accelerator physics general resonance theory and...

Graduate Accelerator Physics Fall 2015

Accelerator Physics

General Resonance Theory and BBU

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 16


Chromatic Errors and Correction


Superperiod Resonances


Action-Angle Variables

22

0

0

2 22

0

22

0 0

0

Use coordinates in phase space analogous to polar

coordinates, but including the non-linearities.

1 1 1

2 2 2

has units of action. For harmonic oscillator

2 2

1 1/

2 2

H

wJ pdq w

wwH

wJ w H

0

0

amilton equations of motion

0 is constant

2 sin 2 cos

dJ HJ

dt

d H

dt J

w J w J


Real Pendulum

0

0

2 2 2

1 cos for small displacements2 2 2

sin

sin 0 as should be

Original phase space coordinates ,

Introduce action

2 212 cos

2

p pH mgR mgR

m m

p dpd H HmgR

dt p m dt

g

R

p

gRJ p d m

0

0

0

0 0

0 00

cos

where

1 cos

Solvable in elliptical integrals

1 cos 1 cos1 1 1 12 1 cos , ;1

frequency now depends on amplitud

; 2 , ;1;2 2 2 2 2 2

key p e (actoint: ion)

d

H mgR

J m gR F F

θ R


Nonlinear Hamiltonian

/22 2 2 0

0 /2

/2

0

22

0

0

0

/2

0

1 1ˆ

2 2 2

ˆ2

Introduce action-angle variables in unperturbed

Hamiltonian

1 1 1

2 2 2

2cos

cos

betatron phase arb

nn

w n n

n

n

n

n n

n

H w w p w

pp

n

wJ pdq w

Jw

H J p J

itrary phase


Action-Angle Coordinates


Definitions

/2 /2

1 0 0 1 1 1

0

0

/2 1 /2 1

0 0

0 0

/2 2

0

th order Perturbed Hamiltonian

cos

Amplitude Ratio

Detuning

Tune Spread Parameter

n n

r n n nr r

rr

n n

nr nr

n n

n

nr

n

H c p J p J m

JR

J

p J p J

c p

p J


Phase Space Structure Near

Resonance


Resonance Structure


Fourth Order Resonances


Stop Bands


Resonant Extraction


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

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