accelerator physics statistical and collective effects iixx yy x x y y x x x y y y c x x y y c de de...

$: Accelerator Physics Statistical and Collective Effects IIxx yy x x y y x x x y y y C x x y y C DE DE H E H E \G ... ZZ ww ww ww ww w w w w w w: USPAS Accelerator Physics June 2016$
USPAS Accelerator Physics June 2016

Accelerator Physics

Statistical and Collective Effects II

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

Jefferson Lab

Old Dominion University

Lecture 17


Beam Temperature

• K-V has single value for the transverse Hamiltonian

No Temperature!

• Temperature in beam introduces thermal spreads

– Transverse Temperature

• Debye length

– Longitudinal Temperature

• Landau Damping

22 221 1

, , , 1

y yx x

x x y y

y y yx x xC

x x y y C


Waterbag Distribution

• Lemons and Thode were first to point out SC field is

solved as Bessel Functions for a certain equation of state.

Later, others, including my advisor and I showed the

equation of state was exact for the waterbag distribution.

2 2 2202 2

0

2 2 2

0

0 0 0

0 0

2 2 2 2 2 2

02 0

20 0

2 2

1 /2

12

zT SC

T

SC

z SCv

m x ypH x y e

m

A H H

m x yn r dx dy n H e

H

p x y dx dy m x yH

m Hm dx dy


2 2 2

02 0

2

0 0

0 0 0 0

2 2

0 0

2 2 2

0

0

Self-consistent potential solves

12

Debye Length

Analytic solutions in terms of Modified Bessel Functions

/ 12

SCSC

D

vD

p

D

m x yen

H

m H H

e n m e n

m x ye r A I r BK

0

0

/

0 by boundary condition

chosen so that solution without solution to inhomogeneous eqn.

Dr

B

A I


Equation for Beam Radius

2

2 2 2

0

2 2 2

0 0 0

2

02 2

0

Now

12

2

2

At the density vanishes

2 1 /

1 /2

D p

b

D p b D

p

b D

p

rr

r r r

A m

r r

H m I r

I r

0 0

0

/ /ˆ

/ 1

b D D

b b

b D

I r I rn r n

I r


Debye Length Picture*

*Davidson and Qin


Collisionless (Landau) Damping

• Other important effect of thermal spreads in accelerator

physics

• Longitudinal Plasma Oscillations (1 D)

