Single Particle Beam Dynamics Codes
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DESY –MPYHHH Workshop CERN 2004
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Overview
• Introduction
• Code repository
– Models used
– Programming philosophy
• Examples
• Summary
This is a workshop contribution:
The description of methods and their implementation in the various
beam dynamics codes is not complete, not always accurate and
maybe wrong at all.
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Definition
• Describes the motion of a particle in the 6 dimensional
phase space under the influence of external fields
• Linear Motion
– Fitting of linear optic functions etc.
– Definition of magnets and alignment
– Definition of geometry
• Nonlinear Motion
– Nonlinear perturbations
– Dynamic Aperture
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Point of view
• The physicist who cares only about the
methods/assumptions used
• The programmer who wants to implement the newest
programming techniques
• The user (also a physicist/programmer) who doesn’t care
about methods and programming but likes a well
documented, usable, cross-checked code to get the work
done
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Program Layout
1. Get the data/lattice into the code - the lattice parser
2. Calculate
• Linear optics functions
• Tracking
• Construct Map
3. Analyze the result
• Display optics function
• Calculate DA, frequency map, nonlinear distortions
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A legacy of beam dynamics codes
• Many beam dynamics codes written over the years
• Here is a – surely – not complete list:
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
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Model – Ray Tracing
•
•
•
•
•

P
Orbit vector
X  ( x, x, y, y, z,  ) ,  
P0

Transport through elements 
X f  RX i
using R matrix
Linear optics calculations
Concatenated by Matrix multiplication
Extended to higher order (TRANSPORT)
x j , f   R jk x j,i   T jkl x j,i xl,i   U jklm x j,i xl,i xm,i  
k
kl
klm
• NOT symplectic
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC,
RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT,
TRACY, TRANSPORT, TURTLE, UAL
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Symplecticity
f
i
Given a transfer Map z  Mz
And its Jacobian
J is symplectic if
f

z
J ab ( z i )  ai
zb
J T SJ  S
• A Hamiltonian system is symplectic, i.e. a map which
fulfills the symplectic conditions describes a Hamiltonian
system
• Important test to verify validity of chosen approximations
• If violated, artificial damping/excitation of motion
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Model – Element by Element Kick Code
• Elements described by thin lens kicks and drifts
• Always symplectic
• Long elements to be sliced =>slow
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
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Model – Lie Maps
• Lie map methods allow extension of linear concepts in the
non-linear Regime


• Lie transformation x f  exp(: f :) xi
• Lie maps can be factorized and truncated without loosing
symplecticity
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
• Concatenation formulae exist
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Example – Different Transfer functions in BMAD
The computation method
used for tracking and
calculating transfer maps
can be set individually for
each element.
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Model – Differential Algebra
• Differential Algebra techniques allow computation of Taylor
maps
• Basic idea is to “track” a power-series element-by-element
• Taylor Map is again not symplectic but can be used to
obtain factorized Lie map
• AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
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Application of One Turn Maps
• One turn map tracking
– One-turn Taylor Map
– Construct mixed-variable generating functions
– Cremona maps
• Normal form analysis
– Extracts higher order lattice function perturbations and
their parameter dependence as well as driving terms out
of one-turn map
– Very useful
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PTC-TWISS MAD-X Module
E. Courant et al, “A comparison of
several Lattice Tools for
Computation of Orbit Functions of
an Accelerator”, published in
PAC2003 Portland, shown is nx
versus p/p for a simple cyclotron.
Standard MAD-X gives the green
curve which deviates since the
MAD-X (like MAD8) uses the
expanded Hamiltonian. In PTC the
exact attribute allows to the treat
the true Hamiltonian. Note, that
PTC has read-in the structure from
MAD-X input. There is now
PTC_TWISS as attribute of the
PTC MAD-X module (still
rudimentary!) that allows to
produce the Ripken/ Willeke lattice
functions called TWISS3 in
MAD8.
Example for the Afternoon
Normal Form MAD-X Module
There is now also a NORMAL attribute of the
PTC MAD-X module (still rudimentary!) to
calculate dispersion, tune and anharmonicities to
high orders and as function of delta. This module
will be eventually become the replacement of the
DYNAMIC/STATIC of MAD8.
