Accelerator Physics Statistical and Collective Effects I · 2016. 6. 10. · USPAS Accelerator Physics June 2016 Beam rms Emittance Treat the beam as a statistical ensemble as in

USPAS Accelerator Physics June 2016

Accelerator Physics

Statistical and Collective Effects I

S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage

Jefferson Lab

Old Dominion University

Lecture 15


Statistical Treatments of Beams

• Distribution Functions Defined

– Statistical Averaging

– Examples

• Kinetic Equations

– Liouville Theorem

– Vlasov Theory

– Collision Corrections

• Self-consistent Fields

• Collective Effects

– KV Equation

– Landau Damping

• Beam-Beam Effect


Beam rms Emittance

Treat the beam as a statistical ensemble as in Statistical

Mechanics. Define the distribution of particles within the beam

statistically. Define single particle distribution function

, ,x x

where ψ(x,x')dxdx' is the number of particles in [x,x+dx] and

[x',x'+dx'] , and statistical averaging as in Statistical Mechanics,

e. g.

2 2

, , /

, , /

q q x x x x dxdx N

q q x x x x dxdx N


Closest rms Fit Ellipses

22 2

rms x x xx

2 2

x

rms rms

x

rms

xx

2 2

x

rms rms

x

For zero-centered distributions, i.e., distributions that have zero

average value for x and x'


Case: Uniformly Filled Ellipse

2 21 2

, 1x xx x

x x

Θ here is the Heavyside step function, 1 for positive values of

its argument and zero for negative values of its argument

2 2

2 2 2

4

4

14

4

x

x

rms

x

xx

x

Gaussian models (HW) are good, especially for lepton machines


Dynamics? Start with Liouville Thm.

Generalization of the Area Theorem of Linear Optics. Simple

Statement: For a dynamical system that may be described by a

conserved energy function (Hamiltonian), the relevant phase

space volume is conserved by the flow, even if the forces are

non-linear. Start with some simple geometry!

2 1 3 1 3 1 2 1Area

2 2

acute angle has line 1 2 clockwise wrt line 1 3

q q p p q q p p

1 1,q p 2 2,q p

3 3,q p

p

q

In phase space

Area Before=Area After 1

2


Liouville Theorem

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0

, , , ,

, , , ,

, , , ,

,

H Hq p q q p t p q p t

p q

H Hq q p q q q q p t p q q p t

p q

H Hq p p q q p p t p p q p p t

p q

q q p p

0 0 0

0 0 0

0 0 0 0 0 0 0 0

1

0 0 0 0 0 0 0 0

, ,

,

, , , ,1

Area det2

, , , ,

Hq q q q p p t

p

Hp p q q p p t

q

H H H Hq q q p q p t q q p q p t

p p q q

H H H Hq p p q p t p q p p q p

p p q q

2 2

0 0 0 00

11 , ,

2 2t

t

q pH Hq p q p q p t

p q p q


Likewise

0 0 0 0 0 0 0 0

2

0 0 0 0 0 0 0 0

2 2

0 00

, , , ,1

Area det2

, , , ,

11 ,

2t

H H H Hq q q p q q p p t q q p q q p p t

p p q q

H H H Hq p p q q p p t p q p p q q p p t

p p q q

H Hq p q p p p

p q

0 0,

2

q pq p p p t

p q

Because the starting point is arbitrary, phase space area is

conserved at each location in phase space. In three dimensions,

the full 6-D phase volume is conserved by essentially the same

argument, as is the sum of the projected areas in each individual

projected phase spaces (the so-called third Poincare and first

Poincare invariants, respectively). Defeat it by adding non-

Hamiltonian (dissipative!) terms later.


Phase Space

• Plot of dynamical system “state” with coordinate along

abscissa and momentum along the ordinate

2 22

2 2

xp xH m

m

x

xp

Linear

Oscillator


Liouville Theorem

• Area in phase space is preserved when the dynamics is

Hamiltonian

x

xp

1t t

2t t

Area = Area


1D Proof

, 0

, 0

lim

, ,

, ,

, ,

, ,

,

t

x

t x

xx x x x

t

L

x x

c s t

L

x x

c s t t

x x

V t t V tdV

dt t

dx Hx s t t x s t x s x s p s t

dt p

dp Hp s t t p s t p s x s p s t

dt x

dxV t p dx p s t s t ds

ds

dxV t t p dx p s t t s t t ds

ds

Hp s x s p s t

x

0

,

L

x

x

d Hx s x s p s t ds

ds p


0

0

,

, ,

, ,

(why is boundary term of integration by parts zero?)

