−−σσσσy−−−−σσσσz−−−−

n++++ x, y, z, t(((( )))) ==== N++++e

−−−−x2

2σσσσx++++2 −−−−

y2

2σσσσy++++2 −−−−

z++++vt(((( ))))2

2σσσσz++++2

2ππππ(((( ))))3

2 σσσσx++++σσσσ y++++σσσσz++++

σσσσ x±±±± ,σσσσy±±±± ≡≡≡≡ constants

L ==== f RN++++ N−−−−

2ππππ σσσσ x++++2 ++++ σσσσ x−−−−

2(((( )))) σσσσy++++2 ++++ σσσσ y−−−−

2(((( ))))σσσσ x++++ ==== σσσσx−−−− σσσσ y++++ ==== σσσσ y−−−−

L ==== f RN++++N−−−−

4ππππ σσσσxσσσσy

10


Two Main EffectsTwo Main EffectsColliders &Luminosity

F. Sannibale

Charge Related Effects:Charge Related Effects:Or Or beambeam--beambeam effects. effects.

Charge plays a major role, Charge plays a major role, limiting the achievable limiting the achievable

luminosity in most of luminosity in most of storage ring storage ring colliderscolliders..

How the “geometry” of How the “geometry” of the interaction point (IP) the interaction point (IP)

and the size of the and the size of the beams affect luminositybeams affect luminosity

Geometric Effects:Geometric Effects:

11


Geometric Luminosity:Geometric Luminosity:ColliderCollider ParametersParameters


F. Sannibale

σσσσ y∗∗∗∗ ==== κκκκ

1++++ κκκκεεεε ββββy

∗∗∗∗ σσσσx∗∗∗∗ ====

11++++ κκκκ

εεεε ββββx∗∗∗∗

ηηηηx ==== ηηηηy ==== 0

L ==== f RN++++ N−−−− 1 ++++ κκκκ(((( ))))

4ππππ εεεε κκκκ ββββx∗∗∗∗ ββββy

∗∗∗∗ ≅≅≅≅ fRN++++ N−−−−

4ππππ εεεε κκκκ ββββx∗∗∗∗ ββββy

∗∗∗∗

L ==== f RN++++N−−−−

4ππππ σσσσxσσσσy** **

12


Geometric Luminosity Geometric Luminosity Colliders &Luminosity

F. Sannibale

L = 2 f Rv

ccos α c2 − v2 sin2 α dxdydzdt f + f −

f + = N+e−

x cos α −z sin α( )2

2 σ x+2 x,z( )

− y2

2σ y+2 x ,z( )

− x sin α + z cosα −vt( )2

2σ z+2

2π( )32 σ x+ x, z( ) σ y + x, z( ) σ z+

σ k +2 = σ k +

∗2 1+x sin α + z cos α − ∆w+( )

βk +∗2

k = x, y

f − = N−e−

x − ∆x( ) cosα + z sin α[ ]2

2 σ x−2 x,z( )

− y− ∆y( )2

2σ y−2 x ,z( )

− − x − ∆x( )sin α + z cosα +v t −∆t( )[ ]2

2 σ z−2

2π( )32 σ x − x, z( ) σ y − x, z( ) σ z−

σ k −2 = σ k −

∗2 1 +−x sin α + z cos α − ∆w−( )

βk −∗2

k = x, y

•• Very low currentsVery low currents

•• NegligibleNegligiblebeambeam--beam effectsbeam effects

••Crossing angle Crossing angle αααααααα

••Horizontal & verticalHorizontal & verticalOffset Offset ∆∆∆∆∆∆∆∆xx and and ∆∆∆∆∆∆∆∆yy

••IR position IR position ∆∆∆∆∆∆∆∆tt

••Different beta Different beta star for the two star for the two

beamsbeams

13


BeamBeam--Beam EffectsBeam EffectsColliders &Luminosity

F. Sannibale

∆y

z

For a For a gaussiangaussian charge distribution:charge distribution:

∆ ′y = − 2N re ∆yγ

0

∞exp − ∆x 2

2σ x2 + w

− ∆y 2

2σ y2 + w

2σ y2 + w( )

3

2 2σ x2 + w( )

1

2

dw

∆ ′x = − 2N re ∆xγ

0

∞exp − ∆ x 2

2σ x2 + w

− ∆ y 2

2σ y2 + w

2σ y2 + w( )

1

2 2σ x2 + w( )

3

2

dw

[m]∆y

03

03

-100 µm 100 µm

10 mrad

-10 mrad

∆x = 0N = 8.9 1010

σy=20 µmE= 510 MeV

y∆

y′∆

14


Linear BeamLinear Beam--BeamBeamColliders &Luminosity

F. Sannibale

Q Fk ≡

1 0

− 1f k

1

k = x, y

Focusing Focusing QuadrupoleQuadrupole (thin lens):(thin lens):

BeamBeam--BeamBeamDeflectionDeflection

(off(off--center particles)center particles)

Linear BeamLinear Beam--Beam Tune ShiftBeam Tune Shift

ξ y+ =

N− re β y∗+

2π γ σ y− σ y

− + σ x−( )= ∆Qy ξ x

+ =N− re β x

∗+

2π γ σ x− σ y

− + σ x−( )= ∆Qx

∆y << σy ∆x << σx* *

∆ ′ y ≅ − 1f y

∆y f y = 2N re

γσ y σ x + σ y( )

∆ ′ x ≅ − 1f x

∆x f x = 2N re

γσ x σ x + σ y( )

-1

-1

* * *

* * *

15

4

4.5

5

5.5

6

4 4.5 5 5.5 6

Qy

Qx


Choice of the Working PointChoice of the Working PointColliders &Luminosity

F. Sannibale

m + n = resonance order

mQx + nQy = p m,n , p ∈NTune Tune ResonancesResonances

Working PointWorking PointBeamBeam--Beam TuneBeam Tune

FootprintFootprint

16


Nonlinear BeamNonlinear Beam--BeamBeamColliders &Luminosity

F. Sannibale

Nonlinear BeamNonlinear Beam--beam Kickbeam KickSynchroSynchro--betatronbetatron EffectsEffectsRadiation DampingRadiation DampingQuantum Fluctuations in Quantum Fluctuations in Synchrotron Radiation EmissionSynchrotron Radiation EmissionLattice Lattice NonlinearitiesNonlinearities((sextupolessextupoles, higher order , higher order multipolesmultipoles))RF RF NonlinearitiesNonlinearities

--------

--

--

Simulation CodesSimulation Codes

For example:For example:LIFETRACK by LIFETRACK by ShatilovShatilov

ororBBC by HirataBBC by Hirata

[m]∆y

03

03

-100 µm 100 µm

10 mrad

-10 mrad

y∆

y′∆

No analytical solutions for the nonNo analytical solutions for the non--linear beamlinear beam--beam case.beam case.

17


BeamBeam--Beam SimulationsBeam SimulationsColliders &Luminosity

F. Sannibale

5.14, 5.20 5.14, 5.21 5.14, 5.22

5.15, 5.20 5.15, 5.21 5.15, 5.22

5.16, 5.20 5.16, 5.21 5.16, 5.22

BeamBeam--beam simulations for beam simulations for the DAthe DAΦΦΦΦΦΦΦΦNE NE collidercollider..

Qx

Qy QQxx = 5.09 = 5.09 QQyy = 5.07= 5.07

QQxx = 5.15 = 5.15 QQyy = 5.21= 5.21Qx

Qy

18


Maximum Linear BeamMaximum Linear Beam--BeamBeamTune ShiftTune Shift


F. Sannibale

The beamThe beam--beam interaction actually sets the maximum beam interaction actually sets the maximum achievable luminosity in practically all the existing achievable luminosity in practically all the existing colliderscolliders..

No consistent and exhaustive theory exists. No consistent and exhaustive theory exists.

Estimate of the maximum achievable linear tune Estimate of the maximum achievable linear tune shift:shift:

•• Statistical analysis of the maximum linear tune Statistical analysis of the maximum linear tune shifts achieved in existing shifts achieved in existing colliderscolliders

•• SimulationsSimulations

The linear beamThe linear beam--beam parameter is used a measure of the beam parameter is used a measure of the strength of the beamstrength of the beam--beam interaction.beam interaction.

19


Single Bunch LuminositySingle Bunch LuminosityIn Max Linear Tune Shift RegimeIn Max Linear Tune Shift Regime


F. Sannibale

N+ = N− = N

σ x =1

1+ κε β x σ y =

κ1 + κ

ε β y

L = π γre

2

f R ε 1+κ( ) ξ2

βy∗

Equal tune shift Equal tune shift designdesign

L = f R

N + N −

4 π σ x σ y* * ξ w =

N re β w∗

2π γ σw σ y + σ x( ) w = x, y* * *

L = π γre

2

f Rε

1+κ1βy

+κβx

2

ξx ξy* *

β y∗

β x∗ = κ ⇔ ξ x = ξ y = ξ

20


Low Beta SchemeLow Beta SchemeColliders &Luminosity

F. Sannibale

-5

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3z (m)

βy

βx

(m)

Few centimeters vertical beta @ IP are “ routinely” obtained.Few centimeters vertical beta @ IP are “ routinely” obtained.

21


Hourglass EffectHourglass EffectColliders &Luminosity

F. Sannibale

In a low beta insertion, the beta function between the IP and thIn a low beta insertion, the beta function between the IP and the first e first quadrupolequadrupole is given by:is given by:

( )

+== ∗

∗

2

2 1y

yxyxy

zz

ββεβεσ( )

+= ∗

∗

2

1y

yy

zz

βββ

The The hourglass effecthourglass effect can can limit the luminosity limit the luminosity

achievable in low beta achievable in low beta schemes.schemes.

The use of short bunches The use of short bunches can reduce the effect on can reduce the effect on

luminosity.luminosity.

βy

(m)

-5

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3z (m)

““ Hourglass”Hourglass”shaped beamshaped beam

L = π γre

2

f R ε 1+κ( ) ξ2

βy∗

22


Low Beta Scheme ImplicationsLow Beta Scheme ImplicationsColliders &Luminosity

F. Sannibale

Large Vertical Beta Functions in D Quads @ IRLarge Vertical Beta Functions in D Quads @ IR

Larger Negative Values of Vertical Larger Negative Values of Vertical ChromaticitiesChromaticitiesStronger Correcting Stronger Correcting SextupolesSextupoles

Smaller Dynamic ApertureSmaller Dynamic ApertureDecrease of Beam LifetimeDecrease of Beam Lifetime

Short Bunches for Minimizing the Hourglass EffectShort Bunches for Minimizing the Hourglass EffectIncrease of Increase of TouschekTouschek EffectEffectDecrease of Beam LifetimeDecrease of Beam Lifetime

Higher Frequency Components in the Beam SpectrumHigher Frequency Components in the Beam SpectrumPossible Coupling with High FrequencyPossible Coupling with High FrequencyVacuum Chamber Modes: InstabilitiesVacuum Chamber Modes: Instabilities

Higher Peak RF Voltages: Larger Number of CavitiesHigher Peak RF Voltages: Larger Number of CavitiesRF RF NonlinearitiesNonlinearities, Stronger High Order Modes, Stronger High Order Modes

Coherent Synchrotron RadiationCoherent Synchrotron Radiationwith High Current per Bunchwith High Current per Bunch

. . . . . . . . . . . .

23


Round Beam vs Flat BeamRound Beam vs Flat BeamColliders &Luminosity

F. Sannibale

Round Beam (k~1):Round Beam (k~1):A Factor 2 of Luminosity GainA Factor 2 of Luminosity Gain

Both the Beta Functions @ IP Must Be Small:Both the Beta Functions @ IP Must Be Small:Technically Difficult to ObtainTechnically Difficult to Obtain

Large Negative Large Negative ChromaticitiesChromaticities in Both Planesin Both PlanesStrong Strong SextupoleSextupole CorrectionCorrection

Small Dynamic ApertureSmall Dynamic ApertureStrong BeamStrong Beam--beam Effectsbeam Effects

Flat Beam (k<<1):Flat Beam (k<<1):A Factor 2 of Luminosity LossA Factor 2 of Luminosity Loss

Chromaticity Handling not Critical:Chromaticity Handling not Critical:It is Possible to Arrange theIt is Possible to Arrange the

ColliderCollider Parameters in Order toParameters in Order toObtain Better Luminosity PerformancesObtain Better Luminosity Performances

24


MultibunchMultibunch LuminosityLuminosityColliders &Luminosity

F. Sannibale

L = NBLSBLarge Number of Bunches:Large Number of Bunches:Separate RingsSeparate RingsSmall Distance Between 2 Small Distance Between 2 adiacentadiacent bunchesbunchesMultibunchMultibunch InstabilitiesInstabilitiesLow Impedance Vacuum ChamberLow Impedance Vacuum ChamberHOM ‘Free’ Ring ComponentsHOM ‘Free’ Ring ComponentsLongitudinal Feedback SystemLongitudinal Feedback SystemHorizontal Crossing Angle @ IP Required Horizontal Crossing Angle @ IP Required in Order to Avoid Parasitic Crossingin Order to Avoid Parasitic CrossingSynchroSynchro--betatronbetatron ResonancesResonances. . . . .. . . . .Larger Stored CurrentLarger Stored CurrentVacuum System LimitationsVacuum System LimitationsLarge RF Power Large RF Power Vacuum Chamber Large Heating LoadVacuum Chamber Large Heating Load. . . . .. . . . .

--------------

--

--------

25


Detector ImplicationsDetector ImplicationsColliders &Luminosity

F. Sannibale

The The SolenoidalSolenoidal Field Introduces Field Introduces CouplingCouplingbetween the Vertical and Horizontal Planesbetween the Vertical and Horizontal Planes

that Must Be Carefully Corrected.that Must Be Carefully Corrected.

Experimental Requirements ConcerningExperimental Requirements ConcerningSolid Angle Stay Clear Forced to HaveSolid Angle Stay Clear Forced to HavePermanent IR Permanent IR QuadrupolesQuadrupoles and a Veryand a Very

Reduced Configuration of Beam Reduced Configuration of Beam DignosticsDignostics

26


Detector Effects Compensation:Detector Effects Compensation:DADAΦΦΦΦΦΦΦΦNE exampleNE example


F. Sannibale

IP

KLOE Solenoid

Compensating Solenoids

IR Quadrupoles

θS =1

2 Bρ( ) Bzz1

z2

s( )ds

Solenoid Frame Rotation Angle:Solenoid Frame Rotation Angle:

BzComp.1 s( )ds + Bz

KLOE s( )ds + Bz

Comp.2 s( )ds = 0

Field Integral Compensation:Field Integral Compensation:

Rotated IR Rotated IR QuadrupolesQuadrupoles to correct Coupling:to correct Coupling:

θnQ =

12 Bρ( ) Bz

IP

Cn

s( )ds n =1,2,3 Cn ≡ n − th quad center position

Bz = 0.6 T Bρ( )= 1.70 T m

C1 = 0.53 m C2 = 1.04 m C3 = 1.59 m

θ1Q = 5.35 deg θ2

Q = 10.5 deg θ3Q =16.1 deg

27


Integrated LuminosityIntegrated LuminosityOptimization Optimization


F. Sannibale

-0.1

0

0.1

0.2

0.3

0.4

0.5

-500 0 500 1000 1500 2000 2500 3000 3500

Average Luminosity vs Run DurationComplete Injection Scheme

<L>/L0 (beam tau 3000 s Td 1200 s)<L>/L0 (beam tau 4000 s Td 1200 s)<L>/L0 (beam tau 5000 s Td 1200 s)<L>/L0 (beam tau 6000 s Td 1200 s)

<L>

/Lo

Run Duration (s)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-500 0 500 1000 1500 2000 2500 3000 3500

Average Luminosity vs Run DurationCurrent Flip Flop Top up Scheme

<L>/L0 (beam tau 3000 s Td 300 s)<L>/L0 (beam tau 4000 s Td 300 s)<L>/L0 (beam tau 5000 s Td 300 s)<L>/L0 (beam tau 6000 s Td 300 s)

<L>

/Lo

Run Duration (s)

28


Examples of Examples of CollidersCollidersColliders &Luminosity

F. Sannibale

29


Examples of Examples of CollidersCollidersColliders &Luminosity

F. Sannibale

KEKKEKBB--FactoryFactory

1



Historical Overview,Historical Overview,Accelerator Examples and ApplicationsAccelerator Examples and Applications

David RobinDavid Robin

2


ContentContentHistorical Overview,Examples & Applications

D. Robin

• Why Accelerators

• History

• Examples and Applications

3


Why AcceleratorsWhy AcceleratorsHistorical Overview,Examples & Applications

D. Robin

Particle accelerators are devices producing beams of energetic particles such as ions and electrons which are employed for many different purposes.

This includes:• Ultra-precise Electron Microscopy• Creation of New Particles• High Brightness Photon Sources for Material Analysis and

Modification, Spectrometry, … .• Ion Implanters, for Surface Modification and for Sterilization and

Polymerization• Radiation Surgery and Therapy of Cancer

• …

4


UltraUltra--PrecisePreciseElectron MicroscopyElectron Microscopy

Historical Overview,Examples & Applications

D. Robin

Probing particles such as electrons and protons provided by particle accelerators are required for studies of atomic constituents. The

associated de Broglie wavelength of a probing particle defines the minimum object size that can be resolved.

15

, where

4 10 eVs (Planck's Constant) (Particle Momentum)

hp

hp

λ

−

=

= ×

Resolving Smaller Objects Requires Higher Momentum Probe Particles

Example 1 : An electron with a 1 GeV/c momentum will have a de Brogliewavelength of 10-15m (10-14 m ~ nucleus size, 10-15 m ~ proton size).

Example 2 : An electron with a 1 keV/c momentum will have a de Brogliewavelength of ~ 4.0 x 10-12 m. A photon with 1 keV energy has a wavelength

λ = ch/e ~ 1.2 x 10-9 m. This implies ~ 300 times better resolution and shows why electron microscopes have much better resolution than optical ones.

5


UltraUltra--PrecisePreciseElectron MicroscopyElectron Microscopy


D. Robin

Sub-micron objects such as the constituents of a living cell are often investigated in electron microscopes where electrons, accelerated typically to a few hundred kilovolts, are used to hit the objects and scatter from them.

Quarks and leptons can be sensed down to distances of10-18 meters by means of particles from giant accelerators with particle energies of ~100 GeV.

• Objects with dimensions down to the size of a living cell are investigated by optical microscopes and those down to

atomic dimensions by electron microscopes.

The living cell is commonly studied by means of an optical microscope which receives scattered photons of visible light.

• To penetrate the interiors of atoms and molecules, it is necessary to use radiation of a wavelength much smaller than

atomic dimensions.

6


Creation of New ParticlesCreation of New ParticlesHistorical Overview,Examples & Applications

D. Robin

Particles from accelerators colliding with target particles may lead to the creation of new particles, which acquire their mass from the

collision energy according to the formula E=mc2. It is thus by conversion to mass of excess kinetic energy in a collision that

particles, antiparticles and exotic nuclei can be created.

7


Other ApplicationsOther ApplicationsHistorical Overview,Examples & Applications

D. Robin

Particle accelerators are not only unique as tools for the exploration of the subatomic world, but are also used in many different applications such as material analysis and modification and spectrometry especially in environmental science.

• About half of the world's 15,000 accelerators are used as ion implanters, for surface modification and for sterilization and polymerization.

• The ionization arising when charged particles are stopped in matter is often utilized for example in radiation surgery and therapy of cancer. At hospitals about 5,000 electron accelerators are used for this purpose.

• Accelerators also produce radioactive elements that are used as tracers in medicine, biology and material science.

• Of a great and increasing importance in material science are ion and electron accelerators that produce abundant numbers of neutrons and photons (light sources) over a wide range of energies.

8


What Are Accelerators What Are Accelerators Used ForUsed For


D. Robin

110 Nuclear and particle physics research70 Synchrotron radiation sources20 Hadron therapy

200 Medical isotopes production5,000 Radiotherapy1,000 Accelerators in non-nuclear research1,500 Accelerators in industry7,000 Ion implanters and surface modifications

NumberCategory

World wide inventory of accelerators, in total 15,000. The data have been collected by W. Scarf and W. Wiesczycka(See U. Amaldi Europhysics News, June 31, 2000)

9


HistoryHistoryHistorical Overview,Examples & Applications

D. Robin

Cathode Ray TubesLate 1800s

Multipole GapsCockcroft Walton (1920)

Time Varying FieldsIsing (1924) and Wideroe

CyclotronLawrence (1930)Van Der Graff (1930)

Alvarez LinacMcMillan (1946)

Synchrotron Oliphant (1943)

Synchrocyclotron and BetatronMcMillan and Veksler (1944)

Strong FocusingCourant and Snyder (1952)

Electrostatic Field BasedElectrostatic Field Based

Time Varying Field BasedTime Varying Field Based

10


Discovery of XDiscovery of X--RaysRaysHistorical Overview,Examples & Applications

D. Robin

Wilhelm Conrad Röntgen (1845-1923)

Bertha Röntgen’s Hand 8 Nov, 1895

Modern radiograph of a hand

Crooke’s Tube

Among the most important discovery for medicine.

From:hyperphysics.phy-astr.gsu.edu

11


Discovery of ElectronDiscovery of ElectronHistorical Overview,Examples & Applications

D. Robin

In 1896 Joseph John Thomson investigated the nature of the cathoIn 1896 Joseph John Thomson investigated the nature of the cathode de rays which were found to be charged and to have a precise chargerays which were found to be charged and to have a precise charge--toto--mass ratio. This discovery of the first elementary particle, themass ratio. This discovery of the first elementary particle, the electron, electron,

marks the start of a new era, the electronic age.marks the start of a new era, the electronic age.

Original vacuum tubeOriginal vacuum tubeused by J. J. Thomsonused by J. J. Thomson

12


Beyond Cathode Ray TubesBeyond Cathode Ray TubesHistorical Overview,Examples & Applications

D. Robin

Cathode Ray Tubes are Single Gap Devices Cathode Ray Tubes are Single Gap Devices Small Energy (10s of Small Energy (10s of KeVKeV))

The existing different types of accelerators beyond Cathode The existing different types of accelerators beyond Cathode Ray Tubes were invented during a time span of nearly Ray Tubes were invented during a time span of nearly

four decades 1920 four decades 1920 -- 1960 1960

Many of the items mentioned here will be discussed in more Many of the items mentioned here will be discussed in more detail in Lecture 4.detail in Lecture 4.

13


PotentialPotential--Drop AcceleratorsDrop AcceleratorsHistorical Overview,Examples & Applications

D. Robin

• The first high-voltage particle accelerator had a potential drop of the order of 100 kilovolts and was

conceived by and named CockcroftWalton Accelerator in 1920.

• The most common potential-drop accelerator in use today is named

after its inventor, the American Robert Jemison Van de Graaff. Nowadays most van de Graaff

accelerators are commercial devices and they are available with terminal voltages ranging between one and

25 million volts (MV)

One of the biggest tandem One of the biggest tandem accelerators was used for many accelerators was used for many years at years at DaresburyDaresbury in the United in the United Kingdom. Its acceleration tube, Kingdom. Its acceleration tube,

placed vertically, was 42 meters long placed vertically, was 42 meters long and the centre terminal could hold a and the centre terminal could hold a

potential of up to potential of up to 20 million volts20 million volts..Photo: CCLRC Photo: CCLRC

Potential Drop Accelerators Employ Electrostatic Fields

In comparison the potential in clouds just before they are discharged by lightning is

about 200 MV.

14


TimeTime--varying voltagevarying voltageHistorical Overview,Examples & Applications

D. Robin

• In 1924, the Sweden G. Ising suggested that the maximum energy could be

increased by replacing the single gap holding a DC voltage by placing along a straight line several hollow cylindrical electrodes holding pulsed voltages.

• The Norwegian Rolf Wideröe realized that, if the phase of the alternating

voltage changed by 180 degrees during a particle’s trip between gaps, the particle could gain energy in each gap. Based on

this idea he built a three-stage accelerator for sodium ions.

The principle of repetitive acceleration conceived in the 1920s is an important milestone in the quest for higher and higher energies. According to this

principle, acceleration is achieved by means of a time-varying voltage instead of a static voltage as used in e.g. van de Graaff accelerators.

Ising's first suggestion for a linac

Rolf Wideroe

15


CyclotronCyclotronHistorical Overview,Examples & Applications

D. Robin

The first circular accelerator of practical importance based on The first circular accelerator of practical importance based on the the principle of repetitive acceleration was the cyclotron, inventedprinciple of repetitive acceleration was the cyclotron, invented by by

Ernest Orlando Lawrence.Ernest Orlando Lawrence.

