electron dynamics with synchrotron radiationdlr/center/... · synchrotron radiation qphoton...
TRANSCRIPT
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 1
qIntroduction to electron dynamicsCoordinates and Hamiltonian for a storage ring.
Stochastic differential equations including radiation.
Robinson’s Theorem
qModes of oscillationBetatron and synchrotron motion
Radiation damping and quantum fluctuations
Fokker Planck equation and distribution functions
qAdvanced topics by illustrationDynamical effects from radiation
Non-linear resonances
Limit cycles
Not too much detail or rigour in these talks!
See references for further details and justifications,formalism for more general cases, etc.
Electron Dynamics withSynchrotron Radiation
John M. JowettAccelerator Physics Group, SL Division, CERN, CH-1211 Geneva 23
[email protected]://wwwslap.cern.ch/jowett/JMJ.html
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 2
Suggested Reading
&M. Sands, The Physics of Electron Storage Rings, SLAC-121(1970).
Very clear for basic single particle dynamics and effects of synchrotronradiation.
&J.D. Jackson, Classical Electrodynamics, Wiley, New York1975.
Classic text on electrodynamics, includes relativistic dynamics and treatments ofradiation from accelerated charges.
&A.W. Chao, J. Appl. Phys., 50 (2), p. 595 (1979).Simple but general approach to linear theory.
&A.W. Chao, “Equations for Multiparticle Dynamics”, JointUS/CERN School, South Padre Island 1986, Springer LectureNotes in Physics No. 296 (1986).
Treatment of coupled motion, including quantu-lifetime calculations.
&D.P. Barber, K. Heinemann, H. Mais, G. Ripken, “Fokker-Planck treatment of stochastic Particle motion ….”, DESY 91-146 (1991)
General linear theory including spin. See also references therein.
&J.M. Jowett, “Introductory Statistical Mechanics for ElectronStorage Rings”, US Particle Accelerator School, AIPConference Proceedings, No. 153 (1985).
Base for present talk. Contains further details and many references.
&J.M. Jowett, “Electron Dynamics with Radiation and Non-linearWigglers”, CERN Accelerator School, Oxford 1985, CERNReport 87-3 (1987).
Similar treatment, contains some other material.
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 3
Motivation
qProton (hadron) synchrotrons & storage ringsFor single-particle dynamics, an adequate mathematicalmodel is that of a classical particle moving in an appliedexternal electromagnetic field.
State of system is , canonical variables.
Rich field for application of Hamiltonian dynamics.
N.B. Hamiltonian dynamics is a very special case.
qElectron synchrotrons and storage rings
Hamiltonian model, and concepts and techniques derivedfrom it remain useful but do not strictly apply becauseelectrons generate significant electromagnetic fields oftheir own: radiation reaction.
Electron continually loses energy as synchrotron radiation.
Global system of particle + EM field remains Hamiltonianbut, for purposes of describing particle dynamics in thering, we are not interested in the dynamical variables of theEM field.
Employ reduced description in terms of canonicalvariables of classical particle with addditional dissipativeterms in the equations of motion. These terms arestochastic and are second order in the EM coupling
( )q p, ∈R6
“electron” ¢“positron or electron” in this talk.“electron” ¢“positron or electron” in this talk.
α = e c2 / h
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 4
Synchrotron Radiation
qPhoton emission (incoherent only)Intrinsically quantum-mechanical phenomenonÆ emission times and energies of quanta of energy arerandom quantities.
Provided the energies and magnetic fields are not too high,certain average quantities, such as the mean emission rateand the mean radiated power, may be calculated to goodaccuracy within classical electrodynamics.
Only when the magneticfields and energies become so highthat the mean of the classical synchrotron radiationspectrum becomes comparable to the electron energy doesit become necessary to include quantum corrections to thetotal radiated power and the frequency spectrum.
q Semi-classical pictureOrbital quantum numbers of electron very large.
Change during photon emission also large.
Approximate as instantaneous jump in energy since
characteristic frequencies of motion
(critical frequency)
means that freqeuncy spectrum is locally well - defined.
