Foundation of Accelerator Beam Dynamics- Part I and Part II -
THz image
KAERI-2019-092
Yujong Kim
Future Accelerator R&D Team
KAERI & UST, Korea
[email protected], [email protected]
October 19-20th, 2019, HIP, Japan
2
Outline
Acknowledgements & My Other Lecture Notes
Relativistic Particle Motion and Weak Focusing
Transfer Matrix, Betatron Oscillation, and Tune
Dispersion and Momentum Compaction
Magnets (dipole, quadrupole, sextupole, multipole expression, solenoid)
Strong Focusing
Beam Emittance
Chromaticity and Chromatic Effects
Short-Range Wakefields
Bunch Compressor
Accelerator Beamline Lattices (FODO, DBA, TBA, and MBA) and Brightness
Textbook:
An Introduction to the Physics of Particle Accelerators by Mario Conte
Other Reference:
Accelerator Physics by S. Y. Lee
Accelerator Physics Lecture Notes by Y. Kim
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
3
Acknowledgements
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Y. Kim gives his sincere thanks to KAERI + UST colleagues and this school
chair, Prof. Masao Kuriki for their allowance of this lecture, and also to
following friends, references, and former & current supervisors:
KAERI, UST, and BMI: I. Jeong, S. Kim, Dr. J. Lee, and Dr. C. KimPAL & POSTECH: Prof. W. Namkung, Prof. I. S. Ko, and Prof. M. H. ChoSPring-8: Prof. T. Shintake (now at OIST), Prof. Kitamura, Dr. H. Tanaka,
Dr. T. Hara, Dr. T. Tanaka, and Dr. H. TomizawaKEK: Prof. K. Yokoya and Prof. H. MatsumotoPSI: Dr. S. Reiche, Dr. M. Pedrozzi, Dr. H. Braun, and Dr. T. Garvey,DESY: Dr. K. Floettmann, Dr. S. Schreiber, Dr. R. Brinkmann,
Prof. J. Rossbach, Dr. Y. ChaeAPS: Dr. J. Byrd, Dr. M. Borland, and Prof. Kwang-Je KimLANL: Dr. B. CarlstenIndiana University: Prof. S. Y. LeeINFN: Dr. M. FerrarioJefferson Lab: Dr. A. Hutton, Dr. H. Areti, and Dr. S. BensonDuke University: Prof. Y. WuIdaho State University & Idaho Accelerator Center: Prof. D. WellsHiroshima University: Prof. M. KurikiIBS RAON: Dr. D. Jeon, Dr. J. Kim, and Dr. M. Kwon
4
My Other Lecture Notes
Yujong Kim's Other Lecture Notes at Idaho State University, KAERI WCI,
POSTECH, KAIST, VITZRONEXTech, Korea-Japan Joint Summer School,
and ISBA.
Basic Accelerator PhysicsMagnets and Transverse Motion in Accelerators
RF System and Longitudinal Motion in Accelerators
Advanced Accelerator Physics Tutorial for XFEL Projects
Accelerator Beam Diagnostics
Linux Basic for Physicists
Laser Compton Scattering (LCS)
RF Technology and Electron Linear Accelerators
There are my lecture notes on beam dynamics for KoPAS2015.
There is also my lecture note on RF system for ISBA2018.
You can obtain them by sending an email to Yujong Kim:
[email protected], [email protected]
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
5
Short Review of Relativistic Particle Motion
Particle Accelerator Physics is a region of applied Special Relativity:
Law of physics may be expressed in equations having same form in all frames.
Speed of light in free space is the same value for all observers.
electronfor51099906.0/)MeV(/2
UmcU
please note that acceleration in linac gives
a growth of kinetic energy. At an electron gun exit,
energy gain = 500 keV → W = 0.5 MeV
U ~ 0.5 MeV + 0.511 MeV → γ ~ 1.97847358
U ~ pc for Ultra-Relativistic case.
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c= 𝟐. 𝟗𝟗𝟕𝟗𝟐𝟒𝟓𝟖 × 𝟏𝟎𝟖𝐦/𝐬
= 𝑊 +𝑚𝑐2
6
Energy of Rest Mass - mc2 & Isotope
Unified Atomic Mass Unit (u or also known as amu)
1 u = 1.66053892173 ×10-27 kg = 931.49406121 MeV/c2 = 1822.88839 me ~ 1 mp or 1 mn
me = 9.1093897× 10-31 kg = 0.51099906 MeV/c2 for electron
mp = 1.6726231× 10-27 kg = 938.2723 MeV/c2 for proton
~ 1836.152725 me
mec2 = 0.51099906 MeV for electron
mpc2 = 938.2723 MeV = 1836.152725 mec
2 for proton
mnc2 = 939.5656 MeV for neutron
mdc2 = 1875.6134 MeV for deuterium (D or 2H) with one proton & one neutron
Mass and Rest Mass Energy
1 uc² = (1.66053892173 × 10−27 kg) × (2.99792458 × 108 m/s)²
≈ 1.49242 × 10−10 kg (m/s)² ≈ 1.49242 × 10−10 J × (1 MeV / 1.6021773 × 10−13 J) ~ 931.49 MeV
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7
Short Review of Relativistic Particle Motion
protonfor2723.938/)MeV(
electronfor51099906.0/)MeV(
U
U
Ele
ctro
n K
inet
ic E
ner
gy
(M
eV)
Pro
ton
Kin
etic
En
erg
y (
MeV
)
MeV57139386651~ [email protected]~
2
c U - mW U
W
W
p
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𝑈 = 𝛾𝑚𝑐2 = 𝑊 +𝑚𝑐2
𝛾 =𝑊 +𝑚𝑐2
𝑚𝑐2
8
β of IBS RAON Accelerator ?
IBS RAON Accelerator
600 MeV for proton
β ~ 0.8
(particle A)
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HWR + QWR
SSR1 + SSR2
Site: 952,066㎡ (~ 290,000 pyeong)
9
Particle Motion in an Uniform Magnetic Fields
Under a constant magnet field in time, a positive charge q of a design particle
is performing a circular motion with a radius along a design orbit:
Lorentz force = centripetal force (for a reference particle on the design orbit)
betatron
B
&
xBRxBB yy ))(/(
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
B
10
Particle Motion in an Uniform Magnetic Fields
magnet (or momentum) rigidity
[Measured Magnetic field of a dipole for Bunch Compressor of DESY FLASH Facility ]
B = B(I)
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a relation to find beam energy or momentum in a dipole
for a singly charged particle (q = Ne = e with N =1)
11
Particle Motion in an Uniform Magnetic Fields
magnet (or momentum) rigidity
a relation to find beam energy or momentum in a dipole
for a singly charged particle (q = Ne = e with N =1)
B = B(I)
[Measured Magnetic field of a dipole for Bunch Compressor of DESY FLASH Facility ]
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
12
Working Principle of Spectrometer
→
we can measure momentum or energy
from bending angle, effective length, magnetic field.DESY TN-04 02 by P. Castro
ρ
ρˑ
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dl
B(l)
electrons dipole magnet
B field
13
Magnet Regidity for Heavy Ion Beams
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N : charge number 𝑞 = 𝑁𝑒
𝑈nucleon GeV/u : energy per one proton or neutron
A: atomic mass number = number of proton
= v/c
𝐴
𝑁𝛽𝑈nucleon GeV/u ≅ 0.3𝐵 T 𝜌 m
R
q
Under a constant magnet field in time, a positive charge q of a design particle
is performing a circular motion with a radius along a design orbit.
Local Cartesian Coordinate System (x, y, z) : moving along a design particle
x : radial outward, x ≡ R - (continuous changing direction)
y : vertically up
z : direction of motion of the design particle
s : total traveling distance along the design orbit, s = s(θ ) = s(ωt)
negative charge : left-handed coordinates
Trajectory of general particles are slightly
different from the design orbit due to
energy spread and magnetic field gradient
, hence a change of slope of particle trajectories
after s moving (-oscillation).