The inventor of the The inventor of the cyclotron, Ernest cyclotron, Ernest Orlando Lawrence Orlando Lawrence

(left), and his student (left), and his student Edwin Edwin MattisonMattison

McMillanMcMillan

In a cyclotron, the charged particles circulate in a strong magnIn a cyclotron, the charged particles circulate in a strong magnetic field etic field and are accelerated by electric fields in one or more gaps. Afteand are accelerated by electric fields in one or more gaps. After having r having passed a gap, the particles move inside an electrode and are scrpassed a gap, the particles move inside an electrode and are screened eened from the electric field. When the particles exit from the screenfrom the electric field. When the particles exit from the screened area ed area

and enter the next gap, the phase of the timeand enter the next gap, the phase of the time--varying voltage has varying voltage has changed by 180 degrees so that the particles are again acceleratchanged by 180 degrees so that the particles are again accelerated. ed.

16


CyclotronCyclotronHistorical Overview,Examples & Applications

D. Robin

• In 1938 the first European cyclotron at Collège de France in Paris accelerated a deuteron beam up to 4 MeV and by hitting a target, an intense

source of neutrons was produced.

The first cyclotron4.5” diameter (1929).

• A serious problem with the early cyclotrons was the energy limit of about 10 MeV for the acceleration of protons. This limit depends on the slowing

down of protons rotating in a constant magnetic field due to their relativistic increase of mass or equivalent total energy.

Lawrence at the 37” cyclotron (1937)Lawrence at the 37” cyclotron (1937)

17


SynchroSynchro--CyclotronCyclotronHistorical Overview,Examples & Applications

D. Robin

To overcome the energy limitation of a To overcome the energy limitation of a cyclotron, cyclotron, the principle of phase stabilitythe principle of phase stabilitywas invented and proved in 1944/45. The was invented and proved in 1944/45. The

inventors were Vladimir inventors were Vladimir LosifovichLosifovichVekslerVeksler and by Edwin and by Edwin MattisonMattison McMillan, McMillan,

a former student of Lawrence, at the a former student of Lawrence, at the University of California in Berkeley.University of California in Berkeley.

They showed, independently of each They showed, independently of each other, that by adjusting the frequency other, that by adjusting the frequency

of the applied voltage to the decreasing of the applied voltage to the decreasing frequency of the rotating protons, it frequency of the rotating protons, it

was possible to accelerate the protons was possible to accelerate the protons to several hundred to several hundred MeVMeV. .

The largest synchrocyclotron still in use The largest synchrocyclotron still in use is located in is located in GatchinaGatchina outside St outside St

Petersburg and it accelerates protons to Petersburg and it accelerates protons to a a kinetic energy of 1,000 kinetic energy of 1,000 MeVMeV. The iron . The iron poles are 6 meters in diameter and the poles are 6 meters in diameter and the

whole accelerator weighs 10,000 tons, a whole accelerator weighs 10,000 tons, a weight weight comparable to that of the Eiffel comparable to that of the Eiffel

TowerTower. The energies attained correspond . The energies attained correspond to that of a proton accelerated in a to that of a proton accelerated in a

potential drop of one billion volts. It is potential drop of one billion volts. It is used for nuclear physics experiments used for nuclear physics experiments

and medical applications. and medical applications.

18


SectorSector--Focusing CyclotronFocusing CyclotronHistorical Overview,Examples & Applications

D. Robin

In the early 1960s, a new type of In the early 1960s, a new type of cyclotron, the sectorcyclotron, the sector--focusing cyclotron focusing cyclotron emerged. Iron sectors were introduced in emerged. Iron sectors were introduced in

the pole gap so that an the pole gap so that an azimuthalazimuthal variation variation of the magnetic field was obtained. This of the magnetic field was obtained. This

azimuthalazimuthal variation provides a strong variation provides a strong vertical focusing on the circulating beam vertical focusing on the circulating beam

of ions and it is then not necessary to of ions and it is then not necessary to have the azimuthally averaged field to have the azimuthally averaged field to

decrease with increasing radius as it has decrease with increasing radius as it has to do in the conventional cyclotron in to do in the conventional cyclotron in

order to maintain vertical focusing. Hence, order to maintain vertical focusing. Hence, the average magnetic field as a function of the average magnetic field as a function of

radius, can be increased so that the radius, can be increased so that the rotation frequency of the ion remains rotation frequency of the ion remains

constant in spite of the increase of mass constant in spite of the increase of mass of the accelerating ion. of the accelerating ion.

The separated sector cyclotron The separated sector cyclotron in Vancouver, provides 600 in Vancouver, provides 600 MeVMeVnegative hydrogen ions and negative hydrogen ions and it is it is the largest of all cyclotronsthe largest of all cyclotrons. The . The

picture shows the gap inside picture shows the gap inside which the ions are accelerated.which the ions are accelerated.

19


SectorSector--Focusing CyclotronFocusing CyclotronHistorical Overview,Examples & Applications

D. Robin

Application of Cyclotrons with Heavy Ions:Application of Cyclotrons with Heavy Ions:

•• Sector focusing cyclotrons have been very useful for providing lSector focusing cyclotrons have been very useful for providing lowow--energy heavy ions. energy heavy ions.

•• Using accelerated heavy ions, several new elements have been Using accelerated heavy ions, several new elements have been discovered first in Berkeley and discovered first in Berkeley and DubnaDubna and later in Darmstadt. The and later in Darmstadt. The heaviest element so far discovered, element 110, was first foundheaviest element so far discovered, element 110, was first found in in Darmstadt and the discovery has been confirmed by the groups in Darmstadt and the discovery has been confirmed by the groups in DubnaDubna and Berkeley. The research is still intense and element 112 hasand Berkeley. The research is still intense and element 112 hasbeen claimed in Darmstadt, element 114 in been claimed in Darmstadt, element 114 in DubnaDubna. .

•• Since the maximum energy in a cyclotron is limited by the strengSince the maximum energy in a cyclotron is limited by the strength of th of the magnetic field and its radial extension, superconducting wirthe magnetic field and its radial extension, superconducting wire coils e coils are now used instead of conventional copper coils around the iroare now used instead of conventional copper coils around the iron n poles to provide stronger fields. poles to provide stronger fields.

20


BetatronBetatronHistorical Overview,Examples & Applications

D. Robin

21


SynchrotronSynchrotronHistorical Overview,Examples & Applications

D. Robin

The two other types of accelerators based on the principle of reThe two other types of accelerators based on the principle of repetitive petitive acceleration, the synchrotron and the linear accelerator, are imacceleration, the synchrotron and the linear accelerator, are important in portant in

elementary particle physics research, where highest possible parelementary particle physics research, where highest possible particle ticle energies are needed. energies are needed.

In synchrotrons, the particles are accelerated along a ringIn synchrotrons, the particles are accelerated along a ring--shaped orbit and the shaped orbit and the magnetic fields, bending the particles, increase with time so thmagnetic fields, bending the particles, increase with time so that a at a constant orbit constant orbit

is maintained during the acceleration. is maintained during the acceleration.

The synchrotron The synchrotron concept seems to concept seems to

have been first have been first proposed in 1943 proposed in 1943 by the Australian by the Australian

physicist physicist Mark Mark Oliphant.Oliphant.

22


WeakWeak--Focusing SynchrotronsFocusing SynchrotronsHistorical Overview,Examples & Applications

D. Robin

The first synchrotrons were of the so called The first synchrotrons were of the so called weakweak--focusing typefocusing type. The vertical . The vertical focusing of the circulating particles was achieved by sloping mafocusing of the circulating particles was achieved by sloping magnetic fields, gnetic fields,

from inwards to outwards radii. At any given moment in time, thefrom inwards to outwards radii. At any given moment in time, the average average vertical magnetic field sensed during one particle revolution isvertical magnetic field sensed during one particle revolution is larger for larger for

smaller radii of curvature than for larger ones. smaller radii of curvature than for larger ones. •• The first synchrotron of this type was the Cosmotron at the BrooThe first synchrotron of this type was the Cosmotron at the Brookhaven khaven

National Laboratory, Long Island. It started operation in 1952 aNational Laboratory, Long Island. It started operation in 1952 and nd provided protons with provided protons with energies up to 3 energies up to 3 GeVGeV. .

•• In the early 1960s, the world’s highest energy weakIn the early 1960s, the world’s highest energy weak--focusing focusing synchrotron, the synchrotron, the 12.5 12.5 GeVGeV Zero Gradient SynchrotronZero Gradient Synchrotron (ZGS) started its (ZGS) started its

operation at the Argonne National Laboratory near Chicago, USA.operation at the Argonne National Laboratory near Chicago, USA.•• The The DubnaDubna synchrotron, the largest of them all with a radius of synchrotron, the largest of them all with a radius of 28 meters28 meters

and with a weight of the magnet iron of and with a weight of the magnet iron of 36,000 tons36,000 tons

Cosmotron

Weak Focusing

23


StrongStrong--Focusing SynchrotronsFocusing SynchrotronsHistorical Overview,Examples & Applications

D. Robin

In 1952 Ernest D. Courant, Milton Stanley Livingston In 1952 Ernest D. Courant, Milton Stanley Livingston and Hartland S. Snyder, proposed a scheme for and Hartland S. Snyder, proposed a scheme for strong focusing of a circulating particle beam so strong focusing of a circulating particle beam so that its size can be made smaller than that in a weakthat its size can be made smaller than that in a weak--focusing synchrotron. focusing synchrotron.

•• In this scheme, the bending magnets are made In this scheme, the bending magnets are made to have alternating magnetic field gradients; to have alternating magnetic field gradients; after a magnet with an axial field component after a magnet with an axial field component decreasing with increasing radius follows one decreasing with increasing radius follows one with a component increasing with increasing with a component increasing with increasing radius and so on. radius and so on.

•• Thanks to the strong focusing, the magnet Thanks to the strong focusing, the magnet apertures can be made smaller and therefore apertures can be made smaller and therefore much less iron is needed than for a weakmuch less iron is needed than for a weak--focusing synchrotron of comparable energy.focusing synchrotron of comparable energy.

•• The first alternatingThe first alternating--gradient synchrotron gradient synchrotron accelerated electrons to 1.5 accelerated electrons to 1.5 GeVGeV. It was built at . It was built at Cornell University, Ithaca, N.Y. and was Cornell University, Ithaca, N.Y. and was completed in 1954.completed in 1954.

Size comparison between the Cosmotron's weak-focusing magnet (L) and the AGS alternating gradient focusing magnets

24


StrongStrong--Focusing SynchrotronsFocusing SynchrotronsHistorical Overview,Examples & Applications

D. Robin

Soon after the invention of the principle of alternatingSoon after the invention of the principle of alternating--gradient focusing, the gradient focusing, the construction of two nearly identical very large synchrotrons, whconstruction of two nearly identical very large synchrotrons, which are ich are still in still in

operationoperation, started at the European CERN laboratory in Geneva and the , started at the European CERN laboratory in Geneva and the Brookhaven National Laboratory on Long Island in New York. Brookhaven National Laboratory on Long Island in New York. At CERN protons At CERN protons

are accelerated to 28 are accelerated to 28 GeVGeV and at Brookhaven to 33 and at Brookhaven to 33 GeVGeV. The CERN proton . The CERN proton synchrotron (PS) started operation in 1959 and the Brookhaven Alsynchrotron (PS) started operation in 1959 and the Brookhaven Alternating ternating

Gradient Synchrotron (AGS) in 1960. Gradient Synchrotron (AGS) in 1960.

Brookhaven AGSCERN PS

25


The Largest SynchrotronThe Largest SynchrotronHistorical Overview,Examples & Applications

D. Robin

Inside the 6.9 km long tunnel of the Inside the 6.9 km long tunnel of the CERN 450 CERN 450 GeVGeV super proton super proton

synchrotron. The blue magnets synchrotron. The blue magnets focus, and the red magnets bend the focus, and the red magnets bend the

particles.particles.Photo: Photo: CernCern

Aerial view of the CERN laboratory situated between Aerial view of the CERN laboratory situated between Geneva airport and the Geneva airport and the JuraJura mountains. The circles mountains. The circles

indicate the locations of the SPS and LEP accelerators indicate the locations of the SPS and LEP accelerators placed in underground tunnels. After the LEP placed in underground tunnels. After the LEP

accelerator has stopped operation at the end of the accelerator has stopped operation at the end of the year 2000, it was dismounted and the large year 2000, it was dismounted and the large HadronHadron

ColliderCollider (LHC) is currently being installed in the 27 km (LHC) is currently being installed in the 27 km long tunnel.long tunnel.

Photo: CERN Photo: CERN

26


Linear AcceleratorsLinear AcceleratorsHistorical Overview,Examples & Applications

D. Robin

•• After After IsingIsing and and WideroeWideroe scheme, an improved version of a linear accelerator was scheme, an improved version of a linear accelerator was conceived in 1946 by Luis Walter Alvarez who generated the radioconceived in 1946 by Luis Walter Alvarez who generated the radio frequency frequency

voltage differently; standing radiovoltage differently; standing radio--frequency waves inside cylindrical cavities. frequency waves inside cylindrical cavities. These so called Alvarez structures are still used for ion and prThese so called Alvarez structures are still used for ion and proton acceleration. oton acceleration.

Alvarez was awarded the 1968 Nobel Prize in Physics for his deciAlvarez was awarded the 1968 Nobel Prize in Physics for his decisive sive contributions to elementary particle physics.contributions to elementary particle physics.

KEK

•• A big boost to the development of linear accelerators came whenA big boost to the development of linear accelerators came when Hansen and Hansen and the Varian brothers (1937) developed the first klystron (frequenthe Varian brothers (1937) developed the first klystron (frequencies up to 10 GHz) cies up to 10 GHz)

an efficient and high power source of radio frequency.an efficient and high power source of radio frequency.•• In parallel, newer and more efficient RF structures were obtainIn parallel, newer and more efficient RF structures were obtained by coupling ed by coupling

together many pillboxtogether many pillbox--like cavities.like cavities.

Stanford Linear AcceleratorStanford Linear Accelerator

•• Very high energy accelerators became a feasible reality and sevVery high energy accelerators became a feasible reality and several machines eral machines where constructed.where constructed.

27


AcceleratorAccelerator--BasedBasedLight SourcesLight Sources

Historical Overview, Examples & Applications

D. Robin

Modern synchrotron light sources are accelerators optimized for Modern synchrotron light sources are accelerators optimized for the the production of synchrotron radiation.production of synchrotron radiation.

ESRFESRF-- FranceFrance

APS APS -- USAUSA

More in Lecture 14!More in Lecture 14!

28


Colliding BeamsColliding BeamsHistorical Overview,Examples & Applications

D. Robin

In the continuous race for higher energies, required in the searIn the continuous race for higher energies, required in the search for ch for undiscovered heavy particles and for the exploration of smaller undiscovered heavy particles and for the exploration of smaller distances, particle distances, particle colliderscolliders have been found to be superior to other types of acceleratorhave been found to be superior to other types of accelerator--based based experiments (fixed target). experiments (fixed target).

•• The first electronThe first electron--positron positron collidercollider in operation was ADA in 1960 in in operation was ADA in 1960 in FrascatiFrascati. . This little storage ring with a little more than 1 m diameter waThis little storage ring with a little more than 1 m diameter was conceived s conceived and designed by Bruno and designed by Bruno ToshekToshek and operated at 250 and operated at 250 MeVMeV. ADA was the . ADA was the proof of principle that allowed to set the theoretical and experproof of principle that allowed to set the theoretical and experimental basis imental basis for the later construction of accelerators such as the LEP at CEfor the later construction of accelerators such as the LEP at CERN with RN with almost 27 Km circumference.almost 27 Km circumference.

•• At the same time, pioneering work on how to collide two beams ofAt the same time, pioneering work on how to collide two beams of electrons electrons circulating in two synchrotrons was done in Novosibirsk at the circulating in two synchrotrons was done in Novosibirsk at the BudkerBudkerinstitute. institute.

•• The first The first collidercollider to be used for experiments was the intersecting storage to be used for experiments was the intersecting storage rings (ISR), used at CERN from 1971 to 1983.rings (ISR), used at CERN from 1971 to 1983.

•• Several Nobel prizes were assigned for results obtained by accelSeveral Nobel prizes were assigned for results obtained by accelerators (B. erators (B. Richter and S. Ting 1976, C. Richter and S. Ting 1976, C. RubbiaRubbia and S. van and S. van derder MeerMeer 1984)1984)

29


Other AcceleratorsOther AcceleratorsHistorical Overview,Examples & Applications

D. Robin

The continuous electron beam facility The continuous electron beam facility (CEBAF) at the Jefferson Laboratory, (CEBAF) at the Jefferson Laboratory, Virginia, USA, accelerates electrons Virginia, USA, accelerates electrons up to 6 up to 6 GeVGeV in a racein a race--track track microtronmicrotron

with a circumference of 1.4 km. with a circumference of 1.4 km. Acceleration takes place in 338 Acceleration takes place in 338

hollow shells (cavities) placed in the hollow shells (cavities) placed in the straight sections inside straight sections inside cryomodulescryomodulesand the beam is bent 180 degrees in and the beam is bent 180 degrees in

five different arcs. During the first five different arcs. During the first revolution, the electrons move in the revolution, the electrons move in the

upper arcs, they descend upper arcs, they descend successively and after five successively and after five

revolutions of acceleration they have revolutions of acceleration they have reached the bottom arcs. Experiments reached the bottom arcs. Experiments are situated in three different halls, A, are situated in three different halls, A,

B and C. In the future, a new hall D B and C. In the future, a new hall D will be added and the energy will be will be added and the energy will be

increased to 12 increased to 12 GeVGeV..Illustration: DOE/Jefferson Lab. Illustration: DOE/Jefferson Lab.

Superconducting RFSuperconducting RF

30


Other AcceleratorsOther AcceleratorsHistorical Overview,Examples & Applications

D. Robin

Cooler Storage RingsCooler Storage Rings•• Cooling a circulating particle beam means reducing the momentum Cooling a circulating particle beam means reducing the momentum

spreads and the transverse dimensions of the beam. spreads and the transverse dimensions of the beam. •• Electron cooling was invented in Novosibirsk in the late 1970s aElectron cooling was invented in Novosibirsk in the late 1970s and nd

Electron cooling is useful for improving the quality of beams ofElectron cooling is useful for improving the quality of beams ofprotons, antiprotons and ionsprotons, antiprotons and ions

Meson FactoriesMeson Factories•• During the 1960s, three accelerators were built to provide intenDuring the 1960s, three accelerators were built to provide intense se

fluxes of beams of mediumfluxes of beams of medium--energy, several hundred energy, several hundred MeVMeV, charged p, charged p--mesons.mesons.

Neutron SourcesNeutron Sources•• When a highWhen a high--energy proton penetrates a target of heavy material such energy proton penetrates a target of heavy material such

as lead, tungsten or uranium, numerous neutrons are knocked out.as lead, tungsten or uranium, numerous neutrons are knocked out. For For example, one proton of 800 example, one proton of 800 MeVMeV stopped in a target of uranium gives stopped in a target of uranium gives rise to about 30 neutrons on the average.rise to about 30 neutrons on the average.

•• At present, the most powerful pulsed neutron source is located aAt present, the most powerful pulsed neutron source is located at the t the Rutherford Appleton Laboratory near Oxford, U.K., where a 70 Rutherford Appleton Laboratory near Oxford, U.K., where a 70 MeVMeVlinear accelerator is the injector to a synchrotron that providelinear accelerator is the injector to a synchrotron that provides protons s protons of 800 of 800 MeVMeV with an intensity of 200 microampereswith an intensity of 200 microamperes

31


ConclusionConclusionHistorical Overview,Examples & Applications

D. Robin

Exponential growth of energy with timeExponential growth of energy with time•• Increase of the energy by an order of magnitude every 6Increase of the energy by an order of magnitude every 6--10 years10 years•• Every new idea evolves up to a point of saturation and than is Every new idea evolves up to a point of saturation and than is

replaced by another new ideareplaced by another new idea

Energy is not the only interesting parameters where Energy is not the only interesting parameters where there has been phenomenal improvementsthere has been phenomenal improvements

•• Exponential growth in Brightness (for example) of 13 orders of Exponential growth in Brightness (for example) of 13 orders of magnetudemagnetude in only 40 years!in only 40 years!

With clever new ideas these advances will surely With clever new ideas these advances will surely continue into the future!continue into the future!

32


ThanksThanksHistorical Overview,Examples & Applications

D. Robin

Wish to thank Y. Wish to thank Y. PapaphilippouPapaphilippou and and N.CatalanN.Catalan--LasherasLasherasfor sharing the for sharing the tranparenciestranparencies that they used in the that they used in the USPAS, Cornell University, Ithaca, NY 20th June USPAS, Cornell University, Ithaca, NY 20th June –– 1st 1st July 2005July 2005

Wish to acknowledge the web based article Accelerators Wish to acknowledge the web based article Accelerators and Nobel Laureates by Sven and Nobel Laureates by Sven KullanderKullander which can be which can be viewed at viewed at http://http://nobelprize.org/physics/articles/kullandernobelprize.org/physics/articles/kullander//

1



Maxwell EquationsMaxwell Equationsand Special Relativity in Acceleratorsand Special Relativity in Accelerators


2

• Need to know how particles will move in the presence of electric and magnetic fields. Present a basic review of classical physics*– Equations of Motion– Calculations of the Fields– Special Relativity

• Give a couple examples

* Will ignore quantum mechanical effects for now


MotivationMotivationMaxwell Equations& Special Relativity

D. Robin

3

• Feynman Lectures by R. Feynman, R. Leighton, and M Sands

• Introduction to Electrodynamics by D. Griffith

• Spacetime Physics by E. Taylor and J. Archibald

• Particle Accelerator Physics, Basic Principles and Linear Beam Dynamics by H. Wiedemann


ReferencesReferencesMaxwell Equations& Special Relativity

D. Robin

4

amF rr=

Newton’s Law’s of Motion

( )BvEqFrrrr

×+=

Lorentz Force Equation – Force on a charged particle traveling with velocity, v, in the presence of an electric, E, or magnetic, B, field

To determine the particle motion one needs to know the electric and magnetic fields –Maxwell’s Equations


Equations of MotionEquations of MotionMaxwell Equations& Special Relativity

D. Robin

5

( ) daAdA ⋅=⋅∇ ∫∫ τ

I. Divergence Theorem

II. Curl Theorem

( ) dIAdaA ⋅=×∇ ∫∫


Fundamental TheoremsFundamental TheoremsMaxwell Equations& Special Relativity

D. Robin

6


VectorialVectorial AlgebraAlgebraChanging the

Particle EnergyF. Sannibale

Curl Theorem (Curl Theorem (Stoke’sStoke’s Theorem)Theorem)

Divergence TheoremDivergence Theorem

Volume IntegralVolume Integral

Surface Integral (Flux)Surface Integral (Flux)

Line Integral (Line Integral (CircuitationCircuitation))

∫∫ ⋅∇=⋅VS

dVFdSnF

( )∫∫ ⋅×∇=⋅Sl

dSnFldF

( ) FF ∀=×∇⋅∇ 0

( ) uu ∀=∇×∇ 0 uFF ∇=⇔=×∇ 0oror((FF is conservative if curl is conservative if curl FF is zero)is zero)

( ) ( ) FFFF ∀∇−⋅∇∇=×∇×∇ 2

7

o

Eερ

=⋅∇

I. Gauss’ Law(Flux of E through a closed surface) = (Charge inside/ε0)

0=⋅∇ B

II. (No Name)(Flux of B through a closed surface) = 0

2212 Nm/C1085.8 −×=oεpermittivity of free space


Maxwell’s EquationsMaxwell’s EquationsMaxwell Equations& Special Relativity

D. Robin

8

tBE∂∂

−=×∇

III. Faraday’s Law(Line Integral of E around a loop) =

-d/dt(Flux of B through the loop)

0 0 0EB jt

µ µ ε ∂∇× = +

∂

IV. (Ampere’s Law modified by Maxwell)(Integral of B around a loop) = (Current through the loop)/ε0

+d/dt(Flux of E through the loop)

7 24 10 / Ao Nµ π −= ×permeability of free space


Maxwell’s EquationsMaxwell’s EquationsMaxwell Equations& Special Relativity

D. Robin

9

22

0 0 2

EEt

µ ε ∂∇ =

∂

In free space combining Gauss’ Law and Faraday’s Law

22

0 0 2

BBt

µ ε ∂∇ =

∂

In free space combining Ampere’s Law and the last Maxwell Equations

22

2 2

1 where is the velocity of the waveff vv t

∂∇ =

∂

Equation of a wave in three dimensions is


Maxwell’s Equations and LightMaxwell’s Equations and LightMaxwell Equations& Special Relativity

D. Robin

10


From Maxwell EquationsFrom Maxwell Equationsto Wave Equationto Wave Equation

Changing theParticle EnergyF. Sannibale

11

22

2 2

1 where is the velocity of the waveff vv t

∂∇ =

∂

Equation of a wave in three dimensions is

8

0 0

1 3.00 10 /v m sµ ε

= = ×

Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves traveling at the speed of light

Perhaps Light is an electromagnetic wave

“We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena” - Maxwell


Maxwell’s Equations and LightMaxwell’s Equations and LightMaxwell Equations& Special Relativity

D. Robin

12

0

0 0 0

1

0

E dS dV

B dS

dBE dl dSdt

dEB dl j dS dSdt

ρε

µ ε µ

⋅ =

⋅ =

⋅ = − ⋅

⋅ = − ⋅ + ⋅

∫ ∫

∫

∫ ∫

∫ ∫ ∫

rr

rr

rr rr

rr r rr r


Maxwell’s Equations in Integral FormMaxwell’s Equations in Integral FormMaxwell Equations& Special Relativity

D. Robin

13

1.The principle of relativity. The laws of physics apply in all inertial reference systems.