0

0z

zz

z

nv n

t

d eE

dt m

E en

z


Linearized

0

0

22

0 0

2

0

2

0

0

0

p

z

zz

z

z

i t

p

vnn

t z

eE

t m

E e n

z

en e nEnn

t m z m

e nn e

m

In fluid limit plasma oscillations are undamped


Vlasov Analysis of Problem

2

2

0

0 0

0

2

2

0

, , ,

0

0th order solution

, 0

linearized

e z x y

z e

z

e z i

e z

z e

z

e z

F z p t z p dp dp

v e Ft z z p

eF dp n

z

F F p

Fv F e

t z p z

eF dp

z


Initial Value Problem

• Laplace in t and Fourier in z

0

0

2 /

0

2

20

ˆ Im large enough to converge

1 ˆ2

ˆ 0

ˆ, ,

, , 0 2 / ˆ, , ,2 / 2 /

ˆ ,2

i t

i t

C

i t

ilz L

l

e z ze z

z z

F dte F t

F t d e F

ddte F t i F F t

dt

z t l t e

i F l p t e l F pF l p l

v l L L v l L

e

Ll

l

0

0

2 / ˆ ,2 /

, , 0

2 /

z z

z

z

e z

z

z

l F p pdp l

L v l L

F l p teidp

v l L


Dielectric function

• Landau (self-consistent) dielectric function

• Solution for normal modes are

20

0

ˆ, , ,

/, 1

2 2 /

z z

z

z

D l l N l

F p pe LD l dp

l v l L

20

2

0

0 02

2

, 0

, 12 /

/1

2 /

z

z

z

z

p z

z

D l

F peD l dp

m v l L

F p ndp

v l L


Collisionless Damping

• For Lorentzian distribution

• Landau damping rate

0

2 20

2

2 2 2

2

2

11

2 /

2 /

z

p z

zz

p

F

n p

dppp l Lm

i l Lm

2p

li

L


Negative Mass Instability

• Simplified argument: assume longitudinal clump on

otherwise uniform beam

• Particles pushed away from clump centroid

• If above transition, come back LATER if ahead of clump

center and EARLIER if behind it

• The clump is therefore enhanced!

• INSTABILITY; particles act as if they have negative mass

(they accelerate backward compared to force!)

θ


Longitudinal Impedance

longitudinal wake function

distance between exciting charge and test charge

1, / units V/C

trailing particle (singly charged) picks up voltage per turn

z q arrival

ring

W

q

W E z t c dzq

" "

" "

of

total energy loss

z

z

V z e z W z z dz

U e z dz e z W z z dz


Frequency Domain

" "

" "

/

/

/

, , note the coordinate moves with beam

1, ,

Fourier Transform

1

1

1

2

z

i c

z

i c

i z c

I z t c z t z

z zV z t I z t W z z dz

c c

V I e W dz Z Ic

Z e W dc

W z e Z

2

2 2

0

Loss factor

2Re

d

Uk Z I d

q q


NMI Simple Analysis

0

2

0

0 0

0

2

00 2

revolution frequency of particle

2 2

oscillation frequency of disturbance

2

c

i n t

zn n

i n t

n

cn

d

dt t t

d d dE dE

dt dE dt E dt

dEqV qZ I e

dt

e

Zqn i

0

nI

E


Linearized Continuity Equation

2

0 0

0 0

0

1

0

z b z

z

z

n n

I v r v

vt z

vt R

t

n I nI


Oscillation Frequency

2

22 0 00 2

02

Re 0 1 mode has positive imaginary part

instability

Resistive impedance has positive real part

"Resistive wall instability"

If Re 0 (e.g. space charge impedance at long

cnq In i Z

E

Z

Z

wavelengths)

stability/instability depends on sign of RHS

Im 0 (inductive, stable if 0, unstable if 0)

Im 0 (capacitive, space charge is this way,

c cZ

Z

stable if 0, unstable if 0)

Later case is negative mass instability

c c


NMI Growth time

2

02 2

0

0

0

0

2 2

0 0

0

Impedance?

2 2

2 2

1 1 2ln /4

1 2ln / 1 2ln /2 2

b b

b br

b b

z c b

i n t

n

SC n c b n c b

e r e rr r c r r

r rE B

e er r c r r

r r

eE r r

z

e

in inV r r I r r

c

n

2 2 2

2 0 0 0 0

2 2 2

0 0 0

1 2ln /2 4

c cSC c b

nq I n q Ii Z r r

E c E

Δz

rb

rc


Stabilization by Beam Temperature?

0

0

0

0 0 00

0

Canonical variables , /

0

current perturbation is

n

n

i n t

n

n n i n t

c

n n

p p

t

e

i ne

I q d


Dispersion Relation

0 0 0

2

0

0

2 3

0 0

2

0

0 0

0

2

0

1/

/1

2

recover before

/ 2

/

2

c

c

c

n

b

b

n n

dEE

dt

q Zi d

E n

N

N nd

n n


Landau Damping

00 0 22 2 2

0 0

0 0

2

0 0

2 2 220 0

20 0

22

0 0

0

Use our favorite analytic distribution

ˆ1 1

ˆ

ˆ

ˆ1

2 ˆ

12 ˆ

ˆ

c

c

n

c

n

n

q Z I ni d

E n

q Z I ni

E n ni

n ni V iU


2

2

1 1

2

1

2

i t

i t

b

b

i t

i t

u u Fe

eu F

dN

N d

eu F d

eu F d


LD from another view

2

1

Single Oscillator

1 1

2

Many oscillators distributed in frequency

1

1 1

2

for

i t

i t

N

i

i

i t

i t

u u Fe

Feu t

dN

N d

u

UN

FeU d

FeU d


Resonance Effect

2 2

2 2

. .