τ ργ
τω γ
ρ
γ
γ
≈ <<
<< ≡
c
cc
1
1 32
3
( )
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 5
[ ]
( )
[ ]( )
Power radiated by accelerated particle
in any frame
4 - vector form of the Lorentz force in purely
transverse magnetic field
where is a reference value of the particle momentum and
is a normalised magnetic field (units of m
rad
rad
-
PdE
dt
e
m c
dp
d
dp
d
dp
d
e
mF p
e
mc
Pe e
m c
e r p
m cp b
p
b
e
= −
=
= ⋅ =
= = ×
= × =
23
4
0
0
23
4 23
20
2 3
2 20
4 3
22 2
32 20
0
/
,
,
/( )
( )
πετ τ
τπε
µµ
µµν
µ
E p B 0
p B
p B x
x 1).
[ ]( )
( )
( )
PdE
dt
e
mcc
E c e
Uc
Pr
mc
EC
E
Cr
mc
e
e
rad
rad
-3
Energy lost per turn in a ring of constant bend radius
where
where = m.GeV (electrons)
= − =
≈ =
= = =
= × −
23
4
2 43
43
8 85 10
20
3
2 2
0 2 3
4 4
2 35
/
,
/
.
πε
ρρ
πρ πρ ρ
π
γ
γ
p B
p B
qResults from electrodynamicsRadiation from accelerated charge, quoted withoutderivation (see, e.g., Jackson)
be
p cx y s( ) ( , , )x B≡
0
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 6
Photon energy spectrum
qClassical frequency spectrumre-interpret as energy spectrum of photons
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2 2.5 3 3.5 4
x=u/u c
u uc/
( )S ξ
( )F ξ
u uc2 2/
( ) ( ) ( )
( ) ( ) ( ) ( )
P dP
S
S K z dz F S
ccrad
rad= =
≡ ≡
∞
∞
∫
∫
3 3ω ω ωω
ω ω
ξπ
ξ ξ ξ ξξ
0
5 3
9 38
, /
, //
( )P n u u du n u
u
uX Xrad = =∞
∫ ( ) , ( )/
0
3 h
h
( )
N s
n u s ducr p
b s
p
X
Xe
X
( )
, ( )
.
=∞
∫
photon emission rate
= =5 3
6
independent of 0
0
h
( )
u
u
cp
mcp b s
X
c
X
=
=
mean photon energy
=8
15 3
34
503
2h( )( )
( )( )
u
cp
mcp b s
X
X
2
0
2
64 211
12
=
=
mean - square photon energy
h
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 7
Stochastic Radiation Power
qRandom emission time and photon energy
qInstantaneous radiation power
Density of a given realisation (fixed :
energy spectrum of photons
X s
u s s s u u u s N s f u s c
f s
X j jj
X X X
X
, )
( , ) ( ) ( ), ( , ) ( ) ( ; ) /
( )
Ω Ω= − − =
=
∑δ δ
( )
Stochastic average:
P s N s u
c c p b s
cr p
mcb s
e
p cx y s
X X X
X
eX
( ) ( )
( )
, ( ) ( , , )
==
= =
12 2 2
102
30
2
3B
P s c u s sX j jj
( ) ( ),= −∑ δ
( )Stochastic representation of fluctuating power:
P s c p c b s c b s sX X X( ) ( ) ( ) ( )= +2 21
22
3ξ
See SLAC Summer School for further details.Representation of power can be approximated or simplifiedin various ways of which this is one.
ξ ξ δ( ) ( ) ( )s s s s j′ = −
Simulated photon emissionin a fixed magnetic field
( )c
r p
mce
203
6
5524 3
= h
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 8
Correlation function
qApplication of “Campbell’s Theorem”Classical deterministic radiation power has beensupplemented with a term of order .
Average radiation power and its quantum fluctuationsdepend nonlinearly on the particle’s coordinates throughthe instantaneous momentum and spatial dependences ofthe magnetic field.
qSimulationSimulation of photon emission involves emitting photonsat random times along the particle’s path with probabilitydepending on the local magnetic field and p.
Each photon’s energy must be generated randomly inaccordance with a distribution depending on the samephysical quantities.
( )
( ) ( )
Stochastic representation of fluctuating power:
mean fluctuating part
Two - point correlation function of fluctuations:
P s c p c b s c b s s
P s P s P s
P s P s cN s u s s
X X X
X X X
X X X Xs
( ) ( ) ( ) ( )
( ) ( ) $ ( )
$ ( ) $ ( )
= +
= +
⇒ ′ = − ′
2 21
22
3
2
ξ
δ
h
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 9
Storage Ring Coordinates
q Azimuthal coordinate s (usually) plays role of time(independent variable) in accelerator dynamics.