2. The universal Speed of light. The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source.


Special RelativitySpecial RelativityMaxwell Equations& Special Relativity

D. Robin

14

cvcv

=

−

=

β

γ

2

2

1

1

where v is the velocity of the particle and c is the velocity of light

vmp γ= :momentum particle where m is the rest mass of the particle

( )BvEqdtdpF

rrrr×+==


LorentzLorentz Factor, Factor, γγMaxwell Equations& Special Relativity

D. Robin

15

Rest Energy, Eo : Eo = mc2

Total Energy, E : E = mγc2

Momentum, p : p = mγv

2 22 20 p cE E= +


Energy and MomentumEnergy and MomentumMaxwell Equations& Special Relativity

D. Robin

16

• Two particles have equal rest mass m0.Laboratory Frame (LF): one particle at rest, total energy is E.

Centre of Mass Frame (CMF): Velocities are equal and opposite, total energy is Ecm.

• The quantity is invariant. • In the CMF, we have • In general• In the LF, we have and• And finally


LorentzLorentz Particle collisionsParticle collisionsMaxwell Equations& Special Relativity

D. Robin

17

Two inertial frames moving with respect to each other with velocity,v ( )

−=

==

−=

cxvtt

zzyy

vtxx

2'

'''

γ

γ


LorentzLorentz TransformationTransformationMaxwell Equations& Special Relativity

D. Robin

18

Two celebrated consequences of the transformation are Time dilation and Lorentz contraction

Time dilation. A clock in the primed frame located at x = vt will show a time dilation, t’ = 1/γ

Lorentz contraction. An object in the primed frame with length L’ along the x’ axis and is at rest in the primed frame will be of length L = L’/γ in the unprimed frame


Time Dilation and Time Dilation and LorentzLorentz ContractionContractionMaxwell Equations& Special Relativity

D. Robin

19

( )( )BEE

BEEEE

yzz

zyy

xx

v

v

+=

−=

=

γ

γ'

'

'

−=

+=

=

EBB

EBBBB

yzz

zyy

xx

cvcv

2'

2'

'

γ

γ


LorentzLorentz Transformation of Transformation of Electric and Magnetic FieldsElectric and Magnetic Fields

Maxwell Equations& Special Relativity

D. Robin

20



D. Robin


21


L2 Possible HomeworkL2 Possible HomeworkMaxwell Equations & Special Relativity

D. Robin

Problem 1.Problem 1.Protons are accelerated to a kinetic energy of 200 Protons are accelerated to a kinetic energy of 200 MeVMeV at the end of the at the end of the FermilabFermilabAlvarez linear accelerator. Calculate their total energy, their Alvarez linear accelerator. Calculate their total energy, their momentum and their momentum and their

velocity in units of the velocity of light.velocity in units of the velocity of light.Problem 2.Problem 2.

A charge A charge pionpion has a rest energy of 139.568 has a rest energy of 139.568 MeVMeV and a mean life time of and a mean life time of ττ = 26.029 = 26.029 nsecnsec in its rest frame. What are the in its rest frame. What are the pionpion life times, if accelerated to a kinetic life times, if accelerated to a kinetic

energy of 20 energy of 20 MeVMeV? And 100 ? And 100 MeVMeV? A ? A pionpion beam decays exponentially like beam decays exponentially like ee--ττ/t/t. At . At what distance from the source will the what distance from the source will the pionpion beam intensity have fallen to 50%, if beam intensity have fallen to 50%, if

the kinetic energy is 20 the kinetic energy is 20 MeVMeV? Or 100 ? Or 100 MeVMeV??Problem 3.Problem 3.

A positron beam accelerated to 50 A positron beam accelerated to 50 GeVGeV in the in the linaclinac hits a fixed hydrogen target. hits a fixed hydrogen target. What is the available energy from a collision with a target elecWhat is the available energy from a collision with a target electron assumed to be tron assumed to be at rest? Compare this available energy with that obtained in a lat rest? Compare this available energy with that obtained in a linear inear collidercollider where where

electrons and positrons from two similar electrons and positrons from two similar linacslinacs collide head on at the same collide head on at the same energy.energy.

Rest energy of an electron = 0.511 Rest energy of an electron = 0.511 MeVMeVRest energy of a proton = 936 Rest energy of a proton = 936 MeVMeV

22


L2 Possible HomeworkL2 Possible HomeworkMaxwell Equations & Special Relativity

D. Robin

•• Show that a function satisfying f(x,t)=Show that a function satisfying f(x,t)=f(xf(x--vtvt) automatically ) automatically satisfies the wave equation.satisfies the wave equation.

•• A A muonmuon has a rest mass of 105.7MeV and a lifetime at rest of has a rest mass of 105.7MeV and a lifetime at rest of 2.2e2.2e--6 s. 6 s.

–– Consider a Consider a muonmuon traveling at 0.9c with respect to the lab traveling at 0.9c with respect to the lab frame. What is its lifetime? How far does the frame. What is its lifetime? How far does the muonmuon travel? travel? How does this compare to the distance it would travel if How does this compare to the distance it would travel if

there were no time dilation?there were no time dilation?–– Consider a Consider a muonmuon accelerated to 1GeV. accelerated to 1GeV.

–– What is its velocity? How long does it live?What is its velocity? How long does it live?•• For a nonFor a non--relativistic charge moving in the z direction, relativistic charge moving in the z direction, calculate the general particle trajectory when subjected to a calculate the general particle trajectory when subjected to a

field field BBxx==BBzz=0, and B=0, and Byy=sin(2=sin(2ππz/z/λλ) for 0<z<z) for 0<z<z00, and B, and Byy=0 =0 elsewhere.elsewhere.

1



Charged Particle Optics. MatrixCharged Particle Optics. MatrixRepresentation of the Accelerator ElementsRepresentation of the Accelerator Elements


2

• What are the Optics?– Magnet Definitions– Magnet Functions

• Particle motion in accelerator– Coordinate system– Beam guidance

• Dipoles– Beam focusing

• Quadrupoles• Hill’s equations and Transport Matrices

– Matrix formalism– Drift– Thin lens– Quadrupoles– Dipoles

• Sector magnets• Rectangular magnets

– Doublet– FODO


OutlineOutlineSingle Particle DynamicsMatrix Representation

D. Robin

3

• The Optics are the distribution of elements (typically magnetic or electrostatic) that guide and focus the beam - sometimes referred to as the lattice.

Focusing Elements

Bend Element


What are the Optics?What are the Optics?Single Particle DynamicsMatrix Representation

D. Robin

4

Choice of the design depends upon the goal of the accelerator–Small spot size–High brightness–Small divergence–Obey certain physical constraints (building

or tunnel)–…


Optics or Lattice Design Optics or Lattice Design Determines the Beam PropertiesDetermines the Beam Properties

Single Particle DynamicsMatrix Representation

D. Robin

5

The motion of each charged particle is determined by the electric and magnetic forces that it encounters as it orbits the ring:

• Lorentz Force( ),

is the relativistic mass of the particle, is the charge of the particle, is the velocity of the particle, is the acceleration of the part

F ma e E v Bmeva

= = + ×

icle, is the electric field and, is the magnetic field.

EB


Equations of MotionEquations of MotionSingle Particle DynamicsMatrix Representation

D. Robin

6

1. Given an existing lattice, determine the properties of the beam.

2. For a desired set of beam properties, determine the design of the lattice.

The first problem is in principle straight-forward to solve.

The second problem is not straight-forward – a bit of an art.


Two Problems Two Problems (Inverse Problems)(Inverse Problems)


D. Robin

7

QuadrupolesSextupoles

Dipoles


Magnets to Guide Magnets to Guide and Focus the Beamand Focus the Beam


D. Robin

8

• 2n-pole:dipole quadrupole sextupole octupole …

n: 1 2 3 4 …

• Normal: gap appears at the horizontal plane• Skew: rotate around beam axis by π/2n angle• Symmetry: rotating around beam axis by π/n angle, the field

is reversed (polarity flipped)

N

S

N

S

S

NN

S S

SN N N

N

N

N

S

S

S

S


Magnet DefinitionsMagnet DefinitionsSingle Particle DynamicsMatrix Representation

D. Robin

9

There are several magnet types that are used in storage rings:Dipoles used for guiding

Bx = 0 By = Bo

Quadrupoles used for focussingBx = KyBy = Kx

Sextupoles used for chromatic correction

Bx = 2SxyBy = S(x2 – y2)


Typical Magnet TypesTypical Magnet TypesSingle Particle DynamicsMatrix Representation

D. Robin

N

S

N

S

S

N

NS S

SN N

10


Functions of the Functions of the Magnetic ElementsMagnetic Elements


D. Robin

N

S

N

S

S

N

NS S

SN N

11

In the Lorentz Force Equation as written below, the dependent variable is time, t

• Lorentz Force

( ), is the relativistic mass of the particle, is the charge of the particle, is the velocity of the particle, is the acceleration of the part

F ma e E v Bmeva

= = + ×

icle, is the electric field and, is the magnetic field.

EB


Dependent VariableDependent VariableSingle Particle DynamicsMatrix Representation

D. Robin

12

• Consider a storage ring for particles with energy Ewith N dipoles of length l

• The bending angle is

• The bending radius is

SNS ring dipole

The integrated dipole strength will be

By fixing the dipole field, the dipole length is imposed and vice versa

The highest the field, shorter or smaller number of dipoles can be used

Ring circumference (cost) is influenced by the field choiceFundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

DipolesDipolesSingle Particle DynamicsMatrix Representation

D. Robin

13

• Magnetic element that deflects the beam by an angle proportional to the distance from its centre (equivalent to ray optics) provides focusing.

• For a focal length f the deflection angle is

• A magnetic element with length l and with a gradient g has a fieldso that the deflection angle is

y

α

f

focal pointThe normalised focusing strength

In more practical units, for Z=1

The focal length becomes and the deflection angle is


Focusing ElementsFocusing ElementsSingle Particle DynamicsMatrix Representation

D. Robin

14

• Quadrupoles are focusing in one plane and defocusing in the other

• The field is

• The resulting force

• Need to alternate focusing and defocusing in order to control the beam, i.e. alternating gradient focusing

• From optics we know that a combination of two lenses with focal lengths f1 and f2 separated by a distance d

• If f1 = -f2, there is a net focusing effect, i.e.

v

F

B

F

Bv


QuadrupolesQuadrupolesSingle Particle DynamicsMatrix Representation

D. Robin

15

Coordinate system used to describe the motion is usually locally Cartesian or cylindrical

Typically the coordinate system chosen is the one that allows the easiest field representation

y xs

Change dependent variable from time, t, to longitudinal position, s


Coordinate SystemCoordinate SystemSingle Particle DynamicsMatrix Representation

D. Robin

16

Integrate through the elements

Use the following coordinates*

*Note sometimes one uses canonical momentum rather than x’ and y’

0

, ' , , ' , ,dx dy p Lx x y yds ds p L

δ τ∆ ∆

= = = =

y xs


IntegrateIntegrateSingle Particle DynamicsMatrix Representation

D. Robin

17

• The equations of motion within an element is

• The fields have to be defined


General equations of motionGeneral equations of motionSingle Particle DynamicsMatrix Representation

D. Robin

18

• The equations become

• Inhomogeneous equations with s-dependent coefficients

• Note that the term 1/ρ2 corresponds to the dipole weak focusing

• The term ∆P/(Pρ) is present for off-momentum particles


Equations of motion Equations of motion –– Linear fields Linear fields


D. Robin

19

• Solutions are combination of the ones from the homogeneous and inhomogeneous equations

• Consider particles with the design momentum. The equations of motion become

with

• Hill’s equations of linear transverse particle motion• Linear equations with s-dependent coefficients (harmonic oscillator

with time dependent frequency)

• In a ring or in transport line with symmetries, coefficients are periodic

• Not feasible to get analytical solutions for all accelerator

George Hill


Hill’s equationsHill’s equationsSingle Particle DynamicsMatrix Representation

D. Robin

20

u

u

• Consider K(s) = k0 = constant

• Equations of harmonic oscillator with solution

with

for k0 > 0

for k0 < 0

Note that the solution can be written in matrix form


Harmonic oscillator Harmonic oscillator –– springspringSingle Particle DynamicsMatrix Representation

D. Robin

21

• General transfer matrix from s0 to s

• Note that

which is always true for conservative systems

• Note also that

• The accelerator can be build by a series of matrix multiplications

from s0 to s1

from s0 to s2

from s0 to s3

from s0 to sn

…S0

S1 S2 S3 Sn-1

Sn


Matrix formalismMatrix formalismSingle Particle DynamicsMatrix Representation

D. Robin

22

• System with mirror symmetry

S

• System with normal symmetry

S


Symmetric linesSymmetric linesSingle Particle DynamicsMatrix Representation

D. Robin

23

to get a total 4x4 matrix

• Combine the matrices for each plane


4x4 Matrices4x4 MatricesSingle Particle DynamicsMatrix Representation

D. Robin

24

0 L

x’

x

x’⋅L

s

• Consider a drift (no magnetic elements) of length L=s-s0

• Position changes if there is a slope. Slope remains unchanged


Transfer matrix of a driftTransfer matrix of a driftSingle Particle DynamicsMatrix Representation

D. Robin

25

0 f

x’

x

x’

x

0 f

• Consider a lens with focal length ±f

• Slope diminishes (focusing) or increases (defocusing). Position remains unchanged


Focusing Focusing -- defocusing thin lensdefocusing thin lensSingle Particle DynamicsMatrix Representation

D. Robin

26

• Consider a quadrupole magnet of length L. The field is

• with normalized quadrupole gradient (in m-2)

The transport through a quadrupole is

x’

x0 L s


QuadrupoleQuadrupoleSingle Particle DynamicsMatrix Representation

D. Robin

27

• Consider a dipole of length L. By setting in the focusing quadrupole matrix

the transfer matrix for a sector dipole becomes

with a bending radius

• In the non-deflecting plane

• This is a hard-edge model. In fact, there is some edge focusing in the vertical plane

θ

L


Sector DipoleSector DipoleSingle Particle DynamicsMatrix Representation

D. Robin

28

• Consider a rectangular dipole of length L. At each edge, the deflecting angle is

It acts as a thin defocusing lens with focal length

• The transfer matrix is with

• For θ<<1, δ=θ/2.

• In deflecting plane (like drift) in non-deflecting plane (like sector)

θ


Rectangular DipoleRectangular DipoleSingle Particle DynamicsMatrix Representation

D. Robin

29

x

L

• Consider a quadrupole doublet, i.e. two quadrupoles with focal lengths f1 and f2 separated by a distance L.

• In thin lens approximation the transport matrix is

with the total focal length

• Setting f1 = - f2 = f• Alternating gradient focusing seems overall focusing

• This is only valid in thin lens approximation!!!Fundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

QuadrupoleQuadrupole Doublet Doublet and AG Focusingand AG Focusing


D. Robin

30

• Consider a defocusing quadrupole “sandwiched” by two focusing quadrupoles with focal lengths f.

• The symmetric transfer matrix from center to center of focusing quads

with the transfer matrices

• The total transfer matrix is

L L


FODO CellFODO CellSingle Particle DynamicsMatrix Representation

D. Robin

31

• From Gauss law of magnetostatics, we construct a vector potential

• Assuming a 2D field in x and y, the vector potential has only one component As

• The Ampere’s law in vacuum (inside the beam pipe)

• Using the previous equations one finds the conditions which are Riemann conditions of an analytic function.

There exist a complex potential of z = x+iy with a power series expansion convergent in a circle with radius |z| = rc (distance from iron yoke)

x

yiron

rc


Magnetic Magnetic MultipoleMultipole ExpansionExpansionSingle Particle DynamicsMatrix Representation

D. Robin

32

• From the complex potential we can derive the fields

• Setting we have

• Define normalized units

on a reference radius, 10-4 of the main field to get

• Note: n’=n-1 the American conventionFundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

Magnetic Magnetic MultipoleMultipole ExpansionExpansionSingle Particle DynamicsMatrix Representation

D. Robin

33



D. Robin


34


L3 Possible HomeworkL3 Possible HomeworkSingle Particle Dynamics Matrix Representation

D. Robin

•• Derive the thin matrix representation for a focusing Derive the thin matrix representation for a focusing quadrupolequadrupole starting from the starting from the “thick” element matrix. Hint: calculate the limit for the matrix“thick” element matrix. Hint: calculate the limit for the matrix when the when the quadrupolequadrupole

length approaches zero while the integrated magnetic field is kelength approaches zero while the integrated magnetic field is kept constant. pt constant.

•• Suppose that a particle traverses, first, a thin focusing lens Suppose that a particle traverses, first, a thin focusing lens with a focal length with a focal length F; second, a drift of length L; third, a thin defocusing lens wiF; second, a drift of length L; third, a thin defocusing lens with focal length F; th focal length F;

and, fourth, another drift of length L. Calculate the matrix forand, fourth, another drift of length L. Calculate the matrix for this cell.this cell.

•• Consider a system made up of two thin lenses each of focal lengConsider a system made up of two thin lenses each of focal length th FF, one , one focusing and one defocusing, separated by a distance focusing and one defocusing, separated by a distance LL. Show that the system . Show that the system

is focusing if is focusing if |F|>L|F|>L..

1



Changing the Particle EnergyChanging the Particle Energy


2


Why Accelerate Particles?Why Accelerate Particles?Changing the


In both the relativistic and nonIn both the relativistic and non--relativistic case “ accelerating” relativistic case “ accelerating” a particle means to modify (increase) its energy.a particle means to modify (increase) its energy.

•• In In colliderscolliders tuning the tuning the c.mc.m. energy on resonance allows . energy on resonance allows to create new particles to create new particles

In most of accelerator applications the particle energyIn most of accelerator applications the particle energyis one of the fundamental design parameter and tuning knob:is one of the fundamental design parameter and tuning knob:

•• In light sources the energy defines the spectrum of the In light sources the energy defines the spectrum of the emitted radiation emitted radiation

•• The energy defines the penetration depth of a particle The energy defines the penetration depth of a particle inside materials (cancer therapy, …)inside materials (cancer therapy, …)

In relativistic particles storage rings the energy losses needIn relativistic particles storage rings the energy losses needto be restored in order to keep the particles stored. to be restored in order to keep the particles stored.

3


How to Accelerate Particles?How to Accelerate Particles?Changing the


Charged particles:Charged particles: Electric fields mainly.Electric fields mainly.

Neutral particlesNeutral particles can be accelerated by:can be accelerated by:scattering, ‘scattering, ‘spallationspallation’’

Much more efficient and much more controllableMuch more efficient and much more controllable

Large number of schemes and techniques used to Large number of schemes and techniques used to generate the required electric fields.generate the required electric fields.

Continuous R&D going onContinuous R&D going on

4


Electromagnetic FieldsElectromagnetic FieldsChanging the


Maxwell Equations in vacuum (SI Units Maxwell Equations in vacuum (SI Units –– differential form):differential form):

dt

BdE −=×∇

dt

EdJB 000 εµµ +=×∇

0ερ=⋅∇ E

0=⋅∇ B

Time variable Time variable magnetic fieldsmagnetic fields

are are always always associated with associated with electric fieldselectric fields

(and vice versa)(and vice versa)

Coulomb’s or Gauss’ law for electricityCoulomb’s or Gauss’ law for electricity

Gauss’ law for magnetismGauss’ law for magnetism

Faraday’s lawFaraday’s law

Ampere’s lawAmpere’s law

18311831--18791879

5


LorentzLorentz EquationEquationChanging the


B fields can change the trajectory of a particleB fields can change the trajectory of a particleBut But cannotcannot do do work work and thus change its energyand thus change its energy

( )BvEqF ×+=

EqF =

ldFW ⋅= ( ) ⋅×+⋅= ldBvqldEq

ldEqW ⋅=18531853--19281928

6


Electric Field RepresentationElectric Field RepresentationChanging the


( )∞

−∞=

−=n

kstninoeEE 0ω

( ) ( ) ( )[ ]kstikstEeEE oksti

o −+−== − ωωω sincos

( ) ( )∞

∞−

−= kstiefdtE ωωπ2

1

We will use in our calculations this representation.We will use in our calculations this representation.Such a representation is quite general.Such a representation is quite general.

In fact, arbitrary electric fields can be represented as:In fact, arbitrary electric fields can be represented as:

Plane wave representation:Plane wave representation:

Periodic Case Periodic Case NonNon--periodic Case periodic Case

7


From DC to ….From DC to ….Changing the


( )kstieEE −= ω0

In actual accelerators we often deal with a single frequency:In actual accelerators we often deal with a single frequency:

02 ≈πω

Hz310102 −≈πω

Hz116 10102 −≈πω

Hz1812 10102 −≈πω

Electrostatic AcceleratorsElectrostatic Accelerators

Induction, Induction, BetatronsBetatrons

Radio Frequency (RF) Radio Frequency (RF) acceleratorsaccelerators

Laser Laser ponderomotiveponderomotive accelaccel ..

Present dominant technologyPresent dominant technology

8


Electrostatic Accelerators:Electrostatic Accelerators:The Simplest SchemeThe Simplest Scheme


Still one of the most used schemes for electron sourcesStill one of the most used schemes for electron sources

- --

-- -

Cathode Anode

E HVVqW −=

HVV

BudkerBudker InstituteInstitute

Diode PierceGeometry

9


Electrostatic Accelerators:Electrostatic Accelerators:The The CockcroftCockcroft--Walton SchemeWalton Scheme


James James CockcroftCockcroft and Ernest Walton in and Ernest Walton in 1932 accelerated protons to 800 1932 accelerated protons to 800 keVkeVand produced fission of Lithium in and produced fission of Lithium in

Helium Helium (Nobel Prize 1951)(Nobel Prize 1951)

HeLip 2→+

FERMILABFERMILAB

Still used as the first Still used as the first accelerator stage for protons accelerator stage for protons

and ionsand ions

V = V0 sin(ωωωωt)

2 V0 4 V0

VOut = 2 V0 NCELLS

VOut

CASCADE GENERATOR (1914)CASCADE GENERATOR (1914)

10


Electrostatic Accelerators:Electrostatic Accelerators:The Van de The Van de GraaffGraaff


7MV in 19337MV in 1933~ 20 MV nowadays~ 20 MV nowadays

••The needle transmits the charge to The needle transmits the charge to the belt by glow discharge and/or the belt by glow discharge and/or

field emissionfield emission

••The electric field inside the sphere The electric field inside the sphere is zero permitting the passage of the is zero permitting the passage of the

charge from the belt to the spherecharge from the belt to the sphere

••The maximum voltage is limited by The maximum voltage is limited by voltage breakdown. Inert gasses voltage breakdown. Inert gasses

((FreonFreon, SF6) help., SF6) help.

11


Van de Van de GraaffGraaff Accelerator:Accelerator:ApplicationsApplications


Tandem SchemeTandem Scheme

••Negative ions (HNegative ions (H-- for example) are for example) are created and accelerated through the created and accelerated through the

first stagefirst stage••At the end of the first stage the At the end of the first stage the

electrons are ‘stripped’ out from the electrons are ‘stripped’ out from the ions (by a gas target for example)ions (by a gas target for example)

••In the second stage the positive ions In the second stage the positive ions (protons in our example) are (protons in our example) are

accelerated. The net energy gain is accelerated. The net energy gain is twicetwice the voltage of the Van de the voltage of the Van de GraaffGraaff

+

++

+++ +

1st Stage

2nd Stage

12


Low Frequency Accelerators:Low Frequency Accelerators:Induction Induction LinacsLinacs & & BetatronsBetatrons


dt

BdE −=×∇

Induction Accelerators allows for very high currents (~ 1kA) at Induction Accelerators allows for very high currents (~ 1kA) at relatively moderate energies (few relatively moderate energies (few MeVMeV))

Beam

( )tII ωsin0=InductionLinac

First Induction Accelerators in ~ 1935First Induction Accelerators in ~ 1935

Betatron: ring for beta particles (electrons)

Hz60502 ÷≈πω

13


EM Fields in Free Space:EM Fields in Free Space:The Wave EquationThe Wave Equation


zyxit

E

cz

E

y

E

x

EE iiii

i ,,1

2

2

22

2

2

2

2

22 =

∂∂=

∂∂+

∂∂+

∂∂=∇

From Maxwell equations it is possible to derive for free space From Maxwell equations it is possible to derive for free space electromagnetic waves:electromagnetic waves:

2

2

2

2

2

22

:

zyx

Laplacian

∂∂+

∂∂+

∂∂=∇

( ) ( ) ( )( ) ( ) ( )( )[ ]kztikztiB

kztikztiFzz eeAeeAyxEE +−+−−− ±+±= ωωωω,0

For accelerating particles we need For accelerating particles we need a nona non--zero field component in the zero field component in the

z direction (for example)z direction (for example)2

2

22 1

t

E

cE z

z ∂∂−=∇

Forward waveForward wave Backward waveBackward wave

And we will look for a solution in the shape:And we will look for a solution in the shape:

14


EM Fields in Free Space:EM Fields in Free Space:Solution in Cylindrical CoordinatesSolution in Cylindrical Coordinates


011

02

2

2

20

2

20

20

2

=

−+

∂∂+

∂∂+

∂∂

zzzz Ek

c

E

rr

E

rr

E ωθ

( ) ikttizz eerEE ±±= ωθ,0 2

2

22 1

t

E

cE z

z ∂∂−=∇

2

2

2

2

22

22 11

zrrrr ∂∂+

∂∂+

∂∂+

∂∂=∇

θ

( ) ( ) θθ inzz erErE ±= 00

~, 0

~~

1~

02

22

2

20

20

2

=

−−+

∂∂+

∂∂

zzz E

r

nk

cr

E

rr

E ω

( ) ( )rkBrkAE CnCnz YJ~

0 += 22

22 k

ckC −= ω

The typical RF accelerating structures present axial symmetry.The typical RF accelerating structures present axial symmetry.The natural coordinates for this case are the cylindrical coordiThe natural coordinates for this case are the cylindrical coordinates.nates.

