Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o s ee t h is pi ct u re .
SPACE CHARGE EFFECTS
IN PHOTO-INJECTORS
Massimo Ferrario
INFN-LNF
Madison, June 28 - July 2
• Space Charge Dominated Beams
• Cylindrical Beams
• Plasma Oscillations
Space Charge: What does it mean?
Space Charge Regime ==> The interaction beween particles is dominated
by the self field produced by the particle distribution, which varies
appreciably only over large distances compare to the average separation of
the particles ==> Collective Effects
A measure for the relative importance of collective effects is the
Debye Length lD
Let consider a non-neutralized system of identical charged particles
We wish to calculate the effective potential of a fixed test charged particle
surrounded by other particles that are statistically distributed.
 r  
C
C
r
 s r   ?
e
4o
N total number of particles

n average particle density
The particle distribution around the test particle will deviate from a uniform
distribution so that the effective potential of the test particle is now defined as
the sum of the original and the induced potential:
Poisson Equation
 2s r  
e
o
r  
Local deviation from n

e
o
nr 
n r   n m r   n
Local microscopic
distribution


1 N
n m r    e r  ri 
N i 1
Presupposing thermodynamic equilibrium,nm will obey the
following distribution:
n m r   ne

n r   n m r   n  n e
e s r  / kB T
e s r  / kB T
e s r 
1 
kB T


Where the potential energy of the particles is assumed to be much smaller
than their kinetic energy

 s r  e
  s r   2   r 
lD
o
2
The solution with the boundary condition that s vanishes at infinity is:
C r / lD
s r 
 e
r
lD
lD 
o k B T
e2 n
Conclusion: the effective interaction range of the test particle is
limited to the Debye length
The charges sourrounding the test particles have a screening effect
 s r    r  for r  lD
C r / lD
s r   e
r

 s r    r  for r  lD
lD

Smooth functions for the charge and field distributions can be used
Important consequences
If collisions can be neglected the Liouville’s theorem holds in the 6-D
phase space (r,p). This is possible because the smoothed space-charge
forces acting on a particle can be treated like an applied force. Thus the
6-D phase space volume occupied by the particles remains constant
during acceleration.
In addition if all forces are linear functions of the particle displacement
from the beam center and there is no coupling between degrees of
freedom, the normalized emittance associated with each plane (2-D
phase space) remains a constant of the motion
Continuous Uniform Cylindrical Beam Model

I
a 2 v
a
J
I
a 2



  E  dS   dV
Gauss’s law
2rlo E r   2r 2 l

Ampere’s law
2lB  oJlr
Linear with r
o


Er 
r
Ir

for r  a
2
2o 2o a v
B 
 B  dl  o  J  dS
oJr
Ir
B 
 o
2
2a2
for r  a


c
Er
Lorentz Force
Fr  eE r  cB   e1  E r 
2
eE r
2
has only radial component and

is a linear function of the transverse coordinate
The attractive magnetic force , which becomes significant at high
velocities, tends to compensate for the repulsive electric force.
Therefore, space charge defocusing is primarily a non-relativistic
effect
Equation of motion:
d 2 r eE r
eI
m 2  2 
r
2
2
dt

2 o a v
2
d2r
2 2 d r
 c
dt 2
dz 2

K
eI
2I

2m 3
ov 3 Io 3  3
4o mc 3
Io 
e
d2r
eI
K

r 2 r
2
3
2 3
dz
2m o a v
a
Generalized perveance
Alfven current
Laminar Beam
If the initial particle velocities depend linearly on the initial coordinates
dr0 
 Ar0 
dt
then the linear dependence will be conserved during the motion, because of the
linearity of the equation of motion.

This means that particle trajectories do not cross ==> Laminar Beam
What about RMS Emittance (Lawson)?
x 2  2xx  x 2  rms
x’
x’rms
x
xrms
x 2  rms and
    1
2
x2  rms
 

rms 

1
2
2rms

x
2
xx 
d 2
x 
dz
rms
x   xx 
2
2

In the phase space (x,x’) all particles lie in the interval bounded by the points (a,a’).
x’
a’
a
x
What about the rms emittance?


