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9FERM ILAB-PUB-09-281-AD-CD

Fully 3D M ultiple B eam D ynam ics Processes Sim ulation for the

Tevatron

E.G.Stern,� J.F.Am undson,P.G.Spentzouris,and A.A.Valishev

Ferm i NationalAccelerator Laboratory

(Dated:July 22,2013)

Abstract

W epresentvalidation and resultsfrom a sim ulation oftheFerm ilab Tevatron including m ultiple

beam dynam ics e�ects. The essentialfeatures ofthe sim ulation include a fully 3D strong-strong

beam -beam particle-in-cellPoisson solver,interactionsam ong m ultiplebunchesand both head-on

and long-range beam -beam collisions,coupled linearopticsand helicaltrajectory consistentwith

beam orbitm easurem ents,chrom aticity and resistivewallim pedance.W evalidateindividualphys-

icalprocessesagainstm easured data where possible,and analytic calculationselsewhere.Finally,

wepresentsim ulationsofthee�ectsofincreasing beam intensity with singleand m ultiplebunches,

and study the com bined e�ect oflong-range beam -beam interactions and transverse im pedance.

The resultsofthe sim ulationswere successfully used in Tevatron operationsto supporta change

ofchrom aticity during thetransition to colliderm odeoptics,leading to a factoroftwo decreasein

proton losses,and thusim proved reliability ofcollideroperations.

PACS num bers:29.27.-a

�Electronicaddress:egstern@ fnal.gov

1

I. M O T IVAT IO N

TheFerm ilabTevatron[1]isap-�pcollideroperatingatacenter-of-m assenergyof1:96TeV

andpeaklum inosityreaching3:53� 1032cm �2 s�1 .Thecollidingbeam sconsistof36bunches

m oving in a com m on vacuum pipe. Forhigh-energy physics operations,the beam scollide

head-on at two interation points (IPs) occupied by particle detectors. In the intervening

arcsthebeam sareseparated by m eansofelectrostaticseparators;long-range(also referred

to as parasitic) collisions occur at136 other locations. E�ects arising from both head-on

and long-rangebeam -beam interactionsim poseseriouslim itationson m achineperform ance,

hence constante�ortsare being exerted to betterunderstand the beam dynam ics. Due to

theextrem ecom plexity oftheproblem anum ericalsim ulation appearstobeoneofthem ost

reliablewaysto study theperform anceofthesystem .

Studiesofbeam -beam interactions in the Tevatron Run IIm ainly concentrated on the

incoherente�ects,which werethem ajorsourceofparticlelossesand em ittancegrowth.This

approach wasjusti�ed by thefactthattheavailableantiproton intensity wasa factorof10

to 5 lessthan theproton intensity with approxim ately equaltransverseem ittances.Several

sim ulation codeswere developed and used forthe optim ization ofthe colliderperform ance

[2,3].

W ith the com m issioning ofelectron cooling in the Recycler,the num berofantiprotons

availabletothecollidersubstantionally increased.Duringthe2007and 2008runstheinitial

proton and antiproton intensities di�ered by only a factor of3. M oreover,the electron

coolingproducesm uch sm allertransverseem ittanceoftheantiproton beam (’ 4� m m m rad

95% norm alized vs.’ 20� m m m rad forprotons),leading to the head-on beam -beam tune

shiftsofthe two beam sbeing essentially equal. The m axim um attained totalbeam -beam

param eterforprotonsand antiprotonsis0.028.

Underthese circum stances coherent beam -beam e�ects m ay becom e an issue. A num -

beroftheoreticalworksexistpredicting the lossofstability ofcoherentdipole oscillations

when the ratio ofbeam -beam param etersis greaterthan ’ 0:6 due to the suppression of

Landau dam ping[4]. Also,the com bined e�ectofthe m achine im pedance and beam -beam

interactionsin extended length bunches coupleslongitudinalm otion to transverse degrees

offreedom and m ay producea dipoleorquadrupolem odeinstability [5].

Understandingtheinterplaybetween allthesee�ectsrequiresacom prehensivesim ulation.

2

Thispaperpresentsa m acroparticlesim ulation thatincludesthem ain featuresessentialfor

studying the coherent m otion ofbunches in a collider: a self-consistent 3D Poisson solver

for beam -beam force com putation,m ultiple bunch tracking with the com plete account of

sequence and location oflong-range and head-on collision points, and a m achine m odel

including ourm easurem entbased understanding ofthecoupled linearoptics,chrom aticity,

and im pedance.

In SectionsII{V wedescribe thesim ulation subcom ponentsand theirvalidation against

observed e�ects and analytic calculations. Section VIshows results from sim ulation runs

which presentstudiesofincreasing thebeam intensity.Finally,in Section VIIwestudy the

coherent stability lim its forthe case ofcom bined resistive wallim pedance and long-range

beam -beam interactions.

II. B EA M B EA M 3D C O D E

The Poisson solver in the Beam Beam 3d code is described in Ref.[6]. Two beam s are

sim ulated with m acroparticles generated with a random distribution in phase space. The

acceleratorringisconceptuallydivided intoarcswith potentialinteraction pointsattheends

ofthearcs.The opticsofeach arcism odeled with a 6� 6 linearm ap thattransform sthe

phasespacefx;x0;y;y0;z;�gcoordinatesofeach m acroparticlefrom oneend ofthearctothe

other. There issigni�cantcoupling between the horizontaland verticaltransverse coordi-

natesin theTevatron.ForourTevatron sim ulations,them apswerecalculated usingcoupled

lattice functions[7]obtained by �tting a m odel[8]ofbeam elem entcon�guration to beam

position m easurem ents.The longitudinalportion ofthe m ap producessynchrotron m otion

am ongthelongitudinalcoordinateswith thefrequency ofthesynchrotron tune.Chrom atic-

ityresultsinanadditionalm om entum -dependentphaseadvance��x(y) = �0Cx(y)�p=pwhere

Cx(y) isthenorm alized chrom aticity forx (ory)and �0 isthedesign phaseadvanceforthe

arc.Thisisageneralization ofthede�nition ofchrom aticity toapply toan arc,and reduces

to the norm alized chrom aticity (��=�)=(�p=p)when the arc encom passesthe whole ring.

