ORBIT: A Code for Collective Beam Dynamics
in High-Intensity Rings
J.A. Holmes*, V. Danilov*, J. Galambos*, A. Shishlo*, S. Cousineau®,
W. Chou*, L. Michelottf, J.-R Ostiguy*, J. Wei&
*Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
@
Indiana University, Bloomington, IN 47405, USA
#
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
&
Brookhaven National Laboratory, Upton, NY, 11973, USA
Abstract. We are developing a computer code, ORBIT, specifically for beam dynamics calculations in highintensity rings. Our approach allows detailed simulation of realistic accelerator problems. ORBIT is a particle-incell tracking code that transports bunches of interacting particles through a series of nodes representing elements,
effects, or diagnostics that occur in the accelerator lattice. At present, ORBIT contains detailed models for strip-foil
injection, including painting and foil scattering; rf focusing and acceleration; transport through various magnetic
elements; longitudinal and transverse impedances; longitudinal, transverse, and three-dimensional space charge
forces; collimation and limiting apertures; and the calculation of many useful diagnostic quantities. ORBIT is an
object-oriented code, written in C++ and utilizing a scripting interface for the convenience of the user. Ongoing
improvements include the addition of a library of accelerator maps, BEAMLINE/MXYZPTLK; the introduction of
a treatment of magnet errors and fringe fields; the conversion of the scripting interface to the standard scripting
language, Python; and the parallelization of the computations using MPI. The ORBIT code is an open source,
powerful, and convenient tool for studying beam dynamics in high-intensity rings.
INTRODUCTION
High-intensity proton ring beams are characterized by
low energy, high intensity, and low loss requirements.
Satisfying the beam-loss requirements necessitates a
detailed understanding of beam dynamics in this regime.
At high intensity, collective effects due to space charge
and wakefields strongly affect the beam behavior, and
single particle models do not suffice. Also, because of the
complexity of collective phenomena for bunched beams
in high-intensity rings, a computational approach is
productive for theoretical studies and indispensable in
solving detailed design and engineering problems.
Recognizing this, the SNS Accelerator Physics Group
at ORNL, with help from colleagues at BNL, undertook
the development of an object-oriented general-purpose
code, ORBIT [1,2]. ORBIT began as a C++ rewrite of
ACCSIM [3], developed under the SuperCode driver shell
[4], but has since undergone extensive independent
development, which will be described here.
ORBIT is a particle tracking code in 6D phase space.
Its basic classes are herds, which are groups of particles,
and nodes, which operate on the herds. ORBIT has been
designed to simulate real machines: it has detailed models
for transport through various lattice elements; injection
foil and painting; rf and acceleration; 2.5D and 3D space
charge; longitudinal impedance and space charge;
transverse impedance; apertures: collimation; and beam
diagnostics.
Present work on ORBIT focuses on the exploitation of
some of the more recent and computationally demanding
physics modules and on the enhancement of overall
capabilities. With the completion of the transverse
impedance, 3D space charge, and collimation routines, the
physics for new classes of problems is available.
However, meaningful calculations will require high
performance computing techniques, such as the
implementation of parallel computing. Also, because
ORBIT has developed with an emphasis on collective
effects, single particle transport models are still
rudimentary. In order to assess losses to the accuracy
demanded in recent and future high intensity rings, high
order maps and a comprehensive treatment of magnet
errors are necessary. Finally, although SuperCode is an
innovative tool, it is neither standard nor supported.
Comparable tools, based on the Python programming
language, now are standard and present an attractive
option.
Consequently, we are now developing parallel
algorithms for collective beam dynamics, including higher
order single particle maps, creating magnet error routines,
and planning to implement a new driver shell based on
Python. We now describe the models in ORBIT.
COMPLETED ORBIT MODELS
The coordinate representation in ORBIT utilizes the
usual accelerator expansion. The horizontal direction is
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
2002 American Institute of Physics 0-7354-0097-0
241
representedby
by x,x , xx′, , the
the vertical
vertical direction
direction by
by yy ,, /,
y′ ,
represented
φ , and
and
andlongitudinal
longitudinalphase
phasespace
spaceby
bythe
thephase
phase angle
angle (/>,
and
energy deviation,
deviation, dE
from aa specified
specified closed
closed orbit
orbit
dE, , from
energy
reference particle.
particle. The
The independent
independent variable
variable isis the
the
reference
machinelocation
locations.
s.
machine
ORBITcan
canconstruct
construct lattices
lattices by
by reading
reading output
output files
files
ORBIT
fromMAD
MAD[5]
[5]ororDIM
DIMAD
[6],by
bydirect
direct specification,
specification, or
or
from
AD [6],
specifying aa uniform
uniform focusing
focusing channel.
channel. Linear
Linear
byby specifying
transport isis carried
carried out
out through
through symplectic
symplectic matrix
matrix
transport
multiplication.
