A multigrid-based beam dynamics code for high current proton linacs

Abstract

A beam dynamics code has been developed for the design assessment
of a superconducting high current proton linac for the ENEA/INFN TRASCO
Project (intended for nuclear waste transmutation in a subcritical
reactor). The code deals with the dynamics of a moderate energy (above
tens of MeV) and high current proton beam in a beamline composed of
elliptical superconducting cavities and quadrupoles. A 3D Poisson solver
based on multigrid techniques has been chosen for the space charge
calculations. The simulation results for the reference linac design,
with particular care to emittance growth and halo generation, are
reported

Full text

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A MULTIGRID-BASED BEAM DYNAMICS CODE
FOR HIGH CURRENT PROTON LINACS
Paolo Pierini
†, Giovanni Bellomo and Carlo Pagani,
INFN Milano LASA, Via Fratelli Cervi, 201 I-20090 Segrate (MI) Italy

† Email: Paolo.Pierini@mi.infn.it
Abstract
A beam dynamics code has been developed for the design
assessment of a superconducting high current proton linac
for the ENEA/INFN TRASCO Project (intended for
nuclear waste transmutation in a subcritical reactor). The
code deals with the dynamics of a moderate energy (above
tens of MeV) and high current proton beam in a beamline
composed of elliptical superconducting cavities and
quadrupoles. A 3D Poisson solver based on multigrid
techniques has been chosen for the space charge
calculations. The simulation results for the reference linac
design, with a particular care to emittance growth and
halo generation, are here reported.
1 GENERAL FEATURES OF THE CODE
The code[1] has been specifically developed for the
simulation of high current proton beams in a
superconducting linac that uses "elliptical" cavities, as is
the case for TRASCO[2]. The beam is advanced along the
linac in phase steps (where the phase is defined as
?
??
ω=φ
, and
??
ω is 2π the frequency of the linac). In
case a particle crosses an element boundary during a
timestep, a substep is performed up to the element end and
the step is then continued in the following element.
Particles crossing an element boundary where output is
desired are collected in buffers for disk storage.
The beamline is composed of a sequence of elements of
different types: quadrupoles, drift spaces and cavities. An
"ideal" 3D uniform focussing element can also be used for
analytical checks of the code.
For the propagation in the quadrupoles and drifts
analytical maps are used (which include chromatic effects
for the quadrupole), whereas for the RF cavities a direct
integration in the cavity field is performed. A cavity is
described either with an analytical field shape of an
"ideal" reduced β cavity or by an on-axis field map
(provided by any cavity electromagnetic code). Off-axis
expansion, to second order, of the electric and magnetic
fields is computed for particle tracking.
Space charge is computed in multiples of the particle
tracking phase steps. The space charge kicks can be
evaluated either using a direct point to point method or
with a fast 3D Poisson solver in the reference particle rest
frame, based on a multigrid algorithm.
1.1 RF Cavities Modeling
The codes used for the design of the TRASCO linac
replace the elliptical RF cavities with a properly chosen
"ideal" cavity with an analytical field given by:
(
ϕ+ω=
???????
?????
sin2),(
)