See pages 60 & 66, An Introduction to the Physics of High Energy Accelerators by Edwards14
Equations of Motions for Weak Focusing
B
xBRxBB yy ))(/(
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
)1(
x
xR
15
Magnetic Fields and Unit Vectors
From Page 66, An Introduction to the Physics of High Energy Accelerators by Edwards
→
R
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1616
Magnetic Fields and Unit Vectors
From An Introduction to the Physics of High Energy Accelerators by Edwards
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1717
Now return to Page 20 in Conte's Book
Equations of Motions for Weak Focusing
r @ Edwards' book → R = + x @ Conte's book
ignore any energy change
from previous page and
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Equations of Motions for Weak Focusing
define field index n
by applying the paraxial approximation, i. e.
18Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
1919
Equations of Motions for Weak Focusing
similar to equations of simple harmonic oscillator!
Therefore, x & y motion can be stable and focused simultaneously only when
0 < n < 1 : weak focusing
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2020
Weak Focusing - Lorentz Force
forces for coming out positive charge
center
horizontal defocusing for
radially opening poles
horizontal focusing for
radially closing poles
center
vertical focusing for
radially opening poles
vertical defocusing for
radially closing poles
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2121
Weak Focusing - Resultant Force
Horizontal motion in Betatron is controlled by a resultant force of
centrifugal force (mv2/R) and centripetal Lorentz force (qvBy), Fres :
0
2
qvBv
m
betatron
B
here n: field index
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2222
Weak Focusing
Horizontal motion in Betatron is controlled by a resultant force of
centrifugal force (mv2/R) and Lorentz force (qvBy), Fres , which acts to push
the particles towards the reference or equilibrium orbit under 0 < n < 1.
R
q
n = 0.5
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𝒎𝒗𝟐
𝑹
qvBy
2323
Solutions of Motion Equations
with the condition 0 < n < 1 gives a possible solution:
If there are initial conditions at = 0; x(0) = x0, x'(0) = x0' = dx/ds = (1/)(dx/d) =0
or in transfer matrix form in the horizontal plane:
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2424
Transfer Matrix, Betatron Oscillation, & Tune
From similarly, the transfer matrix in the vertical plane:
From solutions of equation of motion, we can find the fact that motion of particles
shows sinusoidal behaviors. This type of transverse oscillation is called the betatron
oscillation, which is induced due to the nonzero magnetic field gradient n.
horizontal and vertical betatron tunes QH and QV are defined as the numbers of
cycles of horizontal and vertical betatron oscillation which are made by a particle in
one turn circulation. QH & QV < 1 for weak focusing (0 < n < 1).
Define x & y betatron phase: &
for weak focusing
snn1
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2525
Examples of Betatron Oscillation and Tunes
betatron oscillation & tune for strong focusing:
QH > 1
QV > 1
What is difficulty in weak focusing accelerator?
Can we observe the betatron oscillation in a region with quadrupoles?
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2626
Transfer Matrix for Field Free Drift Space
x = x0 + Lx0'
x' = x0'
Transfer Matrix of a Drift Space with Length L
See also page 32 for other cases:
uniform magnetic field but no field gradient (n = 0)
90 degree, and opposite sign of n
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old particle position
new particle position
particle angle
Lx0'
design orbit
then, the horizontal motion is described by:
2727
Momentum Dispersion - Beam Spreading
So far, we assumed that all particles have a same energy (mono-energy).
But in the real situation, there is a horizontal beam broadening in a dipole due to the
momentum spread. There is the beam broadening only in the horizontal or radial plane.
Why there is no beam broadening vertically?
0Bqp
ppmcp
ppmcp
)1(
x
xR
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2828
Momentum Dispersion - Beam Spreading
by using definition of the field index
by ignoring nonlinear terms in x.
horizontal equation of motion under nonzero momentum spread
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2929
Momentum Dispersion - Beam Spreading
by inserting
Similarly, the vertical motion is described by an equation at page 18 :
0Bqp
ignoring x2 term
)1(p
ppppmcp
)1(
x
xR
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3030
Momentum Dispersion - Beam Spreading
By keeping linear terms, the equation of motion in the vertical plane is given by
vertical equation of motion under nonzero momentum spread
= same as previous case, without any momentum spread. Why?
Possible particular solutions of equations of motions in the x plane:
If there are initial conditions at = 0; x(0) = x0, x'(0) = x0' = dx/ds = (1/)(dx/d) =0
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
3131
Transfer Matrix under Momentum Spread
Therefore, solutions of motion equation in the horizontal plane under a momentum
spread can be given by
horizontal transfer matrix under momentum spread for weak focusing (0 < n < 1)
vertical transfer matrix is same as before without momentum spread (see page 24).
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2p
p 22
p
p 22
3232
Examples - Electron Motion in Dipole Magnet
If a particle is in a uniform magnetic field such as in a wide dipole or cyclotron, n =0.
orbit of electrons under
entering uniform B field
If
If are initial conditions for x, x', p/p, then final x, x' and p/p of the particle:
beam broadening = p
p4
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3333
Examples - Weak Focusing Synchrotron
Synchrocyclotron is a machine with a constant radius but with a varying magnetic field
in time according to the increasing momentum at an RF cavity.
cell or superperiod = a periodic block of elements
here cell number = 4 (one dipole + one drift space l0)
3 ways to make the transfer matrix for one cell in synchrotron above.
drift between dipoles = l0
drift = (l0/2)/=l0/2
synchrotron is a virtual circular machine:
= 2R
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l0
3434
Examples - Weak Focusing Synchrotron
Find the transfer matrix for one cell in synchrotron!
First of all, let's ignore the momentum spread.
We will consider it when we study momentum compaction factor.
drift = (l0/2)/=l0/2
per cell
total tune for the synchrotron = 4 cells QH
see details in textbook
0
H
H
21,
lR
Q
R
sin(x+y) = sin(x).cos(y) + sin(y).cos(x) & cos(x+y) = cos(x).cos(y) - sin(x).sin(y)
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a b
ab
3535
Examples - Weak Focusing Synchrotron
total vertical tune for the synchrotron = 4 cells QV
Similarly, for the vertical plane (try to drive these)
VQ
R
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3636
Momentum Compaction Factor p
p is defined as the fractional difference in circumference of the reference orbit
with respect to the fractional difference in the particle momentum. That is,
in synchrotron, circumferences are different for particles with different momenta.
]m[]T[2998.0)GeV/c(,/
/ BpBqp
pdp
LdLp
From the weak focusing synchrotron as shown in page 33,
paths in straight section lo are same for all particles with different momenta.
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3737
Momentum Compaction Factor p
For a weak focusing synchrotron, the momentum compaction factor p is larger than 1
because QH is smaller than 1. Therefore, the difference of circumference for particles
with a momentum spread is large. Hence the dimension of vacuum chamber becomes
large to build a high energy weak focusing synchrotron (problem).
But for the strong focusing machine, momentum compaction factor is small
enough because QH is much higher than 1.
Example of PLS storage ring: QH ~ 14.28, p << 1
→ no big circumference difference for particles with different energies.
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(1 + 𝑥)−1≅ 1 − 𝑥 𝑓𝑜𝑟 𝑥 ≪ 1
3838
Liouville's Theorem
A particle density function f(x, y, z, px, py, pz, t) can be defined as the number of particle
per volume of six dimensional phase space (x, y, z, px, py, pz) at a given time t.
Here, x, y, z is the three spatial coordinates, and px, py, pz are their corresponding
momentum components.
Liouville's Theorem:
In the local region of a particle, the particle density in phase space is constant in time,
provided that the particles move in a general field consisting of magnetic fields and of
fields whose forces are independent of velocity (or non-dissipative systems
such as no radiation loss, space charge force, and wakefields).)(vFF
0dt
df
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Typically, magnets in accelerator supply transverse or longitudinal magnetic fields.
From the Lorentz force, magnets are used to change of the direction of motion of a
particle; . Example, bending magnet.