The additionalphase advance isapplied to each particle in the decoupled coordinatebasis

so thatsym plecticity ispreserved.

The Tevatron includes electrostatic separators to generate a helicaltrajectory for the

oppositely charged beam s.Them ean beam o�setattheIP isincluded in thePoisson �eld

3

solvercalculation.

Di�erentparticlebunchesareindividually tracked through theaccelerator.They interact

with each otherwith thepattern and locationsthatthey would havein theactualcollider.

Theim pedancem odelappliesa m om entum kick to theparticlesgenerated by thedipole

com ponentofresistive wallwake�elds[9]. Each beam bunch isdivided longitudinally into

slices containing approxim ately equalnum bers ofparticles. Aseach bunch istransported

through an arc,particlesin slice ireceive a transverse kick from the wake �eld induced by

thedipolem om entoftheparticlesin forward slicej:

�~p ?

p=

2

�b3

r

4��0c

�

N jr0< ~rj>

�

Lpzij

(1)

The length ofthe arc is L, N j is the num ber ofparticles in slice j, r0 is the classical

electrom agnetic radius ofthe beam particle e2=4��0m 0c2,zij is the longitudinaldistance

between the particle in slice ithat su�ers the wake�eld kick and slice j that induces the

wake. ~rj is the m ean transverse position ofparticles in slice j,b is the pipe radius,c is

thespeed oflight,� istheconductivity ofthebeam pipeand � areLorentzfactorsofthe

beam .Quantitieswith unitsarespeci�ed in theM KSA system .

III. SY N C H R O B ETAT R O N C O M PA R ISO N S

W ewillassessthevalidityofthebeam -beam calculation bycom paringsim ulated synchro-

betatronm odetuneswith am easurem entperform edattheVEPP-2M 500M eV e+ e� collider

and described in Ref.[12]. These m odesare an unam biguousm arkerofbeam -beam inter-

actionsand provide a sensitive toolforevaluating calculationalm odels.These m odesarise

in a colliding beam acceleratorwherethelongitudinalbunch length and thetransversebeta

function areofcom parablesize.Particlesatdi�erentzpositionswithin abunch arecoupled

through theelectrom agneticinteraction with theopposingbeam leadingtothedevelopm ent

ofcoherentsynchrobetatron m odes.Thetuneshiftsfordi�erentm odeshaveacharacteristic

evolution with beam -beam param eter� = N r0=4� �,in which N isthenum berofparticles,

r0 is the classicalelectrom agnetic radius ofthe beam particle,and � is the unnorm alized

one-sigm a beam em ittance.

There are two coherent transverse m odes in the case ofsim ple beam -beam collisions

between equalintensity beam s without synchrotron m otion: the � m ode where the two

4

beam soscillate with the sam e phase,and the � m ode where the two beam soscillate with

opposite phases [10]. W ithout synchrotron m otion,the � m ode m ode has the sam e tune

as unperturbed betatron m otion while the � m ode frequency is o�set by K �,where the

param eterK isapproxim ately equalto and greaterthan 1 and dependson the transverse

shapeofthebeam s[11].Thepresenceofsynchrotron m otion introducesam orecom plicated

spectrum ofm odeswhosespectroscopy isoutlined in Fig.1 in Ref.[12].

W e sim ulated the VEPP-2M colliderusing Courant-Snyderuncoupled m aps. The hori-

zontalem ittance in the VEPP-2M beam ism uch largerthan the verticalem ittance. The

bunch length (4cm )iscom parableto��y = 6cm soweexpecttoseesynchrobetatron m odes.

In order to excite synchrobetatron m odes,we set an initialy o�set ofone beam sigm a

approxim ately m atching theexperim entalconditions.

Longitudinale�ectsofthebeam -beam interaction weresim ulated by dividing thebunch

into six slices. At the interaction point,bunches drift through each other. Particles in

overlapping slices are subjected to a transverse beam -beam kick calculated by solving the

2D Poisson equation for the electric �eld with the charge density from particles in the

overlapping beam slice.

Sim ulation runswith arangeofbeam intensitiescorrespondingtobeam -beam param eters

ofup to 0.015 were perform ed,in e�ect m im icking the experim entalprocedure described

in Ref.[12].Foreach sim ulation run,m odepeakswereextracted from theFouriertransform

ofthem ean bunch verticalposition.An exam pleofthespectrum from such a run isshown

in Fig.1 with three m ode peaks indicated. In Fig.2,we plot the m ode peaks from the

Beam Beam 3d sim ulation asa function of� asred diam ondsoverlaid on experim entaldata

from Ref.[12]and a m odelusing linearized coupled m odesreferred to asthem atrix m odel

described in that reference and Ref.[13,14]. As can be seen,there is good agreem ent

between theobservation and sim ulation giving uscon�dencein thebeam -beam calculation.

IV . IM P ED A N C E T EST S

W ake�eldsor,equivalently,im pedancein an acceleratorwith a conducting vacuum pipe

gives rise to wellknown instabilities. Our aim in this section is to dem onstrate that the

wake�eld m odelin Beam Beam 3d quantitatively reproduced these theoretically and experi-

m entally wellunderstood phenom ena.Thestronghead-tailinstability exam ined by Chao[9]

5

0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15tune

10-6

10-5

10-4

10-3

10-2

10-1

Fou

rier

pow

er

(arb

. u

nit

s)

A

B

C

FIG .1:Sim ulated m odespectra in theVEPP-2M colliderwith � = :008 showing synchrobetatron

m odes. The line indicated by A is the base tune,B is the �rst synchrobetatron m ode,C is the

beam -beam � m ode.

arisesin extended length bunchesin thepresence ofwake�elds.Forany particularacceler-

atoropticaland geom etric param eters,there isan an intensity threshold above which the

beam becom esunstable.

