From Tracking Code to Analysis: Generalised Courant-Snyder Theory for Any Accelerator Model

351 Pages·2016·3.25 MB·English

by  Etienne Forest (auth.)

Etienne Forest
From Tracking
Code to
Analysis
Generalised Courant-Snyder Theory for
Any Accelerator Model
From Tracking Code to Analysis
Etienne Forest
From Tracking Code
to Analysis
Generalised Courant-Snyder Theory for Any
Accelerator Model
123
EtienneForest
KEK,HighEnergy Accelerator
ResearchOrganization
Tsukuba
Japan
ISBN978-4-431-55802-6 ISBN978-4-431-55803-3 (eBook)
DOI 10.1007/978-4-431-55803-3
LibraryofCongressControlNumber:2016932319
©SpringerJapan2016
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Conscience doth make cowards of us all.
—William Shakespeare
Foreword
Sincethediscoveryofthestrongfocusingprinciplebetween1949and1952,many
acceleratorsfromseveraltensofmetersincircumferencetotensofkilometershave
been proposed and constructed based on this principle. As in many other areas of
physics and engineering, computer simulations became an accurate and efficient
tool for accelerator design and operation. It is essential for designing any modern
high-performance accelerator where engineering tolerance is involved in an early
stage.
The earlier generation of accelerator simulation codes, limited by computing
capabilities, tended to use a simple matrix for so-called linear magnets—solenoid,
dipole,quadrupole—andthinlenskicksforsextupolesandhighermultipoles.These
codes were symplectic by design. Attempts to improve them initially went in the
directionofimprovingthematrixbyderivinghigher-ordercorrections.Icannamea
few codes: Karl Brown’s Transport (second-order), Dragt’s Marylie (third-order),
Berz’s Cosy 5.0 (fifth-order), and Cosy-Infinity (arbitrary order). All these codes
producemapsaccuratearoundasingleorbit,usuallytheso-calleddesignorbit.They
emphasizemodelaccuracybutlackself-consistencysincetheysingleoutonespecial
orbit.Whenanalyzingthedynamicswithawiderrangeofenergy,orbits,andeven
design1orbits,some matrix-basedcodes need special care or are even notusable.
Ron Ruth proposed a different approach: to improve the matrix code using
integrationtechniquesthataresymplectic.ItturnsoutthatformostHamiltoniansin
ring dynamics, this solution is feasible. When a simulation code is developed
following Ruth’s methods, it is more self-consistent, and most extra dynamical
effects can be included with ease. Calculations of lattice functions stay closer to
tracking and to first principles. In the final analysis, it is a more reliable model for
thering.Ofcourse,intheearly1980s,theRuthapproachlostthenonlinearmatrix
representation of the map. This state of affairs changed drastically with the advent
of truncated power series algebra (TPSA, also called differential algebra by Berz
and others).
1Recirculatorsrequireseveraldesignorbitsthroughthesamemagnet.
vii
viii Foreword
Etienne Forest’s pioneer work on normal form analysis has provided a general,
clean, and unified framework for single-particle beam dynamics simulations that
includeRuth’sintegrationwhileprovidingmapswiththeappropriateanalysistools.
All these are realized in his PTC program which is used for the examples in this
book.
In the PTC framework, brute force tracking is still possible because it is a
Ruth-inspired integrator. However, a transfer map can be obtained around any
closed orbit and normalized to an arbitrary order. The lattice function calculations
are then performed in a form close to their definition: βx ¼hx2i=Jx and νx,
@νx=@ðdp=pÞ, ξx are coefficients of the Taylor expansion of νx and higher-order
quantities correspond to high-order coefficients in the properly defined version
pffiffiffiffiffiffiffiffi
ofthesequantities.Coupledlatticefunctionslikehxyi= JxJy areeasiertocompare
with measurement.
My own experience with spin dynamics is limited, but I can see that the spin
tunes and phase advances can be calculated under the same framework in a clean
andunifiedway.After all,thenormalizationprocesshasnoassumptionsaboutthe
details of the dynamics.
When I realized the existence of this modern framework, I developed a TPSA
package in C++ and was introduced to Etienne by my then group leader David
RobinattheLawrenceBerkeleyLaboratory.Withinlessthanaweek,hisPTCwas
abletorunwithmyTPSApackage.However,formostacceleratorphysicistsdoing
single-particle dynamics simulations, it is not necessary to develop a TPSA or
normal form library. Just as it is for the users of singular value decomposition
(SVD), fast Fourier transformation (FFT), or the popular frequency analysis of
Laskar,theonlyessentialthingistoknowhowtousethesemathematicaltoolsand
their processed data. That is the focus of this book. Symplectic integrators, TPSA,
the unique beam-line structure of PTC that fully exploits the lens description of
beam dynamics, although an integral part of Etienne’s general framework, are not
covered in this book. The emphasis here is on the structure of the analysis tools
irrespective of the simulation code one ends up using.