Again, because of the axial symmetry it is convenient to assume Again, because of the axial symmetry it is convenient to assume that that the the azimuthalazimuthal component of the field has periodicity component of the field has periodicity nn

Which has as general solution:Which has as general solution:

wherewhere cutoffcutoffwavenumberwavenumber

15


EM Fields in Free Space:EM Fields in Free Space:Wave Equation General SolutionWave Equation General Solution


( ) ( )[ ] ikztiinCnCnz eeerkBrkAE ±±±+= ωθYJ

( ) ( ) ( ) ( )( ) ( ) ( )( )[ ]kztikztiB

kztikztiFCnz eeAeeAnrkE +−+−−− ±+±= ωωωωθcosJ

( ) kindtheoffunctionsBesselx stn 1J ≡

( ) kindtheoffunctionsBesselx ndn 2Y ≡

And analogously for the longitudinal component of the magnetic fAnd analogously for the longitudinal component of the magnetic f ield:ield:

By using these expressions and the Maxwell curl equations it is By using these expressions and the Maxwell curl equations it is possible to derive similar expressions for possible to derive similar expressions for EErr , , EEθθθθθθθθ, , , , BBrr and and BBθθθθθθθθ

( ) ( ) ( ) ( )( ) ( ) ( )( )[ ]kztikztiB

kztikztiFCnz eeCeeCnrkB +−+−−− ±+±= ωωωωθcosJ

16


EM Fields in Free Space:EM Fields in Free Space:Boundary Conditions & ClassificationBoundary Conditions & Classification


The general solution can now be applied to real geometries.The general solution can now be applied to real geometries.

By imposing the values that the fields assume at the boundaries By imposing the values that the fields assume at the boundaries ((boundary conditionsboundary conditions), the values of the constants in the general ), the values of the constants in the general

solution can be evaluated and the solution for the specific solution can be evaluated and the solution for the specific geometry is found.geometry is found.

It is useful to classify the possible solutions in the It is useful to classify the possible solutions in the following categories:following categories:

•• TM modesTM modes: where the magnetic field is in : where the magnetic field is in the transverse planethe transverse plane

•• TE modesTE modes: where the electric field is in : where the electric field is in the transverse planethe transverse plane

•• TEM modesTEM modes: where both the electric and : where both the electric and magnetic fields are in the transverse planemagnetic fields are in the transverse plane

TMTMnnθθθθθθθθnnrrnnzz

nnθθθθθθθθ : periodicity in : periodicity in θθθθθθθθnnrr : periodicity in r: periodicity in rnnzz : periodicity in z: periodicity in z

Good forGood forAcceleration!Acceleration!

17


EM Fields in Free Space:EM Fields in Free Space:The Cutoff FrequencyThe Cutoff Frequency


22

22

Ckc

k −= ωFrom the definition of cutoff From the definition of cutoff wavenumberwavenumber::

( ) ( )kztiFCz eArkE −= ω

0J

CC ck=ω Cutoff (angular) frequencyCutoff (angular) frequencyBy defining:By defining:

propagates wavethereal is02 >> kkCωω

llyexponentia decreases and

propagatenot does wavetheimmaginary is02 << kkCωω

zkC on dependnot does and propagatesnot does wavethe0== ωω

18


EM Fields in Free Space:EM Fields in Free Space:Pill Box Cavity ExamplePill Box Cavity Example


E

B

beam

a

L

Pill Box Boundary Conditions:Pill Box Boundary Conditions:( ) ( ) 0==== arEarEz θ

( ) ( ) 000 ==== zEzEr θ

( ) ( ) 0==== LzELzEr θ

( ) ( )trkEE CCTMz ωcosJ00

010 =

For example, for the TM010 mode:

0010010 =∂∂=∂∂ θTMz

TMz EzE

akC 405.2=

( ) ( )trkEkc

B CCC

TM ωωθ sinJ102

010 −= acf CC ππω 2405.22 ==

mmaMHzf 5.229500:Example 0 =⇔=

TMmodes

19


Phase Velocity & Group Velocity:Phase Velocity & Group Velocity:DefinitionsDefinitions


Phase VelocityPhase Velocity

( )kztEEz −= ωcos0

0=−=dt

dzk

dt

d ωϕ

kzt −=ωϕ

kvP

ω=

dk

dvG

ω=

( ) ( ) ( ) ( )c

c

cckckv

CCC

P >−

=−

=−

==22222 1 ωωωω

ωω

ωω

22 kkc C +=ω

ckk

c

kk

ck

dk

dv

CC

G <+

=+

==2222 1

ω

For propagating waves For propagating waves vvPP > > cc No acceleration is possible!No acceleration is possible!

Group VelocityGroup Velocity

20


Phase Velocity & Group VelocityPhase Velocity & Group VelocityPhysical MeaningPhysical Meaning


vvPP is the speed at which the phase propagates or at is the speed at which the phase propagates or at which the wave propagates rigidlywhich the wave propagates rigidly

( )kztEEz −= ωcos0

( ) ( )[ ] ( ) ( )[ ] ( ) ( )kztkztzkktzkkt ∆−∆−=∆−−∆−+∆+−∆+ ωωωωωω coscos2coscos

ck

vP >= ω

cdk

dvG <= ω vvGG is the speed at which the energy propagates or is the speed at which the energy propagates or

a variation of the wave envelope propagatesa variation of the wave envelope propagates

For example: wave beating.For example: wave beating.

kvP

ωω =

dk

d

kvP

ωωω ≈

∆∆=∆

21


RF Accelerators:RF Accelerators:WideroeWideroe and Alvarez Schemesand Alvarez Schemes


In 1946 Alvarez overcame to the In 1946 Alvarez overcame to the inconvenient by including the inconvenient by including the WideroeWideroestructure inside a large metallic tube structure inside a large metallic tube forming an efficient cavity where the forming an efficient cavity where the

fields were confined.fields were confined.

In 1925In 1925--28 28 IsingIsing and and WideroeWideroe conceived the first linear accelerator (conceived the first linear accelerator (linaclinac). The ). The revolutionary device was based on the revolutionary device was based on the drift tubes schemedrift tubes scheme..

RFii TvL2

1≅Synchronicity condition:Synchronicity condition:

At high frequency the At high frequency the WideroeWideroe scheme becomes scheme becomes lossylossy due to electromagnetic due to electromagnetic radiation.radiation.

During the decelerating half period of the RF, the beam is shielDuring the decelerating half period of the RF, the beam is shielded inside the ded inside the conductive tubes.conductive tubes.

200 MHz RF200 MHz RFsource from radarssource from radars

The Alvarez structures are still widely used as preThe Alvarez structures are still widely used as pre--accelerator for protons and accelerator for protons and ions. The particles at few hundred ions. The particles at few hundred keVkeV from a from a CockcroftCockcroft--Walton for example, are Walton for example, are

accelerated to few hundred accelerated to few hundred MeVMeV..

22


Accelerating Structure Evolution Accelerating Structure Evolution Changing the


By “ loading” the cylindrical structure by disks By “ loading” the cylindrical structure by disks vvPP can be reduced down to can be reduced down to match the speed of the particle for an efficient accelerationmatch the speed of the particle for an efficient acceleration

KEK PARTICLEv

Pv

-- Traveling wave, constant impedance: Traveling wave, constant impedance: electric field decreases exponentially electric field decreases exponentially with length. The irises have constant radius.with length. The irises have constant radius.

-- Traveling wave, constant gradient: Traveling wave, constant gradient: the electric field is constant along the the electric field is constant along the structure. The iris have decreasing radius.structure. The iris have decreasing radius.

-- Standing wave: Standing wave: no wave propagation. The beam transit time in the cell must no wave propagation. The beam transit time in the cell must be much smaller than the wave period for efficient accelerationbe much smaller than the wave period for efficient acceleration

-- Normal conductive or Super conductiveNormal conductive or Super conductiveLL--band ~ 1.5 GHz, Sband ~ 1.5 GHz, S--band ~ 3 GHz, Xband ~ 3 GHz, X--band ~ 11GHz, ….band ~ 11GHz, ….

Newer and more efficient RF structures were obtained by couplingNewer and more efficient RF structures were obtained by coupling together together many pillboxmany pillbox--like cavities.like cavities.

23


Energy Gain in RF StructuresEnergy Gain in RF StructuresChanging the


Synchronism:Synchronism:

ssKIN eLPrqE ϕτ cos1 20

−−=

ssKIN

eLPrqE ϕ

ττ

τ

cos1

20

−−=

2length th cavity wiaFor

4

4sin

0RF

particle

RF

particle

RF

sKIN L

v

vPRqE

λωλ

ωλ

=

=Standing Standing

WaveWave

Traveling Wave Traveling Wave Constant ImpedanceConstant Impedance

Traveling Wave Traveling Wave Constant GradientConstant Gradient

PARTICLEv

PvTW:TW:

t

Ez

SW:SW:

( ) ( )tErE CTMz ωcos0 0

010 ==Transit Transit

time factortime factor

24

Frequency Frequency PreferencePreference

Frequency Frequency ScalingScaling

PARAMETERPARAMETER


Scaling with FrequencyScaling with FrequencyChanging the


0LPrE sKIN ∝

The table is incomplete. For an exhaustive list see for example:The table is incomplete. For an exhaustive list see for example:The Stanford TwoThe Stanford Two--Mile AcceleratorMile Accelerator (Chapter 6), R.B. Neal Editor (1968).(Chapter 6), R.B. Neal Editor (1968).

Higher Higher frequencies frequencies pushes to pushes to

high energy high energy applicationsapplications

Lower Lower frequencies frequencies pushes to pushes to

high current high current applicationsapplications

Choosing the RF frequency is a very critical step in designing aChoosing the RF frequency is a very critical step in designing an n accelerator. Several parameters depend in opposite way on the accelerator. Several parameters depend in opposite way on the

frequency and the best match to the application will be a frequency and the best match to the application will be a frequency that trading between contrasting requirements:frequency that trading between contrasting requirements:

HighHighωωωωωωωω-- 22RF energy stored in the structureRF energy stored in the structure

HighHighωωωωωωωω1/21/2Maximum possible electric field strength (empirical)Maximum possible electric field strength (empirical)

HighHighωωωωωωωω-- 1/21/2Total RF peak power (Total RF peak power (PP00))

HighHighωωωωωωωω1/21/2Shunt impedance per unit length (Shunt impedance per unit length (rrss))



PARAMETERPARAMETER

LowLowωωωωωωωω-- 11Diameter of beam apertureDiameter of beam aperture

LowLowωωωωωωωω-- 22Maximum RF power available for single sourceMaximum RF power available for single source

LowLowωωωωωωωω 1/21/2Beam Loading (Beam Loading (--dV/didV/di))

LowLowωωωωωωωω-- 1/21/2Peak beam current at maximum conversion efficiencyPeak beam current at maximum conversion efficiency

HighHighωωωωωωωω-- 22RF energy stored in the structureRF energy stored in the structure

HighHighωωωωωωωω1/21/2Maximum possible electric field strength (empirical)Maximum possible electric field strength (empirical)

HighHighωωωωωωωω-- 1/21/2Total RF peak power (Total RF peak power (PP00))

HighHighωωωωωωωω1/21/2Shunt impedance per unit length (Shunt impedance per unit length (rrss))



PARAMETERPARAMETER

25


Large and SmallLarge and SmallLinear AcceleratorsLinear Accelerators


•• The combined development and optimization of the described RF sThe combined development and optimization of the described RF structures tructures jointly to the development of powerful and efficient RF sources jointly to the development of powerful and efficient RF sources (klystrons), (klystrons),

permitted the ambitious design and construction of large and verpermitted the ambitious design and construction of large and very high y high energy linear accelerators.energy linear accelerators.

•• In the International Linear In the International Linear ColliderCollider (ILC) (ILC) project, electron and positron project, electron and positron linacslinacs longer than longer than

30 Km and with energies over 500 30 Km and with energies over 500 GeVGeV are are under consideration.under consideration.

•• Above all, we want to mention the 3Above all, we want to mention the 3--km linear accelerator that started km linear accelerator that started operating in 1966 at the Stanford Linear Accelerator Center and operating in 1966 at the Stanford Linear Accelerator Center and that is that is

capable of accelerating electron and positrons up to more than 5capable of accelerating electron and positrons up to more than 50 0 GeVGeV, with , with an average gradient in the RF structure of ~ 17 an average gradient in the RF structure of ~ 17 MeV/mMeV/m..

•• At the same time, much smaller At the same time, much smaller linacslinacs from few from few MeVMeV to few hundred to few hundred MeVMeV are are the “ backbone” of the injector in most existing electron accelerthe “ backbone” of the injector in most existing electron accelerators.ators.

••R&D on higher frequency RF structures is R&D on higher frequency RF structures is demonstrating gradients larger than 100 demonstrating gradients larger than 100 MeV/mMeV/m..

26


Linear AcceleratorsLinear Acceleratorsvs. Circular Acceleratorsvs. Circular Accelerators


•• RF cavity technology allowed the development of both linear andRF cavity technology allowed the development of both linear andcircular accelerators.circular accelerators.

•• The main advantage of circular accelerators is that a single caThe main advantage of circular accelerators is that a single cavity, vity, where the beam passes many times guided by the confinement actiowhere the beam passes many times guided by the confinement action n of magnetic fields, is capable of very high energy acceleration.of magnetic fields, is capable of very high energy acceleration. This is This is

a very efficient scheme where only a relatively small amount of a very efficient scheme where only a relatively small amount of RF RF power is required.power is required.

•• Unfortunately, for light particles the emission of synchrotron Unfortunately, for light particles the emission of synchrotron radiation radiation can limit the maximum energy achievable (~ 100 can limit the maximum energy achievable (~ 100 GeVGeV for electrons).for electrons).

•• In general, circular accelerators are more efficient with heavyIn general, circular accelerators are more efficient with heavyparticles and medium energy electrons, while linear acceleratorsparticles and medium energy electrons, while linear accelerators are are

preferred with high energy electrons.preferred with high energy electrons.

•• Efficiency is not all. For example, circular machines usually sEfficiency is not all. For example, circular machines usually show how more stable beam characteristics while the beam more stable beam characteristics while the beam emittanceemittance can be can be

(maintained) smaller in linear accelerators. Different applicati(maintained) smaller in linear accelerators. Different applications can ons can find their best match in either one or the other schemes. find their best match in either one or the other schemes.

27


Cyclotron andCyclotron andSynchroSynchro--cyclotroncyclotron


1939 Nobel PrizeProton Source

UniformMagnetic

Field

AcceleratedProtons

ElectricField

cvforeB

m

veB

mv

veB

p

v

rTR <<==== ππππ 2222

For nonFor non--relativistic particlesrelativistic particlesthe revolution periodthe revolution period

does not depend on energydoes not depend on energy

•• If the RF frequency is equal to the particles revolution frequeIf the RF frequency is equal to the particles revolution frequency synchronicity is ncy synchronicity is obtained and acceleration is achieved.obtained and acceleration is achieved.

•• The The synchrosynchro--cyclotron is a variation that allows acceleration also of relaticyclotron is a variation that allows acceleration also of relativistic vistic particles. The RF frequency is dynamically changed to matchparticles. The RF frequency is dynamically changed to match

the changing revolution frequency of the particlethe changing revolution frequency of the particle

The first cyclotron4.5” diameter (1929).

In an uniform magnetic field:In an uniform magnetic field:

•• In 1946 Lawrence built in Berkeley the 184” In 1946 Lawrence built in Berkeley the 184” synchrosynchro--cyclotron with an orbit radius cyclotron with an orbit radius of 2.337 m and capable of 350 of 2.337 m and capable of 350 MeVMeV protons. The largest cyclotron still in operation is protons. The largest cyclotron still in operation is

in in GatchinaGatchina and accelerates protons to up 1 and accelerates protons to up 1 GeVGeV for nuclear physics experiments. for nuclear physics experiments.

E. O. Lawrence

28


MicrotronsMicrotronsChanging the


integeran is nn=∆γSynchronicity conditionSynchronicity condition(energy gain per turn)(energy gain per turn)

0.511 0.511 MeVMeV for electronsfor electrons938 938 MeVMeV for protons for protons

Useful only for accelerating electrons.Useful only for accelerating electrons.The maximum energy is~ 30 The maximum energy is~ 30 MeVMeV (limited by the magnet size)(limited by the magnet size)

B

magnet

RFcavity

magneticshield

Electronsource

VekslerVeksler19441944

MorozMoroz and Roberts 1958and Roberts 1958

29


Synchrotrons &Synchrotrons &Storage RingsStorage Rings


APSAPS

••Achieving higher energies in cyclotrons Achieving higher energies in cyclotrons requires very large magnets.requires very large magnets.

Above ~ 400 Above ~ 400 MeVMeV the realization of cyclotronsthe realization of cyclotronsbecomes inconvenient and expensivebecomes inconvenient and expensive

•• A storage ring is a synchrotron were the particles are not acceA storage ring is a synchrotron were the particles are not accelerated lerated but just stored at a fixed energy for a relatively long time.but just stored at a fixed energy for a relatively long time.

CollidersColliders, synchrotron light sources, …, synchrotron light sources, …

onaccelerati during changemust

beam icrelativist-nonfor

2 00

RFRF

f

m

BeZhfhf ==

γπ

•• The synchronicity condition is given by:The synchronicity condition is given by:

•• Synchrotrons have achieved energy as high asSynchrotrons have achieved energy as high as100 100 GeVGeV for electrons and 1000 for electrons and 1000 GeVGeV for protonsfor protons

•• In a synchrotron the radius is fixed and all the In a synchrotron the radius is fixed and all the fields can be confined only around the orbit.fields can be confined only around the orbit.

βγβγto scalemust constant0 ∝== B

ZeB

cmR

30


RF Sources: The KlystronRF Sources: The KlystronChanging the


•• Invented by Hansen Invented by Hansen and the Varian brothers and the Varian brothers

in 1937in 1937

•• Very powerful source Very powerful source from ~ 100 MHz to more from ~ 100 MHz to more

than 10 GHzthan 10 GHz

•• Widely used in all kinds of acceleratorsWidely used in all kinds of accelerators

31


R&D on FutureR&D on FutureAccelerator SchemesAccelerator Schemes


•• The R&D on new acceleration techniques is extremely active and The R&D on new acceleration techniques is extremely active and addressed addressed towards a very large variety of new accelerating techniques. Restowards a very large variety of new accelerating techniques. Results are ults are

promising especially from the accelerating gradient point of viepromising especially from the accelerating gradient point of view were extremely w were extremely high values have been already obtained.high values have been already obtained.

•• Among the techniques under study, Among the techniques under study, here we want to mention as an here we want to mention as an

example, the one based on the soexample, the one based on the so--called called laser laser wakefieldwakefield accelerationacceleration..Laser

Gas

Gas jet nozzle

e- bunch

Plasmachannel

L’OASISL’OASISLBLLBL

•• A high intensity laser is focused on an A high intensity laser is focused on an atomic gas jet.atomic gas jet.

••The laser ionizes most of the atoms The laser ionizes most of the atoms creating a plasma and also stimulating a creating a plasma and also stimulating a resonant motion of the electrons in the resonant motion of the electrons in the

plasma. plasma.

•• This electron motion breaks the charge balance inside the very This electron motion breaks the charge balance inside the very dense plasma dense plasma inducing extremely high gradients in the plasma area surroundinginducing extremely high gradients in the plasma area surrounding the laser.the laser.

••Electrons in the plasma can find the right phase and can be acceElectrons in the plasma can find the right phase and can be accelerated to high lerated to high energies. Gradients of many tens of energies. Gradients of many tens of GeV/mGeV/m in few mm have been already in few mm have been already

demonstrated.demonstrated.

32


The Cyclotron:The Cyclotron:Different Points of ViewDifferent Points of View


By Dave Judd and By Dave Judd and RonnRonn MacKenzieMacKenzie

From LBNL Image LibraryFrom LBNL Image LibraryCollectionCollection

…the operator

33


Possible HomeworkPossible HomeworkChanging the


••Prove that the magnetic field cannot do workProve that the magnetic field cannot do work

••Explain how the cascade generator in the Explain how the cascade generator in the CockcroftCockcroft--Walton Walton schemes worksschemes works

••Explain why the electric field inside the Van de Explain why the electric field inside the Van de GraaffGraaff sphere sphere is zerois zero

••Show that for free space electromagnetic waves E and B vectors aShow that for free space electromagnetic waves E and B vectors are mutually re mutually orthogonal and have no component in the wave propagation directiorthogonal and have no component in the wave propagation direction.on.

Show also the relation between their modules.Show also the relation between their modules.

••Calculate the maximum energy gain in Calculate the maximum energy gain in MeVMeV vs. input power in MW for a 3.048 vs. input power in MW for a 3.048 m long constant gradient accelerating structure with shunt impedm long constant gradient accelerating structure with shunt impedance for unit ance for unit

length length rrss= 53 M= 53 MΩΩΩΩΩΩΩΩ/m and attenuation factor /m and attenuation factor ττττττττ = 0.57. Calculate also the power = 0.57. Calculate also the power required for accelerating relativistic electrons to 60 required for accelerating relativistic electrons to 60 MeVMeV

•• Derive the expression for the transit time factor for a pillboxDerive the expression for the transit time factor for a pillbox resonating in its resonating in its TMTM010010 modemode

•• Calculate the internal diameter of the external pipe of a 200 MCalculate the internal diameter of the external pipe of a 200 MHz Alvarez Hz Alvarez structure operating in the TM010 mode.structure operating in the TM010 mode.

34


VectorialVectorial AlgebraAlgebraChanging the


Curl Theorem (Curl Theorem (Stoke’sStoke’s Theorem)Theorem)

Divergence TheoremDivergence Theorem

Volume IntegralVolume Integral

Surface Integral (Flux)Surface Integral (Flux)

Line Integral (Line Integral (CircuitationCircuitation))

⋅∇=⋅VS

dVFdSnF

( ) ⋅×∇=⋅Sl

dSnFldF

( ) FF ∀=×∇⋅∇ 0

( ) uu ∀=∇×∇ 0 uFF ∇=⇔=×∇ 0oror

((FF is conservative if curl is conservative if curl FF is zero)is zero)

( ) ( ) FFFF ∀∇−⋅∇∇=×∇×∇ 2

35


From Maxwell EquationsFrom Maxwell Equationsto Wave Equationto Wave Equation


1



Phase Space Representation.Phase Space Representation.Ensemble of Particles, Ensemble of Particles, EmittanceEmittance..


2


From Single Particle to a BeamFrom Single Particle to a BeamPhase Space& EmittanceF. Sannibale

•• The number of particles per bunch in most accelerators is The number of particles per bunch in most accelerators is typically included between 10typically included between 1055 to 10to 101313. .

•• Integrating the particle motion for such a large number of Integrating the particle motion for such a large number of particles along accelerators with length ranging from few particles along accelerators with length ranging from few

meters up to tens of kilometers can prove to be a tough task.meters up to tens of kilometers can prove to be a tough task.

•• Fortunately, Fortunately, statistical mechanicsstatistical mechanics gives us very developed gives us very developed tools for representing and dealing with sets of large number tools for representing and dealing with sets of large number

of particles.of particles.

•• Quite often, the statistical approach can give us elegant and Quite often, the statistical approach can give us elegant and powerful insights on properties that could be hard to extract powerful insights on properties that could be hard to extract

by approaching the set using single particle techniques.by approaching the set using single particle techniques.

3


A Convenient Reference FrameA Convenient Reference FramePhase Space& EmittanceF. Sannibale

•• In accelerators we are interested in studying particles along tIn accelerators we are interested in studying particles along their heir trajectory. A natural choice is to refer all the particles relattrajectory. A natural choice is to refer all the particles relat ively to a ively to a

reference trajectoryreference trajectory ..