By Green's Thm.

=0

L

x x x

x

L

x

x x

x

x

xc s t

dx sdV d H Hp s x s p s x s p s ds

dt ds p x ds

dp s dx sH Hx s p s x s p s ds

ds p x ds

H Hdp dx

p x

when the differential is an exact differential

i.e., , in other words always

note the integrand above is really , so is a "potential"

for phase space!!!

x x

H H

x p p x

dH H


3D Poincare Invariants

• In a three dimensional Hamiltonian motion, the 6D phase

space volume is conserved (also called Liouville’s Thm.)

• Additionally, the sum of the projected volumes (Poincare

invariants) are conserved

Emittance (phase space area) exchange based on this idea

• More complicated to prove, but are true because, as in 1D 2 2

i i i i

H H

q p p q

6

6D x y z

V

V dp dp dp dxdydz

2 2 2

4 4 4

proj proj proj

proj proj proj

x y z

V V V

y z z x x y

V V V

dp dx dp dy dp dz

dp dp dydz dp dp dzdx dp dp dxdy


3 3

1 10

3

10

is a loop in 6D phase space

, , ,

, ,

, , 0

for any surface in

L

i

i i i

i i i it

L

i i

i i i t

t p s t q s t

dx sd d H Hp dx p s x s p s x s p s

dt ds p x ds

dp s dx sH Hx s p s x s p s dH

ds p x ds

2 2 2

2 2

3 3 3

1 1 1 proj

23

1

33

1

6D phase space , with

i i i i i i

i i iV V V

i i y z z x x y

i

i i x y z

i

V V

p dx dp dx dp dx

dp dx dp dp dydz dp dp dzdx dp dp dxdy

dp dx dp dp dp dxdydz


Vlasov Equation

0 as the distibution evolvesd

dt

0

, , , ,lim 0

where the equation for ANY (this is what makes

it hard to solve in general!) individual orbits

through phase space is given by ,

t

t t q t t p t t t q t p td

dt t

q t p t

dq dp

t dt q dt

0

p

By interpretation of ψ as the single particle distribution

function, and because the individual particles in the distribution

are assumed to not cross the boundaries of the phase space

volumes (collisions neglected!), ψ must evolve so that


Conservation of Probability

3 3; , is a conserved quantity

continuity equation for is

0

0

for the Hamiltonian system

q p

q p

N t t q p d xd p

q pt

dq dp H H

t dt q dt p p q

dq dp

t dt q dt

0p

, ,

, ,

q q q

p p p

q

p

, ,

, ,

q q

p p


Jean’s Theorem

The independent variable in the Vlasov equation is often

changed to the variable s. In this case the Vlasov equation is

The equilibrium Vlasov problem, ∂ψ /∂t =0, is solved by any

function of the constants of the motion. This result is called

Jean’s theorem, and is the starting point for instability analysis

as the “unperturbed problem”.

0dq dp

s ds q ds p

If , , , , where , , , are constants of the motionf A B C A B C

dx dp f dx A dp A f dx B dp B

dt x dt p A dt x dt p B dt x dt p

f dx C dp C f dA f dB f dC

C dt x dt p A dt B dt C

0dt


Examples

1-D Harmonic oscillator Hamiltonian. Bi-Maxwellian

distribution is a stationary distribution

2 2 21exp / exp / 2 exp / 2 ,

2 2x

mH kT p mkT mx kT

kT

As is any other function of the Hamiltonian. Contours of

constant ψ line up with contours of constant H

2 D transverse Gaussians, including focusing structure in ring

2 2

2 2

; , ; , exp 2 /

exp 2 /

x x x x

y y y y

s x x y y s x s xx s x

s y s yy s y

Contours of constant ψ line up with contours of constant Courant-

Snyder invariant. Stationary as particles move on ellipses!


Solution by Characteristics

More subtle: a solution to the full Vlasov equation may be

obtained from the distribution function at some the initial

condition, provided the particle orbits may be found

unambiguously from the initial conditions throughout phase

space. Example: 1-D harmonic oscillator Hamiltonian.