Time t becomes the coordinate for the third degree of freedom(different particles pass s at different times). Usually use time-delay w.r.t. reference particle.
q Particles move in a neighbourhood of a reference trajectory(ideally a curve passing through centres of all magnets)
q Each of the coordinates (x,y,ct) has a conjugate momentumvariable
Usually measure in units of reference longitudinal momentum sothese momenta are dimensionless variables.
In these units, px, py are equal to the (small) angles of particletrajectory with respect to reference trajectory.
Particle motion
ρ( ) / ( )s G s= 1
x
y
s
Bending radius
Reference trajectory
Coordinate system for single-particle motion in circular accelerator
G s ds( )∫ = 2π
r x= +ρ
θ
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 10
Applied fields
qVector potential in storage ringFor illustration describe only flat ring with dipoles, uprightquadrupoles, sextupoles and RF cavities. Other terms canbe added to describe other elements.
One relevant component of vector potential in Coulombgauge:
qHamiltonian
( ) ( )( )
( ) ( ) ( )( ) ( )( )
( ) ( )
A x y t s G s x
p c
exG s G s
xK s x y K s x xy
eVs s t
s s
kC k k
k
, , , .
$cos
= +
= − +
+ − + − +
+ − +∑
A e 1
12
30 12 1
2 2 16 2
3 2 K
ωδ ω φ
RFRF
( )G s = curvature of reference orbit
( ) ( ) ( )( )
( ) ( )( )
( ) ( )( ) ( )( )
( )( )
H x y t p p E s e G s x
eA x y t s G s xE
cm c p p
t E z ct m c E p
H x y z p p p s eA x y t z p s
G s x p p p
x x s
s x y
t
t x x s t
x y
, , , , , ; .
, , ,
, / ,
, , , , , ; , , , ,
− = + +
= − − + − − −
− = − −
= −
− + − −
p A e 1
1
1
1
2
22 2 2 2
2 4 2
2 2 2
Canonical transformation to simplify momentum - dependence
a
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 11
Equations of motion
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )∑ +−−≈′
++=′+−−+−−−=′
′≈+≈−−
+−=′
+≈−−
+=′
+≈−−
+=′
kktkC
k
y
x
yx
t
y
yx
y
x
yx
x
czssc
Vep
xyKpyKpp
yxKpxKGpppGp
tcGxppp
pGxz
p
pGx
ppp
pGxy
p
pGx
ppp
pGxx
φωδ /cosˆ
11
11
11
RF
2021
10
22202
11
200
222
222
222
K
K
Note dependences on canonical momenta.
Radiation has not yet been included.
How can we add terms describing the effect of radiationreaction?
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 12
Radiation Reaction Forces
qConsider single photon emissionSweeping many electrodynamic subtleties under the carpet(see e.g., Jackson for further discussion).
Photon emission is essentially collinear (opening angle ofradiation from beam is 1/2g) with particle motion and ismodelled as instantaneous momentum change:
Must also include the Hamiltonian part of the equations ofmotion.
( )
( ) ( )( )( ) ( )
p p u p u 0 p ua
a
a
a
− > × =−
− = − ′′
− = − ′′
= ′ ′ →
= − ≈ − ′
= − ′ ′ = − ′
−
+
∫
/ , . ,
/
/ /
/ /
c
p p u c
p pu
c
p
pp
u
c
x
t
p pu
c
p
pp
u
c
y
t
s
u P s ds
dp P s dt c P s z ds c
dp P s x t dt c P s
x xx
x
y yy
y
X
s
s
X X t
x X X
where
Consider short time interval around emission azimuth
as
to construct stochastic differential equations:
0
0
2
σ
σ
σ
( )( ) ( )x ds c
dp P s y t dt c P s y ds cy X X
/
/ / /
2
2= − ′ ′ = − ′
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 13
General equations of motion
qCombine Hamiltonian and radiation reactionSpecial form applies in these coordinates only.