•• In each point of this trajectory we can define In each point of this trajectory we can define (for example) a Cartesian frame moving with (for example) a Cartesian frame moving with

the reference particle .the reference particle .

•• In this frame the reference particle is always In this frame the reference particle is always at the origin and its momentum is always at the origin and its momentum is always

parallel to the direction of the parallel to the direction of the zz axis.axis.

•• The coordinates The coordinates xx, , yy, , zz for an arbitrary particle represent its for an arbitrary particle represent its displacement relatively to the reference particle along the thredisplacement relatively to the reference particle along the three e

directions.directions.

•• In the lab frame the particle moves on the curvilinear coordinaIn the lab frame the particle moves on the curvilinear coordinate te sswith speed with speed dsds//dtdt. .

•• Such a trajectory is assumed to be the solution of the Such a trajectory is assumed to be the solution of the LorentzLorentz equation equation for the particle with the nominal parameters (reference particlefor the particle with the nominal parameters (reference particle). ).

zy

x

ReferenceReferencetrajectorytrajectory

yxz ˆˆˆ ×=

4


Phase Space RepresentationPhase Space RepresentationPhase Space& EmittanceF. Sannibale

In relativistic classical mechanics, the motion of a single In relativistic classical mechanics, the motion of a single particle is totally defined when, at a given instant particle is totally defined when, at a given instant t,t, the the position position rr and the momentum and the momentum pp of the particle are given of the particle are given

together with the forces acting on that position.together with the forces acting on that position.zzyyxxr iiii ˆˆˆ ++= zpypxpp ziyixi ˆˆˆ ++=

zFyFxFF zyx ˆˆˆ ++=

It is quite convenient to use the soIt is quite convenient to use the so--called called phase spacephase spacerepresentation, a 6representation, a 6--D space where the D space where the ii thth particle assumes the particle assumes the

coordinates: coordinates: ziiyiixiii pzpypxP ,,,,,≡

In most accelerator physics calculations, the three planes can bIn most accelerator physics calculations, the three planes can be e considered with very good approximation as decoupled.considered with very good approximation as decoupled.

In this situation, it is possible and convenient to study the paIn this situation, it is possible and convenient to study the particle rticle evolution independently in each of the planes:evolution independently in each of the planes:

xii px , yii py , zii pz ,

5


Particles Systems & EnsemblesParticles Systems & EnsemblesPhase Space& EmittanceF. Sannibale

wp

w

( )wpwP ,1

wp

w

wp

w

The phase space can now be used for representing particles:The phase space can now be used for representing particles:

The set of possible states for a system of The set of possible states for a system of NN particles particles is referred as is referred as ensembleensemble in statistical mechanics.in statistical mechanics.

In the statistical approach, the particles lose their In the statistical approach, the particles lose their individuality. The properties of the whole system individuality. The properties of the whole system

as individual entity are now studied.as individual entity are now studied.

Important properties of the density functions can now be derivedImportant properties of the density functions can now be derived..Under particular circumstances, such properties allow to calculUnder particular circumstances, such properties allow to calculate the ate the

time evolution of the particle system without going through the time evolution of the particle system without going through the integration of the motion for each single particle.integration of the motion for each single particle.

( ) zyxzyxD dpdzdpdydpdxpzpypxf ,,,,,6( ) zyxwdpdwpwf wwD ,,,2 =

The above expressions indicate the number of particles containedThe above expressions indicate the number of particles contained in the in the elementary volume of phase space for the 6D and 2D cases respectelementary volume of phase space for the 6D and 2D cases respect ively.ively.

zyxw ,,=

zyxwNdpdwf wD ,,2 ==Ndpdzdpdydpdxf zyxD = 6

The system is fully represented by the density of particles The system is fully represented by the density of particles ff66D D and and ff22DD ::

6


Hamiltonian SystemsHamiltonian SystemsPhase Space& EmittanceF. Sannibale

NonNon--Hamiltonian Forces:Hamiltonian Forces:••Stochastic processes (collisions, quantum emission, diffusion, …Stochastic processes (collisions, quantum emission, diffusion, …))

••Inelastic processes (ionization, fusion, fission, annihilation, Inelastic processes (ionization, fusion, fission, annihilation, …)…)••Dissipative forces (viscosity, friction, …)Dissipative forces (viscosity, friction, …)

A system of variables A system of variables qq (generalized coordinates) and (generalized coordinates) and pp (generalized (generalized momentamomenta) is Hamiltonian when exists a function) is Hamiltonian when exists a function

HH((qq, , pp, , tt) that allows to describe the evolution of the system by:) that allows to describe the evolution of the system by:

The function The function HH is called is called HamiltonianHamiltonian and and qq and and pp are referred as are referred as canonical conjugate variables.canonical conjugate variables.

i

i

p

H

dt

dq

∂∂=

i

i

q

H

dt

dp

∂∂−=

In the particular case that In the particular case that qq are the usual spatial coordinates are the usual spatial coordinates xx, , yy, , zz and and p their conjugate p their conjugate momentamomenta ppxx, , ppyy, , ppzz, , HH coincides with the total energy of coincides with the total energy of

the system:the system:

EnergyKineticEnergyPotential +=+= TUH

,,....,,

,,....,,

21

21

N

N

pppp

qqqq

≡≡

7


The Continuity EquationThe Continuity EquationPhase Space& EmittanceF. Sannibale

If there is a flow of matter going inside the If there is a flow of matter going inside the volume then the density inside the volume volume then the density inside the volume

must increase in order to must increase in order to conserve the massconserve the mass..

( ) dzdydxdVvolumetheinmassdzdydxtzyx =≡,,,ρ

dtdSnvdSdtvdm n ⋅−=−= ρρ dSnvdt

dm ⋅−= ρ ⋅−=S

dSnvdt

dM ρ

=V

dVM ρ ⋅−=SV

dSnvdVdt

d ρρ ⋅∇=⋅VS

dVFdSnF

⋅∇=⋅VS

dVvdSnv ρρ ⋅∇−=VV

dVvdVdt

d ρρ

0=⋅∇+∂∂

vt

ρρ

n

v

nvvn ⋅=

dS

The The continuity equationcontinuity equation is a consequence is a consequence of the conservation lawof the conservation law

Let the density Let the density ρ ρ ρ ρ ρ ρ ρ ρ ::

But it is also true that:But it is also true that:

8


The The LiouvilleLiouville TheoremTheoremPhase Space& EmittanceF. Sannibale

1809-1882

( ) xxD pxvpxf ,and,Let 2 ≡≡ρ0=⋅∇+∂∂

vt

ρρ

( ) ( )x

xf

p

pfp

p

fx

x

f

t

f

p

fp

x

fx

t

fvf

t

fD

x

xDx

x

DDD

x

DxDDD

D

∂∂+

∂∂+

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂=⋅∇+

∂∂

22222222

22

02

2

2

222 =∂∂

∂−∂∂

∂=∂∂+

∂∂

xD

xD

x

xDD px

Hf

px

Hf

p

pf

x

xf

td

dfp

p

fx

x

f

t

fvf

t

f Dx

x

DDDD

D 22222

2 =∂∂+

∂∂+

∂∂=⋅∇+

∂∂

02 =td

df D

Let us use the continuity equation with our phase space distribuLet us use the continuity equation with our phase space distributions.tions.For simplicity we will use the 2D distribution, but the same exaFor simplicity we will use the 2D distribution, but the same exact results ct results

apply to the more general 6D case.apply to the more general 6D case.

But our system is HamiltonianBut our system is Hamiltonian

LiouvilleLiouville TheoremTheorem: The phase space density for : The phase space density for a Hamiltonian system is an invariant of the a Hamiltonian system is an invariant of the motion. Or equivalently, the phase space motion. Or equivalently, the phase space

volume occupied by the system is conserved.volume occupied by the system is conserved.

9


Decoupling the Problem:Decoupling the Problem:the Longitudinal Phase Spacethe Longitudinal Phase Space

Phase Space& EmittanceF. Sannibale

0

0

E

EE −=δ

•• In most of existing accelerators the phase space planes are weaIn most of existing accelerators the phase space planes are weakly kly coupled. In particular, we can treat the longitudinal plane indecoupled. In particular, we can treat the longitudinal plane independently pendently

from the transverse one in the large majority of the cases.from the transverse one in the large majority of the cases.

•• In the longitudinal plane we apply our electric fields for acceIn the longitudinal plane we apply our electric fields for accelerating the lerating the particles and changing their energy.particles and changing their energy.

•• It becomes natural to use It becomes natural to use energyenergy as one of the longitudinal plane as one of the longitudinal plane variable together with its canonical conjugate variable together with its canonical conjugate timetime. .

•• The effects of the weak coupling can be then investigated as a The effects of the weak coupling can be then investigated as a perturbation of the uncoupled case.perturbation of the uncoupled case.

•• In accelerator physics, the In accelerator physics, the relative energy variationrelative energy variation δδδδδδδδ and the and the relative relative time ‘distance’time ‘distance’ ττττττττ with respect to a reference particle are often used:with respect to a reference particle are often used:

0tt −=τ

•• According to According to LiouvilleLiouville, in the presence of Hamiltonian forces, the area , in the presence of Hamiltonian forces, the area occupied by the beam in the longitudinal phase space is conserveoccupied by the beam in the longitudinal phase space is conserved.d.

•• More in Lecture 8…….More in Lecture 8…….

10


The Transverse Phase SpaceThe Transverse Phase SpacePhase Space& EmittanceF. Sannibale

ds

dxxpxi =′→

For the transverse planes For the transverse planes xx, , ppxx and and yy, , ppyy, it is usually used a slightly , it is usually used a slightly modified phase space where the momentum components are replaced modified phase space where the momentum components are replaced by:by:

ds

dyypyi =′→

The physical meaning of the new variables:The physical meaning of the new variables:

xcmds

dxvm

dt

dxmp sx ′=== 000 βγγγ

yds

dyy θtan==′

w

s

PROJECTIONWSp

wθxds

dxx θtan==′

ycmpy ′= 0βγ( ) 2121

−−== βγβ and

c

vwhere s

Note that Note that xx and and ppxx are canonical conjugate variables while are canonical conjugate variables while xx and and xx are are not unless there is no acceleration (not unless there is no acceleration (γγγγγγγγ and and ββββββββ constant)constant)

yxw ,=The relation between this new variables and the The relation between this new variables and the

momentum (when momentum (when BBzz = 0) is:= 0) is:

11


Definition of Definition of EmittanceEmittancePhase Space& EmittanceF. Sannibale

We will consider the decoupled case and use the We will consider the decoupled case and use the ww, , ww plane where plane where ww can be either can be either xx or or yy..x′

x

We define as We define as emittanceemittance the phase space area the phase space area occupied by the system of particles, divided by occupied by the system of particles, divided by ππππππππ

yxwA ww

w ,== ′

πε

As we previously shown, As we previously shown, xx and and yy are conjugate to are conjugate to xx and and yy when when BBzz = 0 = 0 and in absence of acceleration. In this case, we can immediatelyand in absence of acceleration. In this case, we can immediately apply apply the the LiouvilleLiouville theorem and state that for such a system the theorem and state that for such a system the emittanceemittance is is

an invariant of the motionan invariant of the motion..

This specific case is actually extremely important.This specific case is actually extremely important.In fact, for most of the elements in a beam In fact, for most of the elements in a beam transferlinetransferline, such as dipoles, , such as dipoles,

quadrupolesquadrupoles, , sextupolessextupoles, …, the above conditions apply and the , …, the above conditions apply and the emittanceemittance is conserved.is conserved.

12


EmittanceEmittance Conservation inConservation inThe Presence of The Presence of BBzz


•• When the When the BBzz component of the magnetic field is present (component of the magnetic field is present (solenoidalsolenoidallenses for example), the transverse planes become coupled and thlenses for example), the transverse planes become coupled and the e phase space area occupied by the system in each of the transversphase space area occupied by the system in each of the transverse e

planes is not conserved anymore.planes is not conserved anymore.

•• Anyway in this situation, the Anyway in this situation, the LiouvilleLiouville theorem still applies to the 4D theorem still applies to the 4D transverse phase space where the transverse phase space where the ipervolumeipervolume occupied by our system is occupied by our system is

still a motion invariant.still a motion invariant.•• Actually, if we rotate the spatial reference frame around the Actually, if we rotate the spatial reference frame around the zz axis by the axis by the LarmorLarmor frequencyfrequency ωωωωωωωωLL = = qBqBzz / / 22γ γ γ γ γ γ γ γ mm00, then the planes become decoupled and , then the planes become decoupled and

the phase space area in each of the planes is conserved again. the phase space area in each of the planes is conserved again.

zx

13


EmittanceEmittance Conservation inConservation inThe Presence of AccelerationThe Presence of Acceleration


When the particles in a beam undergo to acceleration, When the particles in a beam undergo to acceleration, β β β β β β β β and and γγγγγγγγ change change and the variables and the variables xx and and xx are not canonical anymore. are not canonical anymore. LiouvilleLiouville theorem theorem

does not apply and the does not apply and the emittanceemittance is not conserved.is not conserved.

The last expression tells us that the quantity The last expression tells us that the quantity β γ εβ γ εβ γ εβ γ εβ γ εβ γ εβ γ εβ γ ε is a system invariant is a system invariant during acceleration. By defining the during acceleration. By defining the normalized normalized emittanceemittance::

cm

p

p

py y

z

y

0

0tanγβ

θ ===′cm

p

p

py y

z

y

000

0

0

000 tan

γβθ ===′

γβγβ 00

0

=′′

y

y

00

case in thisthat shownbecanIty

y

y

y

′′

=εε

000 yy εγβεγβ =

0yp

0zp0θ

0p 0yp

θ 0200

20

20

2

2

2zz p

cmTT

cTmTp

++=

p

Accelerated by EzAccelerated by Ez

yxwwwn ,== εγβεWe can say that the We can say that the normalized normalized emittanceemittance is conserved during accelerationis conserved during acceleration..

The acceleration couples the longitudinal plane with the transveThe acceleration couples the longitudinal plane with the transverse one:rse one:the 6D the 6D emittanceemittance is still conserved but the transverse is not. is still conserved but the transverse is not.

14


The Real Beam Case:The Real Beam Case:r.m.sr.m.s EmittanceEmittance


222 xxxxrms ′−′=ε

( )( )

′′

′′≅= =

xddxxxf

xddxxxfx

N

xx

D

D

N

nn

,

,

2

22

1

2

2 ( )( )

′′

′′′≅

′=′ =

xddxxxf

xddxxxfx

N

xx

D

D

N

nn

,

,

2

22

1

2

2

( )( )

′′

′′′≅

′=′ =

xddxxxf

xddxxxfxx

N

xxxx

D

D

N

nnn

,

,

2

21

x

x′

rmsrmsrmsrms

xxxx

xx

xx

εεεε

=′′

−′+′

22

2

2

2

For a real beam composed by For a real beam composed by NN particles we can calculate the second particles we can calculate the second order statistical moments of their phase space distribution:order statistical moments of their phase space distribution:

And define the And define the rmsrms emittanceemittance as the quantity:as the quantity:

This is equivalent to associate to the real This is equivalent to associate to the real beam an beam an equivalent or phase ellipseequivalent or phase ellipse in the in the phase space with area phase space with area ππππππππ εεεεεεεεrmsrms and equation:and equation:

2x′

2x

15

0=t

w

w′

w

w′ 01 >= tt

w

w′12 ttt >>=


Nonlinear ForcesNonlinear Forcesand and FilamentationFilamentation


•• In the case of a Hamiltonian system, as a consequence of the In the case of a Hamiltonian system, as a consequence of the LiouvilleLiouvilleTheorem the Theorem the emittanceemittance is conservedis conserved

•• This is true even when the forces acting are on the system are This is true even when the forces acting are on the system are nonlinear nonlinear (space charge, nonlinear magnetic and/or electric (space charge, nonlinear magnetic and/or electric filedsfileds, …), …)

•• This is This is notnot true in the case of the true in the case of the rmsrms emittanceemittance..••In the presence of nonlinear forces the In the presence of nonlinear forces the rmsrms emittanceemittance is not conservedis not conserved

•• Example: Example: filamentationfilamentation . Particles with different phase space coordinates, . Particles with different phase space coordinates, because of the nonlinear forces, move with different phase spacebecause of the nonlinear forces, move with different phase space velocityvelocity

But the But the rmsrms emittanceemittance calculated for increasing times calculated for increasing times increasesincreases..

•• The The emittanceemittance according to according to LiouvilleLiouville is still conserved.is still conserved.

16


The The TwissTwiss ParametersParametersPhase Space& EmittanceF. Sannibale

We saw that a beam with arbitrary phase space We saw that a beam with arbitrary phase space distribution can be represented by an equivalent distribution can be represented by an equivalent

ellipse with area equal to the ellipse with area equal to the rmsrms emittanceemittance divided divided by by ππππππππ. and with equation: . and with equation:

yxwwwww wTwTwTw ,222 ==′++′ εαγβ 1with 2 =− TwTwTw αγβ

The status of the beam at a given moment is totally defined whenThe status of the beam at a given moment is totally defined when the the emittanceemittance and two of the and two of the TwissTwiss parameters are known. parameters are known.

x′

x

yxwwwww

ww

ww

wwwrmsw

,22

2

2

2

==′′

−′+′

εεεε

A convenient representation for this ellipse, often used in acceA convenient representation for this ellipse, often used in accelerator lerator physics, is the one by the sophysics, is the one by the so--called called TwissTwiss Parameters Parameters ββββββββTT, , γγγγγγγγTT and and ααααααααTT ::

By comparing the two ellipse equations, we can derive:By comparing the two ellipse equations, we can derive:

yxwwwww wTwwTwwTw ,22 =−=′=′= εαεγεβ

17


Propagating the Propagating the TwissTwiss ParametersParametersPhase Space& EmittanceF. Sannibale

When the beam propagates along the When the beam propagates along the beamlinebeamline, the eccentricity and the , the eccentricity and the orientation of the equivalent ellipse change while the area remaorientation of the equivalent ellipse change while the area remains ins constant (constant (LiouvilleLiouville theorem). In other words, the theorem). In other words, the TwissTwiss parameters parameters change along the line according to the action of the line elemenchange along the line according to the action of the line elements.ts.x′

x

The single particle matrix formalism can now be extended to the The single particle matrix formalism can now be extended to the TwissTwissparameters. For example for a drift of length parameters. For example for a drift of length LL in the horizontal plane:in the horizontal plane:

0

00

0

0

10

1

xx

xLxx

x

xL

x

x

′=′′+=

′

=

′

( )

( ) 002

0000

20

2

002

022

02

002 2

xxxLxxLxxx

xx

xxLxLxxLxx

′+′=′′+=′′=′

′+′+=′+=

x′

xDRIFTDRIFT

x′

x F QUADF QUAD

x′

x

εαεγεαεγεγ

εαεγεβεβ

00

0

002

0 2

TTT

TT

TTTT

L

LL

−=−=

−+=

−

−=

0

0

02

10

010

21

T

T

T

T

T

T

L

LL

αγβ

αγβ

18


Getting Familiar with theGetting Familiar with theConcept of Concept of EmittanceEmittance


A couple of examples:A couple of examples:

Propagation of beams with Propagation of beams with different different emittanceemittance through a through a

FODO latticeFODO lattice

Propagation of beams with Propagation of beams with different different emittanceemittance through a through a

drift spacedrift space

19


The Concept of AcceptanceThe Concept of AcceptancePhase Space& EmittanceF. Sannibale

Example: Acceptance of a slitExample: Acceptance of a slit

y

y’

-h/2

h/d

-h/d

-h/2h

d

ElectronTrajectories

Matched beam emittance

Acceptance at the slit entrance

Unmatched beam emittance

20


The The EmittanceEmittance RoleRoleIn Accelerator ApplicationsIn Accelerator Applications


The The emittanceemittance is a fundamental parameter in mostis a fundamental parameter in mostaccelerator applications.accelerator applications.

•• Free electron lasers (FEL):Free electron lasers (FEL): Intensity of the radiation Intensity of the radiation strongly depends on strongly depends on emittanceemittance. The smaller the better . The smaller the better

•• Synchrotron light sources:Synchrotron light sources: smaller smaller emittancesemittances gives gives higher brightness higher brightness

•• CollidersColliders:: higher higher emittancesemittances give higher luminosity give higher luminosity (in beam(in beam--beam limited regime)beam limited regime)

•• Electron microscopes:Electron microscopes: High resolution requires lower High resolution requires lower emittancesemittances

•• … …

21


Possible HomeworkPossible HomeworkPhase Space& EmittanceF. Sannibale

•• Calculate the Calculate the TwissTwiss parameter transport matrix for both planes parameter transport matrix for both planes of a focusing of a focusing quadrupolequadrupole in the thin lens approximation.in the thin lens approximation.

•• Prove the relation Prove the relation εεεεεεεε//εεεεεεεε00==yy’/’/yy’’00, where , where εεεεεεεε and and εεεεεεεε00 are the vertical are the vertical emittanceemittance after and before acceleration by a field after and before acceleration by a field EEzz, and , and yy’ and ’ and

yy’’00 are the divergences after and before acceleration.are the divergences after and before acceleration.Tip: use the definition of Tip: use the definition of rmsrms emittanceemittance

1


Lectures No. 6 & 7Lectures No. 6 & 7

Beam Optical FunctionsBeam Optical Functions& & BetatronBetatron Motion.Motion.


2

Want to touch on a number of concepts including:

• Weak Focusing• Betatron Tune• Strong Focusing• Closed Orbit• One-Turn Matrix• Twiss Parameters and Phase Advance• Dispersion• Momentum Compaction• Chromaticity


ConceptsConceptsOptical Functions& Betatron Motion

D. Robin

3

Weak Focusing–V. Veksler and E. M.

McMillan around 1945

Strong Focusing–Christofilos (1950),

Courant, Livingston, and Snyder (1952)


Historical ProspectiveHistorical ProspectiveOptical Functions& Betatron Motion

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4


WeakWeak--Focusing SynchrotronsFocusing SynchrotronsOptical Functions& Betatron Motion

D. Robin

The first synchrotrons were of the so called weak-focusing type.

• The vertical focusing of the circulating particles was achieved by sloping magnetic fields, from inwards to outwards radii.

• At any given moment in time, the average vertical magnetic field sensed during one particle revolution is larger for smaller radii of curvature than for larger ones.

Weak Focusing

5

Uniform field is focusing in the radial plane but not in the vertical plane


Stability of transverse oscillationsStability of transverse oscillationsOptical Functions& Betatron Motion

D. Robin

6

Stability in BOTH PLANES requires that 0<n<1Vertical focusing is achieved at the expense of horizontal focusing

rdrBdBn

//

−=

Focusing in both planes if field lines bend outward


Weak focusingWeak focusingOptical Functions& Betatron Motion

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7

The number of oscillations about the design orbit in one turn

design orbit

design orbit


TuneTuneOptical Functions& Betatron Motion

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8

Expressing these results in terms of derivatives measured along the equilibrium orbit

( )

orbitdesign therespect to with derivative a is ' where

0'' ,01'' 20

20

=+=−

+Rnyy

Rxnx

Stability requires that 0<n<1

The particle will oscillate about the design trajectory with the number of oscillations in one turn being

ly vertical n

radially n-1

The number of oscillations in one turn is termed the tune of the ring.

For stable oscillations the tune is less than one in both planes.


Weak focusingWeak focusingOptical Functions& Betatron Motion

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9

Disadvantage

• Tune is small (less than 1)• As the design energy increased so does the circumference of the

orbit • As the energy increases the required magnetic aperture increases

for a given angular deflection• Because the focusing is weak the maximum radial displacement is

proportional to the radius of the machine.

The result is that the scale of the magnetic components of a highenergy synchrotron become unreasonably large and costly


Disadvantages of weak focusingDisadvantages of weak focusingOptical Functions& Betatron Motion

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10


WeakWeak--Focusing SynchrotronsFocusing SynchrotronsOptical Functions& Betatron Motion

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• The first synchrotron of this type was the Cosmotron at the Brookhaven National Laboratory, Long Island. It started operation in 1952 and provided protons with energies up to 3 GeV.

• In the early 1960s, the world’s highest energy weak-focusing synchrotron, the 12.5 GeV Zero Gradient Synchrotron (ZGS) started its operation at the Argonne National Laboratory near Chicago, USA.

• The Dubna synchrotron, the largest of them all with a radius of 28 meters and with a weight of the magnet iron of 36,000 tons

Cosmotron

11

Solution

Strong focusing

Use strong focusing and defocusing elements (|n| >>1)


Disadvantages of weak focusingDisadvantages of weak focusingOptical Functions& Betatron Motion

D. Robin

12

One would like the restoring force on a particle displaced from the design trajectory to be as strong as possible.

ALS Bend (n~25)

• In a strong focusing lattice there is a sequence of elements that are either strongly focusing or defocusing.

