Space Charge Effect

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
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.
The as represent the 3 half axis in the rest
frame
3
Space Charge Fields in a Bunch (2)
The 2-D ellipsoid case (cigar-like bunch)
The elliptic integral for the longitudinal
direction reduces to
Where the as are respectively the bunch radius
and the half bunch length in the laboratory
system. Solving the integral
4
Longitudinal Space Charge Force
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
In the Lab. system, neglecting transverse motion
Space Charge is a defocusing effect leading to
bunch lengthening
6
Longitudinal Space Charge in a Linac
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
Energy deviation of a particle with respect to
the reference one
Derivation with respect to time leads to
since
and
8
Longitudinal Space Charge in Synchrotron (2)
The second order energy equation can then be
writen
with
showing that - below transition (? gt 0) the
S.C. effect is defocusing - above transition
(? lt 0) the S.C. effect is focusing The later is
often referred to negative mass effect
9
Longitudinal Space Charge in Synchrotron (3)
In the case of a cigar-type beam, following
Reisers 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).
10
Radio-Frequency Gun
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.
Ez
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.
12
Acceleration of Intense Beams
Obviously the accelerated beam gets its energy
from the stored energy in the cavity
PRF Pdiss. Pbeam The
cavity voltage is the vector sum of the voltage
due to the generator and the beam loading
Vt VRF Vbeam ZRF
Ig Zb Ib 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, Qb . The loaded Q
becomes

13
Acceleration of Intense Beams (2)
Equivalent circuit with beam
During acceleration a synchronous phase is
established between the current and the voltage

Vt
Ib
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.