0 0 0 0 0 0

0 0 0 0 0 0

0 0

0 0 0 0

cos sin / cos sin /

sin cos sin cos

, ; ,

Let , ; cos sin / , sin cos

x t t t t t x t x t t t t x

x t t t t t x t x t t t t x

x x t t f x x

x x t f t t x t t x t t x

0

0 0

2

0 0 0 0

0 0

2

0 0

; , ; ,

sin cos cos sin

cos sin

sin / cos

t t x

dx t x x dx t x xf f

t x dt x dt

f ft t x t t x t t x t t x

x x

dx f fx t t t t

dt x x x

dx f fx t t t t

dt x x x

0d

dt


Breathing Mode

The particle envelope “breaths” at twice the revolution

frequency!

x

x'

x

x'

x

x'

x

x'

x

x'

Quarter

Oscillation


Sacherer Theory

Assume beam is acted on by a linear focusing force plus

additional linear or non-linear forces

2

2

2

2 2 3 2 2

2 2

2 2

2 2 2 2 2

0

0

For space charge example we'll see

1

Now

0

0

Assume distributions zero-centered and let

=

x x

y y

x y x y

x y

x x

y y

x k x F

y k x F

qE qEF

mc mc

xx k x F x

yy k y F y

x x x x y y

2 2 2 =y y


2 2

2 2 2

2 2 2 2

2 2 2 2 2

2 2 2

2 2

2

2 2 2

2

3

2 2

2 2

Also

1

2

/ 0

0

x x

x x

x x

x

x

x xx x xx

x x xx xx x

xx x xx x k x F x

x xx xx x x k x F x

xx x xx xx x xx k x F x

x

xx x x F xx k x

x x

x

2

2

30 "Envelope" equation

xrmsx

F xk x

x x


rms Emittance Conserved

22 2

2 2 2 2

2 2 2 2 2

2 2 2 2

2

2

2 2 2

2 2

2 2

x x

x x x x

x x

x x xx

x x x x xx xx

xx x x x x xx x k x F x

x k x x F x xx k x F x

x F x xx F x

For linear forces derivative vanishes and rms emittance

conserved. Emittance growth implies non-linear forces.


Space Charge and Collective Effects

• Collective Effects

– Brillouin Flow

– Self-consistent Field

– KV Equation

– Bennet Pinch

– Landau Damping


Simple Problem

• How to account for interactions between particles

• Approach 1: Coulomb sums

– Use Coulomb’s Law to calculate the interaction

between each particle in beam

– Unfavorable N2 in calculation but perhaps most realistic

– more and more realistic as computers get better

• Approach 2: Calculate EM field using ME

– Need procedure to define charge and current densities

– Track particles in resulting field


Uniform Beam Example

• Assume beam density is uniform and axi-symmetric going

into magnetic field n r

r ar

0 0

2

0 0

Electric Field

1

2

Self-Magnetic Field by Ampere's Law

22

r r

qn qnrE E r

r r

qn crB qn c r B r


Brillouin Flow

2

2

0

2 22

2 2 3

0

Total Collective Force on beam particle

v 12

effective (de)focussing strength

where the non-relativistic "plasma frequency" is 2

By previous work with

p

p

q nF q E B r

q nk

c m

2

2

solenoids in the rotating frame, can have

equilibrium (force balance) when

non-relativistic plasma and cyclotron frequencies 2 2

This state is known as and neglectBrillouin Flo

be

w s

p cc L

am temperature (fluid flow)


Comments

• Some authors, Reiser in particular, define a relativistic

plasma frequency

• Lawson’s book has a nice discussion about why it is

impossible to establish a relativistic Brillouin flow in a

device where beam is extracted from a single cathode at an

equipotential surface. In this case one needs to have either

sheering of the rotation or non-uniform density in the self-

consistent solution.

22

3

0

p

q n

m


Vlasov-Poisson System

• Self-consistent Field

2

0 0

0

3 3

enE B e J e v n

n d q v n q d q

0

/

dq p

dt t x p

q p m

p q E v B


K-V Distribution

• Single value for the transverse Hamiltonian

• Any projection is a uniform ellipse

22 221 1

, , , 1

y yx x

x x y y

y y yx x xC

x x y y C

2 2 2 2 2

0

2 2

22

1 1 0 1 , 1

, , , 1

, , , 1

x x y y

x x

x x

r c dr c c

x yx x y y dx dy

x x xx x y y dydy


Self-Consistent Field

2 2

2 22 2

0 0

2

32 20 0

32 20 00

uniform ellipsoid potential is (Landau and Lifshitz)