• The overall lattice is “stable”• In a strong focusing lattice the

displacement of the trajectory does not scale with energy of the machine

• The tune is a measure of the amount of net focusing.


Strong FocusingStrong FocusingOptical Functions& Betatron Motion

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13


StrongStrong--Focusing SynchrotronsFocusing SynchrotronsOptical Functions& Betatron Motion

D. Robin

In 1952 Ernest D. Courant, Milton Stanley Livingston and Hartland S. Snyder, proposed a scheme for strong focusing of a circulating particle beam so that its size can be made smaller than that in a weak-focusing synchrotron.

• In this scheme, the bending magnets are made to have alternating magnetic field gradients; after a magnet with an axial field component decreasing with increasing radius follows one with a component increasing with increasing radius and so on.

• Thanks to the strong focusing, the magnet apertures can be made smaller and therefore much less iron is needed than for a weak-focusing synchrotron of comparable energy.

• The first alternating-gradient synchrotron accelerated electrons to 1.5 GeV. It was built at Cornell University, Ithaca, N.Y. and was completed in 1954.

Size comparison between the Cosmotron's weak-focusing magnet (L) and the AGS alternating gradient focusing magnets

14

Describing the Motion

In principle knowing both the magnetic lattice and the initial coordinates of the particles in the particle beam is allone needs to determine where all the particles will be in some future time.

Ray-tracing each particle is a very time consuming especially for a storage ring where the particles go around for billions of turns.

Can do much more

Want to understand the characteristics of the ring Maps


StrongStrong--Focusing SynchrotronsFocusing SynchrotronsOptical Functions& Betatron Motion

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15


What Can We Learn?What Can We Learn?Optical Functions& Betatron Motion

D. Robin

• Some parts of the ring the beam is large and in others it is small• The particles oscillate around the ring a number of times

16

• Tune is the number of oscillations that a particlemakes about the design trajectory

Design orbit

On-momentumparticle trajectory


TuneTuneOptical Functions& Betatron Motion

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17

• Use a map as a function to project a particles initial position to its final position.

• A matrix is a linear map• One-turn maps project project the particles

position one turn later


What is a Map?What is a Map?Optical Functions& Betatron Motion

D. Robin

xx’yy’δτ

initial

xx’yy’δτ

final

MAP

18

Begin with equations of motion Lorentz force

Change dependent variable from time to longitudinal position

Integrate particle around the ring and find the closed orbit

Generate a one-turn map around the closed orbit

Analyze and track the map around the ring


Generating a MapGenerating a MapOptical Functions& Betatron Motion

D. Robin

19

A closed orbit is defined as an orbit on which a particle circulates around the ring arriving with the same position and momentum that it began.

In every working story ring there exists at least one closed orbit.

Closed orbit


Find the Closed OrbitFind the Closed OrbitOptical Functions& Betatron Motion

D. Robin

20

A one-turn map maps a set of initial coordinates of a particle to the final coordinates, one-turn later.

The map can be calculated by taking orbits that have a slight deviation from the closed orbit and tracking them around the ring.

Closed orbit

( ) ( )

( ) ( )

, ,

' '' '

, ,

' ' +...'

' ' +... '

f ff i i i co i i co

i i

f ff i i i co i i co

i i

dx dxx x x x x x

dx dx

dx dxx x x x x x

dx dx

= + − + −

= + − + −


Generate a oneGenerate a one--turn Map turn Map Around the Closed OrbitAround the Closed Orbit

Optical Functions& Betatron Motion

D. Robin

21

There are two approaches to introduce the motion of particles in a storage ring

1. The traditional way in which one begins with Hill’s equation, defines beta functions and dispersion, and how they are generated and propagate, …

2. The way that our computer models actually do it

I will begin with the first way


Two approachesTwo approachesOptical Functions& Betatron Motion

D. Robin

22

( )( )

''

''

0

0x

x

x K s x

y K s y

+ =

+ =Fundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

Piecewise FocusingPiecewise FocusingOptical Functions& Betatron Motion

D. Robin

Assume that in a strong focusing synchrotron synchrotron the focusing varies “piecewise around the ring

s

23

Illustration in the simple case of Hill’s Equation – on-energy

Analytically solve the equations of motionGenerate mapAnalyze map

In a storage ring

with periodic solutions

( )( )

''

''

0

0x

x

x K s x

y K s y

+ =

+ =


Case of Hill’s EquationCase of Hill’s EquationOptical Functions& Betatron Motion

D. Robin

24


Phase FunctionPhase FunctionOptical Functions& Betatron Motion

D. Robin

Solution of the second condition

If we select the integration constant to be 1: then

' ' ''

'

0const

β ψ βψ

βψ

+ =

⇒ =

0

( ) (0)( )

s dsss

ψ ψβ

= +∫Knowledge of the function β(s) along the line allows to compute the phase function

' 1βψ =

25


TwissTwiss ParametersParametersOptical Functions& Betatron Motion

D. Robin

Define the Betatron or twiss or lattice functions (Courant-Snyder parameters)

26

• Eliminating the angles by the position and slope we define the Courant-Snyder invariant

• This is an ellipse in phase space with area πε• The twiss functions have a geometric

meaning

• The beam envelope is

• The beam divergence


CourantCourant--Snyder invariantSnyder invariantOptical Functions& Betatron Motion

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27


The Beam BehaviorThe Beam BehaviorOptical Functions& Betatron Motion

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Meaning of Beam Envelope and Beta Function and Emittance

Area of ellipse the same everywere (emittance)Orientation and shape of the ellipse different everywhere (beta and alpha function)

28

( )'' 0u k s u+ =


Solution of Hill’s EquationSolution of Hill’s EquationOptical Functions& Betatron Motion

D. Robin

The general solution of

Can be written as

( ) ( ) cos( ( ) (0))u s s sε β ψ ψ= −

There are two conditions that are obtained

2'' 2 '2 21 1 ' 0

2 2kββ β ψ ββ − − + =

' ' '' 0β ψ βψ+ =

29


ELSAELSAOptical Functions& Betatron Motion

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30


Example from ELSAExample from ELSAOptical Functions& Betatron Motion

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31


How to Compute How to Compute TwissTwiss Parameters at one pointParameters at one point


D. Robin

Steps

1. Compute the one turn transfer matrix2. Extract the twiss parameters and tunes

32

One can write the linear transformation, Rone-turn, between one point in the storage ring (i) to the same point one turn later

1' ' ' '

where = ' '

i i

one turn

x C S xx C S x

C SR

C S

+

−

=

i


One Turn Transfer MatrixOne Turn Transfer MatrixOptical Functions& Betatron Motion

D. Robin

33

The one turn matrix (the first order term of the map) can be written

Where α, β, γ are called the Twiss parameters

and the betatron tune, ν = φ/(2*π)

For long term stability φ is real |TR(R)|= |2cos φ |<2

cos sin sin

' ' sin cos sinone turn

C SR

C Sϕ α ϕ β ϕγ φ ϕ α ϕ−

+ = = − −

22

1

'

,βα

αγβ

= −

+=


One turn matrixOne turn matrixOptical Functions& Betatron Motion

D. Robin

34

One can diagonalize the one-turn matrix, R

This separates all the global properties of the matrix into N and the local properties into A.

In the case of an uncoupled matrix the position of the particle each turn in x-x’ phase space will lie on an ellipse. At different points in thering the ellipse will have the same area but a different orientation.

1one turnone turnN AR A

−

−− =

x

x’

x

x’


Computation of Computation of betabeta--functions and tunesfunctions and tunes


D. Robin

35

The eigen-frequencies are the tunes. A contains information about the beam envelope. In the case of an uncoupled matrix one can write Aand R in the following way:

The beta-functions can be propagated from one position in the ring to another by tracking A using the transfer map between the initial point the final point

This is basically how our computer models do it.

1one turnone turnN AR A

−

−− =

1 0 01

cos sin cos sin sinsin cos sin cos sin

ββϕ ϕ ϕ α ϕ β ϕα

φ ϕ α γ φ ϕ α ϕβ β ββ

+ = −− − −

f fi iA R A=


Computation of Computation of betabeta--functions and tunesfunctions and tunes


D. Robin

36

Transport of the twiss parameters in terms of the transfer matrix elements

Transfer matrix can be expressed in terms of the twissparameters and phase advances

2 2

2 2

21

2 ' '

' ' '' 'f i

C CS SCC C S SSC C S S

β βα αγ γ

− = − + −

−

( )

( )1

cos sin sin

sin cos cos sin

ffi i fi f i fi

ifi

i f i f ifi fi fi f fi

ff i f i

R

βϕ α ϕ β β ϕ

β

α α α α βϕ ϕ ϕ α ϕ

ββ β β β

+

= + − − + −


Transport of the beam ellipseTransport of the beam ellipseOptical Functions& Betatron Motion

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37

Assume that the energy is fixed no cavity or damping• Find the closed orbit for a particle with slightly

different energy than the nominal particle. The dispersion is the difference in closed orbit between them normalized by the relative momentum difference

∆p/p = 0

∆p/p > 0' ' ' '

,

,

x y

x y

p px D y Dp pp px D y Dp p

∆ ∆= =

∆ ∆= =


DispersionDispersionOptical Functions& Betatron Motion

D. Robin

38

Dispersion, D, is the change in closed orbit as a function of energy

Dispersion

∆E/E = 0

∆E/E > 0

xEx DE

∆=

0 0 1

'' ' ' 'x

x

f i

x C S D xx C S D xδ δ

=


DispersionDispersionOptical Functions& Betatron Motion

D. Robin

39

• Dispersion is the distance between the design on-energy particle and the design off energy particle divided by the relative difference in energy spread between the two.

' '

x

x

px Dppx Dp

∆=

∆=


Dispersive SystemsDispersive SystemsOptical Functions& Betatron Motion

D. Robin

40

Momentum compaction, α, is the change in the closed orbit length as a function of momentum.

∆E/E = 0

∆E/E > 0

L pL p

α∆ ∆=


Momentum CompactionMomentum CompactionOptical Functions& Betatron Motion

D. Robin

0

0

LxD dsα

ρ= ∫

41

• Off-momentum particles are not oscillating arounddesign orbit, but around chromatic closed orbit

• Distance from the design orbit depends linearly withmomentum spread and dispersion

Design orbitDesign orbit

On-momentumparticle trajectory

Off-momentumparticle trajectory

Chromatic close orbit


Chromatic Closed OrbitChromatic Closed OrbitOptical Functions& Betatron Motion

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42

Focal length of the lens is dependent upon energy

Larger energy particles have longer focal lengths


Chromatic Chromatic AberationAberationOptical Functions& Betatron Motion

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43

By including dispersion and sextupoles it is possible to compensate (to first order) for chromatic aberrations

The sextupole gives a position dependentQuadrupole

Bx = 2SxyBy = S(x2 – y2)


Chromatic Aberration Chromatic Aberration CorrectionCorrection


D. Robin

44

• No dispersion or dispersion slope at the beginning and end of the line


Achromatic TransportAchromatic TransportOptical Functions& Betatron Motion

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45

• No dispersion or dispersion slope at the end of the line

• Dispersion is negative in the central bends (cuts the corner)


Isochronous TransportIsochronous TransportOptical Functions& Betatron Motion

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46

• No dispersion or dispersion slope at the end of the line• Dispersion is positive in the central bend but the central

bend is inverted


Isochronous and Isochronous and Achromatic TransportAchromatic Transport


D. Robin

47

In an linear uncoupled machine the turn-by-turn positions and angles of the particle motion will lie on an ellipse

0

0 0'

( ) ( ) cos( ( ) )

( ) cos( ( ) ) sin( ( ) )( ) ( )

x s s s

x s s ss s

β

β

ε β ϕ ϕ

α εε ϕ ϕ ϕ ϕβ β

= +

= − + − +

2 22 ' '

Area of the ellipse, :

x xx x

ε

ε γ α β= + +


Beam EllipseBeam EllipseOptical Functions& Betatron Motion

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48

Beam ellipse matrix

Transformation of the beam ellipse matrix

xxbeam

β αε

α γ−

= − ∑

,, , ,

Tx xx i fbeam f beam i x i fR R− −

=∑ ∑



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49


Transport of the Transport of the Beam EllipseBeam Ellipse


D. Robin

50

Transport of the twiss parameters in terms of the transfer matrix elements

Transfer matrix can be expressed in terms of the twissparameters and phase advances

2 2

2 2

21

2 ' '

' ' '' 'f i

C CS SCC C S SSC C S S

β βα αγ γ

− = − + −

−

( )

( )1

cos sin sin

sin cos cos sin

ffi i fi f i fi

ifi

i f i f ifi fi fi f fi

ff i f i

R

βϕ α ϕ β β ϕ

β

α α α α βϕ ϕ ϕ α ϕ

ββ β β β

+

= + − − + −



D. Robin

51

This approach provides some insights but is limited

Begin with on-energy no coupling case. The beam is transversely focused by quadrupole magnets. The horizontal linear equation of motion is

2

2

3 356

( ) ,

where , with ( )

being the pole tip field the pole-tip radius, and [T-m] . [GeV/c]

T

T

d x k s xds

BkB a

BaB p

ρ

ρ

= −

=

≈Fundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

First approach First approach ––traditional onetraditional one


D. Robin

52

The solution can be parameterized by a psuedo-harmonic oscillation of the form

0

0 0'

,

( ) ( ) cos( ( ) )

( ) cos( ( ) ) sin( ( ) )( ) ( )

where ( ) is the beta function, ( ) is the alpha function, ( ) is the betatron phase, andx y

x s s s

x s s ss s

sss

β

β

ε β ϕ ϕ

α εε ϕ ϕ ϕ ϕβ β

βαϕ

= +

= − + − +

is an action variableε

0

s dsϕβ

= ∫Fundamental Accelerator Theory, Simulations and Measurement Lab – Arizona State University, Phoenix January 16-27, 2006

Hills equationHills equationOptical Functions& Betatron Motion

D. Robin

53


L6 & L7 Possible HomeworkL6 & L7 Possible HomeworkOptical Functions & Betatron Motion

D. Robin

•• At the At the azimuthalazimuthal position position ss in an proton storage ring, the in an proton storage ring, the TwissTwiss parameters are parameters are ββxx=10 m, =10 m, ββyy=3 m, and =3 m, and ααxx==ααyy=0. If the beam =0. If the beam emittanceemittance εε is 10 nm for the horizontal is 10 nm for the horizontal plane and 1 nm for the vertical one and the dispersion function plane and 1 nm for the vertical one and the dispersion function ηη at that location at that location is zero for both planes, what is the is zero for both planes, what is the rmsrms beam size (beam envelope) and the beam size (beam envelope) and the rmsrmsbeam divergence for both planes at the location beam divergence for both planes at the location ss? What will be the case for an ? What will be the case for an electron beam?electron beam?

•• Explain what the dispersion function represent in a storage rinExplain what the dispersion function represent in a storage ring. Explain what is g. Explain what is the difference between dispersion and chromaticity.the difference between dispersion and chromaticity.

•• Explain the difference between an Explain the difference between an achromatachromat cell and an cell and an isochronousisochronous one.one.

•• In the horizontal direction, the oneIn the horizontal direction, the one--turn transfer matrix (map) for a storage turn transfer matrix (map) for a storage ring is:ring is:

•• Is the Is the emittanceemittance preserved?preserved?•• Is the motion stableIs the motion stable

7.005.0

15.1

54


L6 & L7 Possible HomeworkL6 & L7 Possible HomeworkOptical Functions & Betatron Motion

D. Robin

1. Show that there are two conditions that can be derived relating

2.

( )'' 0u k s u+ =

( ) ( ) cos( ( ) (0))u s s sε β ψ ψ= −

( ), ( )s sβ ψ

x

x’

Sketch the phase space ellipse at these locations

Focusing quadBeam envelope

1



Longitudinal Dynamics in Storage RingsLongitudinal Dynamics in Storage Rings


2

By


Path Length DependencePath Length DependenceOn TrajectoryOn Trajectory

Storage RingsLongitudinal Dynamics

F. Sannibale

In the example (sector bending magnet) In the example (sector bending magnet) LL > > LL00 so that so that ααααααααCC > 0> 0Higher energy particles will leave the magnet later.Higher energy particles will leave the magnet later.

zz Bq

cm

qB

p 0γβρ ==

20

20

20

20

1cm

E

cm

Ecm

cm

W +=+==γ

p > p0C

electronsforE MeV 511.01 ][+≅γ

protonsforE GeV 938.01 ][+≅γ

B andA between length Trajectory0 =L

C andA between length Trajectory=L

0

0

0

0

p

pp

L

LL −∝−

00 p

p

L

LC

∆=∆ α wherewhere ααααααααCC is constantis constant

p0

AB

O

ρρρρ

000

1E

E

p

p

L

LFor CC

∆≅∆=∆>> ααγ

3


Path Length DependencePath Length Dependenceon Velocityon Velocity


F. Sannibale

Consider two particles with different momentum on parallel trajeConsider two particles with different momentum on parallel trajectories:ctories:

ppp ∆+= 01 L1

L0

( ) ctLctL 0001 βββ =∆+= 00

01

0 ββ∆=−=∆

L

LL

L

L

( ) βγγβγβ ∆=∆=∆= 3000 cmcmpcmp

ββγ ∆=∆

2

0p

p

02

0

1

p

p

L

L ∆=∆γ

•• This path length dependence on momentum applies everywhere,This path length dependence on momentum applies everywhere,also in straight trajectories.also in straight trajectories.

At a given instant At a given instant tt::

ButBut::

•• The effect quickly vanishes for relativistic particles.The effect quickly vanishes for relativistic particles.

•• Higher momentum particles precede the ones with lower momentum.Higher momentum particles precede the ones with lower momentum.

4


Total Path LengthTotal Path LengthDependence on MomentumDependence on Momentum


F. Sannibale

0sss −=∆

002

0

1

p

p

p

p

L

sCC

∆−=∆

−−=∆ ηα

γ

•• In this reference frame we can combine the previous results andIn this reference frame we can combine the previous results andobtain for the obtain for the path length dependence on momentumpath length dependence on momentum::

•• We define as the We define as the reference orbitreference orbit the trajectory of length the trajectory of length LL00 that the that the reference particlereference particle with nominal energy with nominal energy EE00 describes between describes between AA and and BB. . The position The position ss of a generic particle will be referred to of a generic particle will be referred to ss00, the position of , the position of

the reference particle on the reference orbitthe reference particle on the reference orbit ::

•• Let’s consider a particle moving in a region in the Let’s consider a particle moving in a region in the presence of electric and magnetic fields. Under the presence of electric and magnetic fields. Under the

action of such fields, the particle will define a action of such fields, the particle will define a trajectory of length trajectory of length LL between the points between the points AA and and BB..

ds

A

B

particlereferencetheprecedesparticlethesfor 0<∆

Where the constantWhere the constant ηηηηηηηηCC = = γγγγγγγγ--22 –– ααααααααCC isiscalled thecalled the momentum compactionmomentum compaction

5


Energy VariationEnergy VariationStorage Rings

Longitudinal DynamicsF. Sannibale

( ) qVsdtrEqEL

F =⋅=∆ ,0

•• We define as We define as VV the voltage gainthe voltage gain for the particle. for the particle. VV depends only on the particle trajectory and includes the contridepends only on the particle trajectory and includes the contribution of bution of

every electric field present in the area (RF fields, space chargevery electric field present in the area (RF fields, space charge fields, e fields, fields due to the interaction with the vacuum chamber, …)fields due to the interaction with the vacuum chamber, …)

•• The particle can also experience The particle can also experience energy variations energy variations UU((EE) that depend also ) that depend also on its energyon its energy, as for the case of the radiation emitted by a particle under , as for the case of the radiation emitted by a particle under acceleration (synchrotron radiation when the acceleration is traacceleration (synchrotron radiation when the acceleration is transverse).nsverse).

ds

E(r, t)AB

•• The energy gain for a particle that moves from The energy gain for a particle that moves from AA to to BB is given by:is given by:

•• The total energy variation will be given by the sum of the two The total energy variation will be given by the sum of the two terms:terms:

( )EUqVET +=∆

6


The Rate ofThe Rate ofChange of EnergyChange of Energy


F. Sannibale

( ) ( ) ( )000 EUsqVsET +=∆

The energy variation for the reference particle is given by:The energy variation for the reference particle is given by:

For particle with energy For particle with energy E = EE = E00 + + ∆∆∆∆∆∆∆∆EE and orbit position and orbit position s = ss = s00 + + ∆∆∆∆∆∆∆∆ss::

( ) ( ) ( ) ( ) ( ) EdE

dUEUs

ds

dVqsqVEEUssqVsE

EsT ∆++∆+≅∆++∆+=∆

00

0000

Where the last expression holds for the case whereWhere the last expression holds for the case where∆∆∆∆∆∆∆∆s << Ls << L00 (reference orbit length) and (reference orbit length) and ∆∆∆∆∆∆∆∆E << EE << E00..

In this approximation we can express the average rate of change In this approximation we can express the average rate of change of of the energy respect to the reference particle energy by:the energy respect to the reference particle energy by:

( ) ( )0

0

T

sEsE

dt

Ed TT ∆−∆≅∆

∆+∆≅∆

EdE

dUs

ds

dVq

Tdt

Ed

Es 000

1

particlereferencetheofvelocityc

BandAbetweenorbitreferencetheoflenghtLwith

c

LTwhere

==

=0

0

0

00 ββ

7


Towards the LongitudinalTowards the LongitudinalMotion Equation Motion Equation


F. Sannibale

In the present approximation of small In the present approximation of small ∆∆∆∆∆∆∆∆ss and and ∆∆∆∆∆∆∆∆EE, the average rate of change , the average rate of change of the particle position respect to the reference particle positof the particle position respect to the reference particle posit ion is:ion is:

000

1

p

p

TL

s

dt

dC

∆−≅∆ η

c

Ep

c

dEdp

0ββ∆≅∆=

dt

Ed

pdt

sd C ∆−=∆

02

2 η

0000 000

0000002

2 11

EC

s

C

E

C

s

C

dE

dU

Tp

pcs

ds

dV

T

q

pE

dE

dU

Tps

ds

dV

T

q

pdt

sd ∆−∆−=∆−∆−=∆ ηβηηη

∆+∆≅∆

EdE

dUs

ds

dVq

Tdt

Ed

Es 000

1

dt

sd

dE

dU

Ts

ds

dV

T

q

pdt

sd

Es

C ∆+∆−=∆

00 0002

2 1η

00 p

pc

dt

sdC

∆−=∆ ηβdt

pd

p

c

dt

sd C ∆−=∆

0

02

2 ηβ

But:But: and remembering:and remembering:

8


The LongitudinalThe LongitudinalMotion Equation Motion Equation


F. Sannibale

02 22

2

=∆Ω+∆+∆s

dt

sd

dt

sdDα

( )dt

sdptE

C

∆−=∆η

0

Finally, by defining the quantities:Finally, by defining the quantities:

We obtain the equations of motion for the longitudinal plane:We obtain the equations of motion for the longitudinal plane:

0

0

EE

Ls

<<∆<<∆

000

2 1

sC ds

dV

T

q

pη=Ω

002

1

ED dE

dU

T−=α

We will study the case of storage rings where We will study the case of storage rings where dVdV//dsds is mainly due to the is mainly due to the RF system used for restoring the energy lost per turn by the beaRF system used for restoring the energy lost per turn by the beam.m.

9


The Damped OscillatorThe Damped OscillatorEquationEquation


F. Sannibale

02 22

2

=∆Ω+∆+∆s

dt

sd

dt

sdDα

This expression is the well known This expression is the well known damped harmonic oscillator equation, damped harmonic oscillator equation,

which has the general solution:which has the general solution:

( ) ( )titit BeAeets D Ω−Ω− +≅∆ α

noscillatiodampedanti0

noscillatiodamped0

−⇔<⇔>

D

D

αα

motion unstable0

noscillatio stable02

2

⇔<Ω

⇔>Ω ( )titit BeAee D Ω−Ω +α ( )titit BeAee D Ω−Ω− +α

tDe α−

00 2 >Ω> andDα

The stable solution represents an oscillation with frequency The stable solution represents an oscillation with frequency 22ππππππππ ΩΩΩΩΩΩΩΩ and with and with exponentially decreasing amplitude.exponentially decreasing amplitude.

10


Damping in the Case ofDamping in the Case ofStorage RingsStorage Rings


F. Sannibale

•• The case of damped oscillations is exactly what we want for stoThe case of damped oscillations is exactly what we want for storing ring particles in a storage ring.particles in a storage ring.

0>Dα002

1

ED dE

dU

T−=α 0

0

<EdE

dU

•• The synchrotron radiation (SR) emitted when particles are on a The synchrotron radiation (SR) emitted when particles are on a curved curved trajectory satisfies the condition. The SR power scales as: trajectory satisfies the condition. The SR power scales as:

( ) ( ) radiustrajectoryPdtdU SR ≡−−=−∝−= ρργρβγ 22224 1

•• Typically, synchrotron radiation damping is very efficient in eTypically, synchrotron radiation damping is very efficient in electron lectron storage rings and negligible in proton machines.storage rings and negligible in proton machines.