Define radiation coupling functions, note dependences onmomenta (root of Robinson Theorem):
′ = ′ = − −
′ = ′ = − −
′ = ′ = − +
xH
pp
H
x
P s
c
H
p
yH
pp
H
y
P s
c
H
p
zH
pp
H
z
P s
c
H
p
xx
X
x
yy
X
y
tt
X
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
( ),
( ),
( )
2
2
2
( ) ( )
( ) ( )
( ) ( )
′ = − − + +
≡ − +
′ = − − + +
≡ − +
′ = − + + +
≡ − −
pH
xGx pp c b s c b s s
H
x
p
c
pH
yGx pp c b s c b s s
H
y
p
c
pH
zGx p c b s c b s s
H
z
p
c
x x X X
x
y y X X
y
tX X
tt
∂∂
ξ
∂∂∂∂
ξ
∂∂∂∂
ξ
∂∂
1
1
1
12
23
0
12
23
0
21
22
3
0
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
Π
Π
Π Stochastic differentialequations of motion
Stochastic differentialequations of motion
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 14
Robinson’s Theorem
qNon-Liouvillian flowThe equations of motion describe a flow which does notconserve phase space measure.
The divergence of this flow is
where we used the fact that does not depend on , .
∇ ⋅ = −
−
+
= − − +
+
X V∂
∂∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
p
P s
c
H
c p
P s
c
H
p
p
P s
c
H
p
P s
c
H
p
H
p
H
p
P s
c
H
p
P s p p
x
X
y
X
y
X
X
x y
X
X x y
( ) ( )
( )
( ) ( )
( )
2 2
2
2
2
2
2
2
2
2 2
2
( ) ( )X X V X
V
V0 1
1 0 X
0 0
0 K X
K
= ′ =
=−
−
=−
x y z p p p
H P s
c
H
t x y
X
, , , , , , ,
( )
with given by the RHS of the equations of motion.
where
∂∂
∂∂2
1 0 0
0 1 0
0 0 1
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 15
qEvaluate derivatives of H (exactly!)
qCombine terms
qTotal “damping” is half of
This is a local version of Robinson’s Theorem and willhold for all canonical transformations we make.
It also includes fluctuations of the radiation power.
qRobinson’s original version (1-turn average)
( )( )
Neglect synchrotron radiation in places (RF cavities)
where vector potential depends on (through original
time - dependence), so we can evaluate:
p
H
pGx
p p
p p px
y
x y
∂∂
2
2
2 2
2 2 2 3 21= +−
− −/
∇ ⋅ = − +− −
′ = +− −
X V ( )( )
( )
14
1
2 2 2 2
2 2 2 2
GxP s
c p p p
t GxE
c p p p
X
x y
x y
From original Hamiltonian, we have
Phase - space compression
per unit time
= − 4 P s
EX ( )
α tot where valid for ≈ = ′ <<∫2
10 00
0U f
EU P s t ds
U
EX ( ) ,
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 16
Closed orbit with radiation
q“Sawtooth” closed orbit satisfies:
qCanonical transformation:
′ = =
′ = − − = −
′ = =
′ = − + = −
xH
pH y
pH
x
P s
p c
H
pH
P
p cH
zH
H
pH
z
P s
p c
HH
P
p cH
xp
x
X
xx p
t
t
X
z
x
x
t
0
00
00
0
02
00
0
02 0
00
0
0
0
02
00
0
02 0
∂∂
∂∂
∂∂
∂∂ δ
∂∂
∂∂ δ
δ
δ
and similar for
( )
,
( )
( ) [ ][ ][ ][ ]
Take out closed orbit with :
=
from generating function
, ,
energy sawtooth
X ( ( ) ~, , ( ) ~, ( ) ~ , , ( ) )
, ~ , ~ , ( ) ~ ( )
( ) ( )
~~ , ,
x s x z s z p s p s
F x y z p p x x s p p s
z z s s
xF
p
F
t x x
t x y x x
t t
x
0 0 0 0
0 0
0 0
+ + + +
= − + +
+ − +
= = −
K K
K
LLL
δ ε
ε
ε δ∂∂
δ ∂∂ ε
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 17
qNew Hamiltonian:
( )( )
( )[ ] [ ]
~ ( ) ~,
( ), , ( ), ~ ~
~ ~ ~
( ) ~ ( ) ( )
~ ~
~
H H X s X sF
s
H x s p s H x H p
H x H xp H p X
x s p p s x x s p
H x H xp
H p H
x x p x
xx xp x p p x
x x x
xx xp x
p p x xx
x
x x x
x
x x
= + +
= + +
+ + + ++ − ′ + + − ′ −
= +
+ + +
0
0 0 0 0
12 0
2
0
12
2 3
0 0 0 0
12 0
2
0
12
2 12
∂∂
K K
L L
L
O
( )02
0
12 0
3
0
02 0
0
02 0
~ ~
~ ~
x H xp H X
P
p cH x
P
p cH z
xp x p p
p
x x x
x
+ + +
− − +
O
L δ
( )( )( )( )
( )( )( )
B u t
s o c o n ta in s o f th e f o r m
H G s x s p s
H G s x s
H
P
p cG s x s p s x p s y z
p x
x y
x 0 0 0
0 0
0
02 0 0 0
1
1
1
≈ +
≈ − +
− + + −
( ) ( ) , ,
( )~
( ) ( ) ~ ( ) ~ ~
K
δ
r a d ia t io n t e r m s
Ænext slide
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 18
Stable phase angle
Equilibrium gives stable phase angle as a component of the6-dimensional closed orbit.
