G1B An Introduction to Beam Dynamics
Transcript of G1B An Introduction to Beam Dynamics
An Introduction to
Beam DynamicsBeam DynamicsC. Zhang
6th OCPA Accelerator School
July29 - August 7, 2010 • Beijing
G1BG1B
PrefacePrefaceBeam dynamics is the study of particle beams,
their motion in environments, involving external electro-magnetic fields and their interactions, including the interaction of beams with matters, of beams with beams, and of particle beams with radiation. Evolving from concepts and ideas derived from classical mechanics, electromagnetism, statistical physics, and quantum physics. The study of beams is opening up a very rich field, with new effects being discovered and new types of beams with novel characteristics being realized. Basic knowledge of the beam physics is briefly introduced in this lecture for the students who are preparing to work in the field of Particle accelerators.
ContentsContentsBasic ConceptsTransverse MotionLongitudinal MotionCollective EffectsLepton and Hadron
1. Basic Concepts1. Basic ConceptsConstants & Relations
Motion in E-M Fields
Linear accelerators
Synchrotrons
1.1 Constants & RelationsSpeed of lightRelativistic energyRelativistic momentum
E-p relationship
Kinetic energyEquation of motion under Lorentz force ( ) ( )BvEqv
dtdmf
dtpd rrrrrr
×+=⇒= γ0
c=2.99792458×1010 cm/sec
E=mc2=m0γc2
p=mv=m0γβc
β=v/c γ=(1-β2)-1/2
E2/c2=p2+m02c2
Ultra-relativistic particles: β≈ 1, E≈ pc
T=E-m0c2=m0c2(γ −1)
Constants & RelationsConstants & Relations (cont.)(cont.)
Planck constant
Electron charge
Electron volts
Energy in eV
Energy and rest mass
Electron
Proton
[ ]e
cme
mcE2
02
eV γ==
e=1.6021773×10-19 Coulumbs
1eV=1.6021773×10-19 Joule
1eV/c2=1.78×10-36 kg
m0, e=0.51099906 MeV/c2=9.1093897×10-31 kg
m0, p=938.2723 MeV/c2=1.6726231×10-27 kg
h=6.626075×10-34 J s
1.2 1.2 Motion in EMotion in E--M FieldsM FieldsGoverned by Lorentz force
Acceleration along a uniform electric field (B=0)
[ ]BvEqdtpd rrrr
×+=
( ) EpE
qcBvEpE
qcdtdE
dtpdpc
dtdEE
cmcpE
rrrrrr
rr
r
⋅=×+⋅=⇒
⋅=⇒
+=
22
2
420
222
cvtmeEx
vtz<<
⎪⎭
⎪⎬⎫
≈
≈for path parabolic
22
0γ
A magnetic field does not alter a particle’s energy. Only an electric field can do this.
C.R. Prior: The Physics of Accelerators
1.3 1.3 Linear Linear AcceleratorsAcceleratorsSimplest example is a vacuum chamber with one or more DC accelerating structures with the E-field aligned in the direction of motion.
Limited to a few MeVTo achieve energies higher than the highest voltage in the system, the E-fields are alternating at RF cavities.
Avoids expensive magnetsNo loss of energy from synchrotron radiation (q.v.)But requires many structures, limited energy gain/metreLarge energy increase requires a long accelerator
BEPC electron-positron linac
SNS Linac, Oak Ridge
Structure 1:Structure 1:Travelling wave structure: particles keep in phase with the accelerating waveform. Phase velocity in the waveguide is greater than c and needs to be reduced to the particle velocity with a series of irises inside the tube whose polarity changes with time. In order to match the phase of the particles with the polarity of the irises, the distance between the irises increases farther down the structure where the particle is moving faster. But note that electrons at 3 MeV are already at 0.99c.
Structure 2:Structure 2:
A series of drift tubes alternately connected to high frequency oscillator.Particles accelerated in gaps, drift inside tubes .For constant frequency generator, drift tubes increase in length as velocity increases.Beam has pulsed structure.
1.4 1.4 SynchrotronSynchrotronssPrinciple of frequency modulation but in addition variation in time of B-field to match increase in energy and keep revolution radius constant.