Magnets with transverse fields (ex, dipole, quadrupole) are backbone of accelerator
and beam transport system, and magnets with longitudinal fields (ex, solenoid) can be
used to detect colliding beams in colliders or to focus beams in a low energy (ex, gun).
3939
Magnets in Accelerator
IL
NknIknIB
7
0 104]T[
http://hyperphysics.phy-astr.gsu.edu
magnetic field of Solenoid B [T], where
: permeability in material = k0
N : number of coil turns = nL
L : length of solenoid [m]
I : current [A]
k : relative permeability
~ 200 for magnetic iron
~ 20000 for -metal for magnetic shielding
(75% nickel, 2% chromium, 5% copper, 18% iron)
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4040
Magnetic Field of Ferromagnetic Material
)()(000 materialindipolenetcurrent BBHMHMHB
where
H [A/m] is magnetic field strength driven by external driving current (I)
M [A/m] is magnetization, density of net magnetic dipole moments in a material
http://hyperphysics.phy-astr.gsu.edu H due to external driving current I
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coil part iron core part
4141
Hysteresis Loop of Magnets with Iron Core
)(000 MHMHB
where
H [A/m] is magnetic field strength driven by external driving current (I)
M [A/m] is magnetization, density of net magnetic dipole moments in a material
http://hyperphysics.phy-astr.gsu.edu
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coil part iron core part
4242
Hysteresis Loop of Magnets with Iron Core
)(000 MHMHB
http://hyperphysics.phy-astr.gsu.edu
where
H [A/m] is magnetic field strength driven by external driving current (I)
M [A/m] is magnetization, density of net magnetic dipole moments in a material
at a certain driving current I (or H), there are two different M depending on history or
direction of I.
We have to cycle all magnets with iron cores to reproduce the same magnetic fields B at
a certain current I.
M
H or I
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coil part iron core part
4343
Cycling of Magnets with Iron Core
We have to cycle all magnets with iron cores to reproduce the same magnetic fields B at
a certain current I.
Generally, cycling should be done two times (at least) by changing current of power
supply gently to get a good reproducibility in magnetic field:
gently go Imax → waiting (until set value = reading value) → go Imin → waiting →
go Imax → waiting → go Imin → waiting → setting a current
cycling of a magnet with an unipolar (left) and a bipolar (right) power supply
I
M
9: setting a current3, 7
4, 8
1, 5
2, 6
I
M
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4444
2D Magnet Model
In preliminary accelerator design, magnets can be considered with the 2D model
if the considering region is 1.5 times of a gap height d from the end of the magnet.
Specially, the 2D model works well if magnet length is long enough.
But in the real accelerator design, the end fringe field effects should be considered.
Until we learn the end fringe field effect, let's assume magnet with the 2D model.
pole gap height = dGood
2D Region
1.5d
d
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4545
Maxwell Equations in 2D Magnet Model
For static field and if there is no current (ignoring beam current),
Then, magnetic field strength H can be expressed as the gradient of the magnetostatic
potential and it satisfies the Laplace equation.
0
0
0,0
H
B
jt
E
02
H
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4646
Laplace Equation and Multipole Expansion
The general solution of the Laplace equation in the cylindrical coordinates (r, , z)
(see cylindrical harmonics in John R. Reitz's Foundations of Electromagnetic Theory)
can be found in 2D cylindrical coordinates;
Here a is the reference radius of the expansion ~ half of magnet gap.
While, we can also expand the field of bending magnets in a series of multipoles:
02 0
112
2
2
rrr
rr
y
x
a2
(r,)
HB
: normal strength of 2(n+1)th multipole
: skew strength of 2(n+1)th multipole
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4747
Multipole Expansion
Normal Multipole Components:
n = 0 for normal dipole component
n = 1 for normal quadrupole component
n = 2 for normal sextupole component
n = 3 for normal octupole component
n = 4 for normal decapole component
Skew multipole Components :
poles are rotated by /2(n+1),
example,
skew dipole = 90 deg rotation for n =0
skew quadrupole = 45 deg rotation for n = 1
normal dipole & quadrupole
skew dipole & quadrupole
http://pbpl.physics.ucla.edu
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4848
Equipotential Lines of → Pole Face
HB
equipotential lines of = pole face
Similarly to electric fields, magnetic field must be perpendicular to the equipotential
lines of , which gives the pole face of the magnet.
Equation of Optimum Pole Face for Normal Magnets:
Pole Face of Dipole :
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http://pbpl.physics.ucla.edu
n =2 for QM
Example, Normal Dipole Magnet :
b0 = 1, all other bn & an = 0
note that
normal magnets have poles on y-axis for even n (dipole for n =0, sextupole n =2, ...),
normal magnets have no pole on any axis for odd n (quadrupole for n =1, octupole n =3, ...)
4949
Normal & Skew Magnets
for normal magnets for skew magnets
see also current relation in textbook!Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
5050
Dipole Magnet
equipotential lines of = pole face
Apply Ampere's law to estimate the magnetic field in dipole gap
IldH
IldB
0
0
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N turnsI
5151
Quadrupole Magnet
Apply Ampere's law to estimate the magnetic field in quadrupole
S
N
N
S
here note that
current direction is wrong.
it is reversed.
n = 1
r = a
= /4 for south
= -/4 for north
centerQMnear@00 HB
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5252
2(n+1) Multipole Magnet
For sextupole, n = 2
centernear@00 HB
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Here we can ignore Bz in the 2D model because length of QM is long enough (l > a).
By using
near QM center,
And from Ampere's law
5353
Quadrupole Lens
In the absence of electric field, which means no acceleration,
from page 6, motion equation becomes;
centerQMnear00 HB
0 B
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5454
Quadrupole Lens
H
From QM magnetostatic potential at pages 48 & 52, rHB centeQMnear0
QMsperfectfor),(,2
),(
)GeV/c(
)A(2998.0
)GeV/c(
)T/m(2998.0)()(
0100
0
10
gxyxBgyya
Gy
aa
NIHyxB
p
IC
p
gsg
p
q
x
B
p
q
pa
qGsk
yxx
mag
x
y
see distributed paper on measurement g
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
5555
Quadrupole Lens
Here general solutions are used. From initial conditions (x = x0, x' = x0' at s =0):
In transfer matrix form:
After considering opposite sign k in y (k → -k), in vertical plane:
i
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5656
Quadrupole Lens
Note that QM always makes opposite focusing in x & y planes:
horizontal focusing QM (QF) → vertical defocusing
vertical focusing QM (QD) → horizontal defocusing
If
Thin Lens Approximation for QM
here f is the focal length of the QM.
(x0, x'0 = 0)
(x = 0, x'= -x0/f)
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5757
Fringe Field Effects in a Dipole
Up to now, we assumed that the design trajectory is orthogonal to the both ends of a
dipole magnet. But in real situation, there are many cases where the trajectory is not
orthogonal to the ends of dipoles.
→ there are additional focusing or defocusing fringe field effects in a dipole.
n
n
: normal vector with respect to the magnet yokes : rotation angle of the yoke-ends around vertical axis.
> 0 for the normal vector is at outside of the beam trajectory (entering - clockwise).
< 0 for the normal vector is at inside of the beam trajectory (exit - clockwise).