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Programming Philosophies
• Integrate the physics into a high level mathematics software
(like Mathematica, MatLab, …)
• Many existing codes can be ‘operated’ from high level
software
• Only few codes are ‘embedded’ into MatLab
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
• Advantages:
– Input/output dealt with built in math functions
– Easy implementation in control system
• Disadvantage: Slow
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Programming Philosophies
• Provide the user with a toolbox (library) rather than an
existing program which does contain the needed elements
and procedures
• Realized in C++, F90, Pascal
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
• Can be tailored to specific problem, should be easy to
maintain and extend
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Programming Philosophies
• A more or less advanced process control is implemented in
the code itself and allows complicated run logic
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
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Programming Philosophies
• Programs tailored to more or less specific needs
• Not universal, but usually the code is not as complex and
allows easy changes
• Often only used at only one laboratory and closely linked to
the specific facility and control system
AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD,
ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE,
MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK,
SAD, SIXTRACK, SYNCH, TEAPOT, TRACY,
TRANSPORT, TURTLE, UAL
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Input Format
• A common format for input and output does not exist
• A mad-like description of beam lines seems is the most
accepted approach
• Much more information on a beam line element needed
– Errors, Aperture, Wakefields, ….
– Magnet errors
• Systematic
• Random
– Time dependent variation of magnet strength, magnet positions
– Correlation between errors important
• Girder motion
• Correlated systematic errors
• Attempt for unified input (SXF)
Status?
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Input Format
(Typical) Life Cycle of an Accelerator Project (from N. Walker)
• Simple models (sequences of elements)
Basic Lattice
Design
• Tend to work with smaller modules
• Fitting, constraints etc. Lattice matching
• ‘Generic’ magnet families
Performance
Simulation
construction
Modelling of
real machine
• Definition of basic parameters
• (may have more than one possible
optics)
Quadrupole
L
K1 value or range of K1 values
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Input Format
(Typical) Life Cycle of an Accelerator Project (from N. Walker)
• More complex (complete) models
Basic Lattice
Design
• Tolerance studies
• Simulation of a range of “errors”
• Refinement of parameter specifications
Performance
Simulation
• Definition of prototype component
• Tuning algorithms, diagnostics specs.
construction
Modelling of
real machine
• Power supplies (circuits), Klystrons etc.
Quadrupole
L
Pole tip radius
max / min pole-tip field
Tolerances (used to generate random errors)
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Input Format
(Typical) Life Cycle of an Accelerator Project (from N. Walker)
Basic Lattice
Design
• Engineering design of accelerator
components
Performance
Simulation
construction
Modelling of
real machine
Quadrupole prototype (family)
L
Pole tip radius
max / min pole-tip field
Tolerances
Documentation (drawings, cad files etc)
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Input Format
(Typical) Life Cycle of an Accelerator Project (from N. Walker)
• Complete model of real machine
Basic Lattice
Design
• Used for
• Online modelling
• Continued performance studies
Performance
Simulation
construction
Modelling of
real machine
• Tuning studies etc.
Quadrupole QF10 (now unique)
L
Pole tip radius
max / min pole-tip field
Measured field map
Other unique physical attributes
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Input Format
(Typical) Life Cycle of an Accelerator Project (from N. Walker)
Basic Lattice
Design
Input Format should support all
stages of the project
Performance
Simulation
construction
Modelling of
real machine
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Example – HERA-e Synchro-Betatron Resonances
• Working Point close to integer was found to have low
lifetime
• Incomplete cancellation of chromatic perturbations in the
two IR’s north and south
• Analytic treatment with perturbation theory by F. Willeke
(EPAC04)
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Example – HERA-e Synchro-Betatron Resonances
• Tracking with SIXTRACK
– constant initial amplitude in x, y, d
– change of coherent tune with arc FODO quads
