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...
TRANSCRIPT
![Page 1: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/1.jpg)
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
Debye Length Picture*
*Davidson and Qin
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
![Page 24: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/24.jpg)
USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
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
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USPAS Accelerator Physics June 2016
Multipass BBU Instability
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USPAS Accelerator Physics June 2016
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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USPAS Accelerator Physics June 2016
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
![Page 33: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/33.jpg)
USPAS Accelerator Physics June 2016
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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USPAS Accelerator Physics June 2016
CEBAF (Design) Simulations
Bunch Number
Stable
![Page 35: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/35.jpg)
USPAS Accelerator Physics June 2016
Bunch Number
Close to
threshold
![Page 36: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/36.jpg)
USPAS Accelerator Physics June 2016
Unstable
Bunch Number
![Page 37: 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](https://reader030.vdocuments.site/reader030/viewer/2022040115/5e819a50e54da6353e2d5e13/html5/thumbnails/37.jpg)
USPAS Accelerator Physics June 2016
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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