Magnetic field produced by several bending magnets (dipoles), increases linearly with momentum. For q=e and high energies:
Practical limitations for magnetic fields => high energies only at large radius
e.g. LHC: E = 7 TeV, B = 8.36 T, ρ = 2.7 km
ωnf =
ρω v
EqBc
==2
qBp
=ρ
[m] [T] 0.3[GeV]so ρBEceE
epBρ ≈≈=
Types of Types of SynchrotronSynchrotronssStorage rings: accumulate particles and keep circulating for long periods; used for high intensity beams to inject into more powerful machines or synchrotron radiation factories.Colliders: two beams circulating in opposite directions, made tointersect; maximises energy in centre of mass frame.
2. Transverse Motion2. Transverse Motion(Prof. S.Y.Lee)(Prof. S.Y.Lee)
Motion Description
Bending
Focusing
Hill’s Equation
Phase Space
Lattice
Dispersion
Orbit Distortion
Coupling
Non-linearity
2.12.1 Motion DescriptionMotion Description
x- horizontaly- verticals- longitudinal
Transverse
2.2 Bending2.2 Bending(Prof. (Prof. C.ChC.Ch. Wang). Wang)
In a synchrotron, the confining magnetic field comes from a system of several magnetic dipoles forming a closed arc. Dipoles are mounted apart, separated by straight sections/vacuum chambers including equipment for focusing, acceleration, injection, extraction, collimation, experimental areas, vacuum pumps.
qBp
=ρ
BEPCII dipole
x
By
By increasing E (hence p) and B together in a synchrotron, it is possible to maintain a constant radius and accelerate a beam of particles.
A sequence of focusing-defocusing fields provides a stronger net focusing force.
x
By
BEPCII quadrupole
Quadrupoles focus horizontally, defocus vertically or vice versa. Forces are linearly proportional to displacement from axis.A succession of opposed elements enable particles to follow stable trajectories, making small (betatron) oscillations about the design orbit.
2.3 Focusing 2.3 Focusing (Prof. (Prof. C.ChC.Ch. Wang). Wang)
2.4 Hill2.4 Hill’’s Equations EquationEquation of transverse motion
Drift:
Quadrupole:
Dipole:
Sextupole:
Hill’s Equation:
0,0 =−′′=+′′ ykyxkx
02,0)( 22 =−′′=−+′′ xykyyxkx ss
0,0 =′′=′′ yx
0,012 =′′=+′′ yxx
ρ
0)(,0)( =+′′=+′′ yskyxskx yx
Solution of the HillSolution of the Hill’’s Equations EquationHill’s equation (linear-periodic coefficients)
Property of machine
β(s) – envelope function
Property of the beam
ε − EmittancePhysical meaning (H or V planes)
))(sin()()( 0φφεβ += sssu
0)(2
2=+ usk
dsud
dxdB
Bsk y
ρ1)( −=
∫=)(
)(s
dssβ
φ
)()( ssu εβ=env
)(/)( ssu βε=′
like restoring constant in harmonic motion, the solution is
ConditionMaximum excursions
Envelope
and u denotes x and ywhere
General equation of ellipse is
2.5 Phase Space2.5 Phase SpaceUnder linear forces, any particle moves on an ellipse in phase space (x, x´).
Ellipse rotates in magnets and shears between magnets, but its area is preserved:
Emittance
x
x´
x
x´
εγαβ =+′+′ 22 2 xxxx
222 xxxxrms ′−′=εAcceptance: Ax,y > εx,y
max
2
max
2
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
yy
xx
YAXAββ
,
α, β, γ are functions of distance (Twiss parameters), and ε is a constant. Area = πε.
For non-linear beams can use 95% emittance ellipse or RMS emittance
(statistical definition)
2.6 2.6 LatticeLattice(Prof. C.C. Kuo)(Prof. C.C. Kuo)
The pattern of focusing magnets, bending magnets and straight sections connecting in between;It has a strong influence on aperture of vacuum chambers and thus other systems of the accelerator.
Matrix formalismMatrix formalism
[ ]⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
Δ−ΔΔ++Δ−
−
ΔΔ+Δ=
φαφββ
ββφααφαα
φββφαφβ
β
sin)(cos)()(
sin))(1(cos))((
sin)()sin(cos)(
)/(0
0
00
000
12
sss
ss
ss
ssM
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
+=
μαμμγμβμαμsincossin
sinsincos)/(
0
0000 ssM
As a consequence of the linearity of Hill’s equations, we can describe the evolution of the trajectories in a lattice by means of linear transformations
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛)(')(
)/()(')(
)(')()(')(
)(')(
0
012
0
0
sysy
ssMsysy
sSsSsCsC
sysy
In terms of the amplitude and phase function the transfer matrix is
where α(s)= dβ/ds .