– Frequency Map analysis of 2048 turns
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Example – Frequency Map at BESSY II
• Model incorporates:
– Linear coupled model based on orbit-response measurements
– Dipole and quadrupole fringe fields
– Longitudinal variation of sextupole field
– Systematic octupole in quadrupole
– Sextupole and decapole components created by steering magnets
Measurement
from P. Kuske (BESSY) – EPAC04
Model
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Measurement ALS Off-Momentum Aperture
• Excitation of increasing
horizontal beam centroid motion
with single turn kicker
• (Static) momentum variation by
change of RF frequency
• Single turn BPM’s provide tune
of kicked beam (lower figure)
• Beam intensity monitor records
beam losses (upper figure, dot
size corresponds to relative
intensity loss)
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ALS Off-Momentum Aperture
• Tracking with ALS model after orbit response measurement
• Beam loss on 6th order resonance reproduced
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Early Indicators – Survival Plots
• Huge number of turns
• Early indicators for chaotic
motion
– Frequency maps
– Lyapunov coefficient
– Survival plot
extrapolation
– Diffusion rates in
amplitude space
HERA-p , 2000 Model
• Difficulty with time
dependent process (ripple,
events)
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Beam Dynamics Measurements with Hadrons
• Measurements at SPS, FermiLab, Hera-p, …
• Optional control off non-linearity with additional sextupoles
and octupoles
• Optional additional application of ps-ripple to simulate tune
variations
• Phase space reconstruction with kicker and single-turn
BPM’s
• Resonance driving term analysis from single-turn BPM data
• Model verified by tune-shift with amplitude measurements
(often difficult if aperture restriction does not allow large kick
amplitudes)
• Measurement of DA agrees with predictions within 20 %
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Measurements at SPS 2002
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Resonance driving terms at HERA
• Resonance line amplitudes
obtained from single-turn
PM data
• Line amplitude can be
calculated form normal
form theory
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Conclusion
• The ideal beam dynamics code looks like
– Library type architecture
– Linear and nonlinear matching
– Symplectic map tracking through arbitrary fields with time
and s dependence
– Normal form analysis
– Element by element kick tracking as alternative to the
above
• Pre processing
– Connection to data base and control system
– Uniform format only important for big collaborations
• Post processing
– Includes FMA, early indicators etc.
– Can be done by external math program
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Conclusion
• Single particle dynamics is mature field with many available
tools
• Codes have been benchmarked against experiments with
good agreement if the accelerator model is refined
enough
• Modern tools like Lie algebra and normal form should be
promoted and used more frequently
• But: For today’s accelerator physics problems the single
particle approach is less and less valid:
– Beam-beam, space charge, geometric wakes, ion-and ecloud, coherent synchrotron radiation, ….
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Model
Usage
AT(Matlab)
C
Spear,ALS
BETA
S
Kick
ESRF, BESSY
BMAD
C,S Matrix, Kick, Lie, PTC Cornell
COMFORT
Matrix
DESY
COSY-INFINITY
Taylor, Lie, DA
World
DIMAD
Matrix, GF
SLAC
ELEGANT
L,S
Matrix, Kick
APS, World
LEGO
C
Da, Lie
PEP-II
LIAR
L
Matrix
SLAC
LUCRETIA
L
SLAC
MAD
C,S,L Matrix, Kick, Lie, PTC World
MARYLIE
Lie, GF
World
MERLIN
L
Matrix, Kick
World
ORBIT
C
PETROS
C
Matrix
DESY
PLACET
L
Matrix
CERN, World
PTC
C,S Kick
World (BMAD, MAD)
RACETRACK
S
Kick
Elettra
SAD
C,S,L Matrix, Kick
KEK
SIXTRACK
C,S Kick
World
SYNCH
TEAPOT
C
Kick
(UAL)
TRACY
S
Kick
SLS, Soleil, ALS
TRANSPORT
C,S,L Matrix
World
TURTLE
L
Matrix
UAL
C
Matrix, Kick(Teapot)
RHIC
Comments
beam-beam
wakefield
project started
particle interaction
wakefield
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AT(Matlab)
BETA
BMAD
COMFORT
COSY-INFINITY
DIMAD
ELEGANT
LEGO
LIAR
LUCRETIA
MAD
MARYLIE
MERLIN
ORBIT
PETROS
PLACET
PTC
RACETRACK
SAD
SIXTRACK
SYNCH
TEAPOT
TRACY
TRANSPORT
TURTLE
UAL
EXE SOURCE
U,L,W
DOC
y
Optimisation
Input
U,L
y
y
y
madlike
y
U,L
y
U,L,W y
y
y
y
y
y
y
U,L,W y
y
n
U,L,W y
y
U,L,W
y
y
y
y
y
U
y
U,L,W y
y
(y)
n
n
U,L
U,L
y
y
y
n
y
y
y
y
y
y
y
y
Language
MatLab, DLL
F90, C++, Lib
F77
mad,sxf
C++
madlike,sxf F90
madlike
C
C++, Lib
F90
MatLab, DLL
mad
C,F77,F90
F77
madlike
C++, Lib
C++
F77
C
F90
F77
madlike
F90
sxf
F77
C++
F77
F77
C++
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