β
ω
?
?
??
?
sin
where Eacc is the accelerating field,
synchronous velocity and
are defined so that the cavity gives the same energy gain
of the "real" cavity (as designed by a program as
Superfish[3]) in all the energy range of the linac where the
cavity is used. Note that
?
β does not depend from the
inner cell length, it rather comes from a "dynamical"
equivalence of a real cavity with an ideal sine-like cavity.
This equivalence has been tested against a much simpler
(although much quicker in terms of computational costs)
model of a cavity: the RF gap. The transfer matrix
elements for the longitudinal phase space (∆φ,∆E) are
plotted in Figure 1 for three cavity models: the direct
integration in the
?
? field, the sine-like approximation
defined above and the RF gap equivalence, where the
whole 5 cell cavity is replaced by a single RF gap, giving
the right energy gain and phase advance at each energy.
The plots were computed for the β range corresponding to
the first section of the TRASCO linac (100-190 MeV).
?
β the cavity
?
ϕ the phase. Both Eacc and
?
β
Figure 1: Longitudinal transfer matrix elements for the
three cavity models: direct integration (solid line), sine-
like equivalent (squares, see text) and RF gap (triangles).
Phase space units are deg and MeV.
30
−=ϕ?
deg.
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The transfer matrix of the transverse plane (
shown in Figure 2, for the same cavity models of Figure 1.
It is worth noting that the largest errors are found on the
elements on the main diagonal of the transverse and
longitudinal transfer matrices, where the sine-like
equivalent has an error of 10
gives a difference of the order of 10
In both the transverse and longitudinal matrix cases,
when the cavity is operated near its nominal velocity (here
β=0.5) the three models give approximately the same
values for the elements (to few parts in 10,000).
)
?? ???,
is
-3-10
-4, whereas the RF gap
-2.
Figure 2: Transverse transfer matrix elements for the three
cavity models: direct integration (solid line), sine-like
equivalent (squares, see text) and RF gap (triangles).
Phase space units are mm and mrad.
30
−=ϕ?
deg.
1.2 The Space Charge Multigrid Algorithm
Since the code advances in time (phase), when the space
charge calculation is needed the particle coordinates are
Lorentz-transformed to the rest frame of the reference
particle, the charge distribution is evaluated using a cloud
in cell (CIC) assignment scheme and the 3D Poisson
equation is solved.
The fields computed from the solution potential are then
transformed back in the laboratory frame and the force is
interpolated on the particle position using trilinear
interpolation[4].
The Poisson equation is solved iteratively using a
multigrid solver[5], a fast iterative method that uses
standard relaxation algorithms (in our case we chose the
Gauss-Seidel) and combines it with the discretization of
the problem on a series of nested grid levels (in our case
obtained by step doubling of the finest level).
The code implements both the V-cycle and the full
multigrid[5] cycling strategies, but the V-cycle allows to
make use of the potential map computed at a previous
space charge computation in order to start from a better
estimate for the solution potential. This strategy leads to a
speed up of the space charge calculations (reducing the
time by a typical factor of 30%), in order to achieve the
specified convergence[1].
The restriction operator (needed to transfer quantities to
a coarser mesh level) is the trilinear averaging of the 27
neighboring nodes and the prolongation operator (needed
to transfer the solution to a finer mesh level) is the
trilinear interpolation.
The space charge routine based on the multigrid solver
has been thoroughly tested, both with a comparison with
the point to point space charge calculation (which,
however, is limited to a small number of interacting
particles, due to the time scaling with the squared number
of particles) and with simple analytical cases. A report of
the tests can be found in Reference 1.
Typical TRASCO simulations are performed with 10
10
space charge step every 5-15 phase steps using a mesh of
33×33×33 points, extending to ±5-6 rms beam sizes. The
whole linac runs in a few hours on a desktop system.
4-
5 particles, a phase step of a few (2-4) RF degrees, a
2 THE TRASCO LINAC SIMULATIONS
The 352 MHz superconducting linac accelerates a 25 mA
proton beam from 100 MeV to a maximum energy of
1.6 GeV. The linac is divided in three sections, with five
cell cavities designed for a synchronous β of 0.5,0.65 and
0.85, grouped in cryostats containing 2, 3 and 4 structures
each. The focussing is provided by a periodic array of
quadrupole doublets between the cryostats and the lattice
cells are 8.03, 11.152 and 15.34 meters each.
Beam dynamics studies of the TRASCO linac using the
code described here, with a number of simulation particles