βαγ
21 +=
For a periodic machine the transfer matrix over a turn reduces to
Example: FODO LatticeExample: FODO LatticeThe matrix for one period between mid-planes of F magnet in thin lens approximation is
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−±−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛±⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
μαμμγμβμαμsincossin
sinsincos
2/1)2/1(2/2/1(22/1
12/101
101
1/101
101
12/101
222
22
,
fLfLfLfLLfL
fL
fL
fM yx
m
mm
0
1 sin
)2/sin(12
)2
1(cos
,
,,
2
21
,
=
=±
=
−= −
yx
yxyx
yx
L
fL
α
γμμβ
μ
πμν 2/)( cN=
2.7 2.7 DispersionDispersionThe dispersion has its origin that a particle of higher momentum is deflected through a lesser angle in a bending magnet.
The solutions of the equation can be written in terms of the optics functions
02
1)(1''ppxskx Δ
=⎟⎟⎠
⎞⎜⎜⎝
⎛−+
ρρ
)())(cos()()(0
0 sDppsssx Δ
+−= φφεβ
It was shown that the equations of motion of a charged particle is a linear Hill’s equation
[ ] sdsssss
D )()(cos)()(
sin2)(
φφπνρβ
πνβ
−−= ∫
2.8 2.8 OrbitOrbit DistortionDistortionDipole errors may cause orbit distortion of particle beam.
Element Source Field Error Deflection Direction
Drift space (l=d) Stray field ⟨ΔBs⟩ ⟨ΔBs⟩/Bρ x, y
Field error ⟨ΔB/B ⟩ θ⋅⟨ΔB/B ⟩ x Dipole (angle=θ )
Tilt ⟨Δθz ⟩ θ⋅⟨Δθz ⟩ y
Quadrupole (Kl) Displacement ⟨Δx,y ⟩ Kl⋅⟨Δx,y ⟩ x, y
Similar to dispersion case, the equation of motion is written as
BsB
Bkyy )(1'' Δ
=+ρ
The solutions are )())(cos()()( 0 sysssy cod+−= φφεβ
[ ] sdsssB
ssBsyCOD )()(cos
)()()(
sin2)(
φφπνρ
βπν
β−−
Δ= ∫
2.9 Linear 2.9 Linear CouplingCouplingCoupling is the phenomena that energy exchange between two oscillators. In accelerators, the horizontal and vertical motions are also coupled:
xMxMkykyyMyMkxkx
y
x′−′−−=+′′
′−′+−=+′′
)2/()2/(
, 2
1
0⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∂∂
−=y
Bx
BB
k yxρ ρB
BM s−=Where
x-y coupling can be compensated with skew quadrupoles and anti-solenoids.Measured with tune approaching.
22
2
)(2)()(
νννν
εε
κΔ+−
Δ==
yxx
y
2.10 2.10 NonNon--linearitylinearity
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
+=⎟
⎟⎠
⎞⎜⎜⎝
⎛−+ ∑
=
M
n
nnn izxn
sijskxsksds
xd
2122
2)(
!)()(Re)(
)(1
ρ
Hill’s equation with higher order terms in the expansion of magnetic field
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
+−=+ ∑
=
M
n
nnn izxn
sijskzskds
zd
212
2)(
!)()(Im)(
The Hill’s equations acquire additional nonlinear terms:
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
+=+ ∑
=
M
n
nnnxz izx
nsijskBiBB
100 )(
!)()(
ρ )0,0(00
1n
yn
n x
BB
k∂
∂=
ρ Normal multipoles
)0,0(00
1nx
n
n xB
Bj
∂
∂=
ρSkew multipoles
There is no analytical solution available in general, and the equations have to be solved by tracking or perturbative analysis.
Analytical and Tracking StudiesAnalytical and Tracking StudiesStable and unstable fixed points appears which are connected by separatrices.Islands enclose the stable fixed points.On a resonance the particle jumps from one island to next and the tune is locked at the resonance value.Region of chaotic motion appear.The region of stable motion, called dynamic aperture, is limited by the appearance of unstable fixed points.