1, 1

4

2

field is

2

x

x y dsx y ab

a s b s a s b s

ds

a a ba s b s

ab ds x bE x

a ba s b s

Iz

cXY

3

003 3

0

0 0

42

x y

mcIK I

I q

I x I yE E

c X X Y c Y X Y


K-V Envelope Equation

2

3

2

3

particle trajectories

20

20

Envelope Equation

20

20

No temperature

y

y

xx

y

y

Kx k x x

X X Y

Ky k x x

Y X Y

KX k X

X Y X

KY k Y

X Y Y


Luminosity and Beam-Beam Effect

• Luminosity Defined

• Beam-Beam Tune Shift

• Luminosity Tune-shift Relationship (Krafft-Ziemann

Thm.)

• Beam-Beam Effect


Events per Beam Crossing

• In a nuclear physics experiment with a beam crossing through a thin

fixed target

• Probability of single event, per beam particle passage is

• σ is the “cross section” for the process (area units)

l

P n l

Beam Target

Number density n


Collision Geometry

• Probability an event is generated by a single particle of

Beam 1 crossing Beam 2 bunch with Gaussian density*

Beam 1 Beam 2

2 2 2 2

2 2 2 2 2

23/2

2 2 2

2 2 2 2

2 2 2

2 2

exp / 2 exp / 2exp / 2

2

exp / 2 exp / 2

2

x y

z

x y z

x y

x y

N x yP z dz

N x y

* This expression still correct when relativity done properly

xy


Collider Luminosity

• Probability an event is generated by a Beam 1 bunch with

Gaussian density crossing a Beam 2 bunch with Gaussian density

• Event rate with equal transverse beam sizes

• Luminosity

1 2

2 2 2 2

1 2 1 22 x x y y

N NP

1 2

4 x y

fN NdN

dt

L

33 1 21 2

10

1 2

~ 10 sec cm ,4

for 100 MHz, 10 , 10 microns

x y

x y

fN N

f N N

L


Beam-Beam Tune Shift

• As we’ve seen previously, in a ring accelerator the number

of transverse oscillations a particle makes in one circuit is

called the “betatron tune” Q.

• Any deviation from the design values of the tune (in either

the horizontal or vertical directions), is called a “tune

shift”. For long term stability of the beam in a ring

accelerator, the tune must be highly controlled.