= 0 : sector dipole (no pole rotation).
beam trajectory
[top view of three different dipoles]
sector dipole
n
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
n
beam trajectory
5858
Fringe Field Effects in a Dipole
For > 0,
particles located at positive x take shorter paths in the dipole & to be bent weakly
particles located at negative x take longer paths in the dipole & to be bent strongly
→ horizontal defocusing & vertical focusing
For < 0,
particles located at positive x take longer paths in the dipole & to be bent strongly
particles located at negative x take shorter paths in the dipole & to be bent weakly
→ horizontal focusing & vertical defocusing
x > 0
x < 0
x > 0
x < 0
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
5959
Slanting Dipole = Normal Dipole + Two Wedges
Horizontal Transfer Matrix for a Slanting Dipole: Mslanting = MwedgeMsectorMwedge
Wedge is a kind of focusing or defocusing quadrupole magnet:
horizontal trajectory of positive charges in a wedge
we can consider this
wedge as a defocusing QM!
slanting dipole = wedge + sector dipole + wedge
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
From Conte's book page 84 or Reiser's book page 135-138,
Horizontal Transfer Matrix of a thin Wedge without any dispersion:
Horizontal Transfer Matrix of a thin Wedge with a dispersion p/p:
From Conte's book pages 52 & 56, Eq (3.84) and Eq (3.102), the 6 dimensional transfer
matrix (x, x' y, y', z, p/p) of a sector dipole without any wedge:
Note that a sector dipole can supply a focusing in the horizontal plane.6060
Transfer Matrix for Wedge
0
'
100
01tan
001
'
p
px
x
p
px
x
0'1
tan01
'
x
x
x
x
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
6161
Transfer Matrix for Wedge
Then, the total horizontal transfer matrix of a slanting dipole with wedges and a
dispersion can be obtained from a transfer matrix for wedges (Mw in page No. 60)
and the horizontal transfer matrix of a sector dipole without any wedge (= MH above):
00
H '
100
sincossin1
)cos1(sincos
')(M'
p
px
x
p
px
x
p
px
x
From the 6 dimensional matrix of a sector dipole, let's extract components of only
horizontal position, horizontal angle, and a dispersion (x, x', = p/p) →
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
6262
Transfer Matrix for a Rectangular Dipole
Similarly, from Reiser's book page 135-138 and the vertical components the transfer
matrix of a sector dipole magnet, the total vertical transfer matrix for a slating dipole
with wedges but without a dispersion (y, y') can be given by:
For a Rectangular Dipole Magnet with :
yw,Vyw,V M)(MMM θ
Note that a rectangular dipole can supply a focusing in the vertical plane.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
6363
Angles of Rectangular Dipoles for Chicane
SCSS Bunch Compressor, details can be found from Y. Kim's NIMA 528 (2004) 421.
For the first dipole:bending angle = + 4 deg
edge angle1 for entrance = 0 deg
edge angle2 for exit = + 4 deg
For the second dipole:bending angle = - 4 deg
edge angle1 for entrance = - 4 deg
edge angle2 for exit = 0 deg
For the third dipole:bending angle = - 4 deg
edge angle1 for entrance = 0 deg
edge angle2 for exit = - 4 deg
For the fourth dipole:bending angle = + 4 deg
edge angle1 for entrance = + 4 deg
edge angle2 for exit = 0 deg
bending angle ~ 4 deg
dipole length ~ 0.2 m
beams go from left to right
note clockwise is positive
θedge1 + θedge2 = θbending
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
To keep beam size with a certain limits (< inner diameter of vacuum chamber), we need
high horizontal and vertical tunes (QH & QV > 1) : concepts of Strong Focusing.
QM gives the focusing only in one plane.
But continuous doublets can give the horizontal and vertical focusing:
From two thin lens combination, we can get a focal length of the two combined lens:
If fD = - fF,
f > 0, doublet gives a focusing system.
→ backbone of strong focusing!
6464
Strong Focusing
DFDF ff
l
fff
111
l
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Separated function magnet has each own one function (bending or focusing)
Combined function magnet has more than one functions (bending and focusing)
[separated function dipole] [combined function dipole]
Generally, transfer matrix M(s) for a periodic cell can be made by multiplying transfer
matrices of machine components (drifts, quadrupoles, bending magnets, and so on).
For a generalized coordinate z
Note that the trace of matrix M(s) should be smaller than 2 for the stable periodic beam
motion in ring! (see details from Conte's book pages 97-100 & resonance conditions) .
6565
Twiss Parameters - x,y(s), βx,y(s), γx,y(s)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
For a stable focusing beamline region satisfying , generally, the transfer matrix
can be parameterized with the Twiss parameters ((s), β(s), γ(s)) (see Conte's book pages
97-100) :
6666
Twiss Parameters - x,y(s), βx,y(s), γx,y(s)
Most transverse beam parameters can be expressed by the Twiss parameters, phase advance
emittance ε, and others. Ex, vertical rms beam size & rms divergence with no dispersion:
Design of a stable focusing machine lattice → optimization of the Twiss parameters!
)()()(,)()()( ' ssssss yyyyyy
2
1
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
67676767
Twiss Parameters for PSI 250 MeV injector
pickup
tuners
High Energy Transverse Deflecting Structure (TDS2)
developed with collaboration with INFN and PSI
resonance frequency : ~ 2997.912 MHz
deflecting mode : TM110
type : five cell SW cavity
physical length / average iris diameter : 441 mm / 36 mm
max available klystron power : 7.5 MW
max deflection voltage : 4.5 MV for about 4.1 MW
max slice number for 10 pC (200 pC) : 3 (12) slices
operation energy : ~ 250 MeV (gun region)
rms time resolution for 10 pC (200 pC) : 11 fs (16 fs) @ 4.5 MV TDS2 with five cells
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
6868
Diagnostic Section for PSI 250 MeV Injector5QMs TDS2 5QMs 3FODO 1QM DIPOLE
3FODO Cells
OTRs : 1 2 3 4 5 6 7 8
horizontal phase advance @ 3FODO = 55 deg per cell
vertical phase advance @ 3FODO = 25 deg per cell
cell length = 3.0 m
rms beam size @ 7FODO screens ~ 60 m for 200 pC
max dispersion by a dipole in DIAG1 ~ 0.2 m
With this special DIAG1,
we can measure followings
without change any optics:
- slice emittance
- slice energy spread
- longitudinal phase space
- bunch length
- arrival timing jitter
- projected emittance
- Twiss parameters
- optics matching\
only by turning on and/or
off TDS2 and a dipole in
DIAG1.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
6969
Phase Advance along the Diagnostic Section
5QMs TDS2 5QMs 3FODO 1QM DIPOLE
3FODO Cells
OTRs : 1 2 3 4 5 6 7 8
Slice Emittance Measurement
βy should be high at TDS2 = 20 m
ψy = (nπ+π/2) at OTRs
=107.9 deg @ OTR6
ψx = 0 ~180 deg at OTRs
=189.2 deg @ OTR6
βy should be small at OTRs
= 6.5 m @ OTR6
βx should be small at OTRs
= 4.3 m @ OTR6
σx = 60 µm @ OTRs for 200 pC
Long. Phase Space Reconstruction
βy should be high at TDS2 = 20 m
ψy = (nπ+π/2) at OTR8
= 140.2 deg @ OTR8
high ηx @ OTR8
= 0.2 m @ OTR8
small β-beamsize σx at OTR8
= 26 µm without dipole
βx should be small at OTR8
= 0.96 m @ OTR8
εn, resolution ~ 0.05 µm for x,y = 20 µm
σδ,resolution ~ 1.3×10-4
σt,resolution ~ 16 fs
4.5 MV & 12 slices, 200 pC
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
7070
Beam Profiles along the Diagnostic Section
TDS2 Off for Projected Emittance Measurements
TDS2 = 4.5 MV, DM = 6 deg for Slice Emittance & Long. Phase Space Measurements
K1@Q18 = 4
X-band on
σδ,resolution ~ 1.3×10-4
σt,resolution ~ 16 fsQ = 200 pC
reconstruction @ OTR6
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
7171
Analytical Approach of Twiss Parameters
From Hamiltonian equation of charge particles in a sector dipole without any fringe field
(see Conte's book pages 42-54),
0p
p
''x
''y
consider linear terms only!
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
7272
Analytical Approach of Twiss Parameters
After adding quadrupole focusing effects (see Conte's book page 82)
and bending effects of a sector dipole together, a generalized equation of motion of charge
particles in magnets supplying bending and focusing effects can be given by
here
and energy spread was considered.
0B
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
7373
Analytical Approach of Twiss Parameters
From the generalized equation of motion of charge particles in magnets
which supply bending and focusing effects
Let's ignore the energy spread term (later we will consider it). Then we can get a
generalized Hill's equation:
here z represents x or y and k(s) represents kx(s) and ky(s).
