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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
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Graduate Accelerator Physics Fall 2015
Chromatic Errors and Correction
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Graduate Accelerator Physics Fall 2015
Superperiod Resonances
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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
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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
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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
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Graduate Accelerator Physics Fall 2015
Action-Angle Coordinates
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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
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Graduate Accelerator Physics Fall 2015
Phase Space Structure Near
Resonance
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Graduate Accelerator Physics Fall 2015
Resonance Structure
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Graduate Accelerator Physics Fall 2015
Fourth Order Resonances
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Graduate Accelerator Physics Fall 2015
Stop Bands
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Graduate Accelerator Physics Fall 2015
Resonant Extraction
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Graduate Accelerator Physics Fall 2015
Multipass BBU Instability
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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
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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
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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
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Graduate Accelerator Physics Fall 2015
CEBAF (Design) Simulations
Bunch Number
Stable
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Graduate Accelerator Physics Fall 2015
Bunch Number
Close to
threshold
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Graduate Accelerator Physics Fall 2015
Unstable
Bunch Number
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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