This Hamiltonian system was constructed explicity viacanonical transformations.
Not the same as the “classical” part of the radiation.
( ) ( )
( )( )( )
( ) ( )
( )
~ $cos ~ ( )
( ) ( )~ ( )~ ~
~
~$
sin ~ ( )
H eV
p cs s
cz z s
P
p cG s x s p s x p s y z
H
z
eV
p cs s
cz z s
P
p cG s x
C kk
x y
C kk
=− + −
− + + −
⇒ ′ = − = − + −
− +
∑
∑
Usual terms, including
RF
RF
RF
RF RF
00
0
02 0 0 0
02 0
0
02 0
1
1
ωδ ω
ε ∂∂
δ ω
L
L
( )
( )
( )
$sin ~ ( )
s
Q
dseV
p c cz z s
U
p c
s
kk
Integrating over one turn (in limit of smallish , etc.):
RF RF∆ ε ε ω= ′ = + −
−∫ ∑
02 0
0
0
L
MAD command:BEAM,particle=positron,-radiateTRACK
In optics and tracking programs: if radiation terms are notincluded in the Hamiltonian, the quadratic terms (whichdetermine the betatron and synchrotron oscillations, tunes, etc.)will be wrong. Focussing functions (which include the RF) mustbe defined on the true closed orbit even if damping terms aredropped.
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 19
Energy sawtooth
qExample from LEPLEP2: w05v6, 90 GeV, e+ orbit, ideal RF
Ideal LEP2 optical and RF configuration with 192 SCcavities, 120 Cu cavities.
Example of energy sawtooth and closed orbit, whichcontains additional terms beyond that given directly byenergy sawtooth (Bassetti effect):
Horizontal closed orbit
-3-2-101234
IP1
PU.Q
S16
.L2
PU.Q
D46
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PU.Q
D24
.R3
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R4
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.L6
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D30
.R6
PU.Q
L9.
R7
PU.Q
S7.
L8
PU.Q
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.L1
x / m
m
Energy sawtooth
-4
-20
2
4
IP1
PU.Q
S15
.L2
PU.Q
D42
.L3
PU.Q
D30
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PU.Q
S8.
R4
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L5
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D44
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PU.Q
L18
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PU.Q
S1A
.R8
PU.Q
L14
.L1
Pt/
10^
-3
x s D s x sc x s B( ) ( ) ( )= + +0 δ
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 20
Consequences of LEP2 energy
qSynchrotron radiation ⇒tRadiation damping
tQuantum fluctuations
tLocal variation of energy (down in magnets, up in RFcavities)⇒ “energy sawtooth”, different for e+ and e-⇒ different bending, focussing, nonlinearities, locally⇒ different orbits, Twiss functions, dynamic aperture
Radiation damping per turn:
Quantum excitation per turn:
T P
E
E
N u T
E
E
x
X
X
Tx
03
20
2
2 5
2
4
0
τ ρ
σρ
ετ
= ∝
= ∝
,
Radiation effects in LEP and PEP (no wigglers in either)
10
100
1000
10000
20 40 60 80 100
[E/GeV for LEP], [5E/GeV for PEP]
τ x/T
0
0.00001
0.0001
0.001
0.01
0.1
1
10^
6 [
4 T
0 s e
2 / t x
]
taux/T0 (LEP)taux/T0 (PEP)Q.T0 (LEP)Q.T0 (PEP)
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 21
Tracking Radiating Particles′ = ′ = − −
′ = ′ = − −
′ = ′ = − +
xH
pp
H
x
P s
p c
H
p
yH
pp
H
y
P s
p c
H
p
zH H
z
P s
p c
H
xx
X
x
yy
X
y
tt
X
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂ δ
δ ∂∂
∂∂ δ
( ),
( ),
( )
02
02
02
N.B. Momentum deviation on closed orbit δs(s) reflectssystematic radiation losses and gains from RF (sawtooth).