Nonlinear
elements
can
be
evaluated
in
multiplication. Nonlinear elements can be evaluated in
the
thin
lens
approximation,
and
higher
order
single
the thin lens approximation, and higher order single
particle transport
transport terms,
terms, such
such as
as chromaticity,
chromaticity, can
can be
be
particle
evaluatedusing
usingsecond
secondorder
ordertransport
transportmatrices.
matrices. There
There is
is
evaluated
specificfacility
facilityfor
for the
thetreatment
treatment ofof errors
errors or
or fringe
fringe
nonospecific
fields. Field
Field errors
errors inin straight
straight lattice
lattice elements
elements can
can be
be
fields.
calculated
in
MAD
and
included
in
the
MAD
output.
calculated in MAD and included in the MAD output.
ORBIT can
can inject
inject particles
particles turn-by-turn
turn-by-turn or
or utilize
utilize aa
ORBIT
completedistribution
distribution specified
specified atat the
the start.
start. AA variety
variety of
of
complete
distributions
can
be
generated
internally.
Any
externally
distributions can be generated internally. Any externally
generateddistribution
distributioncan
canbe
beimported.
imported. Injection
Injection painting
painting
generated
treatedwith
with user-defined
user-defined time-dependent
time-dependent closed
closed orbit
orbit
isistreated
bumps. ORBIT
ORBIT contains
contains an
an injection
injection foil
foil model
model that
that
bumps.
keepstrack
trackofoffoil
foilhits
hitsand
andapplies
appliestransverse
transverse kicks
kicks based
based
keeps
multipleCoulomb
Coulomb scattering.
scattering. Particles
Particles that
that miss
miss the
the
ononmultiple
foil
at
injection
are
removed
from
the
beam.
foil at injection are removed from the beam.
The RF
RF cavity
cavity model
model provides
provides longitudinal
longitudinal kicks
kicks
The
based on
on aa specified
specified time-dependent
time-dependent waveform
waveform with
with
based
multiple
harmonics.
For
nonaccelerating
cases,
the
multiple harmonics. For nonaccelerating cases, the
synchronousphase
phaseisis assumed
assumed toto be
be zero,
zero, and
and only
only the
the
synchronous
harmonics and
and time-dependent
time-dependent voltages
voltages need
need to
to be
be
harmonics
specified. For
Foraccelerating
acceleratingcases,
cases,the
theharmonics,
harmonics, voltages,
voltages,
specified.
andtime-dependent
time-dependentdipole
dipolefields
fieldsmust
must be
be specified.
specified. The
The
and
modeluses
usesthis
thisinformation
informationtotocalculate
calculate the
the synchronous
synchronous
model
phaseand
andthe
theresulting
resultingkicks.
kicks. The
Thetransverse
transversephase
phase space
space
phase
alsoadjusted
adjustedtotoconserve
conservenormalized
normalizedemittance.
emittance.
isisalso
The 2.5D
2.5D space
space charge
charge model
model isis implemented
implemented as
as aa
The
seriesofofkicks
kicks separated
separated by
by other
other transport
transport operations.
operations.
series
Particles are
are binned
binned inin aa 2D
2D rectangular
rectangular grid
grid using
using aa
Particles
secondorder
orderdistribution
distribution scheme.
scheme. The
The potential
potential for
for the
the
second
distributedcharges
chargesisisthen
then solved
solved on
on the
the transverse
transverse grid
grid
distributed
using aa fast
fast FFT
FFT solver.
solver. Conducting
Conducting wall
wall (circular,
(circular,
using
elliptical,ororrectangular
rectangular beam
beam pipe)
pipe) boundary
boundary conditions
conditions
elliptical,
arethen
thenimposed
imposedusing
using aa method
method described
described in
in Ref.
Ref. [7].
[7].
are
Particle kicks
kicks are
are obtained
obtained from
from second
second order
order
Particle
interpolationofofthe
thepotential,
potential, completing
completing what
what might
might be
be
interpolation
calledaa"quasi-symplectic"
“quasi-symplectic”evaluation.
evaluation. Finally,
Finally, the
the kicks
kicks
called
areweighted
weightedby
bythe
thelocal
locallongitudinal
longitudinal density
densityto
to account
account
are
forbunch
bunchfactor
factoreffects.
effects. This
This isis the
the reason
reason we
we call
call the
the
for
model 2.5D.
2.5D. There
There isis also
also an
an alternative
alternative direct
direct force
force
model
(momentum-conserving) solver
solver without
without beam
beam pipe
pipe that
that
(momentum-conserving)
usesaamethod
methoddescribed
describedininRef.
Ref. [8].
[8].
uses
Figure 1.
1. 3D
3D space
Figure
space charge
charge benchmark
benchmark for
for triangular
triangular
charge distribution:
distribution: a)
charge
a) tune
tune shifts
shifts vs.
vs. position,
position, b)
b)
longitudinal force
force vs.
vs. position.
position. 2.5D
longitudinal
2.5D and
and 3D
3D results
results agree
agree
with each
each other
other and
and with
with
with analytic
analytic calculations.
calculations._______
The 3D
3D spacecharge
spacecharge model
model is
The
is aa simple
simple generalization
generalization
of
the
2.5D
routine.
Particles
are
of the 2.5D routine. Particles are distributed
distributed to
to aa 3D
3D
rectangular
grid
using
a
second
order
scheme.
Typically,
rectangular grid using a second order scheme. Typically,
for rings,
rings, the longitudinal spacing greatly exceeds
for
exceeds the
the
transverse spacing. The potential is solved
transverse
solved as
as aa 2D
2D
problem using the distributed charges and
problem
and fast
fast Fourier
Fourier
transforms on the transverse grid for
transforms
for each longitudinal
longitudinal
slice.
Conducting
wall
boundary
conditions
slice.
conditions (circular,
(circular,
elliptical, or rectangular beam pipe)
elliptical,
pipe) are
are used
used to
to “tie
"tie
together” the transverse solutions into aa 3D
together"
3D potential.
potential.
Particle kicks are obtained by interpolating the
Particle
the potentials
potentials
in 3D using a second order
in
order “quasi-symplectic”
"quasi-symplectic"
interpolation scheme.
interpolation
ORBIT treats longitudinal impedances
impedances and/or
and/or space
space
charge in a fashion similar to ESME [9].
charge
[9]. The
The longitudinal
longitudinal
impedance is represented as harmonics of
impedance
of the
the fundamental
fundamental
ring
frequency.
Particles
are
binned
longitudinally
ring frequency.
longitudinally and
and
the binned
binned distribution is Fourier transformed. The
the
The space
space
charge contribution to the impedance is combined
charge
combined with
with
the external
external impedance.
The Fourier
the
Fourier transformed
transformed
distribution is multiplied by the impedance
distribution
impedance to
to give
give the
the
longitudinal kicks to the particles. For
longitudinal
For many
many rings,
rings, the
the
synchrotron period is much longer than
than aa turn,
turn, and
and itit isis
sufficient to evaluate the longitudinal impedance
sufficient
impedance and
and
space charge kicks once each turn.
turn. More
More frequent
frequent
evaluations may be required for
evaluations
for rings having
having higher
higher
synchrotron frequencies.
Transverse impedances are also treated
Transverse
treated as
as localized
localized
nodes in ORBIT. The validity of this approach
nodes
approach requires
requires
the element
element length to be short compared to
the
to the
the betatron
betatron
oscillation wavelength. Otherwise, multiple
oscillation
multiple impedance
impedance
nodes are required. As with the longitudinal
nodes
longitudinal impedance,
impedance,
the transverse
transverse impedance is represented by
the
by its
its Fourier
Fourier
components, but now the appropriate frequencies are
components,
are the
the
betatron sidebands of the ring frequency harmonics.
betatron
harmonics. For
For
both impedance models, the formulation incorporates
both
incorporates
velocities below the speed of light. Particle
velocities
Particle kicks
kicks are
are
obtained by convolution of the beam current
obtained
current dipole
dipole
moment with the impedance. In evaluating
moment
evaluating the
the beam
beam
current
moment evolves
evolves from
from
current it is assumed that the dipole moment
the
betatron oscillation.
oscillation.
