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Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings
J. GaoLaboratoire de L’Accélérateur Linéaire
CNRS-IN2P3, FRANCE
Journées Accélérateurs
Porquerolles, Oct. 5-7, 2003.
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Contents
Dynamic Apertures of Multipoles in a Storage Ring
Dynamic Apertures limited by WigglersLimitations on Luminosities in Lepton
Circular Colliders from Beam-Beam Effects
Nonlinear Space Charge Effect Nonlinear electron cloud effect
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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
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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 Mapping
Near the nonlinear resonance, simplify the difference equations to the form of STANDARD MAPPING
sin0KII
I
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Stochastic motions
When 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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General Formulae for the Dynamic
Apertures of Multipoles 2
1
1
)2(21
2,||))2((
1)(2
m
m
m
mx
xmdynaLbmsm
sA
kkdecadyna
i jjoctdynaisextdyna
totaldyna
AAA
A...111
1
2,,
2,,
2,,
,
2,,
2
1
1,,
)()( xA
ssA
xsextdynay
xysextdyna
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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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Wiggler
Ideal wiggler magnetic fields
)cos()sinh()sinh(0 ksykxkBkkB yx
y
xx
)cos()cosh()cosh(0 ksykxkBB yxy
)sin()sinh()cosh(0 ksykxkBkk
B yxy
z
2222 2w
yx kkk
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One cell wiggler
One cell wiggler Hamiltonian
One cell wiggler limited dynamic aperture
iw
yw iLsykyHH )(
1241 4
2
22
201,
2/1
2
2
,13
)()(
)(
wy
w
wy
yy ks
ssA
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Full wiggler and multi-wigglersDynamic aperture for a full wiggler
or approximately
where is the beta function in the middle of the wiggler
w
wwiy
wN
i yw
ywN
i yiywN
NLs
sk
AsA)(
)(31
)(1
,2
1 2
2
1 2,
2
,
wy
w
my
y
ywN LkssA
2,,
)(3)(
my,
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Full wiggler and 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 ny034.0)(, mA ay17584.0)( mlw
067.0)(, mA ny067.0)(, 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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Maximum Beam-Beam Tune Shift in Circular Colliders
Luminosity of a circular collider
ee
IPye
yce
yx
ce
rfNfNL
242
)(2,
yxy
IPyeey
rN
where
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Beam-beam interactionsKicks from beam-beam interaction at IP
),,( ,'' yxee yxfrNxiy
22
2),,,(yx
yxyxf
)(222exp
)(2 222
2
2
2
22 yx
y
x
y
y
yxyx
yixw
yxiyxw
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Beam-beam effects on a beam
We study three cases
2
2
'4
exp12 r
rrNr ee
xx
xx
ee duuxrNx
2
0
22
2
' exp4
exp2
yx
xx
ee yerfxrNy
22exp2
2
2
'
(RB)
(FB)
(FB)
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Round colliding beam
Hamiltonian
22)(
22
2eeyy rNyskpH
kkLsyyy )(......
11521
641
41 6
64
42
2
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Flat colliding beams
Hamiltonians
kxxxkLsxxx )(......
1801
1211 6
64
42
2
kyxyxyxkLsyyy )(......
1201
1211 6
54
32
22)(
22
2eexx
xrNxskpH
22
)(2
22
eeyyy
rNyskpH
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Dynamic apertures limited by beam-beam interactions
Three cases
Beam-beam effect limited lifetime
)(16
)()( 2
2
2,8,
IPyee
ydyna
srNssA
)(6
)()( 2
2
2,8,
IPxee
x
x
xdyna
srNssA
)(23
)()(
2
2,8,
IPyee
yx
y
ydyna
srNssA
(RB)
(FB)
(FB)
)()(exp
)()(
2 2
2,,
1
2
2,,
,s
sAs
sAy
ybbdyna
y
ybbdynayybb
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Recall of Beam-beam tune shift definitions
)(2,
yxx
IPxeex
rN
)(2,
yxy
IPyeey
rN
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Beam-beam effects limited beam lifetimes
Round beam
Flat beam H plane
Flat beam V plane
xx
xxbb
3exp32
1
,
yy
yybb
4exp42
1
,
yy
yybb
23exp
23
2
1
,
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Important finding
Defining normalized beam-beam effect limited beam lifetime as
An important fact has been discovered that the beam-beam effect limited normalized beam lifetime depends on only one parameter: linear beam-beam tune shift.
y
bbbbn
,
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Theoretical predictions for beam-beam tune shifts
msy 30
FByFByRBy ,max,max,max 89.1324
0843.0)1(,max hourbbRBy
0447.0)1(,max hourbbFBy
For example
Relation between round and flat colliding beams
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The Limitation from Space Charge Forces to TESLA Dog-Borne Damping Ring
Total space charge tune shift
Differential space charge tune shift
Beam-beam tune shift
zyxy
yaveesc
LNr
22
,
2)(2
zyxy
yeesc
ssssNrs
22000
00
21
))()()((2)()('
))()()((2)()(
IPyIPxIPy
IPyeeIPbb
ssssNrs
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Space charge effect
Relation between differential space charge and beam-beam forces
Gsfds
sdfIPbb
sc )()(
zG
2221
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Space charge effect limited dynamic apertures
GrNssssA
ee
yx
y
yysc
s
)()(23)()())'(( 0
30
20
2 ,
L
s ysc
ysctotal
sA
sA
002,
,,
)')((1
1)(
scy
ysctotaly
sAR 23)(
2
,,2
Dynamic aperture limited by differential space charge effect
Dynamic aperture limited by the total space charge effect
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Space charge limited lifetime
Space charge effect limited lifetime expressions
Particle survival ratio
2122 , exp2
yyy
ysc RR
yscysc
yysc
,
1
,,
23exp
23
2
)(exp1)(
, scysc
stscR
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TESLA Dog-Borne damping ring as an example
Particle survival ratio vs linear space charge tune shift when the particles are ejected from the damping ring.
TESLA parameters
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Nonlinear electron cloud effect
Relation between differential electron cloud and beam-beam forces
)21(
))()()((2)(
0,0,0,
0,'
LSSS
SNr
yxy
yeeec
)(21)(
IPbbec sF
Ldssdf
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Nonlinear electron cloud effect
Normalized dynamic aperture due to electron cloud
Lyavxav
eec
N
,,,,2
Lecyavey
yecyec
r
AR
,,
,2
2,
23)(
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Combined nonlinear beam-beam and electron cloud effectNormalized dynamic aperture due to
combined beam-beam and electron cloud effects
ybbIPy
IPybbybb
AR
,,,,
,,2
2,,
23)(
RR
R
yecybb
ytotal
2,,
2,,
2,, 11
1
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Combined nonlinear beam-beam and electron cloud effectBeam lifetime due to the combined
effect
where is the damping time of positron in the vertical plane
)exp(2
2,,
2,,
1,,, RR ytotalytotal
yytotal
y,
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PEP-II positron ring as an example
Machine parameter
msy 30, 6067
kmL 2.2
myav 18,,
msy 30,
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PEP-II positron ring as an example
Machine parameter
6067
kmL 2.2myav 18,,
msy 30,
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PEP-II positron ring as an example
If the beam-beam alone limited maximum beam-beam tune shift is
with
the maximum beam-beam tune shift will be reduced to
045.0,max,, ibby
015.0,max,, ibby
101225.3 ec
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Conclusion
Various nonlinear effects are the main limiting factors to the performance of storage rings.
In addition to numerical simulations, analytical treatments are very helpful in understanding the physics behind the phenomena, are very economic.