*

*

*

* *

1 0 cos sin

1/ 1 sin / cos

cos sin

cos / sin / cos / sin

totMf

f f


*

**

Trcos cos sin

2 2

2 4

totM

f

Q ff


Bessetti-Erskine Solution

• 2-D potential of Bi-Gaussian transverse distribution

• Potential Theory gives solution to Poisson Equation

• Bassetti and Erskine manipulate this to

2 2

2 2, exp exp

2 2 2x y x y

Q x yx y

2

0

2 2

2 2

2 20 0

,

exp exp2 2

,4 2 2

x y

x y

x y

x y

q qQx y dq

q q


2 2

2 22 2 2 2 2 2

0

2 2

2 22 2 2 2

0

Im exp2 22 2 2 2

Re exp2 22 2 2

y x

x y

x

x yx y x y x y

y

x yx y x y

x iyQ x iy x x

E w w

Q x iy x xE w w

2 22

Complex error function

y x

x y

x y

x iy

w z

• We need 2-D linear field for small displacements

3/22 20 0

1,0

22 2

x

x y

Q xE x dq

xq q


• Can do the integral analytically

• Similarly for the y-direction

2 2

2 2

2 2

2 2

3 3 32 2 2 2 2 2

0

2 22 2

3/2 1/2 1/22 2 2 2

2 2 2 2 2 2 2 2 2 2 2

2 2

2 2 2 2

1

2 2

1

1 1

2

x y

x y

x y

y x

x y x y y x

y xy x

y x y x y x y x

y x

y x y x x

qdq dq

q q q q

qqdq

q q q

2 2 2 2

2 2

21 1

2 2

x y y x y x

y x y x y xy x x y

0

10,

2y

y x y

Q yE y

y


Linear Beam-Beam Kick

• Linear kick received after interaction with bunch

1 1 1 2 12

1 1 1 2 2 1

2

222 2

0 2

v ,

by relativity, for oppositely moving beams

1 ,

Following linear Bassetti-Erskine model

1 1,0, , exp

2 22

x xx

x z z x

z

x

zx x y

mc q E B x t t dt

mc q E x t t dt

z ctq xE x z t

1 1 moves with ,0, zq x t x ct


Linear Beam-Beam Tune Shift

1 2 21 1

1 2 0

21 22 1

1 2

1 1 2 0 1

2 1

1

1 1

2

121/

4

21/ Both beams relativistic

From linear Bassetti-Erskine model, and replacing the bea

z z

x

z z x x y

z z

z z x x y

x x y

q xmc q

c

N r ef r

m c

N rf

1 12 1 2 1

11 1

m size

1 1

2 21 / 1 / /

Argument entirely symmetric wrt choice of bunch 1 and 2

1 1

2 21 / 1 / /

x y i

x y x y y x x y

i ii ii ix yi i

i ix y x y y x x y

N r N r

N r N r


Luminosity Beam-Beam tune-shift

relationship • Express Luminosity in terms of the (larger!) vertical tune

shift (i either 1 or 2)

• Necessary, but not sufficient, for self-consistent design

• Expressed in this way, and given a known limit to the

beam-beam tune shift, the only variables to manipulate to

increase luminosity are the stored current, the aspect ratio,

and the β* (beta function value at the interaction point)

• Applies to ERL-ring colliders, stored beam (ions) only

* *1 / 1 /

2 2

i i

i y i y iiy x y x

i iy i iy

fN I

r e r

L


Luminosity-Deflection Theorem

• Luminosity-tune shift formula is linearized version of a

much more general formula discovered by Krafft and

generalized by V. Ziemann.

• Relates easy calculation (luminosity) to a hard calculation

(beam-beam force), and contains all the standard results in

beam-beam interaction theory.

• Based on the fact that the relativistic beam-beam force is

almost entirely transverse, i. e., 2-D electrostatics applies.


2-D Electrostatics Theorem

2

0

2 22 121 12 2 2 1 1 1 2

0 2 1

1 1 1 1 1 2 2 1 2 1

2 2

2 2

121 12

2

4

1 1 on 2

2

/ / zero centerred

/i i

Q x xE x

x x

x xF F x x d x d x

x x

n x x Q n x x b Q

Q x d x b x x d x Q

QF F

2 22 1 22 2 1 1 1 22

01 2

2

Q x b xn x n x d x d x

x b x


1 22 1 2 12

1 2

2

21 2 1

0

2

2

0

1 2 1 21 2

0 1 2

2

1

Generalizes take

Transverse interaction in the beam-beam problem

2

x yb

b

x b xx b x y b y

x b x

F x b x d x

E x x b

q q x xp

c x x


1 1 2 2 2 1

2 21 2 1 22 2 1 1 1 222

11 2

22

2 2 1 2

0

2

1 2 2 1

1

22 2

/

4 4

4

4

e eb

b

e

b

e

D b m m

q q x x bn x n x d x d x

m c x x b

eD b N r n x b n x d x r

mc

L b N N n x b n x d x

NL b D b

r

NL b

r


1

1

*

/ 0

0 12

1 / as before2

Maximum when

0, 0

x xy x

y y

y x

e

x

x x y y

D b

D bf

NL

r

D D

b b b b


Luminosity-Deflection Pairs

• Round Beam Fast Model

• Gaussian Macroparticles

• Smith-Laslett Model

2

2 1 2

22 2 2 2

2 eN r b N N

D b L bb b

2 2 2 2

_ 1 2 1 2

22

1 2

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2

; ;

exp exp2

Bassetti Erskine x x y y

yx

x x y y x x y y

D b D b

bbN NL b

2 42 3

1 12

3/22 2 4 2 4

2 2 2 23

1 11 2

2 5/22 4 2 4

ˆ ˆ4 2 ˆ ˆ ˆ ˆ2 4 3sinh sinh

ˆ ˆ ˆ 2 2 2ˆ ˆ4 4

ˆ ˆ ˆ ˆ2 4 4 1 ˆ ˆ ˆ3sinh sinh

2 2 2ˆ ˆ ˆ ˆ4 4

eb bN r b b b b b

D bb AB b b b b

b b b bN N b b bL b

AB b b b b

22

2ˆ yxbb

bA B

Documents

Accelerator Physics Statistical and Collective Effects I · 2016. 6. 10. · USPAS Accelerator Physics June 2016 Beam rms Emittance Treat the beam as a statistical ensemble as in