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Analytical Estimation of Dynamic Aperture Limited by Wigglers
in a Storage Ring
J. GAO 高杰
Institute of High Energy Physics
Chinese Academy of Sciences
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ContentsDynamic Apertures of Limited by
Multipoles in a Storage RingDynamic Apertures Limited by Wigglers
in a Storage RingApplication to TESLA damping ringConclusionsReferences
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Dynamic Aperturs of Multipoles
Hamiltonian of a single multipole
Where L is the circumference of the storage ring, and s* is the place where the multipole locates (m=3 corresponds to a sextupole, for example).
k
mm
zm
kLsLxx
BBmx
sKpH )*(!1
2)(
2 1
12
2
Eq. 1
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Important Steps to Treat the Perturbed Hamiltonian
Using action-angle variablesHamiltonian differential equations should be
replaced by difference equations
Since under some conditions the Hamiltonian don’t have even numerical solutions
pH
dtdq
qH
dtdp
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Standard MappingNear the nonlinear resonance, simplify
the difference equations to the form of STANDARD MAPPING
sin0KII
I
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Some explanationsDefinition of TWIST MAP
)1)(mod(xg
)(Kfxx
)()1( ff
xdx
xdg ,0
)(
where
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Some explanationsClassification of various orbits in a Twist Map, Standard Map is a special case of a Twist Map.
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Stochastic motions
For Standard Mapping, when global stochastic motion starts. Statistical descriptions of the nonlinear chaotic motions of particles are subjects of research nowadays. As a preliminary method, one can resort to Fokker-Planck equation .
97164.00K
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m=4 Octupole as an example
Step 1) Let m=4 in , and use canonical variables obtained from the unperturbed problem.
Step 2) Integrate the Hamiltonian differential equation over a natural periodicity of L, the circumference of the ring
Eq. 1
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m=4 Octupole as an example
Step 3) 111 4sin AJJ
111 JB
ABK 40
LbsJ
Amx 34
21
2
)(
LbsB mx
34
2 )(2
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m=4 Octupole as an example
Step 4) )97164.0(140 ABK
LbsJ
mx ||)(2
1
3421
2/1
34
2
4
2/1
2/11,, ||
)(2)(
)())(2(
Lbs
s
ssJA mx
mx
x
xxoctdyna
One gets finally
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General Formulae for the Dynamic
Apertures of Multipoles
2
1
1
)2(2
1
2, ||))2((
1)(2
m
m
m
mx
xmdyna LbmsmsA
i j kkdecadynajoctdynaisextdyna
totaldyna
AAA
A
...111
1
2,,
2,,
2,,
,
2,,
2
1
1,,
)()( xA
ssA
xsextdynay
xysextdyna
Eq. 2
Eq. 3
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Super-ACOLattice Working point
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Single octupole limited dynamic aperture simulated by using BETA
x-y plane x-xp phase plane
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Comparisions between analytical and
numerical results
Sextupole Octupole
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2D dynamic apertures of a sextupole
Simulation result Analytical result
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Wiggler
Ideal wiggler magnetic fields
)cos()sinh()sinh(0 ksykxkBk
kB yx
y
xx
)cos()cosh()cosh(0 ksykxkBB yxy
)sin()sinh()cosh(0 ksykxkBkk
B yxy
z
2
222 2
wyx kkk
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Hamiltonian describing particle’s motion
)))sin(())sin(((2
1 222 ksApksAppH yyxxzw
))cosh())cosh(1
ykxkk
A yxw
x
where
y
xyx
wy k
kykxk
kA ))sinh())sinh(
1
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Particle’s transverse motion after averaging over one wiggler period
)3
2(
22232
22
2
2
2
xykxkxk
k
ds
xdx
w
x
)3
2(
2 2
22232
22
2
2
2
y
xy
w
y
k
kkyxyky
k
k
ds
yd
In the following we consider plane wiggler with Kx=0
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One cell wigglerOne cell wiggler Hamiltonian
After comparing with one gets
one cell wiggler limited
dynamic aperture
i
wy
w iLsyk
yHH )(124
1 42
22
201,
2/1
2
2
,13
)()(
)(
wy
w
wy
yy ks
ssA
Eq. 4
Eq. 4 Eq. 1
2
2
3
3 w
wykL
b
Using one getsEq. 2
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A full wiggler Using one finds dynamic aperture for a
full wiggler
or approximately
where is the beta function in the middle of the wiggler
,
22
,2 2 21 1,
1 1( )
( ) 3 ( )
w w
w y
N Ny w
y i wi ii y w y wN
ks
A s A s N
wy
w
my
y
ywN LkssA
2,,
)(3)(
my,
Eq. 3
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Multi-wigglers
Many wigglers (M)
Dynamic aperture in horizontal plane
M
j ywjy
ytotal
sAsA
sA
1 2,,
2
,
)(1
)(1
1)(
2,,
2
,
,,, yAA
ywigldynamx
myxwigldyna
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Numerical example: Super-ACO
Super-ACO lattice with wiggler switched off
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Super-ACO (one wiggler)7.2)( mw 017.0)(, mA ny 019.0)(, mA ay
13)(, mmy 17584.0)( mlw 5168.3)( mLw
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Super-ACO (one wiggler)3)( mw 023.0)(, mA ny 024.0)(, mA ay
7.10)(, mmy 17584.0)( mlw 5168.3)( mLw
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Super-ACO (one wiggler)4)( mw 033.0)(, mA ny 034.0)(, mA ay
5.9)(, mmy 17584.0)( mlw 5168.3)( mLw
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Super-ACO (one wiggler)
4)( mw
016.0)(, mA ny 017.0)(, mA ay
5.9)(, mmy
08792.0)( mlw
5168.3)( mLw
033.0)(, mA ny 034.0)(, mA ay17584.0)( mlw
067.0)(, mA ny 067.0)(, mA ay35168.0)( mlw
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Super-ACO (two wigglers)6)( mw 032.0)(, mA ny 03.0)(, mA ay
75.13)(, mmy 17584.0)( mlw 5168.3)( mLw
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Application to TESLA Damping Ring
E=5GeV Bo=1.68T 0.4w m
12wN ,1 9y m
,2 15y m (at the entrance of the wiggler)
(at the exit of the wiggler)
The total number of wigglers in the damping ringis 45.
, 2.1total yA cmThe vertical dynamic aperturedue to 45 wiggler is
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Conclusions1) Analytical formulae for the dynamic apertures limited by multipoles in general in a storage ring are derived.2) Analytical formulae for the dynamic apertures
limited by wigglers in a storage ring are derived.
3) Both sets of formulae are checked with numerical simulation results.4) These analytical formulae are useful both for
experimentalists and theorists in any sense.
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References1) R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky,
“Nonlinear Physics, from the pendulum to turbulence and chaos”, Harwood Academic Publishers, 1988.
2) R. Balescu, “Statistical dynamics, matter our of equilibrium”, Imperial College Press, 1997.
3) J. Gao, “Analytical estimation on the dynamic apertures of circular accelerators”, NIM-A451 (2000), p. 545.
4) J. Gao, “Analytical estimation of dynamic apertures limited by the wigglers in storage rings, NIM-A516 (2004), p. 243.