SinceIfirstmetEtienne,Irecommendedthathewrite anewbookfortheusers
ofbeamdynamicssimulation.Manyreadersofhislastbookmaysharewithmethe
impression that the simplicity of the framework is overshadowed by the mathe-
maticaldetails.Theexample-orientedapproachinthisbookisabetterintroduction
to his framework. Despite its small size, this book contains new results: new
parameterization of the linear map, discussion on spin and spin resonances and,
surprisingly, a numerically general method to compute correctly the Hamiltonian
that enters into the standard perturbation theory found in traditional textbooks: the
methods of Guignard.
Less is more. When a simulation is based only on fundamental principles, its
results are more precise and general and lead to new discoveries.
Upton, New York Lingyun Yang
January 2015
Preface
Pre-BoardingAnnouncement
Alltheexamplesofthisbookarederived fromacodeinwhichIdeliberatelyput
bugs and nonphysical Hamiltonians: the drifts are wrong, the magnets have mis-
takes,physicalconstantsarewrong,thespinequationsarewrong,etc…
Actually I did not have the cojones to confuse readers who might actually
comparetheexamplesofthisbookwithastandardacceleratorcode.Butitmustbe
the conclusion of the reader who understands this book, that the exactness of the
modelistotallyirrelevanttothegeneralstructureofacceleratortheory.
Garbagein;garbageout!
If the style of this book is more akin to the pamphleteer style of V.I. Lenin’s
“StateandRevolution”orThomasPaine’s“CommonSense,”itisnotanaccident.
I am describing shamelessly my own position and conviction.
ThroughmostofthebookItalkinthefirstpersonsingular.Iavoidthescientific
“we”orJuliusCaesar’s“he/one.”Myreasonsaresimple:themainideaofthisbook
—theexistenceofanaturalhierarchy—ismineandIwillnotbeacoward.Itisnot
a review article. I am not interested in giving credit to people whose work is
irrelevant to mine or, as it is often the case, antithetical to my work. I do not care
thattheyaregreatacceleratorphysicistsorhavecontributedvaluableinsightstothe
field.IdescribehereaverynarrowandspecifictopicasIseeit.Somepeoplehave
hadsomeinfluenceonmeinrelationtothisbook.Icannamethemrighthere:Alex
Dragt, Martin Berz, Richard Talman, Ron Ruth, and recently, Desmond Barber.
Other people have contributed to the software I used: Martin Berz (again), David
Sagan,FrankSchmidt,andEricMcIntosh.Finally,AlexanderMolodozhentsevhas
actuallyusedmyworkinphysicalapplicationswhichinsomewayshavemotivated
me to go further.
When I use the collective “we” in this book, it really means “the reader and I.”
OfcourseIamsurethatIoccasionallyusethepassiveandthe“we”whosepurpose
istodiffuseresponsibility—afterallIliveinJapanwherediffusingresponsibilityis
a cultural imperative! I apologise in advance.
ix
x Preface
This book originates in my conviction that, since 1980 at least, computer sim-
ulations of accelerators are ubiquitous and unavoidable. Therefore, as soon as
I entered the field I decided that theory and simulation should be adapted to each
other.
Acceleratortheory,whichinitssimplestincarnationiscalled“Courant-Snyder”
theory, should be designed with the simulation code in mind. Integrators in
accelerator physics, often called “kick codes,” simply push particles around the
machine.However,circa1986,thetoolsoftruncated powerseries algebra(TPSA)
havebeenmadeavailabletotheacceleratorcommunityprimarilythroughthework
of Berz. Thanks to operator overloading, available in C++ and Fortran90, it is
possible to write an integrator which can produce automatically2 not only brute
force tracking but approximate Taylor maps.
Taylor maps can be analyzed by a process called normalization. This process
allowstheCourant-Snydertheorytobeextendedeffortlesslytononlinearities,spin,
radiation, modulated magnets, etc., provided that a normalization library, akin to a
diagonalization library, exists and is available.
Oncethesetoolsareinplace,itispossibletocovertheentirefieldofaccelerator
perturbation theory, provided one respectsand abides by a strict hierarchical code.
The word code here means “computer program” and well as behavior.