Since k(s) is a periodic function with a period of L, the Hill's equation can have a quasi-
periodic solution, which is related to the Twiss parameters:
here , , w(s) is an amplitude function with a
period of L, w(s+L) = w(s), A & B are constants, (s) is a non-periodic phase.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Since zA(s) and zB(s) are solution of the Hill's equation, let's substitute zA(s) in the Hill's
equation:
To valid this equation for all values of , coefficients of sine and cosine functions should
be zero:
→
→
If we insert the last equation to the first equation:
7474
Analytical Approach of Twiss Parameters
1''ln2'ln0'2
'
'' 22
wCwCw
w
w
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
From a general solution of the Hill's equation, we can get the slope z'(s):
☺
☺
here, we used .
By using initial conditions ( ), we can find two
constants A & B:
→
by inserting these A & B in z(s) and z'(s) above (see ☺ region), we can find transfer matrix
M(s) connecting two points z and z0 →7575
Analytical Approach of Twiss Parameters
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Transfer matrix M(s) connecting two points z and z0:
here
If we define following things (Twiss parameters)
7676
Analytical Approach of Twiss Parameters
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
If we insert these relationships to a(s), b(s), c(s) , and d(s), then Transfer matrix M(s)
becomes:
From , , and → , then M(s) can be
written as:
and from
we can find that M(s) above is coincides with:
7777
Analytical Approach of Twiss Parameters
Lss 0 00 ')(' wsw
)(/))(1()(
)()()(
)()()(
2
000
000
sss
sLss
sLss
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
From , the total phase advance per cell and total tune can be given by:
here, N is the number of cells, and QH and QV are the horizontal and vertical tunes.
Note the fact that if the transfer matrix of position and slope is given by
then, transfer matrix for the Twiss parameters is given by:
7878
Analytical Approach of Twiss Parameters
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
From
→
If we re-arrange z(s):
Note that z(s) is a form of . Let's re-write z(s) as
we find 7979
Beam Phase Space and Emittance
from cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Similarly,
→
By squaring and adding following two equations, we can get constant W:
→
W is related to initial phase space coordinates (z0 and z'0).
Similarly, by squaring and adding z(s) and z'(s),
8080
Beam Phase Space and Emittance
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
→
W is also related to new phase space coordinates (z and z').
Therefore constant W is an invariant of motion, so called Courant-Snyder invariant, and it
is related to the area of a rotated ellipse equation of z and z' and beam emittance(see Conte's book page 105). Here W is similar to the total energy of a harmonic oscillator.
Since the area of a generalized rotated ellipse is given by
the total area of z and z' phase space ellipse is πWif we use .
At a certain accelerator location s1, initially, a charge particle is at a
point on a phase space ellipse (z, z'), the particle is continuously
located at some other point on the same ellipse in successive cycles
or turn of the periodic motion if tune is an irrational number.
8181
Beam Phase Space and Emittance
phase space (z, z') area of 100% particles = πW
z'(s1)
z(s1)
1st turn
2nd turn
3rd turn
4th turn
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
The phase space ellipse (z, z') in previous page corresponds to the phase space at a
point (s = s1) in the beamline. Therefore, the shape of the ellipse is not changed even
beams circulate many turns at s1. But its shape can be changeable at different locations
along the beamline as shown below. Even though its shape is changed along thebeamline, its area will be same (πW) at all locations if there is no any dissipative actions
such as space charge force, wakefields, or coherent synchrotron radiation.
8282
Beam Phase Space and Emittance
area @ s1= πW
z'(s1)
z(s1)
1st turn
2nd turn
3rd turn
4th turn
area @ s2= πW
z'(s2)
z(s2)
area @ s3= πW
z'(s3)
z(s3)
s = s1 s = s2
s = s3
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Let's define a new smaller ellipse of the phase space (z and z'), which contains some fraction
of particles (example, 90%). If the area of the smaller ellipse is πε, we define ε as the beam
geometrical (or projected) core emittance of the fractional particles.
Geometrical (Projected) Core EmittanceIf the rms emittance is εrms, (see next page),
and the core emittance ε is nεrms, then, the
percentage of particles contained within the
ellipse of the core emittance is 100(1-e-n/2) (%).
8383
Beam Phase Space and Emittance
total area = πWcontained particles = 100%
z'
z
z'
z
total area = πε
contained particles
= some fraction, ex, 95%
see S.Y. Lee's book
square of 1D
z
z'
πεrms (= area in 39%)
πε = nπεrms
πW (100%)
n =
100(1-e-n/2)(%)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
8484
Beam Phase Space and Emittance
The rms emittance rms for a normalized beam distribution ρ(y,y') with
is defined as:
here, <y> and <y'> are average of position and angle.
σy & σy' are the rms values of beam size and angle.
and r is the correlated coefficient.
rms is estimated for 100% particles but its value
is corresponding to area having about 39% particles.
see more details in S.Y. Lee's book
max amplitude of β-tron motion =
max divergence (angle) of β-tron motion =
example of (y,y'):
2
'
2'0
2
20
2
)'(
2
)(
)',(yy
yyyy
Aeyy
2
20
2
)(
22
1)( y
yy
y
ey
)(s
)(s
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
8585
Beam Phase Space and Emittance
In linac where beams are accelerated, generally, we use the projected normalized beam
emittance, which is an invariant and given by (see Conte's book pages 107-108).
Adiabatic damping of beamsize, angle, and emittance in linac:
Increased energy or γ → geometrical emittance ε becomes smaller, and beamsize and
divergence become smaller. But note that the normalized emittance is an invariant
without any dissipative actions such as space charge force,
wakefields, or coherent synchrotron radiation.
See Sadiq's & Chris's term projects
on QM scanning based emittance measurement method. http://www.physics.isu.edu/~yjkim/course/2010fall/2010fall_ap_term_project04.pdf
http://www.physics.isu.edu/~yjkim/course/2011spring/2011spring_ap_project07.pdf
http://www.physics.isu.edu/~yjkim/course/2011spring/2011spring_ap_project08.pdf
2
0/
1~for
cmU
c
v
γ
n
, σx,y, σ'x,y
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
The generalized equation of motion of charge particles in magnets supplying bending
and focusing effects is given by:
Since the generalized solution of the homogeneous equation with = 0 is given by
→ .
Then the generalized solution of the inhomogeneous equation with ≠ 0 can be written
as
→
Here D(s) is a particular solution with 0 = 1 (see Problem 5-8 in Conte's book).
8686
Dispersion Function (s)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
8787
Dispersion Function (s)
From the initial conditions (x0, x'0) at s = 0:
Since no change in energy spread is assumed, trajectory equations can be written in
matrix form for ≠ 0 :
Here, the trajectory x(s) has two parts: a part due to betatron oscillation, xβ(s) and
the other part due to dispersion or
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
If we insert the trajectory due to the periodic dispersion in the matrix
form:
For a periodic cell, x0 = (s0) 0=(s0+L) = (no energy spread change here).
If we solve equation above, we can find a periodic dispersion function (s) and '(s).
here we used tr(M)= C +S' = 2cos for .
8888
Dispersion Function (s)
(see Problem 5-8 in Conte's book & S. Y. Lee's book)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
8989
RMS Beamsize
In a horizontal dispersion region, two things contribute to the rms horizontal beamsize.
one is due to the betatron amplitude and the other is due to the dispersion function η
and the rms momentum spread σp:
Here we assumed that there is a horizontal bending.
Without any vertical dispersion, the rms vertical beamsize can be written:
222
Epp
Exx
nxp
xxnxp
xxxx
yyy
beamsizermsfor)()(,)()(p
ssxssxp
xp
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
9090
For Gaussian beam distribution,
probability for -1 < x < +1: 68.3%
probability for -2 < x < +2: 95.4%
probability for -3 < x < +3: 99.7%
Therefore, the beam full width is close to a region from -3 to +3 = 6 (99.7%).
x
beam full width = beam diameter ~ 6
full width at half maximum (FWHM)
35482.22ln22FWHM
Full Width & FWHM for Gaussian Distribution
courtesy of Wikipedia
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
If we focus beam very strongly to get a tiny beamspot at a undulator or radiator, can we
reduce beam emittance? How about beam divergence after the point?
If there is no any special action, the beam emittance is not reduced even though we focus
beam very strongly. Here, only -function (instead of emittance) is reduced or minimized
for the strongly focused beam;
for no dispersion case.
In this case, the beam divergence is dramatically increased, and there is a big beam loss
in a undulator gap with a tiny gap if we focus beam too strongly with a poor emittance;
To reduce the loss, we have to improve beam emittance or reduce strength of focusing.9191
Mis-concept with Strongly Focused Beam
)1
(
2
y,x
y,x
y,xy,xy,x'y,'x
y,xy,xy,x
undulator gap ~ 4-6 mm
THz-Radiator gap = 1.2 mm
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
9292
Dispersion Control by Quadrupole
By control phase advance or tune with QMs, we can control dispersion in the nono-zero
dispersion area.
See following pages on dispersion control in the HRRL beamline and Bunch Compressor.
See also dispersion suppressor at pages 122-125 in Conte's book.
0)(for0)( 0 ss
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
93
Dispersion Suppression @ HRRL Beamline
2.525 m
2.795 m
15 cm long QM (Quad2T) with 2 inch ID
24 cm long QM (Q1B) with 2 inch ID
1st and 2nd target
Kiwi dipole ~ 0.25 m (pole edges)
movable hole for transverse collimator
movable slit for energy collimation
beam pipe with ID > 36.1 mm
Faraday cup
movable screen
LINAC T1 TCOL1 1STTG T2 TCOL2
0.3 m
KIW
I T
3 2N
DT
G
dispersion controlling QM
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
94
Dispersion Suppression @ HRRL Beamline
LINAC T1 TCOL1 1STTG T2 TCOL2
dispersion controlling QMxxx
2
p
p
xxxx
After dipoles, horizontal dispersion (x) and energy spread
are the main contributions in the beam size!
Before dipoles, horizontal emittance (x) and beta function
are the main contributions in the beam size!
x controlled
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
95
Dispersion Suppression @ HRRL Beamline
LINAC T1 TCOL1 1STTG T2 TCOL2
dispersion controlling QMxxx
2
p
p
xxxx
After dipoles, horizontal dispersion (x) and energy spread
are the main contributions in the beam size!
Before dipoles, horizontal emittance (x) and beta function
are the main contributions in the beam size!
x controlledx uncontrolled
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
96
Dispersion Suppression @ HRRL Beamline
initial assumed beam parameters
normalized emittance = 16 m
beam energy = 3 - 10 MeV
energy spread = 13% (FWHM) 5.52% (rms), 33.12% (FW)
Q = 50 pC
bunch length = 25 ps (FWHM)
dispersion controlling QM
KIWI KIWI dipole
controlled dispersion
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
97
Dispersion Suppression @ HRRL Beamline
initial assumed beam parameters
normalized emittance = 16 m
beam energy = 3 - 10 MeV
energy spread = 13% (FWHM) 5.52% (rms), 33.12% (FW)
Q = 50 pC
bunch length = 25 ps (FWHM)
dispersion controlling QM
KIWI KIWI dipole
controlled dispersion
uncontrolled dispersion
2
p
p
xxxx
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
9898
Dispersion Control @ Bunch Compressor
Two QMs in Bunch Compressor can control the residual dispersion after
Bunch Compressor.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Trajectories of a design particle and an off-momentum particle in an infinitesimal arc
is shown in figure below.
From the definition of the momentum compaction factor
(see Chapter 2 in Conte's book),
the circumference of a design particle can be given by
and the circumference of an off-momentum particle can be given by
The infinitesimal azimuthal angle d is common for both particles:
9999
Momentum Compaction Factor in Ring
d
p p+dp
xp
ds d
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
After using , then ∆L can be given
The momentum compaction factor can be a function of dispersion and bending radius.
100100
Momentum Compaction Factor in Ring
d
p p+dp
xp
ds d
→
→
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
101101
Momentum Compaction Factor R56 in BC
→
3
2
2
10
2
210
2
011566
0056
3
5666
2
56656
2
10
2
10
2
)('
)(
)(
)(
)(
here
p
p
p
p
p
p
p
px
p
p
p
p
dss
s
sLT
dss
sLR
p
pU
p
pT
p
pRdzdz
p
pL
p
pLL
p
p
p
p
L
L
p
if
Momentum Compaction Factors in Chicane Type Bunch Compressor
A. Nadji et al., NIMA 378 (1996) 376
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
102102
Momentum Compaction Factor R56 in BC
dz
Momentum Compaction Factors in Chicane Type Bunch Compressor
ELEGANT code supplies R56, T566, U5666 in BC
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
103103
Chromaticity
H. Wiedemann, Particle Accelerator Physics
Focal property of a lattice with QMs depends on the normalized strength k of QMs,
which, in turn, depend on the charge particle's momentum (chromatic effects).
→ beamsize growth hence beam emittance dilution due to energy spread.
0
0
00
0
0
0
11)(
)(
1
)()()(
p
pk
p
p
x
sB
p
e
x
sB
p
pp
e
x
sB
pp
e
x
sB
p
esk
y
yyy
Chromatic Effects becomes Stronger: larger β-function
larger energy spread = p/p
stronger QM normalized strength k
longer QM length LQM
βkLQM << 1 to ignore chromatic effects
see Sadiq's term project on chromatic effects!
1212
22
11
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
104104
Chromaticity Correction
H. Wiedemann, Particle Accelerator Physics
SF
SD
Natural Chromaticity ξN (ksi): the fractional difference in tune with respect to the fractional
difference in the particle momentum. The natural chromaticity can be corrected by putting
focusing and defocusing sextupoles as shown figures below.
before correction
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
105
Example of Chromatic Effects
SB03 SB04 XB 4QMs BC 5QMs 3FODO LOLA 2QMs DUMP
strong focusing optics against CSR in BC
x-function ~ 0, x-function ~ 6
Chromatic Effects becomes Stronger:
larger β-function
larger energy spread
stronger QM strength k
longer QM length LQM
~ 2%
β ~ 60
K1@QM ~ 15
PSI 250 MeV Injector Version-1
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
106
SB03 SB04 XB 4QMs BC 5QMs 3FODO LOLA 2QMs DUMP
~ 2%
β ~ 60
K1@QM ~ 15
Example of Chromatic Effects
Chromatic Effects becomes Stronger:
larger β-function
larger energy spread
stronger QM strength k
longer QM length LQM
PSI 250 MeV Injector Version-1
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
107
Interaction of charged beams & discontinuous surroundings
moving charged beam fields from the beam response from surroundings
fields from surroundings response of the beam changes in energy or emittance of
following bunches (or at the next turn of the same bunch) (long-range wakefields) /
changes in energy and angle (hence emittance) of following electrons in the same bunch
(short-range wakefields).
Wakefields - Definition
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
108
Short-Range Wakefields in Linac Accelerators
If an electron bunch moves in a periodic linac structure, there are interactions
between the electrons in a bunch and the linac structure, which induce
changes in beam energies and beam divergences (x' and y') of electrons in the
same bunch. We call these interactions between electrons in the same bunch
and the linac structure as the short-range wakefields, which change beam
energy spread and emittance of the bunch.
blue: an interaction between an electron at the head
region and a linac structure.
pink: short-range wakefield from the linac structure
to a following electron at the tail region.
2a
A. Chao's Handbook of Accelerator Physics & Engineering, p. 252
SLAC-AP-103 (LIAR manual)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Energy loss Ei of a test electron (or slice) i in a bunch due to the short-range
longitudinal wake function WL(s), which is induced by all other preceding
electrons j located at s = |i - j| distance from the test electron i is given by
Here qi and qj are charge of electron (or slice) i and j, and L is the length of the
linac structure. i or j = 1 means the head electron in the bunch, and the sum
term is only evaluated for i > 1.
