High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
B.3.1DIMAD TheprogramDIMAD2studiesparticlebehaviorincircularmachinesandinbeamlines.Thetra- B.3TheSimulationTools chargedparticlecomputercodes. B.3.2TLIE likeitspredecessorDIMAT,istheresultofmanyyearsofexperimentingwithseveraldierent jectoriesoftherelativisticparticlesarecomputedaccordingtothesecondordermatrixformalism. Itdoesnotprovidesynchrotronmotionanalysisbutcansimulateit.Theprogramprovidestheuser withthepossibilityofdeningarbitraryelementstotailortheprogramtospecicuses.DIMAD, tosecondorderlikedimad.ThePhysicsbehindthiscodeisbasedontheuseoftheLieAlgebra vectorspacewithaproductverifyingtheproperties(1)(xy)=(x)y=x(y)and(2) operatortopropagatetransfertmapalongabeamlinesection.ALiealgebraisanalgebra(i.e.a Tlie3isageneral6DrelativisticdesigncodewithaMADcompatibleinputlanguage.The particularityofTLieisitsabilitytocomputetransfermapatanarbitraryorderandnotonlyup Pi@f TheLiealgebraoperatorusedinBeamDynamicsisthePoissonbracketdenedas:[f;g]= forsuchachoiceisthefactthatwiththehelpofthecanonicalHamiltonequations,wecanwrite y(x1+x2)=yx1+yx2)thatalsoveriestheJocobiidentity:x(yz)+y(zx)+z(xy)=0. forafunctionf(pi;qi): @qi@g @pi@g @qi@f @piwheregandfarefunctionsofthegeneralizedvariablespiandqi.Thereason thatfisnotanexplicitfunctionoftimei.e.@f :f:isaLieoperator.ToillustratehowtheTliecodeworks,let'sassume,forthetimebeing, (pi;qi).Instandardnotation,thePoissonbracketoperatorisoftenwritten:f:g=[f;g]where whereHistheHamiltonianthatgovernstheevolutionofthedistributionfintheconjugatespace df dt=[f;H]=:f:H=:H:f,andusingpurelysymbolicequationwehaveintermofoperator: df dt=@f @t+[f;H] @t=0inEqn.(B.8).ThenEqn.(B.8)becomes: (B.8) e:H:f=f+[H;f]+1=2[H;[H;f]]+:::. ThebeautyofLieAlgebratechniqueresidesinthatthecomputationofthefunctionfatatime dierentialequationise:H:fwheretheexponentiationofLieoperatorisdenedastheseries order,thisistheprincipleonwhichTlieisbased:thehamiltonianHalongwiththecorresponding operatore:H:arecomputedforeachbeamlinepieceandareconcatenedusingtheCampbell-Baker- calculationcanbecarriedatanyorderbycomputingthePoissonbracketseriesatthedesired t=t0+,knowingthefunctionfattimet0,justconsistsofcomputingft=e:H:ft0;such dt=:H:sotheLieoperator:H:reducestoasimpletimederivative.Thesolutionofthis REPORT285SLAC-StanfordUniversityCA-USA(1985) Hausdortheorem,thatstatese:HA!C:=e:HA!B:e:HB!C:,toobtaintheLieoperatorforawhole beamlinesection[A,C]. 3ThecodewaswrittenbyJohannesvanZeijtsandFilippoNeri 2R.Sevranckx,K.L.Brown,L.Scachinger,andD.Douglas,\UsersGuidetotheProgramDIMAD",SLAC
B.3.3PARMELA FeaturesofJeersonLabVersion \PhaseandRadialMotioninElectronLinearAccelerators."Itisaversatilemulti-particlecode thattransformsthebeam,representedbyacollectionofmacroparticles,throughauser-specied linacand/ortransportsystem.Itincludesa2-Dspace-chargecalculationandanoptional3-D point-to-pointspace-chargecalculation.PARMELAintegratestheparticletrajectoriesthrough theelds.Thisapproachisespeciallyimportantforelectronswheresomeoftheapproximations nothold.PARMELAworksequallywellforeitherelectronsorions.PARMELAcanreadeld usedbyothercodes(e.g.the"drift-kick"methodcommonlyusedforlow-energyprotons)would AtJeersonLab,amodiedversionofparmelahasbeenproducedbyH.Liu.Itincorporatesa3D distributionsgeneratedbyeitherFISHforrfproblemsorPOISSONformagnetproblems. ModiedSpacechargealgorithm point-by-pointspacechargealgorithmfromK.T.Mc.Donald4.Anoutlineofthealgorithmisas follows.parmelauseatwo-stepmethodtogenerateaspacechargeimpulseoneachmacroparticle: (!)itdeterminesthenetelectromagneticspacechargeeldatthelocationofeachmacroparticle duetoallothermacroparticle,(2)applythespacechargeimpulsetoeachmacroparticle.Then trackthemacroparticlethroughaslice(widthdenedbytheuser)ofthebeamline(forsimple elementthetrackingisperformedusingsecondordertransfermatrix,buttheusercanifdesired eachslice).Thisspacepoint-by-pointalgorithmisverysimplebutbecauseofthe1=r2dependence denethe3Dmapoftheelectromagneticeld.Insuchcasetheequationofmotionisintegratedin (whichcanleadtosingularityornumericalnoise)itmustbeimplementedcarefully.Forinstancein theeventualitytwomacroparticlescomeveryclosetoeachotherthechargeofthemacroparticles inthealgorithmisreduced.Thealgorithmtoreducethemacroparticlechargeisdiscussedindetail elsewhere5 AsimplemodelforCoherentSynchrotronRadiation chargeQiandlengthiorbitingonacirculartrajectoryofradiusRidetectedatthepresenttime byanobserverelectronjderivedandexpressedas: TheimplementationoftheCSRinteractionintothePARMELAcodecloselyfollowsthemethod describedbyCarlsten6wheretheelectriceldgeneratedataretardedangle0byalineiofuniform forshortelectronbunchesincircularmotionusingtheretardedGreen'sfunctiontechnique"Phys.RevE54num1, TN-94-040,JeersonLab,NewportNews,VA-USA(1994) pp838-845(1996) 4K.T.McDonald,IEEETrans.Elect.Dev.35p2052(1988) 6B.E.Carlsten,\Calculationofthenoninertialspace-chargeforceandthecoherentsynchrotronradiationforce 5H.Liu,\ConceptofVariableParticleSizeFactorforaPoint-by-PointSpaceChargeAlgorithm",CEBAFreport Ei;j=Qi i"1 r(1!ibnij)"12i2ixj Ri+2i(1cos(0))##fr (B.9)
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B.3.1DIMAD TheprogramDIMAD2studiesparticlebehaviorincircularmachines<strong>and</strong>inbeamlines.Thetra- B.3TheSimulationTools<br />
chargedparticlecomputercodes. B.3.2TLIE likeitspredecessorDIMAT,istheresultofmanyyearsofexperimentingwithseveraldierent jectoriesoftherelativisticparticlesarecomputedaccordingtothesecondordermatrixformalism. Itdoesnotprovidesynchrotronmotionanalysisbutcansimulateit.Theprogramprovidestheuser withthepossibilityofdeningarbitraryelementstotailortheprogramtospecicuses.DIMAD,<br />
tosecondorderlikedimad.ThePhysicsbehindthiscodeisbasedontheuseoftheLieAlgebra vectorspacewithaproductverifyingtheproperties(1)(xy)=(x)y=x(y)<strong>and</strong>(2) operatortopropagatetransfertmapalongabeamlinesection.ALiealgebraisanalgebra(i.e.a Tlie3isageneral6DrelativisticdesigncodewithaMADcompatibleinputlanguage.The particularityofTLieisitsabilitytocomputetransfermapatanarbitraryorder<strong>and</strong>notonlyup Pi@f TheLiealgebraoperatorusedin<strong>Beam</strong>DynamicsisthePoissonbracketdenedas:[f;g]= forsuchachoiceisthefactthatwiththehelpofthecanonicalHamiltonequations,wecanwrite y(x1+x2)=yx1+yx2)thatalsoveriestheJocobiidentity:x(yz)+y(zx)+z(xy)=0. forafunctionf(pi;qi): @qi@g @pi@g @qi@f @piwhereg<strong>and</strong>farefunctionsofthegeneralizedvariablespi<strong>and</strong>qi.Thereason<br />
thatfisnotanexplicitfunctionoftimei.e.@f :f:isaLieoperator.ToillustratehowtheTliecodeworks,let'sassume,forthetimebeing, (pi;qi).Inst<strong>and</strong>ardnotation,thePoissonbracketoperatorisoftenwritten:f:g=[f;g]where whereHistheHamiltonianthatgovernstheevolutionofthedistributionfintheconjugatespace df dt=[f;H]=:f:H=:H:f,<strong>and</strong>usingpurelysymbolicequationwehaveintermofoperator: df dt=@f @t+[f;H] @t=0inEqn.(B.8).ThenEqn.(B.8)becomes: (B.8)<br />
e:H:f=f+[H;f]+1=2[H;[H;f]]+:::. ThebeautyofLieAlgebratechniqueresidesinthatthecomputationofthefunctionfatatime dierentialequationise:H:fwheretheexponentiationofLieoperatorisdenedastheseries order,thisistheprincipleonwhichTlieisbased:thehamiltonianHalongwiththecorresponding operatore:H:arecomputedforeachbeamlinepiece<strong>and</strong>areconcatenedusingtheCampbell-Baker- calculationcanbecarriedatanyorderbycomputingthePoissonbracketseriesatthedesired t=t0+,knowingthefunctionfattimet0,justconsistsofcomputingft=e:H:ft0;such dt=:H:sotheLieoperator:H:reducestoasimpletimederivative.Thesolutionofthis<br />
REPORT285SLAC-StanfordUniversityCA-USA(1985) Hausdortheorem,thatstatese:HA!C:=e:HA!B:e:HB!C:,toobtaintheLieoperatorforawhole beamlinesection[A,C]. 3ThecodewaswrittenbyJohannesvanZeijts<strong>and</strong>FilippoNeri<br />
2R.Sevranckx,K.L.Brown,L.Scachinger,<strong>and</strong>D.Douglas,\UsersGuidetotheProgramDIMAD",SLAC
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