In this book I display with examples, runnable on any computer platform, the
full gamut of this hierarchy: one-turn maps, normalization of one-turn maps, uni-
versal Twiss3 loop and Fourier transform of the Hamiltonian (or Lorentz force)
expressed in approximate Floquet variables. This final Hamiltonian is the
Hamiltonian often found in standard accelerator textbooks. Here you are told how
to really compute it in a complex and arbitrary ring; better, you are given some
tools that exemplify that power.
I will review the normalization of linear maps in n-degrees of freedom under
most regimes: symplectic, radiative, spin, and modulated magnets. I will then
proceed to the nonlinear normalization. This will include some examples with a
singleresonanceandalimitcycle.Ihavealsoadiscussiononspin.Iwillfinishthe
book with Guignard theory which is the final output of the universal Twiss loop.
It is notable that Guignard theory, which is often the starting point of pertur-
bation theory, is here a final product and thus at the bottom of the hierarchical
structure.Thisisnotamatteroftaste:anyseriouscalculationwhichincludeserrors,
fringe fields, and the effect of earthquakes must follow that hierarchy. Of course if
one pontificates from a throne in the rarefied air of pure mathematics, then it is a
matter of taste. But here I provide real examples, not hot air.
This book is novel in two respects. First, I made an effort to produce examples
on which the reader can check his understanding. Unlike most books where
examplesaretrivialorsimplyabsent,hereallthetoolsinthebookareavailableto
thereaderwith16examplesintheappendicesthatcanberunwithfreelyavailable
2Almostautomatically!
3Somewouldprefertheterm“universallatticefunctionloop.”
Preface xi
compilers. The actual ring used is an old design of the Advanced Light Source at
theLawrenceBerkeley NationalLaboratorywhereIonceworked.Thelattice isin
Appendix A.
Secondly,there arenovel topicsthat arenotfoundintheliterature.Thechapter
onspin containsa fewnovel ideas mostly duetoDesmond Barber andDan Abell.
The chapter on Guignard-like Hamiltonian is totally new and is based on one’s
ability to take the logarithm of a nonlinear map. The idea is not new to me but
I took the time and trouble to implement enough of it for the sake of this book.
OfcourseIhopethatsomeacceleratorphysicistswillreadthisbookandtrythe
examples; they are the intended audience.
ThereaderisinvitedtojumptoChap.9andreadtheconclusionofthisbook.It
mirrors this preface and it is a good way to avoid reading a book!
Finally,IwouldliketostatethatIaminspiredbythelinguistNoamChomskyin
my approach to this field.
First of all, and this is mostly pure coincidence, Chomsky is famous in science
for the concept of a universal grammar and the concept of the Chomsky Normal
Form(CNF)appliedtocontextfreeformalgrammars.Thesecontextfreegrammars
are a little akin to our integrable systems and the CNF to our normal forms.
Ofcoursetherealuniversalgrammarwhichispostulatedtoexistinachild’sbrain
is more complex as is the real dynamics of a symplectic map.
Secondly, and more importantly, I am inspired by Chomsky’s political analysis
methods.Inparticular,onecandeducealotofthingsbylookingatthestructureofa
system4 from far above. And, when our logical deductions are wrong, it is
instructive since it is often the sign of specific human manipulations. Accelerators
are man-made after all. It is logical to assume generically, if we look far above
them,thattheyhavelinearstability.Withoutanyadditionalknowledge,Icanderive
all of Courant-Snyder theory. The details are irrelevant: a marble in a frictionless
gutter or a proton in a synchrotron all lead to the same theory. So if one discovers
that the stable system is linearly unstable, it teaches us a lot about human manip-
ulations.Also,therearemanynumericalmethodsthatarebasedonthequasi-failure
ofintegrability:Laskar’sfrequencyanalysisisperhapsthemostfamous.Generally,
there is a lot to learn from discovering that certain naive expectations are usually
wrong.
Politicallythesameistrue.Iexpectacceleratorlaboratoriestoproducegraduate
students who are partial slaves simply because of the purpose of accelerator phy-
sics. Graduate students are assigned easily to supervisors on the basis of needs
because we are a glorified service station. When it is not the case, we can expect
that some special mechanism was put in place to prevent exploitation.
AveryrespectedJapanesecolleagueofmineoncetoldmethat“myapproachto
thetheory”isnotgoodbecauseitdoesnotpermitthestudenttojumpimmediately
into useful thesis work. I thought that students are supposed to learn and not to be
4Tolookfromfarabove,keepinglesserdetailsaway,isalsoamethodpreachedbyFeynman,who,
withChomsky,ismyintellectualhero.