2
rms
2
rms
 x
C
2
2
x
2
x 2  xx 
x
2n
 x
2
n 1 2

x  Cx n
When n = 1 ==> r = 0

When n = 1
==> r = 0
The presence of nonlinear space charge forces can distort the
phase space contours and causes emittance growth
 r 3 
r
I
Er 

r  2 
2 
2o 2o a v  a 
I  r 2 
r  2 1 2 
a v  a 
x’

a’
a

2
rms
C
2
x
2
x
2n
 x
x
n 1 2
 0
Space Charge induced emittance oscillations
Bunched Uniform Cylindrical Beam Model
L(t)
(0,0)
R(t)
z
l
v z  c
Longitudinal Space Charge field in the bunch moving frame:
Q
˜ 2

R L˜


˜

E z ˜z ,r  0  
4o
˜

E z ( ˜z,r  0) 
20

R 2

0
0
˜l  ˜z
 ˜ ˜ 
l  z   r 



L˜
2
0
3/2
rdrdd˜l
2

R 2  ( L˜  ˜z )2  R 2  ˜z 2  2˜z  L˜ 
Radial Space Charge field in the bunch moving frame
by series representation of axisymmetric field:

r
˜

r3
E r (r, ˜z )    E z ( ˜z ,0)     
16
0 ˜z
2


r
˜  ( L˜  ˜z )
˜z



E r (r, ˜z ) 

2
2
2
2
20 
R  ˜z 
 R  ( L˜  ˜z )
2
Lorentz Transformation to the Lab frame
L
L˜
˜
   

E r  E˜ r


Fr  e
 ˜ 
B   E r  E r
c
c

Er
2


E z (z) 
 20


R 2   2 (L  z) 2  R 2   2 z 2   2z  L
 R 2

z 2
R2
z 2  z 

L 2 2  (1 )  2 2  2  2  1
 20
L
 L L  L 

  L


L

20
z
L
A
R
L


A  (1  ) 2  A   2  2  1
Beam Aspect Ratio
r
  (1  )



E r (r,z) 

2
2
2
2
20 
A   
 A  (1  )
2
Ir

g 
2
40 R v

Q
I

R2 L R 2v
It is still a linear field with r but with a longitudinal correlation 
=1
=5
Ar,s  Rs  s L
L(t)
Rs(t)
t
 = 10
Simple Case: Transport in a Long Solenoid
ks 
qB
2mc
K   
2Ig 
Io  
3
0.9
g
g(
0.7
0.8
0.6
0.5
0
0.0005 0.001 0.0015 0.002 0.0025
metri

K  
R k R 
R
2
s
R 0 ==> Equilibrium solution ? ==> Req   

K  
ks
Small perturbations around the equilibrium solution
K  
R k R 
R
R   Req    r 
2
s
K  
r k Req  r
 Req  r
2
s
K  
K    r 
r k Req  r

1



 r  Req  Req 


Req 
1 R 


eq 
2
s

r   2ks2r   0
r   2ks2r   0

 2k z
R   r  sin  2k z
R   Req    r cos
s
s

Plasma frequency
k p  2k s
Emittance Oscillations are driven by space charge differential
defocusing in core and tails of the beam
px
Projected Phase Space
x
Slice Phase
Spaces
Envelope oscillations drive Emittance oscillations
2
1.5
R()
envelopes
1
0.5
0

0

-0.5
0
1
2
80
metri
3
4
5
60
(z)emi 40
qB
ks 
2mc
20
0
0
z 
1
R
2
2
metri
3
4
5
R2  RR  sin
2
 2k z
s
Perturbed trajectories oscillate around the equilibrium
with the
same frequency but with different amplitudes
X’
X
HBUNCH.OUT
HOMDYN Simulation
of a L-band photo-injector
6
Q =1 nC
R =1.5 mm
L =20 ps
th = 0.45 mm-mrad
Epeak = 60 MV/m (Gun)
Eacc = 13 MV/m (Cryo1)
B = 1.9 kG (Solenoid)
5
sigma_x_[mm]
4
n
[mm-mrad] 3
sigma_x_[mm]
enx_[um]
I = 50 A
E = 120 MeV
n = 0.6 mm-mrad
2
1
6 MeV
0
0
5
3.5 m
10
Z [m]
z_[m]
15