the previous
previous turn through a simple betatron
242
1-3
emittances at any point/node in the ring; histograms of
emittances
atat any
the
ring;
histograms
ofof
emittances
any point/node
point/node
theand
ring;
histograms
particle
distributions
in x, y, inin
phi,
emittance;
rms
particle
distributions
in
x,
y,
phi,
and
emittance;
rms
particle
distributions
in
x,
y,
phi,
and
emittance;
rms
emittances or beam moments versus turn or versus
emittances
oror beam
moments
versus
emittancesstatistical
beam
momentsof
versus
turn oror versus
versus
position;
calculation
beta turn
functions;
and
position;
calculation
ofofcentroid.
beta
position; statistical
statistical
calculation
beta functions;
functions;
and
longitudinal
harmonics
of the beam
Becauseand
of
longitudinal
harmonics
Because
ofof
longitudinal
harmonicsof
ofthe
thebeam
beamcentroid.
centroid.
Because
the
recent development
of
3D
models
in ORBIT,
we are
the
development
ofof3D
models
ininORBIT,
we
the recent
recent our
development
modelsmore
ORBIT,
weare
extending
diagnostics
to3Dinclude
functions
ofare
extending
our
diagnostics
to
include
more
functions
ofof
extending
our
diagnostics
to
include
more
functions
longitudinal position.
longitudinal
longitudinalposition.
position.
'
m
ONGOING DEVELOPMENTS
ONGOING
ONGOINGDEVELOPMENTS
DEVELOPMENTS
1:1
Figure 2.
Phase space coordinates for benchmark of
Figure
2. impedance
Phase space
for calculation
benchmark
of
Figure
2.
Phase
coordinates
benchmarkfor
of
transverse
model
with analytic
transverse
impedance model with analytic calculation
transverse
impedance
calculation for
for
pencil beam.
pencil beam.__________________________
beam.
pencil
Circular, elliptical, rectangular, or racetrack apertures
Circular,
elliptical,
or can
racetrack
racetrack
apertures
areCircular,
defined inelliptical,
ORBIT. rectangular,
The apertures
be set apertures
either to
are
defined
in ORBIT.
can be tabulate
set
to
are
defined
in
apertures
set either
eitherthe
to
allow
particles
to pass The
through
and simply
allow
particles
andthesimply
tabulate
the
allow
to the
pass
through
tabulate
the
hits, orparticles
to remove
particles
from
beam and
tabulate
hits,
or to
tolocations.
remove the particles from the beam and
hits,
or
remove
and tabulate
tabulate
the loss
the loss
loss locations.
locations.
the
ORBIT contains a detailed collimation model.
ORBIT shapes
contains
collimation
model.
ORBIT
contains
a detailed
collimation
model.
Collimator
include
the aperture
shapes, single
or
Collimator
shapes
include the
aperture
shapes,
single
Collimator
single or
or
multiple edges
at arbitrary
angles,
and also
a rectangular
multiple
edges at that
and beam
also aawindows
rectangular
multiple
edges
arbitrary
rectangular
plate collimator
can beangles,
used for
or
plate
collimator
used for
beam windows
or
plate
can be
windows
or
foils. collimator
Physics that
includes
multiple
Coulomb
scattering,
foils.
Physics
Coulomb
foils.
Physics
includes
multiple elastic
Coulomb
scattering,
ionization
energy
loss, nuclear
andscattering,
inelastic
ionization
energy
elastic
inelastic
ionization
loss, nuclear
inelastic
scattering, energy
and Rutherford
scattering
for and
aand
number
of
scattering,
and
Rutherford
scattering
a for
number
of
scattering,
Rutherford
for
number
of
materials. and
Monte
Carlo algorithms
arefor
used
particle
materials.
Monte
are
used
for
particle
materials.
Monte
Carlo
algorithms
for
particle
transport inside the collimator, and step sizes are carefully
transport
inside
the collimator,
and step sizes are carefully
transport
inside
the
carefully
adjusted near
collimator
boundaries.
adjusted near
near collimator
collimator boundaries.
adjusted
The present emphasis in ORBIT development is on
The
development
isisonon
The present
presentofemphasis
emphasis
inORBIT
ORBIT
development
improvement
both in
physics
and
computational
improvement
both
improvementIn of
of physics
both physics
physics and
andarecomputational
computational
capabilities.
the
category,
we
significantly
capabilities.