•• The The damping timedamping time 1/1/ααααααααDD (~ ms for e(~ ms for e--, ~ 13 hours LHC at 7 , ~ 13 hours LHC at 7 TeVTeV) is usually ) is usually much larger than the period of the longitudinal oscillations much larger than the period of the longitudinal oscillations 1/21/2π π π π π π π π Ω Ω Ω Ω Ω Ω Ω Ω (~(~ µµµµµµµµs). s).

This implies that the damping term can be neglected when calculaThis implies that the damping term can be neglected when calculating the ting the particle motion for particle motion for t t << << 1/1/ααααααααDD ::

022

2

=∆Ω+∆s

dt

sdHarmonic oscillator equationHarmonic oscillator equation

11

RFT


SynchronicitySynchronicityin Storage Ringsin Storage Rings


F. Sannibale

Let’s consider a storage ring with reference trajectory of lengtLet’s consider a storage ring with reference trajectory of length h LL00::

( ) ( )tVtV RFRF ωsinˆ=

c

LT

β0

0 =

RFT

330 == hTT RF

RFCavity

h

ffhTT RF

RF == 00

Synchronicity ConditionSynchronicity Condition

RFRFRF f

Tω

π21 ==

number harmonic

h

the

calledisintegerThe

12


The SynchrotronThe SynchrotronFrequency and TuneFrequency and Tune


F. Sannibale

In a storage ring the at equilibrium:In a storage ring the at equilibrium:

tcs 0β=

000

2 1

sC ds

dV

T

q

pη=Ω ( ) ( ) ( )thVtVtV RFRF 0sinˆsinˆ ωω ==

( )00

0

0

cosˆ1

00

tc

Vh

dt

dV

cds

dVRF

ts

ωβω

β==

( )

frequencynsynchrotro

c

Vh

p

qs

C ϕπβ

ηω cos2

ˆ

00

20

2 =Ω

phasessynchronoutRFs ≡= 0ωϕ

tunensynchrotro

S0ω

ν Ω=

( ) ( ) ( ) 0sinˆ000 =−=+ UVqEUsqV sϕ

Vq

Us ˆ

sin 0=ϕ

Where Where UU00 is the energy lost per turn and is the energy lost per turn and VV is integrated over turn.is integrated over turn.

For our storage ring:For our storage ring:

13


The SynchrotronThe SynchrotronOscillationsOscillations


F. Sannibale

022

2

=∆Ω+∆s

dt

sd ( )dt

sdptE

C

∆−=∆η

0

( )ψ+Ω∆=∆ tss cosˆ

Ω<<Dα

( )ψη

+ΩΩ

∆=∆ tp

sEC

sinˆ 0

pcE ∆=∆ 0β

tRFωφ =sPhase φφϕ −=: tcs 0β=RF

csω

φβ0=

( )ψϕϕ +Ω= tcosˆ

0

:p

pDeviationMomentumRelative

∆=δ

( )ψηω

ϕδ +ΩΩ= th C

sinˆ

0

IfIf Additionally:Additionally:

Synchrotron OscillationsSynchrotron Oscillations

A different set of variables:A different set of variables:

ForFor ∆∆∆∆∆∆∆∆s << Ls << L00 and and ∆∆∆∆∆∆∆∆E << EE << E00..

14


The LongitudinalThe LongitudinalPhase spacePhase space


F. Sannibale

( )ψϕϕ +Ω= tcosˆ ( )ψηω

ϕδ +ΩΩ= th C

sinˆ

0

1ˆˆ

2

022

2

=

Ω+

ϕηωδ

ϕϕ Ch

ϕ

δ0>Cη

Ch ηωϕ

0

ˆ Ω

ϕ

We just found:We just found:

This equation represents an ellipse in theThis equation represents an ellipse in thelongitudinal phase space longitudinal phase space ϕϕϕϕϕϕϕϕ, , δδδδδδδδ

0>Cη

ϕ

δ0>Dα

( )ψϕϕ α +Ω= − te tD cosˆ

( )ψηω

ϕδ α +ΩΩ= − teh

t

C

D sinˆ

0

With damping:With damping:

In rings with negligible synchrotron radiation (or with negligibIn rings with negligible synchrotron radiation (or with negligible le nonnon--Hamiltonian forces, the longitudinal Hamiltonian forces, the longitudinal emittanceemittance is conserved.is conserved.

This is the case for heavy ion and for most proton machines. This is the case for heavy ion and for most proton machines.

15


Phase StabilityPhase StabilityStorage Rings


We showed that for the synchronous phase:We showed that for the synchronous phase:

( ) ( )tVtV RFRF ωsinˆ=

t

V

=

Vq

US ˆ

arcsin 01ϕVq

US ˆ

sin 0=ϕ

−=

Vq

Uor S ˆ

arcsin 02 πϕ

002

00

1

p

p

p

p

L

s

T

tCC

∆−=∆

−−=∆=∆ ηα

γ

ButBut

unstablestableFor SSC21 ,0 ϕϕη >

stableunstableFor SSC21 ,0 ϕϕη <

For positive charge particles:For positive charge particles:

For negative charge For negative charge particles all the phases particles all the phases

are shifted by are shifted by ππππππππ..

2

1

1

=

CTR α

γCrossing the transition energy Crossing the transition energy

during energy ramping requires during energy ramping requires a phase jump of ~ a phase jump of ~ ππππππππ

We define as We define as transition transition energyenergy the energy at the energy at

which which ηηηηηηηηCC changes sign.changes sign.

16

ϕϕϕϕϕϕϕϕSS ≠≠≠≠≠≠≠≠ 00 or or ππππππππ


Large AmplitudeLarge AmplitudeOscillationsOscillations


F. Sannibale

( ) ( ) ( ) ( ) ϕϕϕϕ

ϕϕϕϕϕψϕ

VqVqd

dVqqVVqqVE SSSST

S

ˆˆsinˆ +=+≅+=+=∆

So far we have used the So far we have used the small oscillation approximationsmall oscillation approximation where:where:

In the more general case of larger phase oscillations:In the more general case of larger phase oscillations:

( ) ( ) ( )ϕϕϕϕψ +≅+=∆ SST VqqVE sinˆ

•• Stable and unstable orbits exist. The two regions are separatedStable and unstable orbits exist. The two regions are separated by a special by a special trajectory called trajectory called separatrixseparatrix

•• Larger amplitude orbits have smaller synchrotron frequenciesLarger amplitude orbits have smaller synchrotron frequencies

δ

ϕϕϕϕϕϕϕϕϕSS = = 00 or or ππππππππ Separatrices

RF “ Buckets”

And by Numerical integration:And by Numerical integration:

•• For larger amplitudes, trajectories in the phase space are not For larger amplitudes, trajectories in the phase space are not ellipsis ellipsis anymore.anymore.

17


Momentum AcceptanceMomentum AcceptanceStorage Rings


ϕϕϕϕϕϕϕϕSS ≠≠≠≠≠≠≠≠ 00 or or ππππππππ

The RF bucket is the area of the longitudinal phase space where The RF bucket is the area of the longitudinal phase space where a particle a particle orbit is stableorbit is stable

0

2

0

ˆ2

pch

Vq

p

p

CACCβηπ

=

∆

δ

ϕϕϕϕϕϕϕϕϕSS = = 00 or or ππππππππ

RF Buckets

The The momentum acceptancemomentum acceptance is defined as the maximum momentum that a is defined as the maximum momentum that a particle on a stable orbit can have.particle on a stable orbit can have.

(∆∆∆∆p/p0)ACC.

( )0

2

0

ˆ2

2 pch

Vq

Q

QF

p

p

CACCβηπ

=

∆

( )

−−=

QQQF

1arccos12 2

factorvoltageOver

U

VqQ

s 0

ˆ

sin

1 ==ϕ

18


Bunch LengthBunch LengthStorage Rings


•• In electron storage rings, the statistical emission of synchrotIn electron storage rings, the statistical emission of synchrotron radiation ron radiation photons generates photons generates gaussiangaussian bunches.bunches.

•• The over voltage The over voltage QQ is usually large so that the core of the bunch “ lives” in is usually large so that the core of the bunch “ lives” in the small oscillation region of the bucket. The equation of motithe small oscillation region of the bucket. The equation of motion in the on in the

phase space are elliptical:phase space are elliptical:

1ˆˆ

2

022

2

=

Ω+

ϕηωδ

ϕϕ Ch

0

0 ˆˆp

pcs

h CC ∆Ω

=∆Ω

=ηδηωϕ

•• If If σσσσσσσσpp//pp00 is the is the rmsrms relative momentum spreadrelative momentum spread of the of the gaussiangaussian distribution, distribution, then the then the rmsrms bunch lengthbunch length is given by:is given by:

( ) 02

0

003

0 cosˆ2 pVfh

p

q

c

p

c p

S

CpCS

σϕ

ηβπ

σησ =Ω

=∆

•• In the case of heavy ions and of most of protons machines, the In the case of heavy ions and of most of protons machines, the whole RF whole RF bucket is usually filled with particles. The bunch length bucket is usually filled with particles. The bunch length ll is then proportional is then proportional

to the difference between the two extreme phases of the to the difference between the two extreme phases of the separatrixseparatrix::

( ) πλϕϕ 212 RFl −=

19


Effects of theEffects of theSynchrotron RadiationSynchrotron Radiation


F. Sannibale

•• A charged particle when accelerated radiates.A charged particle when accelerated radiates.

( ) 2

4

3203

2

ρE

cm

rcP

dt

dU eSR −=−=

•• In high energy storage rings transverse acceleration induces siIn high energy storage rings transverse acceleration induces significant gnificant radiation (synchrotron radiation) while longitudinal acceleratioradiation (synchrotron radiation) while longitudinal acceleration generates n generates

negligible radiation (negligible radiation (1/1/γγγγγγγγ22).).

turnperlostenergydtPUfinite

SR=ρ

0

( )[ ]planesallindamping

dtEPdE

d

TdE

dU

T

DYDX

SRE

D

αα

α

,

2

1

2

10

00 0

=−=

emittancesandspreadmomentummequilibriup

YX

p

εε

σ

,0

•• Synchrotron radiation plays a major role in the dynamics of an Synchrotron radiation plays a major role in the dynamics of an electron electron storage ringstorage ring

radiuselectronclassicalre ≡

curvaturetrajectory≡ρ

20


Energy Lost per TurnEnergy Lost per TurnStorage Rings


( ) 2

4

3203

2

ρE

cm

rcP

dt

dU eSR −=−=

c

dsdtctcts =≅= β

•• For relativistic electrons:For relativistic electrons:

turnperlostenergydtPUfinite

SR=ρ

0

( ) ==ρρ ρfinite

e

finite

SR

ds

cm

ErdsP

cU

2320

40

03

21

•• In the case of dipole magnets with constant radius In the case of dipole magnets with constant radius ρ ρ ρ ρ ρ ρ ρ ρ ((isoiso--magneticmagnetic case):case):

( ) ρπ 4

032

0

03

4 E

cm

rU e=

•• The average radiated power is given by: The average radiated power is given by:

( ) ncecircumfereringLL

E

cm

rc

T

UP e

SR ≡==ρ

π 40

3200

0

3

4

21


Damping CoefficientsDamping CoefficientsStorage Rings


By performing the calculation one obtains:By performing the calculation one obtains:

( ) 2

4

3203

2

ρE

cm

rcP

dt

dU eSR −=−= ( )[ ]=−= dtEP

dE

d

TdE

dU

T SRE

D 000 2

1

2

1

0

α

( )DET

UD += 2

2 00

0α

Where Where DD depends on the lattice parameters.depends on the lattice parameters.For the For the isoiso--magnetic separate functionmagnetic separate function case:case: πρ

α2

LD C=

Analogously, for the transverse plane:Analogously, for the transverse plane:

( )DET

UX −= 1

2 00

0α andand00

0

2 ET

UY =α

Sometimes the Sometimes the partition numberspartition numbers are used:are used:

112 =−=+= YXS JDJDJ withwith = 4iJ

22


Quantum Nature of Quantum Nature of Synchrotron RadiationSynchrotron Radiation


F. Sannibale

•• We saw that synchrotron radiation induces damping in all the We saw that synchrotron radiation induces damping in all the planes.planes.

•• Because of that, one would expect that all the particles shouldBecause of that, one would expect that all the particles shouldcollapse in a single point.collapse in a single point.

•• This This does notdoes not happen because of the happen because of the quantum nature of quantum nature of synchrotron radiationsynchrotron radiation..

•• In fact, photons are randomly emitted in quanta of discrete In fact, photons are randomly emitted in quanta of discrete energy and every time a photon is emitted the parent electron energy and every time a photon is emitted the parent electron

undergoes to a “ jump” in energy.undergoes to a “ jump” in energy.

•• Such a process perturbs the electron trajectories exciting Such a process perturbs the electron trajectories exciting oscillations in all the planes.oscillations in all the planes.

•• These oscillations grow until reaching These oscillations grow until reaching equilibriumequilibrium when balanced when balanced by the radiation damping.by the radiation damping.

23


EmittanceEmittance and and Momentum SpreadMomentum Spread


F. Sannibale

•• At equilibrium the momentum spread is given by:At equilibrium the momentum spread is given by:

mCwhereds

ds

J

C

p qS

qp 13

2

320

2

0

1084.31

1−×==

ρ

ργσ

casemagneticiso

J

C

p S

qp

−

=

ργσ 2

0

2

0

•• For the horizontal For the horizontal emittanceemittance at equilibrium:at equilibrium:

( ) ηηαηγηβ ′++′= TTTsH 222

=

ds

dsH

JC

Xq 2

320

1 ρ

ργε where:where:

•• In the vertical plane, when no vertical bend is present, the syIn the vertical plane, when no vertical bend is present, the synchrotron nchrotron radiation contribution to the equilibrium radiation contribution to the equilibrium emittanceemittance is very small and the is very small and the

vertical vertical emittanceemittance is defined by machine imperfections and is defined by machine imperfections and nonlinearitiesnonlinearitiesthat couple the horizontal and vertical planes:that couple the horizontal and vertical planes:

εκ

εεκ

κε1

1

1 +=

+= XY and factorcouplingwith ≡κ

24


Time Scale inTime Scale inStorage RingsStorage Rings


F. Sannibale

At this point we have discussed the motion of a particle in an At this point we have discussed the motion of a particle in an accelerator for all the planes.accelerator for all the planes.

Damping:Damping: several ms for electrons, ~ infinity for heavier particlesseveral ms for electrons, ~ infinity for heavier particles

It can be helpful remarking the time scale for the different It can be helpful remarking the time scale for the different phenomena governing the particle dynamics.phenomena governing the particle dynamics.

BetatronBetatron oscillations:oscillations: ~ tens of ns~ tens of ns

Synchrotron oscillations:Synchrotron oscillations: ~ tens of ~ tens of µµµµµµµµss

Revolution period:Revolution period: ~ hundreds of ns to ~ hundreds of ns to µµµµµµµµss

25


Possible HomeworkPossible HomeworkStorage Rings


•• Calculate the synchrotron frequency and tune for the ALS when tCalculate the synchrotron frequency and tune for the ALS when the ring he ring is operating in the following configuration: RF = 500 MHz, harmois operating in the following configuration: RF = 500 MHz, harmonic nic

number = 328, E = 1.9 number = 328, E = 1.9 GeVGeV, momentum compaction = 0.00137, energy lost , momentum compaction = 0.00137, energy lost per turn = 279 per turn = 279 keVkeV, peak RF voltage = 1.3 MV., peak RF voltage = 1.3 MV.

•• Calculate the ratio between the synchrotron radiation power radCalculate the ratio between the synchrotron radiation power radiated by iated by a particle in the Large a particle in the Large HadronHadron ColliderCollider (LHC), the proton (LHC), the proton collidercollider at CERN, at CERN, and the one radiated by a particle in the Advanced Light Source and the one radiated by a particle in the Advanced Light Source (ALS), the (ALS), the electron storage ring in Berkeley. The magnet bending radius is electron storage ring in Berkeley. The magnet bending radius is ~2810 m ~2810 m

and ~5 m and the particle energy is 7000 and ~5 m and the particle energy is 7000 GeVGeV and 1.9 and 1.9 GeVGeV for the LHC and for the LHC and the ALS respectively. (Remember that the electron mass is 9.1095the ALS respectively. (Remember that the electron mass is 9.1095 1010--3131 Kg Kg

while the proton one is 1.6726 10while the proton one is 1.6726 10--2727 Kg) Kg)

•• Calculate the general solution for the damped harmonic oscillatCalculate the general solution for the damped harmonic oscillator or equationequation

•• Calculate the momentum acceptance for the ALS ring. Compare it Calculate the momentum acceptance for the ALS ring. Compare it with with the acceptance value that the ring would have for zero synchronothe acceptance value that the ring would have for zero synchronous us

phase.phase.

1



Particle SourcesParticle Sources


2


Lecture OutlineLecture OutlineParticle SourcesF. Sannibale

•• Electron SourcesElectron Sources•• Basic information. A brief review and some glossary.Basic information. A brief review and some glossary.

•• How to extract electrons. How to extract electrons.

•• Characteristics of an electron source.Characteristics of an electron source.

•• Examples of existing sources.Examples of existing sources.

•• Performance limiting factors.Performance limiting factors.

•• An example of a new source scheme.An example of a new source scheme.

•• Neutron SourcesNeutron Sources

•• Protons and Heavy Ions SourcesProtons and Heavy Ions Sources

•• AntiAnti --particles Sourcesparticles Sources

3


Electron StoryElectron StoryParticle SourcesF. Sannibale

From the Greek ÈLEKTRON that means “Amber”.

Fundamental particle: lightest lepton.

m = 9.1095 × 10-31 kg or 9.1095 × 10-28 g(1837 times lighter than a proton)

e = 1.6022 × 10-19 C or 4.803 × 10-10 esu

Discovered byJ.J. Thomson in 1897

Cathode Ray Tube

For the first time it was proved that the atom is not indivisible and that is composed by more fundamental components.

4


Where Electrons Can BeWhere Electrons Can BeFound and ProducedFound and Produced

Particle SourcesF. Sannibale

Atom

Solid

Plasma

Pair formation

Heavier Particle Decay Product

Widely used in sources for:accelerators, microscopes,

technological applications, ...

Commonly used inpositrons and anti-protons

sources

Radiation source

Very high gradientaccelerators research.

Proton and heavy ions sources

Low current, high quality beams, microscopes, electron holography,

inverse photoemission, ...

√√√√

√√√√

√√√√√√√√

5


Two Families of Particles:Two Families of Particles:Fermions and BosonsFermions and Bosons


• In quantum physics, all particles can be divided into two main categories according to their spin.• Particles with half-integer spin are called fermions, those with integer spin are called bosons.• Extremely important difference: only fermions, follow the Pauli exclusion principle:

•“No two fermions may occupy the same state”.

•As a consequence, when fermions are introduced into a system, they will occupy higher energy levels when the lower ones are filled up. •On the contrary, bosons will all occupy the lower energy level allowed by the system•Because of the Pauli principle, the two particle categories follow different energy distributions:

( )1

1

−=

kTEBEAe

Ef

Bose-Einstein Distribution:photons, mesons

Bosons( ) ( ) 1

1

+= − kTEEFD

FeEf

Fermi-Dirac Distribution:electrons, protons, neutrons,...

Fermions

6


The The FermiFermi EnergyEnergyParticle SourcesF. Sannibale

We are interested to the case where the system of fermions is a solid with its electrons.The EF value is a property of the particular material. Example: EF for copper is 7 eV.

( )1

1

−=

kTEBEe

Ef

Bose-Einstein Distribution for Bosons

( ) ( )eV7

1

1

=+

= −

F

kTEEFD

Ee

EfF

( ) TEf FFD ∀= 21

Fermi-Dirac Distribution for Fermions

Definition : In a system of fermions the Fermi energy EF is the energy

of the highest occupied state at zero temperature.

We will deal only with electron sources.Being electrons fermions (spin 1/2)

we will concentrate our attention in the Fermi-Dirac distribution

7


Solids and Work FunctionSolids and Work FunctionParticle SourcesF. Sannibale

empty band

populated statesat T = 0 K

Solid

EF

Vacuum (free electrons)

Single Atom

states where electronsare bound to nuclei

States where electronsare bound to the solid,

not to the single nucleus.

StateEnergy

mor

e bo

und

Solid

band

band

The band is a range of energy with a very fine

discrete structure (states).Practically a continuum

IonizationEnergy

Definition: the work function WFis the energy needed to bring an electron from the Fermi level to

the vacuum level(a point at infinite distance away

outside the surface).

WorkFunction

Vacuum

Lastpopulated state

Single Atom

Empty state

Example: for Copper (Cu)EI = 7.7 eVWF = 4.7 eV

8


Insulators and ConductorsInsulators and ConductorsParticle SourcesF. Sannibale

Definition 1: In solids, the valence bandis the band that at T = 0 K, is occupied by the highest energy electrons.

Definition 2: The conduction bandis the higher energy band above the valence band.

INSULATORS. At T = 0 K:• The valence and the conduction bands are separated by a gap with no allowed energy states.• The valence band is completely filled with electrons.• The conduction band is totally empty.

CONDUCTORS. At T = 0 K:• The valence and the conduction bands overlap. The same band is now at the same time of valence and of conduction.• The energy states in such resulting band are only partially filled.

Conduction Band

Valence Band

Energy of electrons

Gap

a. Insulator

Conduction Band

Valence Band

Energy of electrons

b. Conductor

EF

9


The Conduction PhenomenonThe Conduction PhenomenonParticle SourcesF. Sannibale

V

L

E = V/L

Solid

∆le-

lL

VlEEVariationEnergy ∆=∆=∆=

Conduction Band(Empty)

Valence Band(Full)

a. Insulator

∆E

Energy of electrons

No energy state available in the gap.

Energy of electrons

b. Conductor

EF

∆E

Empty energy states are now

available.

No conduction!Conduction!

10

At room temperature,T ~ 300 K


Semiconductors:Semiconductors:a Special Kind Of Insulatora Special Kind Of Insulator


A semiconductor is an insulator with a relatively small gap between the valence and conduction bands.

The gap is small enough that at room temperature (T ~ 300K), such a phase transition has already happened.

Above absolute zero (T = 0K), the atoms in a crystal (solid) start vibrating.

As a result, some electrons scatter with the atoms gaining extra energy(the larger is T, the larger is the extra energy).

For a high enough temperature, in some insulators this extra energy can be larger than the gap and electrons in the valence band are allowed to go in the conduction band.

As a consequence, such a solid undergoes to aphase transition from insulator to conductor when the temperature is increased!

Electrons inConduction Band

Silicon,Germanium,….

11


ThermionicThermionic EmissionEmissionin Conductorsin Conductors


fFD(E)

Energy

EF

EVVacuum

At room temperature,

T ~ 300 K

WF

1.0

Conduction Band

Energy

EF

EVVacuum

At high temperature,T ~ 1,500 K

WF

1.0

fFD(E)

Conduction Band

EmittedElectron

Owen Richardson received a Nobel prize in 1928 "for his work on the thermionicphenomenon and especially for the discovery of the law named after him".

Thermionic emission was initially reported in 1873 by Guthrie in Britain.

12


The Photoelectric EffectThe Photoelectric EffectParticle SourcesF. Sannibale

fFD(E)

Energy

EF

EVVacuum

WF

1.0

Conduction Band

e-

hννhEEnergyPhoton ph ==photon frequency

Planck Constant= 6.626068 × 10-34 m2 kg / s

Fph WEIf ≥

Albert Einstein received the 1921 prize in 1922 for work that he did between 1905 and 1911 on the Photoelectric Effect.

Fphe WET −=−

Max Planck received the 1919 Nobel for the development of the Quantum Theory of the photon.

13


Field EmissionField EmissionParticle SourcesF. Sannibale

Field emission was first observed in 1897 by Robert Williams Wood.

But only in 1928, Fowler and Nordheim gave the first theoretical description of the phenomenon. It was one of the first application of the

quantum mechanics theory.

Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state.

r

eU p

2

04

1

πε−= rEe−

constant=E

Vacuum

e-Vacuum

e- e-Tunneling

Vacuum

14


Secondary EmissionSecondary EmissionParticle SourcesF. Sannibale

Primary ParticlePrimary ParticleSecondary ElectronsSecondary Electrons

Primary ParticlesPrimary Particles : photons, : photons, electrons, protons, neutrons, electrons, protons, neutrons,

ions, ... ions, ...

Physical ProcessesPhysical Processes : ionization, : ionization, elastic scattering, Auger elastic scattering, Auger

Electrons,Electrons, photoelectric effect,photoelectric effect,bremsstrahlungbremsstrahlung and pair and pair

formation,formation, Compton scattering, ... Compton scattering, ...

15


Electron Gun SchematicElectron Gun SchematicParticle SourcesF. Sannibale

ElectronGenerator

ElectronGenerator

Accelerating andFocusing Section

Accelerating andFocusing Section

•• ThermionicThermionic•• PhotoelectricPhotoelectric•• Field emissionField emission•• Secondary emissionSecondary emission•• ........