Tracking using(Taylor) maps +
radiation
Tracking using(Taylor) maps +
radiation
Twiss functions oreigenvectorsfor coupled
synchro-betatronmotion
Twiss functions oreigenvectorsfor coupled
synchro-betatronmotion
Tracking ofoscillations
around closedorbit
Tracking ofoscillations
around closedorbit
( )Closed orbit with :
X (s) = X (s + C)0 0
energy sawtooth
x s y s z s p s p s st x y0 0 0 0 0 0( ), ( ), ( ), ( ), ( ), ( )δ =
x x s D s x
s
x s= + + +
= +
0
0
( ) ( )( ) ,
( )
δ ε
δ δ ε
β
0
x x s x
ss
= +
= +
0( ) %,
( )
M
δ δ ε
Radiation reactionforces (which changephysical momentaonly) are added toHamilton’s equations
P s c u s sX j jj
( ) ( ),= −∑ δ
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 22
Tracking modes (refer to MAD)
qSymplectic tracking with radiationWe proved that is possible to consider symplectic mapsaround the closed orbit (i.e. including stable phase angle,energy sawtooth, etc.) determined by radiation:
( )P s c p s c b x s y s sX ( ) ( ) ( ), ( ),= 20
21 0 0
2
Betatron and synchrotron oscillations, tunes, etc. will be incorrectsince focussing functions (which include the RF) must be defined onthe true closed orbit.
qSymplectic tracking with no radiation P sX ( ) = 0
Symplectic trackingwith no radiation
Symplectic trackingwith radiation
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 23
qRadiation damping arises naturally.qDynamics often becomes relatively simple.qDynamics is very different from symplectic
model!
qTracking with radiation dampingInclude dependences of classical (deterministic) radiationpower on all canonical variables and magnetic elements.
Continuous loss of energy, modification of particlemomenta in all magnetic elements.
( )P s c p c b x y sX ( ) , ,= 2 21
2
LEP290°/60°optics,90 GeV,192 SC +120 Cucavities, nobetatronamplitudes
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 24
Quantum Tracking
qEmission of individual photons gives quantumfluctuations as well as damping
All phenomena related to radiation:closed orbit, damping, energy spread,change of damping partition with fRF,gaussian (or other!) distribution, etc.arise from these photons. Nothinginserted “by hand”!
All phenomena related to radiation:closed orbit, damping, energy spread,change of damping partition with fRF,gaussian (or other!) distribution, etc.arise from these photons. Nothinginserted “by hand”!
∗
∗
∗
Decide how many photons to emit in element of length ,
field according to Poisson distribution with mean
Generate a random energy for each photon according to the
photon distribution for synchrotron radiation (universal form
for distribution in units of critical energy which scales with
momentum and magnetic field .
Modify particle momenta (actually done at entrance and exit
of each element in MAD)
L
b s
N s L c
p b s
X
X
X
( ),
( ) /
( )
Simulating photon emission
P s c u s sX j jj
( ) ( )= −∑ δ
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 25
Amplitude diffusion
q“Random walk” of horizontal action variable
LEP, (108°90°) optics, 87 GeV, “bad” imperfect machinea particle with almost unstable inital horizontal amplitude
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 26
Distribution on resonance (quantum)
LEP2 135°/60° optics, 90 GeV, 192 SC + 120 Cucavities, starts on closed orbit, 10000 turns,
Non-gaussian distribution on 3rd order resonance.
Tracking with quantum fluctuations
Qx ≈ 0 35.