Theresistivewallim pedancem odelappliesan additionalim pulsekick in addition to the

application ofthem ap derived from beam optics.Thetunespectrum iscom puted from the

Fouriertransform ofthebeam bunch positionssam pled attheend ofeach arc.In orderfor

thecalculation tobeagoodapproxim ation ofthewake�eld e�ect,theim pedancekickshould

bem uch sm allerthan thex0ory0changeduetoregularbeam transportsowedividethering

into m ultiple arcs. which bringsup the question ishow m any issu�cient. The di�erence

in calculated im pedance tune shift for a 12 arc division ofthe ring or a 24 arc division

6

FIG .2:Thediam ondsshow sim ulated synchrobetatron m odesasa function ofbeam -beam param -

eter� (diam onds)and ofobserved m odes(points).

isonly 2� 10�4 ,which isless than 3% ofthe synchrotron tune (0.007 in thisstudy),the

relevantscalein thesesim ulations.W eperform thecalculation with 12arcsforcalculational

e�ciency.

In theabsenceofim pedance,wewould expectto seethetunespectrum peak at20.574,

the betatron tune ofthe lattice. W ith a pipe radiusof3cm and a bunch length of20cm ,

resistive wallim pedance produces the spectrum shown in Fig.3 for a bunch of4 � 1012

protonsat150GeV[22].In thissim ulation,thebase tune�� is20:574 and thesynchrotron

tuneis0:007.Threem odepeaksareclearlyevidentcorrespondingtosynchrobetatron m odes

with frequencies�� � �s shifted up by the wake�eld (pointA),�� shifted down (pointB ),

and �� + �s shifted upward (pointC)aswould beexpected in Ref.[15].

In Fig.4,weshow theevolution ofthetwo m odesasa function ofbeam intensity.W ith

the tune and beam environm ent param etersofthissim ulation,Chao’stwo particle m odel

7

20.560 20.565 20.570 20.575 20.580 20.585tune

10-2

10-1

100

101

102

Fou

rier

pow

er

(arb

. u

nit

s)

A

B

C

����s

��

��+ �s

FIG .3: Sim ulated spectrum ofa two slice bunch in the presence ofwake�elds and synchrotron

m otion showing threesynchrobetatron m odesA,B ,and C induced by wake�elds.

predictsinstability developm entatintensitiesofabout9� 1012 particles,which isclose to

where the upperand lowerm odesm eet. W e show two setsofcurvesfortwo slice and six

slice wake�eld calculations. The di�erence between the two slice and six slice sim ulations

is accounted for by the e�ective slice separation, z,that enters Eq. 1. W ith two slices,

the e�ective z islargerthan than the six slice e�ective z,resulting in a sm allerW 0. W ith

the sm allerwake strength,a largernum berofprotonsisnecessary to drive the two m odes

togetherasisseen in Fig.4.

W hen theinstability occurs,them axim um excursion ofthebunch dipolem om entgrows

exponentially asthebeam executesturnsthrough theaccelerator.Thegrowth ratecan be

determ ined by readingtheslopeofagraph oftheabsolutevalueofbunch m ean position asa

function ofturn num berplotted on alogscale.Thegrowth rateperturn ofdipolem otion at

8

0 2 4 6 8 10 12 14 16 18intensity (1012 protons)

�1.0

�0.8

�0.6

�0.4

�0.2

0.0

0.2modetu

nes

(unitsof

�s)

2 slice ��

2 slice ����s

6 slice ��

6 slice ����s

FIG .4: Evolution ofthe base tune and lower synchrobetatron m ode frequenciesasa function of

beam intensity showing thetwo m odesapproaching a com m on frequency dueto im pedance.They

scale isin unitsofthe synchrotron tune.The sim ulationsare shown fora two slice aond six slice

wake�eld calculation.

the threshold ofstrong head-tailinstability hasa parabolic dependence on beam intensity.

The wake�eld calculation reproduces thisfeature,asshown in Fig.5. The growth rate is

slowlyincreasingup totheinstabilitythreshold at5:42� 1012,afterwhich ithastheexplicitly

quadraticdependenceon beam intensity (I)ofgrowth rate= �0:100+ 0:0304I� 0:00207I2.

Chrom aticity interacts with im pedance to cause a di�erent head-tailinstability. W e

sim ulated a rangeofbeam intensitiesand chrom aticity values.Thetwo particlem odeland

them oregeneralVlasov equation calculation [9]indicatethatthegrowth ratescalesby the

head-tailphase � = 2�C��z=c�,where � isthe slip factorofthe m achine and z isroughly

the bunch length. The head-tailphase gives the size ofbetatron phase variation due to

9

2 3 4 5 6 7

intensity (1012

protons)

0.000

0.002

0.004

0.006

0.008

0.010

growth

rate/turn

strong headtail threshold

simulationparabolic fit

FIG .5:Thegrowth rateofdipolem otion in thesim ulated acceleratorwith im pedanceasafunction

ofbeam intensity asthestrong head-tailthreshold isreached superim posed with a parabolic�t.

chrom atice�ectsoverthelength ofthebunch.

Som ediscussion ofthem eaning oftheslip factorin thecontextofa sim ulation isneces-

sary. In a realaccelerator,the slip factorhasan unam biguousm eaning:� = (�C � 1= 2).

Them om entum com paction param eter�c isdeterm ined by thelatticeand istheLorentz

factor. W e sim ulate longitudinalm otion by applying m aps to the particle coordinates z

and � in discretesteps.Thesim ulation param etersspecifying longitudinaltransportarethe

longitudinalbeta function �z and synchrotron tune �s.Note thatthese param etersdo not

m akereference to path length travelled by a particle.However,path length entersinto the

im pedance calculation because wake forces are proportionalto path length. In addition,

analyticcalculationsofthee�ectofwakeforcesdepend on theevolution ofthelongitudinal

particleposition which in turn depend explicitly on theslip factor.Forourcom parisonswith

10

�1.0 �0.8 �0.6 �0.4 �0.2 0.0head-tail phase

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07n

orm

ali

zed

gro

wth

rate

21012

31012

41012

FIG .6:Thenorm alized growth rateofdipolem otion in thesim ulated acceleratorwith im pedance

and chrom aticity asa function ofhead-tailphase � atthree beam intensitiesdem onstrating their

linear relationship close to 0 and the near-universalrelationship for head-tailphase between � 1

and 0.

analyticresultstobem eaningful,weneed touseaslip factorthatisconsistentwith thelon-

gitudinalm apsand thepath lengthsthatenterthewakeforcecalculations.Therelationship

between theslip factor� and thesim ulation param etersis�z = �L =2��s,whereL isthe

length oftheacceleratorand �z = �z=�� isthelongitudinalbeta function[16,17]which m ay

be derived by identifying corresponding term sin the solution to the di�erentialequations

oflongitudinalm otion and a oneterm linearm ap.