FocusingFocusing : usually by : usually by stationary magnetic fields.stationary magnetic fields.

Acceleration:Acceleration: by electric fields.by electric fields.•• ElectrostaticElectrostatic•• PulsedPulsed•• Radio FrequencyRadio Frequency•• WakefieldsWakefields

ApplicationApplication

Higher gradientsHigher gradients

16


Electron SourceElectron SourceMain ParametersMain Parameters


Current:Current:•• Average: from Average: from pApA to several tens of A.to several tens of A.•• Peak: Peak: from from µµAA to thousand of A.to thousand of A.

Energy: Energy: from few from few eVeV to several to several MeVMeVEnergy Spread: Energy Spread: from ~ 0.1 from ~ 0.1 eVeV and up.and up.

Pulse Length: Pulse Length: fromfrom hundreds of hundreds of fsfs to seconds. to seconds. Single electron.Single electron.

Polarization: Polarization: orientation of the electron spinorientation of the electron spin

Time Structure: Time Structure: DCDCPulsed: Pulsed: from single shot to hundreds of kHzfrom single shot to hundreds of kHzCW: CW: from hundreds of MHz to several GHzfrom hundreds of MHz to several GHz

17


The The EmittanceEmittance::An Important Gun ParameterAn Important Gun Parameter


EmittanceEmittance: : volume of the phase space volume of the phase space occupied by the particles of the beamoccupied by the particles of the beam

LiouvilleLiouville Theorem: Theorem: in a Hamiltonian system (nonin a Hamiltonian system (non--dissipative system) the dissipative system) the emittanceemittance is conservedis conserved

x

dz

dxx =′

1p2p

x

dz

dxx =′

x

dz

dxx =′

222 xxxxrms ′−′=εeffective effective ((rmsrms) ) EmittanceEmittance: :

Smaller Smaller emittanceemittance are usually preferred.are usually preferred.It is very easy to increase the It is very easy to increase the emittanceemittance, but very , but very

hard to decrease it!hard to decrease it!

18


Brightness andBrightness andDegeneracy FactorDegeneracy Factor


Short pulses, low energy spread, small Short pulses, low energy spread, small emittancesemittances, high , high current densities, all lead to acurrent densities, all lead to a high degeneracy factor.high degeneracy factor.

Brightness: Brightness: phase space density of particles. I.e. number of phase space density of particles. I.e. number of particles per unit of phase space volume.particles per unit of phase space volume.

This can be interpreted as the fact that the phase space volume This can be interpreted as the fact that the phase space volume occupied by occupied by a particle is given by: a particle is given by: ((λλλλλλλλcc/2/2ππππππππ))3 3 = = elementary phase space volumeelementary phase space volume

HeisenbergHeisenberg uncertainty principle: uncertainty principle: it is impossible to determine with precision and simultaneously, the position and the momentum of a particle.

electronsforpmmchwavelengthCompton

zyxw

c

cw

426.2

,,4

==≡=≥

λπλε

Degeneracy Factor, Degeneracy Factor, δδδδδδδδ : : brightness in units brightness in units of elementary phase space volume. of elementary phase space volume.

Number of particles per elementary volume.Number of particles per elementary volume.

Because of the Because of the PauliPauli exclusion principle the exclusion principle the limit value of limit value of δδδδδδδδ is:is:infinity for bosons and infinity for bosons and 1 for non polarized fermions1 for non polarized fermions ..

Applied to emittances:

19


Examples of Electron GunsExamples of Electron GunsParticle SourcesF. Sannibale

20


More ExamplesMore ExamplesParticle SourcesF. Sannibale

21


Performances LimitationsPerformances LimitationsParticle SourcesF. Sannibale

RF Guns.RF Guns.•• Repetition Rate. Heat load in the RF structures limits.Repetition Rate. Heat load in the RF structures limits.•• Max electric field. Field emission limits. Dark current.Max electric field. Field emission limits. Dark current.

High power High power thermionicthermionic guns.guns.•• Average Current. Limits in the cathodes current density. Average Current. Limits in the cathodes current density. •• Cathode lifetime.Cathode lifetime.•• Cathode thermal Cathode thermal emittanceemittance limitlimit

Field emission guns.Field emission guns.•• Max electric field at the tip. Limits in the minimum size of thMax electric field at the tip. Limits in the minimum size of the tip. e tip. •• Intrinsic low average current.Intrinsic low average current.

Secondary Emission Gun.Secondary Emission Gun.•• Low current densities.Low current densities.•• High energy spread.High energy spread.

22


The Ultimate LimitThe Ultimate LimitParticle SourcesF. Sannibale

•• EmittanceEmittance

•• Energy spreadEnergy spread

•• BrightnessBrightness

DegeneracyDegeneracyfactor factor δδδδδδδδ

•• ThermionicThermionic: : δδδδδδδδ ~ 10~ 10--1414

•• SEM: SEM: δδδδδδδδ ~ 10~ 10--1414

•• PhotoPhoto--RF guns: RF guns: δδδδδδδδ ~ 10~ 10--1212

•• Field emission: Field emission: δδδδδδδδ ~ 10~ 10-- 55

The degeneracy factor inside a metal cathode is ~ 1The degeneracy factor inside a metal cathode is ~ 1How do we loose all of that How do we loose all of that ??

Extraction MechanismExtraction MechanismCoulomb interactionCoulomb interaction

(space charge)(space charge)

Practically, most of the edge applications (accelerators, free Practically, most of the edge applications (accelerators, free electron lasers, microscopes, inverse photoemission, ...) are electron lasers, microscopes, inverse photoemission, ...) are limited by the performance of the electron gun in:limited by the performance of the electron gun in:

!!!!!!

23


A High DegeneracyA High DegeneracyElectron SourceElectron Source


24


Fundamental ConceptsFundamental ConceptsParticle SourcesF. Sannibale

1) Electron Excitation. In the region of well defined and controlled volume (defined by the overlap of the lasers) we ionize on average one alkali atom per laser pulse. The electron in the excited atom will have a total energy close to zero and will start to drift away from the ion.2) Waiting Period. After the laser pulse, we wait the time necessary for the electron to go far enough from the ion loosing most of its kinetic energy and we apply a short pulsed voltage to extract the electron from the ionization region.3) Electron Acceleration. In this step, we accelerate the electron up to the energy required by the considered application.4) Ion Clearing. After the electron acceleration, we apply a “cleaning” field in order to remove the residual ion before the beginning of the following cycle. In this way it is avoided that the residual ion will interact with the electron produced in the next pulse.

The application of all such concepts allows to eliminate the Coulomb interaction between electrons (a single electron per cycle is produced) and to properly control the

interaction between the electron and ions (parent and residual ones).

The degeneracy factor for this source is expected to be: δδδδ ~ 10-2

25


Protons and Ions SourcesProtons and Ions SourcesParticle SourcesF. Sannibale

In most protons and ion sources a gas of neutral atoms In most protons and ion sources a gas of neutral atoms or molecules is “ heated” into a or molecules is “ heated” into a plasma plasma state were ions state were ions and electrons are dissociated and move independently and electrons are dissociated and move independently

as free particles.as free particles.

Heating mechanism can be of various kind: Heating mechanism can be of various kind: thermal, electrical, or light (ultraviolet light thermal, electrical, or light (ultraviolet light

or intense visible light from a laser). or intense visible light from a laser).

In a source, the ions are then extracted from the In a source, the ions are then extracted from the plasma and accelerated.plasma and accelerated.

Neutral gas of practically any specie of atom can be Neutral gas of practically any specie of atom can be produced and used in sources. For example, neutral gas produced and used in sources. For example, neutral gas of metals can be obtained by heating the solid element of metals can be obtained by heating the solid element

inside ovens inside ovens

26


Discharge BasedDischarge BasedIons SourcesIons Sources


In In PenningPenning discharge sources, discharge sources, MagnetronsMagnetrons and and PlasmatronsPlasmatrons a high a high voltage discharge (arc) in 0.001 to 1 voltage discharge (arc) in 0.001 to 1 TorrTorr pressure is used for generating pressure is used for generating

the plasma.the plasma.Figure by C.E. Hill

CERN

The ions then diffuse out from an aperture on the plasma chamberThe ions then diffuse out from an aperture on the plasma chamberand are accelerated by the voltage between cathode and anode.and are accelerated by the voltage between cathode and anode.

The arc electric field The arc electric field accelerates the electrons accelerates the electrons

and a magnetic field and a magnetic field makes them move on makes them move on

spiraling orbits inside the spiraling orbits inside the plasma ionizing more plasma ionizing more

atom along their atom along their trajectory.trajectory.

27


ECR SourcesECR SourcesParticle SourcesF. Sannibale

•• Electron Cyclotron Resonance (ECR) sourcesElectron Cyclotron Resonance (ECR) sources, , exploit this mechanism.exploit this mechanism.

In such sources there is no cathode and the In such sources there is no cathode and the average lifetime and reliability is very good.average lifetime and reliability is very good.

•• NonNon--relativistic particles in a constant magnetic field move relativistic particles in a constant magnetic field move on a circular trajectory at a on a circular trajectory at a constant revolution frequencyconstant revolution frequency

independently from their energy (cyclotron principle):independently from their energy (cyclotron principle):

Figure by NSCL- Michigan State University

m

eB=0ω

•• Let’s consider a plasma immersed in a Let’s consider a plasma immersed in a solenoidalsolenoidal field. Applying an electromagnetic field. Applying an electromagnetic

field with frequency field with frequency ωωωωωωωω00, the electrons in the , the electrons in the plasma will resonate at their cyclotron plasma will resonate at their cyclotron

frequency gaining energy from the field.frequency gaining energy from the field.

•• The electrons will describe spiraling orbits The electrons will describe spiraling orbits with increasing radius and ionizing additional with increasing radius and ionizing additional

atoms along their path. atoms along their path.

28


Negative Ion SourcesNegative Ion SourcesParticle SourcesF. Sannibale

•• The same schemes described for positive ions generation can be The same schemes described for positive ions generation can be used for used for generating negative ions, (penning, magnetrons, …).generating negative ions, (penning, magnetrons, …).

•• In a In a surface sourcesurface source, ~ a mono, ~ a mono--layer of Cesium on the source surface strongly layer of Cesium on the source surface strongly increase the production of Hincrease the production of H--. Collision of the plasma particles with the Cs . Collision of the plasma particles with the Cs

surface generates surface generates desorptiondesorption of ions including the desired Hof ions including the desired H--..

••The physical processes in negative ion sources are still poorly The physical processes in negative ion sources are still poorly understood understood but three types of source are generally recognized; surface, volbut three types of source are generally recognized; surface, volume and ume and

charge exchange. charge exchange.

•• Negative ions find very important applications in Tandems and iNegative ions find very important applications in Tandems and in injecting n injecting into accumulator rings by stripping the charge: the process is ninto accumulator rings by stripping the charge: the process is nonon--

hamiltonianhamiltonian and the beam and the beam emittanceemittance can be reduced.can be reduced.

•• In In Volume SourcesVolume Sources, scattering between the gas, scattering between the gas molecules can generate molecules can generate negative ions. For example, measurements of Hnegative ions. For example, measurements of H-- ions in largeions in large--volume, lowvolume, low--pressure hydrogen discharges indicated densities which were muchpressure hydrogen discharges indicated densities which were much larger larger

than those predicted by theory.than those predicted by theory.•• Double charge exchangeDouble charge exchange of positive (or neutral) ion beams on alkali metal of positive (or neutral) ion beams on alkali metal vapor targets was once a favored method of negative ion productivapor targets was once a favored method of negative ion production. They on. They

are not very efficient in producing high energy Hare not very efficient in producing high energy H-- but are still very useful for but are still very useful for producing “ exotic” species of negative ions.producing “ exotic” species of negative ions.

29


Positron SourcesPositron SourcesParticle SourcesF. Sannibale

•• From quantum field theory, a photon with energy From quantum field theory, a photon with energy larger than twice the electron rest mass (~1.02 larger than twice the electron rest mass (~1.02 MeVMeV) )

“ oscillates” between the phase of photon and the “ oscillates” between the phase of photon and the one of one of pair of virtual electron and positronpair of virtual electron and positron. This . This

virtual particles live for an extremely short time for virtual particles live for an extremely short time for then recombining back into the original photon then recombining back into the original photon

ready for a new cycle to start again.ready for a new cycle to start again.

•• This is a consequence of the This is a consequence of the HeisembergHeisemberg indeterminacy principle indeterminacy principle and these virtual particles cannot be directly detected.and these virtual particles cannot be directly detected.

•• Anyway, if the photon during this “ virtual Anyway, if the photon during this “ virtual particle phase” passes close to an atom nucleus, particle phase” passes close to an atom nucleus,

the interaction between the nucleus fields and the interaction between the nucleus fields and the pair will allow for the virtual particles to the pair will allow for the virtual particles to become real and separate from each other.become real and separate from each other.

30


Positron SourcesPositron SourcesParticle SourcesF. Sannibale

•• In existing positron sources, the high energy photons required In existing positron sources, the high energy photons required for pair for pair production are generated by first impinging a high current electproduction are generated by first impinging a high current electron ron

beam on a high Z metallic target.beam on a high Z metallic target.

1 mm r.m.s.Beam size @ the PC

100 mAPositron current at the Linac end (550 MeV)

250 MeVEnergy at the PC

> 4 ACurrent at the positron converter (PC)

•• The electrons penetrating the material are deflected by the nucThe electrons penetrating the material are deflected by the nuclei fields lei fields and radiate high energy photons. These photons interact with theand radiate high energy photons. These photons interact with the nuclei nuclei

finally generating the pairs.finally generating the pairs.

•• The newborn positrons leaving the target are separated from theThe newborn positrons leaving the target are separated from theelectrons, captured and accelerated to higher energies in a dedielectrons, captured and accelerated to higher energies in a dedicated cated

section of the section of the linaclinac optimized for the task.optimized for the task.

DADAΦΦΦΦΦΦΦΦNENEThe The FrascatiFrascati ΦΦΦΦΦΦΦΦ--factoryfactory

31


Antiproton SourcesAntiproton SourcesParticle SourcesF. Sannibale

Figure from CERN web site

•• Existing sources of antiprotons (Existing sources of antiprotons (FermilabFermilab and CERN) exploit the and CERN) exploit the protonproton--antiproton pair production mechanism when high energy protons antiproton pair production mechanism when high energy protons

scatters on the nuclei of a metallic target generating pairs.scatters on the nuclei of a metallic target generating pairs.

Production rate is very Production rate is very small:small:

~ 10~ 10--55 antiproton/protonantiproton/proton~ 10~ 101111 antiproton/hourantiproton/hour

32


Neutron SourcesNeutron SourcesParticle SourcesF. Sannibale

•• An alternative scheme An alternative scheme fofo generating neutrons generating neutrons is given by the sois given by the so--called called spallationspallation neutron neutron

sourcesource, where a high energy, where a high energy--high power high power accelerator produces pulsed neutron beams by accelerator produces pulsed neutron beams by bombarding a target with intense proton beams.bombarding a target with intense proton beams.

•• The more efficient neutron sources are nuclear reactors. The more efficient neutron sources are nuclear reactors. However, they cannot be built because international treaties However, they cannot be built because international treaties

prohibits civilian use of highly enriched uranium Uprohibits civilian use of highly enriched uranium U235235. .

1 1 GeVGeV Protons at targetProtons at target1.4 MW Proton Power at the Target1.4 MW Proton Power at the Target24 kJ/pulse24 kJ/pulse1.5 x 101.5 x 101414 protons /pulseprotons /pulse

> 1.5 G$> 1.5 G$

33


Possible HomeworkPossible HomeworkParticle SourcesF. Sannibale

•• Calculate the RF frequency for an ECR HCalculate the RF frequency for an ECR H++ source with a source with a solenoidalsolenoidal field of 0.5 T.field of 0.5 T.

•• Calculate the minimum energy in Calculate the minimum energy in eVeV units that a photon units that a photon should have to potentially generate a protonshould have to potentially generate a proton--antiproton pair.antiproton pair.

•• Estimate the number of hours required to store 100 Estimate the number of hours required to store 100 mAmA of of antiprotons in the antiprotons in the TevatronTevatron at the at the FermilabFermilab. The ring circumference . The ring circumference is ~ 6400 m, the beam energy is 980 is ~ 6400 m, the beam energy is 980 GeVGeV. Assume an injection rate . Assume an injection rate of about 6.5 x 10of about 6.5 x 101111 antiprotons/hour. Remember that the antiproton antiprotons/hour. Remember that the antiproton

mass is ~ 1.6726 x 10mass is ~ 1.6726 x 10--2727 kg.kg.


Simulation and Measurement LabSimulation and Measurement Lab

General introduction to the labsGeneral introduction to the labs+ +

HomeworkHomework

SorenSoren PrestemonPrestemon


TwoTwo--dimensional static fields dimensional static fields Simulation and Measurements

Soren Prestemon

Magnetic field described in terms of potentials:Magnetic field described in terms of potentials:In free-space, the magnetic field can be determined from a scalar potential V:

( )( )results from 0 and identity 0, B V B V V= −∇ ∇× = ∇× ∇ = ∀and from a vector potential A:

( )( )results from 0 and identity 0,B A B A A= ∇× ∇ ⋅ = ∇ ⋅ ∇× = ∀

In two dimensions, A is a scalar and we can use complex notationIn two dimensions, A is a scalar and we can use complex notation: :

( )*x y

dF dB iB B i i A iVdz dz

− = = = +

F is an analytic function in a goodF is an analytic function in a good--field region, and can be expanded as a field region, and can be expanded as a TaylorsTaylors series for |z|<|zseries for |z|<|z00|=r|=r00: :

( ) 0( ) ( ) i n

nnF z F x iy F re c zθ

≥= + = = ∑

( , ) ( , ) ( , )

z x iydf f fidz x yf x y u x y iv x y

= +∂ ∂

= = −∂ ∂= +

u vx yu vy x

∂ ∂=

∂ ∂∂ ∂

= −∂ ∂

( ) 1*1

( ) nnn

B z i nc z −

≥⇒ = ∑

CauchyRiemann

10

0

10

0

n n n

nnn

nnn

c in

a rBn

b rB

λ µµ

λ

−

−

= +

=

= −

( )1

*0 1

0

( )n

n nn

zB z B a ibr

−

≥

⎛ ⎞⇒ = − + ⎜ ⎟

⎝ ⎠∑

Note: the coefficients are a function of the reference radius and the characteristic field. By tradition:an: “skew” coefficientsbn: “normal” coefficientsNote: other definitions of an and bn exist!


TwoTwo--dimensional static fields dimensional static fields Simulation and Measurements

Soren Prestemon

Magnetic field described in terms of Magnetic field described in terms of multipolesmultipoles::

( )( )

( )

( )

0

2 22 20 1 1

0 0

2 22 20 1 1

0 0

( , ) ( , ) ( , )

( , ) ...2

( , ) ...2

nn

nF x y A x y iV x y i x iy

b aA x y B b x a y x y xy

r r

a bV x y B a x b y x y xy

r r

λ µ≥

= + = + +

⎡ ⎤⇒ = − + + − + +⎢ ⎥

⎣ ⎦⎡ ⎤

⇒ = − + − + +⎢ ⎥⎣ ⎦

∑ Example: V describes geometry of magnetized surfaces to yield a multipole field; for a pure normal dipole:⇒ only b1 non-zero ⇒ b1y=+/-V0

( )( ) ( )( )0 0

0 01 0

0 01 0

( , ) ( , ) ( , )

( , ) cos( ) sin( )

( , ) cos( ) sin( )

nn in n

n nn

n n

n

n

n n

n

F x y A x y iV x y i x iy i re

b a rA r B r n nn n r

a b rV r B r n nn n r

θλ µ λ µ

θ θ θ

θ θ θ

≥ ≥

≥

≥

= + = + + = +

⎛ ⎞⎛ ⎞⇒ = − − + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⇒ = − − + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑

∑

∑

In cylindrical coordinates:

V0

-V0

g 02y

VB

g=

x

y

Example: A can be used to find current distribution on a cylinder to yield a given multipole field (see homework); for a pure normal dipole:⇒only b1 non-zero

• Things to consider– For iron-dominated magnets, Amperes law

provides much insight:

– When possible, the simulations should correspond to the geometry used in the measurements

– Lab grades will be based on your understanding of the technical issues; if some measurements do not look “clean”, discuss possible reasons and how the measurements could be improved with appropriate equipment


Introduction to lab exercises:Introduction to lab exercises:General issuesGeneral issues


Soren Prestemon

The purpose of the labs and simulations is to provide insight inThe purpose of the labs and simulations is to provide insight into to hardware and design software associated with accelerators hardware and design software associated with accelerators

–– each report should include discussion of issueseach report should include discussion of issues

• Working as a group– Read through the days lab/simulation

assignment and plan your approach for efficiency

– Proceed with the experiments in a safe and orderly manner

• Be sure power supplies are off when connecting leads and/or performing assembly

• Be careful with heavy/clumsy magnet parts (in particular the dipole measurement)

• The Hall probes are sensitive, expensive equipment; handle with care. You can tape them to glass rods, etc, but do not allow the tape to adhere to the probe itself.

– Clean up your experimental area after finishing the days lab; messy areas will be noted!

– Remember that the homework is due after both the lab and simulation component are finished; plan your days accordingly

µ = ∞

g0

2

In air: 1

In iron: 1 0

iron

iron

H dl I Hg H dl

B H

H dl

µµµ

µ

⋅ = = + ⋅

==

>> ⇒ ⋅ ≈

∫ ∫

∫

– Magnetostatics, ,e.g. dipole and quadrupole magnetic fields, can be calculated using the code Poisson. Example scripts (*.am) are available. These are evaluated by right-click,run-autofish). The resulting *.T35 files can be plotted (double-click) or data evaluated (right-click, Interpolate). Multipole data is available in the OUTPOI.txt file.

– The program SynRad provides radiation properties for a wide variety of storage rings and SR sources, including brightness, flux, and power calculations. New storage ring and source parameters can also be analyzed.

– The program BeamOptics_APS.exe provides a forum for the design of a storage ring. The lattices of existing rings can be reviewed, or a new ring can be built “from the ground up”. This program will serve as the foundation for the final project: “Storage Ring Design”.


Introduction to lab exercises:Introduction to lab exercises:SoftwareSoftware


Soren Prestemon

The software is generally intuitive, and is best learned by triaThe software is generally intuitive, and is best learned by trial. Input files l. Input files are available that can be copied and modified as needed for the are available that can be copied and modified as needed for the

problems. problems.

• Software:– Resonant frequencies of pillbox cavities are purely a

function of the geometry. The code TMModeFreq.tcl is in your folder and will evaluate these frequencies.

– Wideroe drift tube lengths can be calculated using the script calcDriftTubeLengths.tcl, given linac rf frequency, initial beam energy, energy gain per gap, and number of drift tubes,

– Cavity fields can be calculated and plotted using the script runUrmel.tcl, which reads a cavity description file *.urmi.

– FODO cells can be analyzed with the script FODOcells.tcl, which plots Twiss functions (β and ηfunctions) and matched and mismatched trajectories.

– Necktie diagrams, which describe FODO cell stability regions, can be generated with the script NecktieDiagrams.tcl. Either the trace of the transfer matrix or the phase advance can be plotted.

–Software location:•Please use your c:\studenti directory for all calculations. If you plan to modify an input file, make a copy with a new name, so that the original can be reviewed in case of problems during running. Example input files are available in the folder C:\LANL\Examples. •The code SynRad is located in the folder C:\Program Files\Stanford University\SynRad. •The code BeamOptics_APS.exe is located in C:\Program Files\APS\Beam Optics (APS version).

1. Show that in free-space, the complex function B* is analytic (hint: a complex function is analytic if it satisfies Cauchy-Riemann)

2. Find the 3rd order (I.e. sextupole) terms in the expansion of A and V in Cartesian coordinates3. Field about an infinite line-current

a) Find the field Br about an infinite line current on a circle of radius rb) Find the field By(x,y=0) for an infinite line located at (x0=0,y0)

4. The vector potential of a line current located at is given by

What is the multipole field associated with the current distribution on the circle r=r0 defined by

Hint: replace I in the expression for A with I(θ) and integrate from 0 to 2π

5. Assume I0=10kA, r0=0.01meters, and m=1. What is the field (Bx,By) at (x=0.005,y=0).Νote: µ0 (permeability of free space) = 4π 10-7


HomeworkHomeworkSimulation and Measurements

Soren Prestemon

0( , )r θ

( )0

1 0

1( , ) ( )2

n

n

I rA r Cos nn r

µφ φ θπ

∞

=

⎛ ⎞= − −⎜ ⎟

⎝ ⎠∑

( )0 0( , )I r r I Cos mθ θ= =

( , )( , ) zA rB rrθθθ ∂

= −∂

Fundamentals of Accelerator Physics and Technology %5Bpresentation Slides%5D %282006%29 WW

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