The transverse trajectory deflection angle change xi' of a test electron i due to
the short-range transverse wake function WT(s), which is excited by all
preceding electrons j is given by
Here the sum term is only evaluated for i > 1.109
Short-Range Wakefields in Linac Accelerators
electron j moving with v ~ c
a test electron i with a distance s away
from preceding electron j and moving with v ~ c
.)(2
)0( 1
1
LqjiWqW
Ei
j
jLiL
i
.)(1
1
'
i
j
Tjji jiLWxqx
SLAC-AP-103 (LIAR manual)
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
L
Longitudinal wake function WL (s) of the test particle in a bunch is the voltage
loss experienced by the test charged particle. The unit of WL (s) is [V/C] for a
single structure or [V/C/m] for a periodic unit length. The longitudinal wake
is zero if test particle is in front of the unit particle (s < 0). For a bunch of
longitudinal charge distribution z, the bunch wake (= voltage gain for
the test particle at position s) is given by
And the minus value of its average gives the loss factor and its rms
value gives energy spread increase:
where L is the length of one period cell, N is the number of electrons in the
bunch.
110
Longitudinal Short-Range Wakefields
a unit charged particle moving with v ~ c
a test charged particle with a distance s away
from the unit charged particle and moving with v ~ c
SLAC-AP-103 (LIAR manual)
SLAC-PUB-11829
SLAC-PUB-9798
TESLA Report 2004-01
TESLA Report 2003-19
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
111
Longitudinal Short-Range Wakefields
a unit charged particle moving with v ~ c
a test charged particle with a distance s away
from the unit charged particle and moving with v ~ c
SLAC-AP-103 (LIAR manual)
SLAC-PUB-11829
SLAC-PUB-9798
TESLA Report 2004-01
TESLA Report 2003-19
red: without short-range wakefield
green: with short-range wakefield
increased nonlinearity in longitudinal
phase space
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
112
Longitudinal Short-Range Wakefields
Longitudinal impedance is the Fourier transformation of the longitudinal
wake function:
Yokoya's wakefield model for periodic linac structure:
L
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
MHI 2π/3 Mode C-band Structure average inner radius a = 6.9535 mm
average outer radius b = 20.10075 mm
period p = 16.6667 mm
iris thickness t = 2.5 mm
cell number for 2 m structure = 119
attenuation constant τ = 0.452
average shunt impedance = 69.5 MΩ/m
filling time = 222 ns
RF pulse length = 0.5 µs
required RF power for 28 MV/m = 38 MW
one 50 MW klystron can drive 3 structures
This structure is used for linac Optimization-XIV
and Optimization-XV with RF Option-IV.
PSI 3π/4 Mode C-band Structure average inner radius a = 6.9545 mm
average outer radius b = 20.7555 mm
period p = 18.7501 mm
iris thickness t = 4.0 mm
cell number for 2 m structure = 106
attenuation constant τ = 0.630
average shunt impedance = 66.1 MΩ/m
filling time = 333 ns
RF pulse length = 0.5 µs
required RF power for 26 MV/m = 28.5 MW
required RF power for 28 MV/m = 33 MW
one 50 MW klystron can drive 4 structures
This structure is used for linac Optimization-XVII,
and Optimization-XVIII with RF Option-VII, VIII. 113113113
Wakefield of Two C-band Linac Structures
disk loaded type linac structure
2a
p
2b
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
MHI 2π/3 Mode C-band Structure (red lines in plots below)
This structure is used for SwissFEL linac Optimization-XIV and Optimization-XV with RF Option-IV.
PSI 3π/4 Mode C-band Structure (black lines in plots below)
This structure is used for SwissFEL linac Optimization-XVII, and Optimization-XVIII with RF Option-VII or
RF Option-VIII.
114114114
both structures have almost same short-range wakefields !
Short-Range Wakefields of Two C-band Structures
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
115
Why we need Bunch Compressor?
Optimization-XVIII with PSI C-band RF Structures for 2.7 kA
To reduce saturation length of XFEL, we need a high peak current (~ kA).
But normal guns can not supply such a high current due to the longitudinal
space charge force.
To generate femtosecond photon beams, the bunch length of electron beams
should also be femtosecond range.
To avoid the longitudinal space charge effect, we have to compress bunch
length at high energies. → We need the bunch compressor(s) at one or two
positions in linac.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
116
Working Prinicple of Bunch Compressor (BC)
Bunch Compressor Layout for SCSS Project - Y. Kim et al, NIMA 528 (2004) 421
chicane. bendr rectangulathefor2
3where
))/(()/()/(
56566
32
56656
RT
EdEEdETEdERdzdz iiiif
)3
2(2
2
56 BB LLR
dt
dETail
Head
from precompressor linac from chicane
dz
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
117117117117
BC for PSI 250 MeV Injector Test Facility
ASTRA up to exit of INSB02 & ELEGANT from exit of INSB02 to consider space chare, CSR, ISR, and wakefields !
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
118118118118
BC for PSI 250 MeV Injector Test Facility
10.5 m
0.75 m4.375 m
~ 0.5 m
0.25 m
QMs
length ~ 0.1 m
max gradient ~ 1.5 T/m
BPM
Collimator
OTR Screen
BM
BM BM
BC1 Dipoles
length = 0.25 m
max bending angle ~ 5 deg @ 250 MeV
0.4
04
m Radiation Port
E = 255.9 MeV
~ 1.673%
R56 = 46.8 mm
= 4.1 deg
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
119119119119
Knobs to minimize Emittance Growth @ BC
From field tolerance studies, we assumed following dipole field errors in BC
dipoles: Δb/b0 = 3.32×10-3 b1/b0 = 8.70×10-5
b2/b0 = 1.83×10-5 b3/b0 = 6.65×10-5 b4/b0 = 4.80×10-5
In this case, the minimum projected emittance can be obtained by
compensating residual dispersion with two small QMs in BC.
After BC1, zero residual dispersion
can be obtained when k1 of QMs
is -0.04 (1/m2). In this case,
horizontal projected
emittance after BC1 also
becomes its minimum.
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Ipeak ~ 352 A
Ipeak ~ 22 A
E ~ 256 MeV, Q = 0.2 nC
~ 1.67%, z = 840 m
nx~ 0.35 m, ny~ 0.35 m
E ~ 256 MeV, Q = 0.2 nC
~ 1.67%, z = 58 m
nx~ 0.38 m, ny~ 0.35 m
120120120120
BC for PSI 250 MeV Injector Test Facility
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
121
FODO Lattice
The most simplest lattice block of modern accelerators = a pair of QF and QD.
FODO cell = a simplest periodic lattice block with QF, O, QD, and O in a cell.
Here O can be a drift space, a bending magnet, undulator, or linac structure.
If the horizontal plane has a FODO cell, the vertical has a DOFO cell.
Betatron phase advances for x and y planes can be different (See SwissFEL DBC1).
Length of One FODO Cell with Two Linac Structures = l = 10.3 m
QF S-band Tube QD S-band Tube
4.3 m long S-band tube 4.3 m long S-band tube
0.7 m long diagnostic section
0.15 m long QM 0.15 m long QM
22 MV/m 22 MV/m
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
122
FODO Lattice
Length of a FODO Cell = l
QF Drift QD Drift
l/2 l/2
To find the maximum and minimum β-function and phase advance of a FODO cell,
let's consider QMs as thin lens ( )
Then,
Total transfer matrix for one FODO cell:
From generalized strong-focusing transfer matrix, we can put;
By taking trace of the matrix → →
1QMlk
1
101
1
01M DF,
fklQM
βmax
βmin
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
123
FODO Lattice
By comparing M12 component:
By using following relationships:
2sin1
sin41
sin
4sin
2
l
f
ll
f
ll
2sin1
sin
2sin1
sin
min
max
l
l
2sin1
2sin1
2sin2
2sin1
2sin2
2cos
2sin2sin
2
2sin4
fl
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
From , we can find a phase advance per FODO cell ().