W hen the growth rate is norm alized by N r0W 0=2�� ��,which includes the beam in-

tensity and geom etricfactors,we expecta universaldependence ofnorm alized growth rate

versushead-tailphasethatbeginslinearly with head-tailphase[18]and peaksaround -1[23].

11

Fig.6 showsthesim ulated growth rateatthreeintensitieswith a rangeofchrom aticites

from �:001 to �0:5 to gethead-tailphasesin the0 to �1 range.Thenorm alized curvesare

nearly identicaland peak close to head-tailphase ofunity.The deviation from a universal

curveisagain dueto di�erencesbetween theidealized m odeland detailed sim ulation.

V . B U N C H -B Y -B U N C H EM IT TA N C E G R O W T H AT T H E T EVAT R O N

Understandingthee�ectofunwanted long-rangecollisionsam ongm ultiplebeam bunches

in the design and operation ofhadron colliders hasreceived attention from otherauthors

[2,19]which underscores the im portance for this kind ofsim ulation. A schem atic ofthe

�llpattern ofproton and antiproton bunches in the Tevatron is shown in Fig.7. There

are three trains oftwelve bunches foreach species. A train occupies approxim ately 81:5�

separated by a gap ofabout38:5�. The bunch train and gap are replicated three tim esto

�llthering.Bunchescollidehead-on attheB0 and D0 interaction pointsbutundergo long

range(electrom agnetic)beam -beam interactionsat136 otherlocationsaround thering[24].

Running thesim ulation with all136 long-rangeIPsturnsoutto bevery slow so weonly

calculated beam -beam forcesatthetwo m ain IPsand and thelong-rangeIPsim m ediately

upstream and downstream ofthem .Thetransversebetafunctionsatthelong-rangecollision

locations are m uch larger than the bunch length,so the beam -beam calculation at those

locationscan beperform ed using only the2D solver.

Oneinteresting consequence ofthe�llpattern and thehelicaltrajectory isthatany one

ofthe 12 bunches in a train experiences collisionswith the 36 bunches in the otherbeam

at di�erent locations around the ring,and in di�erent transverse positions. This results

in a di�erent tune and em ittance growth for each bunch ofa train,but with the three-

fold sym m etry for the three trains. In the sim ulation,em ittance growth arises from the

e�ects ofim pedance acting on bunches that have been perturbed by beam -beam forces.

Thephenom enon ofbunch dependentem ittancegrowth isobserved experim entally[20].

Thebeam -beam sim ulation with 36-on-36 bunchesshowssim ilare�ects.W eran a sim u-

lation of36 proton on 36 antiproton bunchesfor50000turnswith thenom inalhelicalorbit.

The proton buncheshad 8:8� 1011 particles(roughly fourtim esthe usualto enhance the

e�ect)and theproton em ittancewasthetypical20� m m m rad.Theantiproton bunch inten-

sity and em ittance were both halfthe corresponding proton bunch param eter. The initial

12

FIG .7:Schem aticoftheposition ofproton and antiproton bunchesin theTevatron with 36 proton

and 36 antiproton bunches. The diagram shows the positions at a tim e when the lead bunch of

the trainsare at the head-on collision location. Head-on collisions occurat location B0 and D0.

The green shading indicates the part ofthe ring where beam -beam collisions m ay occur in the

sim ulationswith six-on-six bunches.

13

0 5 10 15 20 25 30 35 40bunch

17.4

17.6

17.8

18.0

18.2

18.4

18.6

18.8

(mmmrad)

(a) with long�range

(b) reduced long�range

14

16

18

20

22

24

26

28

store

5792

(mmmrad)

(c) store 5792

FIG .8: The sim ulated and m easured em ittance of each Tevatron proton bunch after running

with 36 proton and 36 antiproton bunches. Curves (a) and (b) which show the em ittance after

50000 sim ulated turns are read with the left verticalaxis. Curve (a) results from a sim ulation

with the nom inalbeam spacing atthe long-range IPs. Curve (b)results from a sim ulation with

the hypotheticalcondition where the beam separation atthe long-range IPsis100 tim esnorm al,

suppressing the e�ect ofthose long-range IPs. Curve (c) is the m easured em ittance ofbunches

after 15 m inutes ofa particular store (# 5792) ofbunches in the Tevatron,and is read with the

rightverticalaxis.

em ittanceforeach proton bunch wasthesam eso changesduring thesim ulation re ectthe

beam -beam e�ect.

Curve(a)in Figure8showstheem ittanceforeach ofthe36proton bunchesin a36-on-36

sim ulation after50000 turnsofsim ulation. The three-fold sym m etry isevident. The end

bunchesofthe train (bunch 1,13,25)are clearly di�erentfrom the interiorbunches. For

14

com parison,curve (c)shows the m easured em ittance taken during acceleratoroperations.

Theobserved bunch em ittancevariation issim ilarto thesim ulation results.Anotherbeam -

beam sim ulation with the beam separation atthe closest head-on IP expanded 100 tim es

itsnom inalvalueresulted in curve(b)ofFigure8 showing a m uch reduced bunch-to-bunch

variation.W econcludethatthebeam -beam e�ectatthelong-rangeIPsistheorigin ofthe

bunch variation observed in therunning m achineand thatoursim ulation ofthebeam helix

iscorrect.