Qx = 1/5
3. Longitudinal Motion 3. Longitudinal Motion (Prof. S.Y.Lee)(Prof. S.Y.Lee)
RF Cavities & Acceleration
Synchrotron Oscillation
Bunches and Buckets
γ-Transition
3.1 RF Cavities & Acceleration3.1 RF Cavities & Acceleration(Prof. R.F. Wang)(Prof. R.F. Wang)
Necessary conditions for acceleration.Both linear and circular accelerators use electromagnetic fields oscillating in resonant cavities to apply the accelerating force.Linac– particles follow straight path through series of cavities.Circular accelerators – particles follow circular path in B-field and particles return to same accelerating cavity each time around.
sqVE ϕsin0=Δ
3.2 Synchrotron Oscillation3.2 Synchrotron Oscillation
( )sEfhV
φφγβ
ηπφ sinsin2
20
200 −−=&&
02 22
0
200 =Δ+=Δ+ φφφ
γβηπφ sΩ
EfhV &&&&
2
cos 02
0
0 fE
hVf ss γβπ
φη=
γβπφη
ν 20
0
2 cos
EhV
ff ss
s ==
This is a biased rigid pendulum
Synchrotron tune
For small amplitudes
Synchrotron frequency
3.3 Bunches and Buckets3.3 Bunches and Buckets
( )ss
s φφφ
φ sinsincos
2−
Ω−=&&
( ) .sincoscos2
22consts
s
s =+Ω
− φφφφ
φ&
( ) ( ) ( )[ ].sincoscos
sincoscos2
222
ssss
ss
s
s φφπφπφ
φφφφ
φ−+−
Ω−=+
Ω−
&
For large amplitudes
Integrated becomes an invariant
The second term is the potential energy function, and the equation of each separatrix is
And the half height when is
( ) ( )21
0max
⎭⎬⎫
⎩⎨⎧
±=Δ ss
s GEh
eVEE φηπ
β ( ) ( )[ ]ssssG φφπφφ sin2cos2 −−=
0=φ&&
Bucket size should be larger than bunch size (ΔE/Es and Δφ (Δτ, Δz))
3.4 3.4 γγ--TransitionTransitionRecall synchrotron focusing strength:
γβφπη
20
2002 cos 2
EfhVΩ s
s =
02
1//
RD
dpdR
Rp
dpdp
pdpfdf
−=+==γ
ββ
η
If η⋅cosφs<0, then Ω2<0, the oscillation gets unstable, where
ppDRR o
Δ⋅+=
The η changes from positive to negative at transition γ :
DR
RD
tr0
02 or 01
==−= γγ
η
The motion is stable for φs∈(0, π/2) , and unstable φs∈(0, π) for γ<γtr, and vice versa for γ>γtr.To cross γtr should be avoided in design or apply the phase-jump technique.
High EnergyLow energy
φs
4. Collective Effects4. Collective Effects(Prof. A.W Chao)(Prof. A.W Chao)
Space ChargeWake Fields and ImpedanceCoherent InstabilitiesBeam-beam EffectsLandau Damping
rEE =→
φBB =→
η…charge density in C/m3
λ... constant line charge π a2ηI…constant current λβc = π a2ηβca…beam radius
Electric Magnetic
0div
εηE =
→ →→= JμB 0curl
→→→→
∫∫∫ = AdBcurldφrB→→→
∫∫∫∫∫ = SdEdVE div
Volume element
Current density (βcη)
cylinder radius r cross sectionlength l radius r
πβcηrμrπBφ2
02 =rππ Eεηπlr r 20
2 =
202 a
rβcπε
IEr = 2202 a
rcπε
IBφ =
s Apply these integrals over
cross section
rx
y
φ
ErBφ
aAdr
φdrI
s
4.1 Space Charge4.1 Space Charge
For β 1 (γ>>1) FE=qEr =-FB=-q⋅Bφ⋅v so ΣF=0
Incoherent Tune Shift in a SynchrotronIncoherent Tune Shift in a Synchrotron
( ) 0)()( =++′′ xsKsKx SC
(s)dsβsKπ
dssβskπ
Δν x
πR
x
πR
xx )()()( SC∫∫ ==2
0
2
0 41
41
νx0 (external) + Δνx (space charge)
For small “gradient errors” kx
x
xxπR
x cγeβRIr
(s)a(s)β
cγeβRIrds
a(s)β
cγeβIr
πΔ
εν 33
0233
02
2
033
0241
−=−=−= ∫
Using I = (Neβc)/(2πR) with N…number of particles in ring εx,yare x, y emittance containing 100% of particles32
0
2 γβπNrΔ
x,yx,y ε
ν −=
“Direct” space charge, unbunched beam in a synchrotronVanishes for γ >>1Important for low-energy machinesIndependent of machine size 2πR for a given N
Beam not bunched (so no acceleration)Uniform density in the circular x-y cross section (not very realistic)
4.2 Wake Fields and Impedance4.2 Wake Fields and Impedance
θθ
θθθθ
mrzWeIzrF
mmrmrzWeIzrFm
mm
mmm
cos)(),,(
)sinˆcosˆ()(),,(
||
1
′−=
−−= −⊥r
Impedance Z = Zr + iZi
Induced voltage V ~ Iw Z = –IB ZV acts back on the beam
Intensity dependant
Wake field will be driven by a beam when there is a discontinuity in the vacuum chamber:
Impedances are just Fourier transforms of wake functions:
)()( , )(/
)( /||/ zWevdzZzWe
vdz
cviZ m
vzimm
vzim ′== −∞
∞−−⊥ ∫∫ ωω ωω
4.3 Instabilities4.3 Instabilities LinacsLinacsEnergy variation along the bunch length
Induced by longitudinal wake fieldsIt can be compensated for by properly phasing.