InInthe
physics
category,
we
capabilities.the
the
physics
category,
weare
aresignificantly
significantly
extending
single
particle
transport
model
by
extending
transport
extending the
the single
single
particle
transport model
model byby
interfacing
ORBIT
and particle
the
BEAMLINE/MXYZPTLK
interfacing
ORBIT
interfacing
ORBIT
and the
the BEAMLINE/MXYZPTLK
BEAMLINE/MXYZPTLK
library
for maps
and and
tracking
[10].
In addition, we are
library
for
maps
and
tracking
[10].
InInaddition,
we
library
for
maps
and
tracking
[10].
addition,
weare
are
including a comprehensive set of
magnet
error routines
including
comprehensive
ofofmagnet
error
routines
including
comprehensive
set
magnet
error
routines
and
a hardaa edge
fringe fieldset
model.
At this
time,
the
and
edge
AtAtthis
time,
the
and aa hard
hard
edge fringe
fringe field
field model.
model.interface
thishas
time,
the
ORBIT
- BEAMLINE/MXYZPTLK
been
ORBIT
-- BEAMLINE/MXYZPTLK
ORBITcompleted
BEAMLINE/MXYZPTLK
interfacehas
hasbeen
been
partly
with some options interface
working,
and
the
partly
completed
with
some
options
partlywill
completed
with
some
optionsworking,
working,and
andthe
the
errors
be included
this
summer.
errors
errorswill
willbe
beincluded
includedthis
thissummer.
summer.
The most urgent computational improvement to
The
computational
improvement
toto
The most
urgent
computational
improvement
ORBIT
ismost
theurgent
introduction
of parallel
processing
ORBIT
is
the
introduction
of
parallel
processing
ORBIT
is
the
introduction
of
parallel
processing
algorithms. We are doing this using the MPI libraries, and
algorithms.
We
are
MPI
libraries,
and
algorithms.
We
aredoing
doingthis
thisusing
the
MPI
libraries,
and
at
present the
parallelization
ofusing
thethe
2.5D
space
charge
atat present
parallelization
ofofthe
space
present
the
parallelization
the 2.5D
2.5D
spacecharge
routines
hasthe
been
carried out and
parallel
versions
ofcharge
the
routines
been
out
and
parallel
versions
the
routines
has
been carried
carried
out
and
parallel
versionsofof
the
3D
spacehas
charge
routines
are
being
developed.
We
3D
charge
are
developed.
3D space
space near-term
charge routines
routines
are being
being
developed.
We
anticipate
completion
of this
work, to We
be
anticipate
ofof this
work,
toto bebe
anticipatebynear-term
near-term
completion
this
work,
followed
creation ofcompletion
parallel versions
of the
diagnostic
followed
by
creation
of
parallel
versions
of
the
diagnostic
followed
by
creation
of
parallel
versions
of
the
diagnostic
routines.
routines.
routines.
In the slightly longer term, we plan to convert the
In
longer
term,
we
In the
the
slightly
longer
term,shell
weplan
plantoSuperCode
toconvert
convertthe
ORBIT
userslightly
interface
and driver
from
tothe
ORBIT
and
SuperCode
toto
ORBITuser
user
interface
andofdriver
driver
shellfrom
from
SuperCode
Python.
Theinterface
feasibility
and shell
methods
for
doing this
Python.
feasibility
doing
this
Python.
The
feasibility
of and
and methods
methods
forbe
doing
this
have
beenThe
examined,
but of
the
work
remainsfor
to
carried
have
have been
been examined,
examined,but
butthe
thework
workremains
remainstotobebecarried
carried
out.
out.
out.
OPEN CODE
OPEN
OPENCODE
CODE
Finally, we emphasize that ORBIT is an open code.
that
isisananopen
Finally,
we emphasize
emphasize
thatORBIT
ORBIT
opencode.
code.
TheFinally,
source we
code
can be obtained
by downloading
from
The
The source
source code
code can
can be
be obtained
obtainedby
bydownloading
downloadingfrom
from
http://www.sns.gov//APGroup/Codes/Codes.htm
and
http://www.sns.gov//APGroup/Codes/Codes.htm
and
http://www.
sns.a gov//APGroup/Codes/Codes
.htmbuilding
and
there
are also
user manual, instructions for
there
also
aa user
there are
are
also
useratmanual,
manual,
instructionsfor
forbuilding
building
ORBIT,
and
examples
that webinstructions
site.
ORBIT,
ORBIT,and
andexamples
examplesatatthat
thatweb
website.
site.
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
We wish to thank our colleagues J. Beebe-Wang, M.
We
Beebe-Wang,
We wish
wish to
tothank
thankour
our
colleagues
Beebe-Wang,
Blaskiewicz,
F.
Jones,
A. colleagues
Luccio,
N.J.J.Malitsky,
and M.
F.M.
\ :-4D -3S *20 -1S 8 10 20 30 4i3 M 8Q
Blaskiewicz,
F.F.their
Jones,
A.
N.N.Malitsky,
and
F.F.
Blaskiewicz,for
Jones,
A. Luccio,
Luccio,
Malitsky,
and
MacLachlan
advice
and
collaboration
during
the
Emlttance (pi rom-wad)
MacLachlan
their
MacLachlan for
theiradvice
adviceand
andcollaboration
collaborationduring
duringthe
the
development
offorORBIT.
development
of
ORBIT.
development
of
ORBIT.
Figure
SNS collimation
collimation study:
study: a)a)
Figure 3.
3. Results from SNS
SNS is managed by UT-Battelle, LLC, under contract
particle
scrapers,
b) effects
effects
of
Figure
Resultsonfrom
collimation
study: of
a)
particle 3.impacts
impacts
beamSNS
scrapers,
b)
SNS
LLC,
under
SNSisismanaged
managedby
byUT-Battelle,
UT-Battelle,
LLC,
undercontract
contract
DE-AC05-00OR22725
for
the U.S.
Department
of
collimation
on emittance
distribution.
particle
impacts
on beam
scrapers, b) effects of
collimation
distribution.
DE-AC05-00OR22725
for
Department
ofof
DE-AC05-OOOR22725
for the
U.S.
Department
Energy.
SNS is a partnership
ofthe
sixU.S.
national
laboratories:
collimation on emittance distribution.
Energy.
ofofsix
laboratories:
Energy. SNS
SNSisisaapartnership
partnership
sixnational
national
laboratories:
Argonne,
Brookhaven,
Jefferson,
Lawrence
Berkeley,
Los
A list of useful diagnostics in ORBIT includes the Alamos,
Oak Ridge.Jefferson,
Argonne,
Brookhaven,
Argonne,and
Brookhaven,
Jefferson,Lawrence
LawrenceBerkeley,
Berkeley,Los
Los
A list ofdumps
useful of
diagnostics
ORBIT includes
following:
particle incoordinates,
tunes,theor Alamos,
and
Oak
Ridge.
Alamos,
and
Oak
Ridge.
A list ofdumps
useful of
diagnostics
ORBIT includes
following:
particle in
coordinates,
tunes, the
or
following: dumps of particle coordinates, tunes, or
243
REFERENCES
1. J. Galambos, J. Holmes, D. Olsen, A. Luccio, and J.
Beebe-Wang,
ORBIT
Users
Manual,
http://www.sns.gOv//APGroup/Codes/Codes.htm
2. J. Galambos, S. Danilov, D. Jeon, J. Holmes, D. Olsen,
J. Beebe-Wang, and A. Luccio, in Proceedings of the
1999 Particle Accelerator Conference, (New York,
1999)3143.
3. F. Jones, Users' Guide to ACCSIM, TRIUMF Design
Note,
TRI-DN-90-17
(1990),
http://www.triumf.ca/compserv/accsim.html
4. S. W. Haney, Using and Programming the SuperCode,
http://www.sns.gov/APGroup/Codes/Codes.htm
5. H. Grote and F. Christoph Iselin, The Mad Program,
Version
8.19,
User's
Reference
Manual,
CERN/SL/90-13, (Geneva, 1996).
244
6. See ftp://csftp.triumf.ca/pub/CompServ/dimad/
7. F. W. Jones, in Proceedings of the 2000 European
Particle Accelerator Conference, (Vienna, 2000)
1381.
8. R. W. Hockney and J. W. Eastwood, Computer
Simulation Using Particles, Institute of Physics
Publishing (Bristol: 1988).
9. J. A. MacLachlan, Longitudinal Phase Space Tracking
with Space Charge and Wall Coupling Impedance,
Fermi National Accelerator Laboratory, FN-446,
(1987).
10.L. Michelotti, MXYZPTLK and BEAMLINE: C++
Objects for Beam Physics, AIP Conf. Proc. No. 255
(Proc. Advanced Beam Dynamics Workshop on
Effects of Errors in Accelerators, Their Diagnosis and
Correction,
Corpus
Christi,
Texas,
1992).