V I. T EVAT R O N A P P LIC AT IO N S

A . Single bunch features

W e looked at the tune spectrum with increasing intensity for equalintensity p and �p

beam scontaining one bunch each. Asthe intensity increases,the beam -beam param eter�

increases. Fig.9 showsthe spectrum ofthe sum and di�erence ofthe two beam centroids

for � = 0:01;0:02;0:04,corresponding to beam bunches containing 2:2� 1011,4:4� 1011

and 8:8� 1011 protons. The abscissa isshifted so the base tune isat0 and norm alized in

unitsofthebeam -beam param eterata beam intensity of2:2� 1011.Thecoherent� and �

m ode peaksare expected to be presentin the spectra ofthe sum and di�erence signalsof

thetwobeam centroids.Thecoherent� m odesareevidentat0,whilethecoherent� m odes

should slightly greaterthan 1,2,and 4 respectively. Increasing intensity also causeslarger

induced wake �elds which broaden the m ode peaks,especially the � m ode,as shown in

Fig.9.

The4D em ittancesathigherintensitiesshow signi�cantgrowth over20000turnsasshown

in Fig.10. The kurtosisexcessofthe two beam srem ainsslightly positive forthe nom inal

intensity,but shows a slow increase at higher intensities indicating the the beam core is

being concentrated asshown in Fig.11.Concentration ofthebunch corewhileem ittanceis

growing indicatesthedevelopm entof�lam entation and halo.

B . Sim ulation ofbunch length,synchrotron m otion and beam -beam interactions

Synchrotron m otion in extended length bunches m odi�esthe e�ectsofthe beam -beam

interaction by shifting and suppressing the coherent m odes. The plots in Fig.12 show

15

�=0.01 sum mode

difference mode

Fourier

power

(arb

.units)

�=0.02 sum mode

difference mode

1 0 1 2 3 4 5normalized tune

�=0.04 sum mode

difference mode

FIG .9:Dipolem odespectra ofthesum and di�erenceo�setsoftwo beam centroidsatthreebeam

intensities corresponding to beam -beam param eter values for each beam of0.01,0.02 and 0.04.

Theverticalscale isin arbitrary units.

sim ulated spectra for sum and di�erence signals ofthe beam centroid o�sets for one-on-

onebunch collisionsin a ring with Tevatron-like optics,with both shortand long bunches,

at three di�erent synchrotron tunes. The sum signalwillcontain the � m ode while the

di�erencesignalwillcontain the� m ode.In thisTevatron sim ulation,thebeam strength is

setsothatthebeam -beam param eteris0.01,thebasetunein theverticalplaneis0.576,and

�y isapproxim ately 30cm . Subplotsa and b ofFig.12 show thatwith sm allsynchrotron

tune both the � and � m ode peaksare evidentwith shortand long bunches. The � m ode

peak isattheproperplace,with the� m odepeak shifted upwardsby theexpected am ount,

butwith longerbunches(subplotsc and d)theincoherentcontinuum isenhanced and the

strength ofthe coherent peaks is reduced. W hen the synchrotron tune is the sam e as or

16

0 5000 10000 15000 20000turns

6

8

10

12

14

16

184D

emittances(10�6mm2mrad2)

�=0.01

�=0.02

�=0.04

�=0.10

FIG .10:Theevolution of4D em ittancesforbeam -beam param etersof0.01,0.02,and 0.04 which

correspond to intensitiesof(a)2:2� 1011,(b)4:4� 1011,(c)8:8� 1011,and (d)1:1� 1012 protons

perbunch.

largerthan the beam -beam splitting (subplotse and f),shortbunchesstillexhibitstrong

coherentm odes,butwith long bunchesthecoherentm odesaresigni�cantly diluted.In the

caseoflong bunches,the� m odehasbeen shifted upwardsto 0.580,and the� m odeisnot

clearly distinguishablefrom thecontinuum .At�s of0:01and 0:02,thesynchrobetatron side

bandsareclearly evident.

C . M ulti-bunch m ode studies

W hen the Tevatron is running in its usual m ode, each circulating beam contains

36 bunches. Every bunch in one beam interacts with every bunch in the opposite beam ,

though only two interaction pointsare usefulforhigh energy physics running. The other

17

0 5000 10000 15000 20000turns

0.1

0.2

0.3

0.4(d)

0.1

0.2

0.3

0.4(a) kurtosisx

kurtosisy

0.1

0.2

0.3

0.4

reducedkurtoses

(b)

0.1

0.2

0.3

0.4(c)

FIG .11:Theevolution of(reduced)kurtosisoftheparticledistribution forintensitiesof(a)2:2�

1011,(b)4:4� 1011,(c)8:8� 1011,and (d)1:1� 1012 protonsperbunch.

136 interaction pointsareunwanted and detrim entalto beam lifetim eand lum inosity.The

beam orbitisde ected in a helicalshape by electrostatic separatorsto reduce the im pact

ofthese unwanted collisions,so the beam s are transversely separated from each other in

allbut the two high-energy physics interaction points. Because ofthe helicalorbit,the

beam separation isdi�erentateach parasiticcollision location.Forinstance,a bunch near

thefrontofthebunch train willundergo m orelong-rangeclosethethehead-on interaction

point,com pared toabunch neartherearofthebunch train.A particularbunch experiences

collisionsatspeci�cinteraction pointswith otherbuncheseach ofwhich hasitsown history

ofcollisions. Thiscausesbunch-to-bunch variation in disruption and em ittance growth as

willbedem onstrated below.

W ewillbegin thevalidation and exploration ofthem ulti-bunch im plem entation starting

18

0.0002

0.0004

0.0006

0.0008

0.0010a) �s=0.001

�z=10cm

b) �s=0.001

�z=43cm

sum

diff

0.0005

0.0010

0.0015

0.0020

0.0025

Fou

rier

pow

er

(arb

. u

nit

s)

c) �s=0.01 d) �s=0.01

0.55 0.56 0.57 0.58 0.59 0.60tune

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010 e) �s=0.02

0.55 0.56 0.57 0.58 0.59 0.60tune

f) �s=0.02

FIG .12:Sim ulated one-on-one bunch y plane� and � m odetunespectra forshortbunches(a,c,

e)and long bunches(b,d,f),forthreedi�erentsynchtrotron tunes,with a Tevatron-like lattice.

with runsoftwo-on-two bunchesand six-on-six bunchesbeforem oving on to investigatethe

situation with the fullTevatron bunch �llof36-on-36 bunches. Two-on-two bunches will

dem onstrate the bunchescoupling am ongsteach other,butwillnotbe enough to dem on-

strate the end bunch versus interior bunch behavior thatcharacterizes the Tevatron. For

that,wewilllook atsix-on-six bunch runs.