Transverse beam break-upInduced by transverse wake fields
zdszxzzWzL
NeszxsksEszxdsdsE
dsd
z′′−′′=+ ∫
∞ ),()()(),()()()],()([ 1
22 λ
It can be cured by the BNS damping: applying stronger focusing on tail than head of the bunch to balance the wake.
i
f
f EE
LEklWNe
kk ln
4)(
21
2−≈
Δ
InstabilitiesInstabilities Circular acceleratorsCircular acceleratorsNegative Mass InstabilityRobinson instability
Caused by fundamental cavity modeBeam will be stabilized by properly tuning the cavity (hω0> ωrf for γ>γt).
Head-tail InstabilityCaused by transverse wake fieldBeam will be stabilized by properly setting chromaticity (ξ> 0 for γ>γt).
Strong head-tail InstabilityCaused by transverse wake as counterpart of beam break-up in a linac.Once the threshold is exceeded the instability tends to grow very fast.
Microwave instability – increasing energy spread and bunch lengthPotential-well distortion – increasing bunch length
Coupled bunch instability multi-bunch effects caused by long-range transverse & longitudinal wakesCure: control of impedance and applying feedback systems
seE
∂∂
−=λ
γπε 204
4.4 Landau Damping4.4 Landau DampingTwo oscillators excited together become incoherent and give zero centre of charge motion after a number of turns comparable to the reciprocal of their frequency difference.
External excitation is outside the frequency range of the oscillators: No damping
External excitation is inside the frequency range of the oscillators The integral has a pole at Ω = ω: Landau damping
Excitation with ω
4.5 Beam4.5 Beam--beam Effectsbeam EffectsW = Energy available in center-of-mass for making new particles For fixed target :
Ec.m. ≅ 2mTEB... and we rapidly run out of money trying to gain a factor 10 in c.m. energy
But a storage ring , colliding two beams, gives:
E c .m . ≅ 2 E B
Problem: Smaller probability that accelerated particles collide .... "Luminosity" of a collider
L = N1N 21A
βc2πR
≈ 1029 .. .1034 cm −2s −1
Beam-beam interactions are non-linear and collective effects22 2/
22)( σ
πσρ rener −= ( )( ) 11
222 2/2
0
2σβ
επr
r er
neF −−±−=
For small amplitude particles, the beam-beam force can be approximated as a focusing lens in both x and y planes with focal length f given by
)(
2)(,
1,
yxyx
eyx
Nrfσσγσ +
=−
The incoherent linear beam-beam tune shifts are
Exhaustive study can be carried out with beam-beam simulation.
