1
Space Charge Effect
Up to now single particle longitudinal dynamics has been considered.
In a real beam (or bunch), with many particles, each particle will
suffer the repulsive forces from the others since they have the
same electrical charge.
This intrinsic effect is however important only at low energies and
vanishes for ultra

relativistic beams where magnetic forces
compensate electric forces.
The space charge forces affect the longitudinal dymamics (as well as
the transverse one).
Since request for higher and higher intensities, at low and medium
energies, is driving the next future the space charge phenomenon
needs particular attention.
2
Space Charge Fields in a Bunch
z
y
x
Assume a uniform particle
distribution inside a 3

D
ellipsoidal beam shape.
In the moving frame of the
bunch the S.C. force is purely
electrostatic; the field
components can be obtained
analytically solving Poisson
equation.
z
y
x
u
M
u
E
u
u
,
,
2
0
0
2
/
1
2
2
2
2
z
y
x
u
z
y
x
u
a
a
a
a
d
a
a
a
M
The a’s represent the 3 half axis in the rest frame
3
Space Charge Fields in a Bunch (2)
r
y
x
a
a
a
0
2
/
3
2
2
2
z
r
z
r
z
a
a
d
a
a
M
2
2
2
2
1
1
1
1
2
1
1
2
z
r
z
a
a
with
Ln
M
case
bunch
short
M
a
a
C
S
transv
pure
E
M
a
a
z
r
z
z
z
z
r
2
.
.
.
0
0
The 2

D ellipsoid case (“cigar

like” bunch):
The elliptic integral for the longitudinal direction reduces to:
Where the a’s are respectively the bunch radius and the half
bunch length in the laboratory system. Solving the integral:
4
Longitudinal Space Charge Force
z
M
e
M
z
e
M
z
e
eE
F
z
z
z
z
z
0
0
0
2
2
2
0
2
/
1
2
2
2
2
2
z
y
x
z
z
y
x
z
a
a
a
a
d
a
a
a
M
with
c
f
frequency
RF
or
bunch
the
f
Ne
I
current
beam
average
the
a
a
a
bunch
the
of
volume
the
b
b
z
y
x
)
(
3
4
z
a
a
a
M
I
c
e
F
z
y
x
z
z
0
8
3
The force acting on a single particle having a longitudinal position z with
respect to the centroid, in the Lab. system, is:
5
Longitudinal Dynamics with Space Charge
dt
d
c
m
dt
d
dt
d
c
m
dt
dm
v
dt
dv
m
dt
dp
F
z
z
z
z
3
0
0
3
0
2
2
m
F
dt
z
d
z
z
a
a
a
M
I
c
m
e
dt
z
d
z
y
x
z
3
0
0
2
2
8
3
0
2
sc
k
In the Lab. system, neglecting transverse motion:
Space Charge is a defocusing effect leading to bunch lengthening
6
Longitudinal Space Charge in a Linac
2
2
2
2
0
2
2
0
2
2
0
s
sc
c
k
with
z
k
k
dt
z
d
)
(
2
5
2
3
2
0
z
sc
M
that
note
k
while
k
Adding the focusing effect from the RF:
At low energy and high current the space charge effect can be
dramatic. Increasing the RF power is expensive.
Particular attention is to be given to the new generation of high
current proton linacs.
7
Longitudinal Space Charge in Synchrotron
sc
z
s
s
s
s
turn
E
R
e
T
eV
T
eV
T
T
E
E
dt
d
2
1
h
R
M
R
dT
dT
dV
T
e
E
dt
d
s
z
s
s
s
0
2
2
E
M
p
R
E
dT
dV
E
T
e
dt
E
d
z
s
s
s
s
s
0
2
2
2
2
2
s
s
E
E
T
T
dt
dT
2
E
R
p
h
s
s
Energy deviation of a particle with respect to the reference one
:
since
and
Derivation with respect to time leads to:
8
Longitudinal Space Charge in Synchrotron (2)
0
2
2
2
2
E
k
dt
E
d
sc
s
0
0
2
2
m
M
e
k
z
sc
The second order energy equation can then be writen:
with:
showing that:

below transition (η > 0) the S.C. effect is defocusing

above transition (η < 0) the S.C. effect is focusing
The later is often referred to “ negative mass effect”
9
Longitudinal Space Charge in Synchrotron (3)
1
1
1
2
1
2
2
2
2
2
0
Ln
M
a
a
g
z
r
z
0
3
3
2
0
2
2
3
g
a
N
c
r
k
z
sc
0
r
In the case of a “cigar

type” beam, following Reiser’s book, one can
introduce a new form factor:
Leading to:
where N is the number of particles and is the classical
electron radius.
As can be seen the S.C. factor varies like while the
corresponding RF factor varies like .
Though the S.C. effect decreases rapidly with energy, special care
has to be taken in the vicinity of transition energy (dilution).
0
r
3
1
10
Radio

Frequency Gun
4
/
2
/
E
z
Photo

cathode
Specifically designed for high intensity, low
energy, electron beam; a multi

cells high Q
cavity provides a large electric field that
rapidly accelerates the beam to ultra

relativistic energy, hence reducing the
space charge effect; it also bunches the
beam but giving large energy spread.
cathode
the
at
particle
the
of
phase
RF
c
k
t
kz
E
E
z
0
0
0
0
2
sin
cos
Generally a short pulse laser hits a photo

cathode to generate short electrons pulses.
11
Radio

Frequency Quadrupole
Specifically designed for intense low velocity protons (or ions) beams; it both
accelerates and focus to control space charge effects (see A. Lombardi lecture)
4 vanes resonator that
provides a quadrupolar
symmetry which gives a
transverse E gradient
for focusing.
Modulated pole shapes provide a longitudinal E
field for acceleration and bunching.
)
(
sin
cos
)
(
2
cos
2
0
2
t
kz
kr
I
A
a
r
X
V
U
)
(
1
;
)
(
/
1
;
2
0
0
0
2
2
ka
AI
X
mka
I
ka
I
m
m
A
k
12
Acceleration of Intense Beams
Obviously the accelerated beam gets its energy from the stored
energy in the cavity:
P
RF
= P
diss.
+ P
beam
The cavity voltage is the vector sum of the voltage due to the
generator and the “beam loading”:
V
t
= V
RF
+ V
beam
= Z
RF
I
g
+ Z
b
I
b
Under proper matching and tuning (cavity on

resonance) the impedance
is just the shunt impedance
R
.
Since the beam loading is just like a power loss one can introduce a
corresponding Q factor,
Q
b
. The loaded Q becomes:
b
e
l
Q
Q
Q
Q
1
1
1
1
0
13
Acceleration of Intense Beams (2)
Equivalent circuit with beam
I
b
V
t
During acceleration a
synchronous phase is
established between the
current and the voltage:
s
b
t
b
I
V
P
sin
2
1
The resulting effect is a detuning of the cavity ; a
feed back system is used to compensate for that.
Optimum power transfer to the cavity and beam is
made by proper matching of the power supply to the
cavity through a feeder and a coupling loop.
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