Download Particle-in-cell simulation study of single and multispecies

PARTICLE-IN-CELL SIMULATION STUDY OF SINGLE AND
MULTISPECIES BEAM
P. Sing Babu*, A. Goswami and V. S. Pandit
Variable Energy Cyclotron Centre, 1- AF, Bidhannagar, Kolkata-700 064, India
Abstract
A self-consistent two dimensional macroparticle model
has been developed and used to study the dynamics of
space-charge-dominated single and multispecies beams in
a solenoid based transport system. We have performed the
simulations for both emittance-dominated and spacecharge-dominated regimes of the beam. For multispecies
beam, particle-in-simulation results shows the formation
of hollow density profiles of subdominant species around
the primary beam and separation is more in the case of
initial K-V distribution of all the species. The evolution
of the beam radius and emittance growth of each species
has been investigated.
INTRODUCTION
The beam envelope equations are generally used for
studying the average beam behaviour but it is not possible
to have any information on emittance growth due to
nonlinear forces. In general case, where nonlinear forces
from both the applied and space-charge fields are present,
the analytical investigation is very difficult. For the selfconsistent description of an intense beam, particle-in-cell
(PIC) simulation methods are widely used [1-7]. In order
to understand the detailed dynamics of space charge
dominated beams with different density distributions, a
two-dimensional PIC code has been developed and used to
investigate the evolution of phase space distribution, rms
emittances, rms size, and centroid in aligned and
misaligned focussing system.
The typical proton fraction from the microwave ion
sources operating presently is of the order of 60-85% of
the extracted beam. The presence of unwanted species
alters the dynamics of the primary beam during the
transport particularly in the cases where the beam
intensity is high. In the PIC method we have included the
effect of unwanted species. The PIC code has also used to
get detailed information about the evolution of the beam
distribution and emittance growth caused due to nonlinear
effects in the case of multispecies beams. The formation
of beam hollow of unwanted species is observed due to
nonlinear space charge effect.
MACROPARTICLE MODEL
We consider a continuous, space-charge-dominated
beam propagating in a beam transport system consisting
of solenoid magnets. In the PIC method we consider both
types of misalignments of solenoids, translational as well
as tilt with respect to the symmetry axis. In this section
we have discussed the PIC method for multispecies beam.
In PIC simulation each species of the beam is
represented by a large number of macroparticles. For a
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*[email protected]
particular species, each macroparticle represents many
ions keeping charge to mass ratio q j m j of a single ion.
At first, the space-charge fields of each species are
calculated in the beam frame where the motion of
particles is non-relativistic. These fields are then Lorentztransformed to the laboratory frame [8]. The equations of
motion of macroparticles of each species are integrated
using a second order leap-frog method [1]. The transverse
calculation region is divided into n x  n y uniform
rectangular meshes to calculate charge density required
for the Poisson solver. For a particular species, the
charge of each macroparticle is distributed among the
four nearest nodes of the computational mesh, using the
area-weighting method. The Poisson equation is then
solved using the fast Fourier transform method to obtain
the space-charge potential on the computational grids
with perfect conducting wall boundary conditions. The
components of the electric field at the grid points are
evaluated numerically by differentiation of the potential at
the grid points using the centered difference scheme.
After that, these electric field components are Lorentztransformed to the laboratory coordinate system [8]. The
process is repeated for all the other species to get the
components of the electric and magnetic fields in the
laboratory frame. The space-charge fields at the position
of macroparticles from the fields at grid points are
obtained utilising an interpolation technique and the same
area weighting scheme as used for the calculation of the
charge density. For the case of beam propagation through
the solenoid focussing fields, it is convenient to work in
the Larmor frame of reference. The Larmor frame
corresponding to different beam species are different.
In 2D PIC simulation, we require the distribution of
macroparticles of each species in the 4D phase
space x, x , y , y  . In the present study, we have
considered five different distributions: K-V, waterbag,
parabolic, Gaussian and semi-Gaussian [4, 6].
RESULTS
The Variable Energy Cyclotron Centre at Kolkata is
developing a high current proton cyclotron. The injection
system consists of a 2.45 GHz microwave ion source and
two solenoid magnets of physical length of 40cm to
transport and match the proton beam at the entrance of
spiral inflector [9]. Generally, microwave ion sources
produce proton fraction only 60-85% of the beam. The
other major unwanted species are H2+ and H3+ beams.
In the simulation, the transverse calculation region
having dimension 12.8 cm ×12.8 cm has been divided
into uniform rectangular meshes with 128 meshes in each
direction. We have used a step size of s  1 mm in the
axial direction of the beam propagation. The number of
macroparticles used for each species is 77000. Typical
values of normalised emittances (in πmmmrad) for p, H2+
and H3+ species used in the simulation are 0.8, 0.4 and
0.27 respectively. The beam energy is 80 keV.
 the growth is very rapid and the emittance growth vs. 
for different combinations of current and emittance are
similar.
Simulation of single species beam
Figure 1 shows the evolution of rms emittance for 10
mA beam. The beam is loaded initially according to five
different distributions: K-V (KV), waterbag (WB),
parabolic (PA), semi-Gaussian (SG) and Gaussian (GA).