In thesestudies,weareonly�llingtheringwith atm ostsixbunchesin abeam .Referring

toFig.7,weseethatonlythehead-on location atB0iswithin thegreen shaded region where

beam -beam collisionsm ay occurwith six bunchesin each beam .Becauseofthebeam -beam

collisions,each bunch isweakly coupled toevery otherbunch which givesrisetom ulti-bunch

collective m odes.

W ebegan theinvestigation ofthesee�ectswith a sim ulation ofbeam swith two bunches

19

each. The bunches are separated by 21 RF buckets as they are are in norm alTevatron

operations. Collisions occur at the head-on location and at parasitic locations 10.5 RF

bucketsdistanton eitherside ofthe head-on location.To m ake any excited m odesvisible,

we ran with 2:2� 1011 particles,which givesa single bunch beam -beam param eterof0:01.

There are fourbunches in thisproblem . W e labelbunch 1 and 3 in beam 1 (proton)and

bunch 2 and 4 in beam 2 (antiproton) with m ean y positions of the bunches y1;:::y4.

By diagonalizing the covariance m atrix ofthe turn-by-turn bunch centroid deviations,we

determ inefourm odes,shown in Fig.13.Fig.13(a)showsthesplitting ofthe� m ode.The

coe�cientsofthe two m odesindicate thatthism ode isprim arily com posed ofthe sum of

corresponding beam bunches(1 with 2,3 with 4)sim ilarto the � m ode in the one-on-one

bunch case.Theothertwom odesin Fig13(b)havethecharacterand location in tunespace

ofthe � m ode,from their coe�cients and also their reduced strength com pared to the �

m ode.

W ith six on six bunches, features em erge that are clearly bunch position speci�c.

Fig.14(a) shows the turn-by-turn evolution of4D em ittance and (b) y kurtosis for each

ofthesix proton bunches.Itisstriking thatbunch 1,the�rstbunch in thesequence,hasa

lowerem ittancegrowth than alltheotherbunches.Em ittancegrowth increasesfasterwith

increasing bunch num berfrom bunches2{5,butbunch 6hasalowerem ittancegrowth than

even bunch 4. The kurtosis ofbunch 1 changes m uch less than that ofany ofthe other

bunches,butbunches2{5 havea very sim ilarevolution,whilebunch 6 ism arkedly closerto

bunch 1.Onedi�erencebetween theoutsidebunches(1and 6)and theinsidebunches(2{5)

isthatthey haveonly onebeam -beam interaction attheparasiticIP closestto thehead-on

collision,while the inside bunches have one collision before the head-on IP,and one after

it.Thetwo parasiticcollision pointsclosestto thehead-on collision pointhavethesm allest

separation ofany ofthe parasitics,so interaction there would be expected to disrupt the

beam m orethan interactionsatotherparasiticlocations.

To testthishypothosis,wedid two additionalruns.In the�rst,thebeam separation at

theparasiticIP im m ediately downstream ofthehead-on IP wasarti�cially increased in the

sim ulation soastohaveessentially noe�ect.Thee�ectofthisisthatthe�rstproton bunch

willnothaveany beam -beam collisionsatan IP closeto thehead-on IP,whilealltheother

buncheswillhave onecollision ata near-head-on IP.Thecorresponding plotsofem ittance

and kurtosisareshown in Fig.15.Thekurtosisdatashowsthatbunches2{5which allsu�er

20

10

20

30

40

50

60F

ou

rier

Pow

er

(arb

. u

nit

s) (a)�mode1

�mode2

�1.0 �0.5 0.0 0.5 1.0 1.5 2.0normalized tune (units of �)

5

10

15

20

25

(b)�mode1

�mode2

FIG .13:M odetunespectrum fora two on two bunch run at2:2� 1011 particles/bunch (� = 0:01).

Figure(a)showsthetwo m odesthatarem ostlike� m odes.� m ode1 is0:53y1+ 0:53y2+ 0:59y3�

0:31y4,� m ode 2 is0:39y1 + 0:49y2 � 0:46y3 � 0:63y4. Figure (b)showsthe two �-like m odes. �

m ode 1 is0:74y1 � 0:66y2 � 0:08y3,� m ode 2 is0:12y1 + 0:20y2 � 0:66y3 + 0:31y4. The absolute

scale ofthe Fourierpowerisarbitrary,butthe relative scalesofplots(a)and (b)arethe sam e.

one beam -beam collision ata close parasitic IP are alltogetherwhile bunch 1 which does

nothavea closeIP collision isseparated from theothers.

Em ittance and kurtosisgrowth in sim ulationswhere the beam separation atthe closest

upstream and downstream parasitic IPswasincreased isshown in Fig.16.In thiscon�gu-

ration no bunch su�ersa strong beam -beam collision ata parasiticIP closeto thehead-on

location so thekurtosisofallthebunchesevolvessim ilarly.

21

0 2000 4000 6000 8000 10000turns

6

8

10

12

14

16

18

20

22

244D

emittances(10�6mm2mrad2)

a)

p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

0 2000 4000 6000 8000 10000turns

�0.2

�0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kurtotis

b)

p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

FIG .14: A six-on-six bunch Tevatron run with 8:8� 1011 particles/bunch: (a)The turn-by-turn

evolution of4D em ittance ofeach ofthesix bunches.(b)Theturn-by-turn evolution ofy kurtosis

ofthesix bunches.