( ) Nr
f yxyxyx
yxeyxyxyx ,
,
,,1,, )(24
ξσσπγσ
βπ
βν =
+==Δ
∗∗−
( )∗
+=
y
yxxc
r
RfL
β
ξξεπ2
0
21
With the maximum beam-beam tune shift, luminosity is
5. Leptons and Hadrons5. Leptons and HadronsLeptons vs. HadronsSynchrotron radiationRadiation dampingRadiation excitationRings vs. Linacs
e+ e-
5.1 Leptons vs. Hadrons5.1 Leptons vs. HadronsHadron Collider (p, ions):
Composite nature of protonsCan only use pt conservationHuge QCD background
LEP:Z0 → 3 jets
LHC: H → ZZ → 4μ
p p
Lepton Collider:
Elementary particlesWell defined initial stateBeam polarizationProduces particles democraticallyMomentum conservation eases decay product analysis
RHIC: Au+Au →……
ppeff
eeeff EE ⋅≈
−+
10
5.2 Synchrotron radiation5.2 Synchrotron radiationThe electromagnetic radiation emitted when the charged particles are accelerated;
ρεγ
0
42
0 3eU =
The particle loses the energy of U0 in one turn;
)()(46.88)(4
0 mGeVEkeVUρ
=For electrons:
For protons:
e.g. LEP2: E=100GeV,ρ=3100 m, U0=1.85 GeV
e.g. LHC: E=7000GeV,ρ= 2804 m, U0= 6.66 keV
)()(1078.7)(4
120 m
GeVEkeVUρ
−×=
5.3 Radiation damping5.3 Radiation dampingSynchrotron radiation loss is compensated by the RF fields:
)sin(00 seVU ϕ=
Longitudinal: higher the energy of particle more SR loses;
This will cause radiation damping.
02 22
2=++ εωε
τε
ss dt
ddtd
Vertical: radiated momentum is with P⊥, while RF compensates P|| ;Horizontal: After the photon emission
δxβ + δxε = 0
5.4 Radiation excitation5.4 Radiation excitation
γρ
σ ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛=
2/1
36455
mcEh
2
32
/1
/
33255
ρ
ργεH
Jmc xx
h=
><
><= 2
3
/1
/
36455
ρ
ρβε
y
yy Jmc
h
With pure damping, the emittance will be zero in x, y and z planes.The radiated energy is emitted in quanta: each quantum carries an energy u = ħω;
22 ''2)( xxxx DDDDsH βαγ ++=Ji are damping partition numbers, H is dispersion invariant
The emission process is instantaneous and the time of emission of individual quanta is statistically independent; Radiation damping combined with radiation excitation determine the equilibrium beam distribution and therefore emittance, beam size, energy spread and bunch length.
Es
z Ωc σασ =
Diffusion effect off
Diffusion effect on
5.5 Rings vs. Linacs 5.5 Rings vs. Linacs (electrons)(electrons)
RingsRingsAccelerate + collide every turn‘Re-use’ RF + ‘re-use’ particles
⇒ efficientFinite beam emittance. Cost:
Linear costs(magnets, tunnel, etc.) : $lin∝ ρRF costs: $RF ∝ U0 ∝ E4/ ρOptimum when: $lin ∝ $RF ⇒ ρ ∝ E2
The size and the optimized cost scale as E2
Also the energy loss per turn scales as E2
⇒ get inefficient when E
LinacsLinacsOne-pass acceleration + collision⇒ need high gradientNO bending magnets ⇒ NO synchrotron radiationSmall beam size to reach a high LuminosityMuch less limited by beam-beam effect A lot of accelerating structuresCost scaling linear with E
e+ e-
Source
Damping ring
Main linac
Beam delivery
Bibliography Bibliography 1. E. Wilson, An Introduction to Particle Accelerators,
OUP, (2001)2. D. Edwards & M. Syphers: An Introduction to the
Physics of High Energy Accelerators3. S.Y. Lee: Accelerator Physics4. A.W.Chao and M.Tigner, Handbook of Accelerator
Physics and Engineering5. M. Conte & W. MacKay: An Introduction to the
Physics of Particle Accelerators6. M. Livingston & J. Blewett: Particle Accelerators7. M. Reiser: Theory and Design of Charged Particle
Beams
QuestionsQuestions(1) A 50 GeV (kinetic energy) proton
synchrotron will receive an injected beam at 4 GeV (kinetic energy) from an injector. The 2 T magnetic field of the bending magnets is excited at top energy of 50GeV. Given that the rest mass of the proton E0 is 0.93827 GeV:a) What is the Bρ at 4 GeV and at 50 GeV?b) What is the bending radius ρ?
(2) A quadrupole doublet (half of the FODO cell from the mid-point of the F to the mid point D) consists of two lenses of focal length f1 and f2 separated by a drift length of l.
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−
−=
2*
1
11
1
fl
f
lfl
M where2121
11*
1ff
lfff
+−=
f f
l
1 2
s
Please show, by writing the three matrices for the lenses, the product matrix is: