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Graduate Accelerator Physics Fall 2015
Accelerator Physics
General Resonance Theory and BBU
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 16
Graduate Accelerator Physics Fall 2015
Chromatic Errors and Correction
Graduate Accelerator Physics Fall 2015
Superperiod Resonances
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Action-Angle Coordinates
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Phase Space Structure Near
Resonance
Graduate Accelerator Physics Fall 2015
Resonance Structure
Graduate Accelerator Physics Fall 2015
Fourth Order Resonances
Graduate Accelerator Physics Fall 2015
Stop Bands
Graduate Accelerator Physics Fall 2015
Resonant Extraction
Graduate Accelerator Physics Fall 2015
Multipass BBU Instability
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
CEBAF (Design) Simulations
Bunch Number
Stable
Graduate Accelerator Physics Fall 2015
Bunch Number
Close to
threshold
Graduate Accelerator Physics Fall 2015
Unstable
Bunch Number
Graduate Accelerator Physics Fall 2015
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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