0 2000 4000 6000 8000 10000turns

10

15

20

25

4D

emittances(10�6mm2mrad2)

a)p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

0 2000 4000 6000 8000 10000turns

0.0

0.1

0.2

0.3

0.4

0.5

0.6

kurtotis

b) p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

FIG .15: In a six-on-six bunch Tevatron run with 8:8 � 1011 particles/bunch, with the beam

spacing atthe�rstparasiticIP downstream ofthehead-on location arti�cially increased:(a)The

4D em ittanceofeach ofthesix bunchesasa function ofturn.(b)they kurtosisofthesix bunches

asa function ofturn.

V II. LO W ER C H R O M AT IC IT Y T H R ESH O LD

During theTevatron operation in 2009 thelim itforincreasing theinitiallum inosity was

determ ined by particle losses in the so-called squeeze phase [21]. Atthisstage the beam s

areseparated in them ain interaction points(notcolliding head-on),and them achineoptics

isgradually changed to decreasethebeta-function attheselocationsfrom 1.5 m to 0.28 m .

22

0 2000 4000 6000 8000 10000turns

6

8

10

12

14

164D

emittances(10�6mm2mrad2)

a)p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

0 2000 4000 6000 8000 10000turns

�0.2

0.0

0.2

0.4

0.6

0.8

kurtotis

b) p bunch 1

p bunch 2

p bunch 3

p bunch 4

p bunch 5

p bunch 6

FIG .16:In a six-on-six bunch Tevatron run with 8:8� 1011 particles/bunch,with theboth nearest

upstream and downstream parasitic IP arti�cially widened: (a)The 4D em ittance ofeach ofthe

six bunchesasa function ofturn.(b)the y kurtosisofthe six bunchesasa function ofturn.

W ith proton bunch intensitiescurrently approaching3:2� 1011 particles,thechrom aticity

oftheTevatron hasto bem anaged carefully to avoid thedevelopm entofa head-tailinsta-

bility. Itwasdeterm ined experim entally thatafterthe head-on collisionsare initiated,the

Landau dam ping introduced by beam -beam interaction isstrong enough to m aintain beam

stability atchrom aticity of+2 units(in Tevatron operations,chrom aticity is��=(�p=p).)

Attheearlierstagesofthecollidercycle,when beam -beam e�ectsarelim ited to long-range

interactions the chrom aticity waskept ashigh as15 unitssince the concern was thatthe

Landau dam ping is insu�cient to suppress the instability. At the sam e tim e,high chro-

m aticity causesparticlelosseswhich areoften largeenough to quench thesuperconducting

m agnets,and henceitisdesireableto keep itata reasonablem inim um .

Ourm ulti-physicssim ulation wasused to determ inethesafelowerlim itforchrom aticity.

The sim ulations were perform ed with starting beam param eters listed in Table I. W ith

chrom aticity setto-2units,and nobeam -beam e�ect,thebeam sareclearlyunstableasseen

inFig.17.W ithbeam sseparated,turningonthebeam -beam e�ectpreventsrapidoscillation

growth during the sim ulation as shown in Fig.18. The bursts ofincreased am plitude is

som etim es indicative ofthe onset ofinstability,but it is not obvious within the lim ited

duration ofthisrun.TheRM S sizeofthebeam also doesnotexhibitany obviousunstable

tendenciesasshown in Fig.19.

Based on these �ndingsthe chrom aticity in the squeeze waslowered by a factoroftwo,

23

TABLE I:Beam param etersforTevatron sim ulation

Param eter value

beam energy 980G eV

p particles/bunch 3:0� 1011

�p particles/bunch 0:9� 1011

p tune(�x;�y) (20.585,20.587)

p (norm alized)em ittance 20� m m m rad

�p tune(�x;�y) (20.577,20.570)

�p (norm alized)em ittance 6� m m m rad

synchrotron tune�s 0.0007

slip factor 0.002483

bunch length (rm s) 43cm

�p=p m om entum spread 1:2� 10�4

e�ective piperadius 3cm

and presently is kept at8-9 units. This resulted in a signi�cant decrease ofthe observed

particlelossrates(see,e.g.,Fig.5 in [21]).

V III. SU M M A RY

Thekeyfeaturesofthedeveloped sim ulation includefullythree-dim entionalstrong-strong

m ulti-bunch beam -beam interactions with m ultiple interaction points,transverse resistive

wallim pedance,and chrom aticity. The beam -beam interaction m odelhas been shown to

reproduce the location and evolution ofsynchrobetatron m odes characteristic ofthe 3D

strong-strong beam -beam interaction observed in experim entaldata from the VEPP-2M

collider. The im pedance calculation with m acroparticlesexcitesboth the strong and weak

head-tailinstabilitieswith thresholdsand growth ratesthatareconsistentwith expectations

from a sim ple two-particle m odeland Vlasov calculation. Sim ulation ofthe interplay be-

tween the helicalbeam -orbit,long range beam -beam interactionsand the collision pattern

qualitatively m atchesobserved patternsofem ittancegrowth.

The new program is a valuable toolforevaluation ofthe interplay between the beam -

24

0 10000 20000 30000 40000 50000 60000 70000 80000turn

�0.4

�0.3

�0.2

�0.1

0.0

0.1

0.2

0.3

0.4xdipole

offset(�m)

FIG .17:Thex dipolem om entin a sim ulation with C = � 2 and no beam -beam e�ectshowing the

developm entofinstability.

beam e�ectsand transverse collectiveinstabilities.Sim ulationshavebeen successfully used

tosupportthechangeofchrom aticity attheTevatron,dem onstratingthateven thereduced

beam -beam e�ectfrom long-rangecollisionsm ayprovideenough Landaudam pingtoprevent

thedevelopm entofhead-tailinstability.These resultswere used in Tevatron operationsto

supporta changeofchrom aticity during thetransition to colliderm odeoptics,leading to a

factoroftwo decrease in proton losses,and thusim proved reliability ofcollideroperations.

A cknow ledgm ents

W ethank J.Qiang and R.Ryne ofLBNL forthe use ofand assistance with theBeam -

Beam 3d program . W e are indebted to V.Lebedev and Yu.Alexahin for usefuldiscus-

sions.Thiswork wassupported by theUnited StatesDepartm entofEnergy undercontract

25

0 10000 20000 30000 40000 50000 60000 70000 80000turn

�0.4

�0.3

�0.2

�0.1

0.0

0.1

0.2

0.3

0.4xdipole

offset(�m)

FIG .18: The x dipole m om entofa representative bunch in a 36-on-36 sim ulation with C = � 2

with beam -beam e�ects and beam sseparated showing no obviousinstability within the lim its of

the sim ulation.

