High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
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spaceabeamthatconsistsofNparticles,isbestdescribed,atagiveninstant,intermsofa d6=dxdpxdydpydzdpzinthevicinityofthepoint(x;y;z;px;py;pz)is: densityfunctionn6(x;y;z;px;py;pz).Thenumberofparticlesinanelementofphasespacevolume Thetotalvolumein6occupiedbythebeam,atagiveninstantis: Thisquantity,generallyreferredas6D-hyper-emittance,iswelldenedprovidedthedensityfunctionn6isacompactfunction. d6N=n6(x;y;z;px;py;pz)d6 V6=ZZZZZZd6 (4.1)<br />
Ausefulsimplication,whendescribingabeam,occurswheneachdegreeoffreedomisindependent ofthetwootherdegreesoffreedom.ThenthesinceHamiltonianwriteasthesumofuncoupled (4.2)<br />
factorizesas: summarizedasfollow: projectedphasespace.Themainpropertiesoftheparticletrajectoriesintheseplanescanbe Insuchcase,thebeamdynamicscanbestudiedseparatelyineachofthethreetwo-dimension sub-hamiltoniancorrespondingforeachofthethreedegreeoffreedom,thedensitydistribution<br />
Thetrajectoriesdependontheinitialvaluesofthecoordinate<strong>and</strong>thetime.Animportant consequenceisthattwotrajectorieswithdierentinitialconditioncannotintersect.Also n6(x;y;z;px;py;pz)=n2;x(x;px)n2;y(y;py)n2;z(z;pz) (4.3)<br />
Inlinearphasespacetransformations,ellipsesmaptoellipses,straightlinestostraightlines. mapintoaboundaryatatimet0whichenclosethesamegroupofparticles. Aboundaryinthephasespacethatencloseagivennumberofparticleatagiventimetwill bifurcation. notethatatrajectoryatagiventimecanhaveseveralvaluetherebyyieldingphase-space<br />
ville'stheoremwhichstatesthatthedensityofparticleintheappropriatephasespaceisinvariantalongthetrajectoryofanygivenpoint.Thistheoremcanalsobeexpressedintermsoftheinvarianceofthephasespacehyper-volumeenclosingachosengroupofpointsastheymoveinthephase space.Theensembleofparticlebehavesasanincompressibleuid: Generallythephasespacedensityfunction,n6(x;y;z;px;py;pz)isLiouvilliani.e.itsatisesLiou- Suchgeometrymightbeappropriatetolimitphasespacedensity.<br />
WeshouldinsistthatLiouville'stheoremappliestoconservativeHamiltoniansystemsi.e.systems theoremcannotbeappliedwhen: inwhichtheforcescanbederivedfromapotential.Inthecaseofchargedparticlebeam,Liouville Emissionofelectromagneticradiation(e.g.synchrotron) dV dt=0 (4.4)<br />
Quantumexcitationeectarenonnegligible<br />
Nonnegligibleselfinteraction(e.g.spacechargeforce,coherentsynchrotronradiation,...)
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