ranging from few thousands to 10
reported in Refs. 6 and 7.
In Figure 3 we present the rms and 90% beam
normalized emittances along the linac. The rms emittance
growth is less than 4% in all planes. The beam distribution
is well matched along the linac, as shown by the nearly
constant behavior of the 90% emittance curves. No sign of
particle drifting away from the beam core can be seen
from the phase space plots.
5, has been already
Figure 3: Normalized emittances along the linac. Bottom
lines rms values, upper lines 90% beam emittance. The x
plane is the solid line, the y plane is the dashed line and
the z plane (on the right axis) is the dotted line. The
simulation was performed with 10,000 particles.
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Figure 4: Ratio of the sizes containing different particle
populations (including, from bottom to top 90%, 99% and
99.9% of the beam, respectively) to the rms beam size.
Another sign of the absence of beam tail formation can
be seen from Figure 4, where we plot the ratio of the size
containing 90%, 99% and 99.9% of the beam to the rms
beam size. This beam size fraction is approximately
constant (averaging the numerical noise due to the poor
statistics on the higher values of beam size fractions)
along the linac, and confirms that particles are not drifting
to the tails of the distribution, as would be the case for a
charge redistribution process, that could give rise to a
large beam halo.
The matched beam conditions at the linac input and the
matching between the three focussing sections of the linac
have been determined with the linear dynamics code,
TRACE3-D[8]. The matching between the sections is
obtained using the gradients of two quadrupole doublets
and the synchronous phase of the cavities in two
cryomodules at the section interface. RF cavities are
modeled in TRACE3-D with a modified tank element
(type 13), consistently with the multiparticle simulations.
The phase advances per cell are always lower than 90
deg in all linac sections. The phase advances per meter, as
computed by the beam dynamics code from the rms beam
parameters, can be seen in Figure 5.
Figure 5: Phase advances per meter (up triangles: x, down
triangles: y, squares: z) along the linac, as computed from
the rms beam during the simulation. The cell lengths of
the three sections are 8.03 m, 11.152 m and 15.34 m,
respectively.
The rms beam envelopes plotted in Figure 6 show a
discontinuity of the beam sizes at the interfaces, and a
small transverse mismatch in the third linac section
(visible also as oscillations in Figure 5 and due to the
approximations of the TRACE3-D matching). However,
no evident emittance growth is produced by these
discontinuities and by the beam mismatch.
Figure 6: rms beam sizes in the TRASCO linac.
3 CONCLUSIONS
A multiparticle space charge code has been developed for
high current superconducting proton linacs with elliptical
cavities. The code uses an accurate cavity modeling, that
reproduces the cavity behavior in the large velocity region
in which the cavities are effectively used. Space charge is
included with a 3D iterative Poisson solver, based on
multigrid methods.
The code has been successfully used since two years to
assess the design of the TRASCO linac.
4 ACKNOWLEDGEMENTS
We thank N. Pichoff (CEA/Saclay) for the many
discussions and his help in the multigrid algorithm testing.
5 REFERENCES
[1]P. Pierini, "A Multigrid-Based Approach to Modeling a High
Current Superconducting Linac for Waste Transmutation", in
Proceedings of the 1998 Int. Computational Acc. Physics Conf.,
September 14-18 1998, Monterey, CA.
C. Pagani, D. Barni, G. Bellomo, R. Parodi, P. Pierini, "Status of
the INFN High Current Proton Linac for Nuclear Waste
Transmutation", in Proceedings of the XIX International Linear
Accelerator Conference, August 23-28 1998, Chicago, IL.
J.H. Billen and L.M. Young, "Poisson Superfish", LA-UR-96-
1834, Los Alamos, NM, revised 1999.
"Computer Simulations Using Particles", by R.W. Hockney and
J.W. Eastwood, Adam Hilger, New York, NY 1988, ISBN 0-
85274-392-0.
"A Multigrid Tutorial", by William Briggs, SIAM, Philadelphia,
PA 1987, ISBN 0-89871-222-1.
G. Bellomo, P. Pierini, "Beam Dynamics In A Proposed 350 MHz
SC Linac For Nuclear Waste Transmutation And Energy
Production", in Proceedings of the 1998 European Particle
Accelerator Conference, Stockholm, June 21-26 1998.
G. Bellomo, P. Pierini, "Beam Dynamics in a High Current SC
Proton Linac for Nuclear Waste Transmutation", in Proceedings
of the XIX International Linear Accelerator Conference, August
23-28 1998, Chicago, IL.
K. Krandall, D. Rushtoi, "TRACE3-D Documentation", LA-UR-
97-886, Los Alamos, NM, 1987.
[2]
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[8]
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Proceedings of the 1999 Particle Accelerator Conference, New York, 1999