Electron Dynamics with Synchrotron Radiationdlr/center/... · Synchrotron Radiation qPhoton...

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


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.


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


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

( )


[ ]

( )

[ ]( )

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


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


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


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


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


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


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?


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= − ′ ′ = − ′


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


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


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 ( ) ,


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


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


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.


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

.L3

PU.Q

D24

.R3

PU.Q

S4.

R4

PU.Q

L12

.L5

PU.Q

D36

.L6

PU.Q

D30

.R6

PU.Q

L9.

R7

PU.Q

S7.

L8

PU.Q

D30

.L1

x / m

m

Energy sawtooth

-4

-20

2

4

IP1

PU.Q

S15

.L2

PU.Q

D42

.L3

PU.Q

D30

.R3

PU.Q

S8.

R4

PU.Q

L7.

L5

PU.Q

D24

.L6

PU.Q

D44

.R6

PU.Q

L18

.R7

PU.Q

S1A

.R8

PU.Q

L14

.L1

Pt/

10^

-3

x s D s x sc x s B( ) ( ) ( )= + +0 δ


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)


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


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


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


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


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


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.


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


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


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


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


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


(same example, continued)


(same example, continued)


Mode 1 distribution (same ex.)

qNon-linear resonances affect distribution



qNon-linear resonances + SBR affectdistribution



qSynchrotron phase-space distribution modifiedby RBSC


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).


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


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!

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