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 27
Fitted distributions
LEP2 135°/60° optics, 90 GeV, 192 SC + 120 Cucavities, starts on closed orbit, 10000 turns,
Fitting of contours to distribution.
3rd order resonance islands visible
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 28
Instability Mechanisms
qChromatic effects, resonancesWell known, variation of Qy with momentum/synchrotronamplitude as in hadron machines.
Synchro-betatron couplings and modulations throughdispersion in RF, non-linear dispersion, etc.
qNon-resonant radiative beta-synchrotroncoupling (RBSC)
Particles with large betatron amplitudes make extra energyloss in quadrupoles so their “stable phase angles” change.
This effect is important in determining the transversedynamic aperture at LEP2 energies.
Determine by radiation integrals:
I K s s ds I K s s ds
U U
U
U
U
I W I W
IW
x x x y x y
s s s
s sx x y y
62
62
0 0
0 06 6
2
= =
≈+
−
≈ =+
∫ ∫( ) ( ) , ( ) ( ) ,
arcsin sin
tan tan
β β
ϕ ϕ ϕ
ϕ ϕ
∆ dipoles quads
dipoles
quads
dipoles
for small
Comparison of radiation integrals for quadrupoles in LEP2 lattices
I6x I6y90°/60° 76.81 221.01
135°/60° 106.79 149.96
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 29
Radiative Beta-synchrotron Coupling
LEP2 90°/60° optics, 90 GeV, etc., fairly largeAx,and initial At=0, 400 turns.
Synchrotron motion is generated from betatron motion andboth damp away.
Tracking with damping
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 30
Losses from Beta-synchrotron Coupling
LEP2 90°/60° optics, 90 GeV, etc., varying Ay ,andinitial At=0, 50 turns. Beam core also shown.
Synchrotron motion is generated from betatron motion andboth damp away.
Lost particle has initial
Tracking with damping
y = 6 mm
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 31
Prevalence of RBSC
qRBSC tends to amplify any effect leading toamplitude growth in transversed planes
e.g., classical synchro-betatron resonance driven byvertical dispersion in RF cavities
LEP, (108°90°) optics, 87 GeV, “bad” imperfect machinea particle with almost unstable inital horizontal amplitude
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 32
(same example, continued)
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 33
(same example, continued)
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 34
Mode 1 distribution (same ex.)
qNon-linear resonances affect distribution
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 35
Mode 2 distribution (same ex.)
qNon-linear resonances + SBR affectdistribution
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 36
Mode 3 distribution (same ex.)
qSynchrotron phase-space distribution modifiedby RBSC
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 37
Coherent excitation of beam
LEP2 90°/60° optics, 90 GeV, 192 SC + 120 Cucavities, starts on closed orbit, 10000 turns,kicker exciting close to tune.
Hopf bifurcation of closed orbit into limit cycleand further structure (crater distribution).
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 38
Tracking Methodology
qAnalysis of tracked orbits Æ physicsFFT spectra, phase space plots (all projections), evolutionof action variables in time, etc.
q4D dynamic aperture scansIn (square roots of) action variables of three normal modesand synchrotron phase
qMost tracking done with radiation dampingOccasionally without, learning to work with quantumfluctuations (tracking becomes a Monte-Carlo).
qFull LEP2 RF system in real layout120 Cu + 192 SC cavities, typical voltages, in real layout
qDynamic aperture independent of number ofturns
(at high energy with enough damping)
qVacuum chamber boundary includedqEnsembles of imperfect machines (Monte
Carlo)qCorrection applied as in control room:
closed orbit, tune, optics at IP
qAll calculations repeated for positron andelectron
Bad Honnef, 11/12/96,J.M.. Jowett, Electron Dynamics with Radiation Page 39
Summary
qElectron dynamics: fluctuating, dissipativesystem
Add stochastic terms to Hamilton’s equations.
Simple problems canbe solved by means of techniques forstochastic systems:
•Fokker-Planck equations
•Integration of truncated moment equations, etc.
qDamping and quantum fluctuationsDamping governed by Robinson’s Theorem.
Damping rates and quantum excitation rate dependstrongly on energy of ring
Distribution functions
qTracking for realistic (complicated) problemsDamping gives rise to new instability mechanisms (RBSC)
Tracking with quantum fluctuations now a feasible meansto calculate core of beam distributions, including allrelevant single-particle effects.
Tails are more difficult!