→ total phase advance for Nc cell FODO lattice = Nc
→ betatron tunes for the Nc cell FODO lattice is given by
If we know length of one FODO cell l, phase advance per cell , then f and βmax and βmin
are automatically determined.
124
FODO Lattice
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
125
Example of FODO Lattice in a Linac
QF TUBE QD TUBE QF TUBE QD TUBE QF
length of one FODO cell in a LINAC
= two 4.3 m long PSI standard S-band tubes
+ two 0.7 m long PSI standard diagnostic sections
+ two 0.15 m long PSI standard QMs = 10.3 m
phase advance per FODO cell = 60 deg
max and min β-function ~ 17.3 m and 5.8 m
normalized QM strength K1 ~ 1.3 m-2
QM gradient @ 1.5 GeV ~ 6.5 T/m
FODO
ignorable chromatic effects at the LINAC
SwissFEL Linac1 Optimization-1
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
126126
5QMs TDS2 5QMs 3FODO 1QM DIPOLE
3FODO Cells
OTRs : 1 2 3 4 5 6 7 8
horizontal phase advance @ 3FODO = 55 deg per cell
vertical phase advance @ 3FODO = 25 deg per cell
cell length = 3.0 m
rms beam size @ 7FODO screens ~ 60 m for 200 pC
max dispersion by a dipole in DIAG1 ~ 0.2 m
With this special DIAG1,
we can measure followings
without change any optics:
- slice emittance
- slice energy spread
- longitudinal phase space
- bunch length
- arrival timing jitter
- projected emittance
- Twiss parameters
- optics matching\
only by turning on and/or
off TDS2 and a dipole in
DIAG1.
FODO Lattice - DIAG1 @ SwissFEL Injector
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
127127
SwissFEL OPT-III & VII
Optimization-III with a longer S-band RF Linacs for Chirp Compensation
Optimization-VII with Shortest C-band RF Linacs for Chirp Compensation
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
128128
SwissFEL - S-band based LINAC2 after BC2
length of one FODO cell in LINAC2
= two 4.3 m long PSI standard S-band tubes
+ two 0.7 m long PSI standard diagnostic sections
+ two 0.2 m long QMs = 10.4 m
pure active length per tube = 4.073032 m
number of cell per tube = 122 including two coupler cells
central cell length = 33.333 mm
iris diameter = 25.4 mm
total cells in LINAC2 = 34 FODO cells
No. of S-band tubes = SB23-SB90 for 34 FODO cells
total needed S-band tubes in LINAC2 = 68
total needed RF stations = 34 with two tubes per station
total needed QMs in LINAC2 = 2x34 = 68
total length of LINAC2 = 353.6 m
One FODO Cell for LINAC2 = 10.4 m
QF 4.3 m long S-band Tube QD 4.3 m long S-band Tube
2998 MHz S-band Tube 2998 MHz S-band Tube
0.7 m long diagnostic section
0.2 m long QM 0.2 m long QM
22 MV/m 22 MV/m
LINAC2 for Optimization-III
Optics for S-band Based LINAC2
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
129
Example of FODO Lattice in a Linac
QF TUBE QD TUBE QF TUBE QD TUBE QF
length of one FODO cell in a LINAC
= two 4.3 m long PSI standard S-band tubes
+ two 0.7 m long PSI standard diagnostic sections
+ two 0.15 m long PSI standard QMs = 10.3 m
phase advance per FODO cell = 60 deg
max and min β-function ~ 17.3 m and 5.8 m
normalized QM strength K1 ~ 1.3 m-2
QM gradient @ 1.5 GeV ~ 6.5 T/m from page 53
FODO
ignorable chromatic effects at the LINAC
SwissFEL Linac1 Version-1
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
130130
Cathode @ 0.0 m
Distance : 0.21 m 0.78 m 1.27 m 2.47 m 19.0 m 25.31 m 28.90 m 30.80 m 32.70 m 34.60 m 38.40 m
ACC2
~ 13 MV/m for first four cavities
~ 17 MV/m for last four cavities
0.0 degree for no compression
E = 127 MeV
R56= 181.3 mm
= 18.0355 deg
~ 0.2% (on crest)
~ 40 MV/m
38 degree
from zero
crossing
e- beams
Q ~ 1.0 nC
z ~ 4.4 ps rms
ACC1GUN BC2
3 FODO cells for emittance measurement
QMs QMs
for matching
Screen : 2GUN 3GUN 4DBC2 6DBC2 8DBC2 10DBC2
Solenoid Laser IDUMP dipole
TTF2 Injector = GUN + Booster (ACC1) + BC2 + 3 FODO Cells
1.3 GHz TESLA Module 1.3 GHz TESLA Module
By optimizing TTF2/FLASH injector properly, we could get an excellent emittance at
the injector. Without any bunch length compression, projected normalized emittance
is about 1.1 µm for 90% beam intensity in 1.0 nC and 4.4 ps (rms) long bunch.
FODO @ DESY FLASH Facility - Diagnostics
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
131
DESY FLASH Facility - 3 FODOs @ DBC2
Q4DBC2H Q5DBC2 Q6DBC2 Q7DBC2 Q8DBC2 Q9DBC2 Q10DBC2H
OTR4DBC2 OTR6DBC2 OTR8DBC2 OTR10DBC2
1.9 m
4 OTR in 3 FODO cells
-OTR4DBC2
-OTR6DBC2
-OTR8DBC2
-OTR10DBC2
Cell phase advance = 45 deg
One cell length = 1.9 m
7 QMs in three FODO Cells
- Q4DBC2 - defocusing
- Q5DBC2 - focusing
- Q6DBC2 - defocusing
- Q7DBC2 - focusing
- Q8DBC2 - defocusing
- Q9DBC2 - focusing
- Q10DBC2 - defocusing
0.46 m 1.9 m 1.9 m
DBC2 section
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
To minimize emittance, light sources use various type dispersion
suppressed achromat lattice.
Lattice for Synchrotron Light Source
132Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
]cm[]T[934.0,2
12
uo
2
2
u
BK
K
Various Lattice for Synchrotron Light Sources
133Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Double-Bend Achromat (DBA)
Triple-Bend Achromat (TBA),
Quadrupole-Bend Achromat (QBA)
Multi-Bend Achromat (MBA) Courtesy of Zhentang Zhao
DBA Vs. MBA based Storage Ring
ringcisomagnetifor108319.3
BMs
213
s
J
MBAfor
81
2
3
BM
2
n .xCN
Somewhat lower energy, longer bending radius, and many dipoles with focusing
QMs are better for us to get a small energy spread and a small emittance.
→ MBA based Diffraction Limited Storage Ring (DLSR),
(electron emittance photon emittance)
4xx,y
134Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
ESRF Hybrid MBA for Large Dynamic Aperture
135
The MBA lattice has both
strong focusing and small
dispersion small
dynamic apertures and
large chromaticities.
To solve this problem, the
Hybrid MBA (HMBA) has
two separate β-functions
and dispersion bumps
located between the ending
dipoles and the first and
last inner dipoles of the
MBA cell, where sextupoles
are placed to ease the
chromaticity correction.
Sextupoles for chromaticity correction
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Courtesy of Zhentang Zhao
Average Spectral Brightness and Coherent Fraction
136
DBA Vs. MBA based Storage Ring
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
Emittance Vs. Synchrotron Generation
MBA based
4th Genertion
SynchrotronDBA/TBA based
3rd Generation
Synchrotron
137
Great Emittance with MBA!
~ 10 pm n ns ~ 0.12 m @ 6 GeV Storage Ring
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
MBA based 4th Generation Synchrotron
4GSLS Group with the Highest Perfomance
New Korean Ring will be close to PEP-X
PLS-II
138Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
3GSLS Vs. MBA based 4GSLS
3GSLS, ex) PLS-II
Structural Images of a Biological Sample with Different Wavelengths(0.5 - 5 Å )
139
4GSLS Group with the Highest Perfomance
New Korean Ring will be close to PEP-X
Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST
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