. .

For our analytic Lorentzian

Energy goes in!

Where does it go?

i t

i t

i t i t

FeU i PV d

U Fe iPV d

Fe FeU i

i


Inhomogeneous Solution

2 2

2 2

2 2

2 2

sin sin

Solution with zero initial excitation

sin sin

No energy flow

sincos

Resonant particles capture energy and oscillation

genera

Fu t a t t

Fa

Fu t t

F tu t t

ted out of phase


Oscillators Similtaneously Excited

2

1

1

1 1

2

Many oscillators distributed in frequency

1

1 1

2

for

i

i t

i t

N

i

i

i t

i t

u t

u u Fe

Feu t

dN

N d

u

UN

FeU d

FeU d


Oscillation Frequency

2

22 0 00 2

02

Re 0 1 mode has positive imaginary part

instability

Resistive impedance has positive real part

"Resistive wall instability"

If Re 0 (e.g. space charge impedance at long

cnq In i Z

E

Z

Z

wavelengths)

stability/instability depends on sign of RHS

Im 0 (inductive, stable if 0, unstable if 0)

Im 0 (capacitive, space charge is this way,

c cZ

Z

stable if 0, unstable if 0)

Later case is negative mass instability

c c


NMI Growth time

2

02 2

0

0

0

0

2 2

0 0

0

Impedance?

2 2

2 2

1 1 2ln /4

1 2ln / 1 2ln /2 2

b b

b br

b b

z c b

i n t

n

SC n c b n c b

e r e rr r c r r

r rE B

e er r c r r

r r

eE r r

z

e

in inV r r I r r

c

n

2 2 2

2 0 0 0 0

2 2 2

0 0 0

1 2ln /2 4

c cSC c b

nq I n q Ii Z r r

E c E

Δz

rb

rc


Multipass BBU Instability


BBU Theory

/2

12

following Krafft, Laubach, and Bisognano

sin2

Single cavity/Single HOM case

On the second pass

With no initial

HOM HOMQHOM HOMtransverse HOM

HOM

t

transverse

r

kRW e

Q

V t W t t I t d t dt

T eV t td t

c

12

displacement

Delay differential (integral) equation

t

transverse r

T eV t W t t I t V t t dt

c


0

0 0

0 0 0 0

0 0 0

0 0

/2

0

2/2 /2

0

Beam current

Normal mode

Sum the geometric series for eigenvalue equation

sin1

1 2 cos

/

HOM HOM

r

HOM HOM HOM HOM

m

i nt

i t t Qi t HOM

i t t Q i t t Q

HOM

HOM

I t I t t mt

V nt V e

e e tKe

e e e e t

K R Q

2

12 0 0

12

/ 2

For sin 0 1 at threshold

HOM

HOM r

k eT I t

T t K


Perturbation Theory Works

0 0 0/2

0

2

12

12

Growth rate

sinIm

2 2

Threshold current

21

/ sin

HOM HOM HOMri t t Q i ti t

HOM r HOM

HOM

HOMth

HOM HOM HOM rHOM

iKe e e e

K t

t Q

Ie R Q Q k T t


CEBAF (Design) Simulations

Bunch Number

Stable


Bunch Number

Close to

threshold


Unstable

Bunch Number


Chromatic (Landau) Damping

12

2

12,

12

12,

12,

When depends on energy offset ,

threshold current is modified to

21

/ sin

If 1, 0

HOMth

HOM HOM eff HOM rHOM

eff

eff

T

Ie R Q Q k T t

T f dT

f d

T

accelerator physics statistical and collective effects iixx yy x x y y x x x y y y c x x y y c de de...

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