DE-AC02-07CH11359 and the Com PASS project funded through the Scienti�c Discovery

through Advanced Com puting program in the DOE O�ce ofHigh Energy Physics. This

research used resourcesoftheNationalEnergy Research Scienti�cCom putingCenter,which

issupported by theO�ceofScienceoftheU.S.Departm entofEnergy underContractNo.

DE-AC02-05CH11231.Thisresearch used resourcesoftheArgonneLeadership Com puting

Facility atArgonneNationalLaboratory,which issupported by theO�ceofScienceofthe

U.S.Departm entofEnergy undercontractDE-AC02-06CH11357.

[1] Run IIhandbook,http://www-bd.fnal.gov/runII

26

0 10000 20000 30000 40000 50000 60000 70000 80000turn

28

30

32

34

36

38

40

42

44xRMS(�m)

FIG .19: The x RM S m om ent ofa representative bunch in a 36-on-36 sim ulation with C = � 2

with beam -beam e�ects and beam sseparated showing no obviousinstability within the lim its of

the sim ulation.

[2] A.Valishev etal.,\O bservations and M odeling ofBeam -Beam E�ects atthe Tevatron Col-

lider",PAC2007,Albuquerque,NM ,2007

[3] M .Xiao etal.,\Tevatron Beam -Beam Sim ulation atInjection Energy",PAC2003,Portland,

O R,2003

[4] Y.Alexahin,\O n theLandau Dam ping and DecoherenceofTransverseDipoleO scillationsin

Colliding Beam s",Part.Acc.59,43 (1996)

[5] E.A. Perevedentsev and A.A. Valishev, \Sim ulation of Head-Tail Instability of Colliding

Bunches",Phys.Rev.ST Accel.Beam s4,024403,(2001)

[6] J. Q iang, M .A. Furm an, R.D. Ryne, J.Com p.Phys., 198 (2004), pp. 278{294, J. Q iang,

M .A.Furm an,R.D.Ryne,Phys.Rev.ST Accel.Beam s,104402 (2002).

27

[7] V.Lebedev,http://www-bdnew.fnal.gov/pbar/

organizationalchart/lebedev/O ptiM /optim .htm

[8] A. Valishev et al., \Progress with Collision O ptics of the Ferm ilab Tevatron Collider",

EPAC06,Edinburgh,Scotland,2006

[9] A.Chao,Physics ofCollective Beam Instabilities in High Energy Accelerators.,pp.56{60,

178{187,333-360 John W iley and Sons,Inc.,(1993)

[10] A.Piwinski,IEEE Trans.Nucl.Sci.N S-26 (3),4268 (1979).

[11] K .Yokoya,Phys.Rev.ST-AB 3,124401 (2000).

[12] I.N.Nesterenko,E.A.Perevedentsev,A.A.Valishev,Phys.Rev.E,65,056502 (2002)

[13] E.A.Perevedentsev and A.A.Valishev, Phys.Rev.ST Accel.Beam s 4, 024403 2001 ; in

Proceedingsofthe7th European ParticleAcceleratorConference,Vienna,2000 unpub-lished

,p.1223;http://accelconf.web.cern.ch/AccelConf/e00/ index.htm l

[14] V.V.Danilov and E.A.Perevedentsev,Nucl.Instrum .M ethodsPhys.Res.A 391,77 1997 .

[15] A.Chao,PhysicsofCollective Beam Instabilitiesin High Energy Accelerators.,Figure4.8 on

p.183,John W iley and Sons,Inc.,(1993).

[16] A.Chao,Physics ofCollective Beam Instabilities in High Energy Accelerators.,p.9,John

W iley and Sons,Inc.,(1993)

[17] D.A.Edwardsand M .J.Syphers,An Introducution to thePhysicsofHigh EnergyAccelerators,

pp.30{46,John W iley and SonsInc.,1993.

[18] A.Chao,Physics ofCollective Beam Instabilities in High Energy Accelerators.,p.351,John

W iley and Sons,Inc.,(1993).

[19] F.W .Jones, W .Herr, T.Pieloni, ParallelBeam -Beam Sim ulation Incorporating M ultiple

Bunches and M ultiple Interaction Regions, THPAN007 in Proceedings of PAC07, Albu-

querque,NM (2007),T.Pieloni,W .Herr,M odelsto Study M ultiBunch Coupling Through

Head-on and Long-range Beam -Beam Interactions,W EPCH095 in Proceedings ofEPAC06,

Edinburgh,Scotland,(2006),T.Pieloniand W .Herr,Coherent Beam -beam M odes in the

CERN Large Hadron Collider(LHC)forM ultiple Bunches,Di�erentCollision Schem esand

M achine Sym m etries,TPAT078 in ProceedingsofPAC05,K noxville,TN,(2005).

[20] V.Shiltsev,etal.,\Beam -Beam E�ectsin theTevatron," PRSTAB,8,101001 (2005).

[21] A.Valishev etal.,\RecentTevatron O perationalExperience",PAC2009,Vancouver,BC,2009

[22] W e are sim ulating a m uch larger intensity than would be possible in the actualm achine in

28

orderto drive thestrong headtailinstability forcom parison with theanalyticalm odel.

[23] The sim ulated m achine isabove transition (� ispositive.)The head-tailinstability develops

when chrom aticity isnegative,thusthehead-tailphaseisnegative.

[24] W ith three-fold sym m etry of bunch trains, train-on-train collisions occur at six locations

around the ring.The collision oftwo trains of12 bunches each results in bunch-bunch col-

lisions at 23 locations which when m ultiplied by six results in 138 collision points.It is a

straightforward com puter exercise to enum erate these locations.Two ofthese locations are

distinguished ashead-on whilethe rem ainderareparasitic.[20]

29