We have seen that the variation of rms beam sizes as a
function of axial distance is almost similar for all the
distributions. It is interesting to note from Fig. 1 that the
rms emittance for all the distributions shows an
oscillating pattern except the KV distribution. There is an
increase in the emittance which is very small in the case
of waterbag and parabolic distributions and relatively
more for the GA and SG distributions. The emittance for
all the distributions increases slowly in the initial part of
the beam line, reaches its first maximum at s ~112 cm just
before the first waist and then decreases with s. It again
starts increasing from the entrance of the second solenoid
and reaches to its maximum before the second waist. The
reduction in the emittance between the region s  0 cm to
s  40 cm for the semi-Gaussian distribution is due to the
relaxation. Figure 2 shows the phase space distributions
of KV and GA beam at s = 0 and s =277cm.
Figure 2: Phase space distributions at s = 0 and s = 277cm
for K-V (KV) and Gaussian (GA) distribution.
Figure 3: The growth in the emittance as a function of
intensity parameter  for Gaussian distribution.
Simulation of multispecies beam
Figure 1: Evolution of rms emittance εrms(s)/ εrms(0) for
different beam distributions.
We have studied the emittance evolution of Gaussian
distribution for different values of  by varying current
from 1mA to 40mA with three values of rms emittances
(in  mmmrad) 3.82, 7.64 and 15.27. The dimensionless
1
intensity parameter   1  8I 0 a 2  n2rms / I  is the ratio
of the space charge force to the external focusing force
defined in smooth focusing approximation [10]. Here
 n rms   rms is the normalised rms emittance and
a  2 X rms (s) is the average of the envelope size over the
path length. The average emittance growth as a function
of  is shown in Fig. 3. It is clear that for higher values of
Figure 4(a) shows the evolution of the beam envelope
of the different species for 10 mA total beam current. The
peak magnetic fields of the solenoids S1 and S2 are 2.8
kG and 2.7 kG, respectively. A slit of radius 5 mm is used
at s  135 cm for the selection of proton beam. In the
simulation p, H2+ and H3+ beams are K-V distributed
initially and fractions (in %) are [60, 20, 20] respectively.
The evolution of emittances of p, H2+ and H3+ beams is
shown in Fig. 4(b). There is a very little growth in the
emittance of proton beam, however the situation is not
same in the case of H2+ and H3+ beams. It can be readily
seen that there is a sharp increase in the emittances of H2+
and H3+ beams after s  100 cm. The main contributor to
this emittance growth is the nonlinear space-charge force
of the converging intense proton beam. At the slit major
portions of the H2+ and H3+ beams are rejected together
with the substantial reduction in their emittances. We see
a sharp increase in the emittance of H2+ and H3+ after the
slit in the region where the proton beam size is small. The
emittance of these species remains almost constant in the
region where the proton beam size is comparable to the
beam sizes of H2+ and H3+ and again it increases as the
proton beam converges. The growth in emittance of H2+
beam is mainly due to the nonlinear space-charge force of
proton beam, whereas the growth in the emittance of H3+
beam is due to the nonlinear space-charge forces of
proton and H2+ beams.
PIC simulation shows the formation of hollow density
profile of H2+ and H3+ beams around the proton beam,
which becomes more distinct after the second solenoid.
We have performed simulations for different species
fraction in total beam current of 10 mA. Figure 5 shows
the real space as well as phase space distributions of p,
H2+ and H3+ beams at s  257 cm, just before the second
waist of the proton beam, where the distributions of the
H2+ and H3+ beams are well separated from the proton
beam. The fractions (in %) of p, H2+ and H3+ beams used
in the simulation are [80, 15, 5]. It is readily seen from the
figure that H2+ and H3+ beams are separated from proton
beam and phase space distributions of H2+ and H3+ beams
are highly distorted for KV distribution. When the initial
distributions of all the species are Gaussian, phase space
distributions of all the species are highly distorted.
CONCLUSION
Figure 4: The evolution of (a) rms beam sizes and (b) rms
emittances of p, H2+ and H3+ beams for I  10 mA. A
circular slit of radius 5 mm is placed at s  135 cm.
A self-consistent PIC method has been developed and
utilized to study the dynamics of space-charge-dominated
single and multispecies beams in a solenoid based
transport system. We have performed the simulations for
both emittance-dominated and space-charge-dominated
regimes (0.3<  <0.99) of the beam. It is shown that for
higher values of  the growth is very rapid and the
emittance growth vs.  for different combinations of
current and emittance are similar. The growth in the
emittance is seen to be more in the converging region of
the beam where the strength of nonlinear space-charge
force is comparatively stronger.
In the case of multispecies beam, the evolution of rms
emittance of proton is very small whereas the rms
emittance growth of subdominant species H2+ and H3+
beams is large due to nonlinear space charge force. It is
found that H2+ and H3+ species, which initially have K-V
distributions, develop hollow density profiles around the
proton beam, downstream the transport line.
REFERENCES
Figure 5: Real space and phase space distributions of H2+
and H3+ at s  257 cm with initial K-V (KV) and
Gaussian (GA) distribution.
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