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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istherenormalizedhorizontalmomentumortheparticledivergence.Thevariablesx<strong>and</strong>x0areno Firstlyitisalwayspreferabletoworkinthetracespacewhichistheplanexx0wherex0=px=pz thereisnocouplingbetweenthesetwosubphase-spaces.Thereforewewillconsiderthehorizontal phasespacexpx,similardiscussionisvalidfortheverticalphasespaceypy. Henceforth,wewillonlyconcentrateonthetransversephase-space,xpx<strong>and</strong>ypy<strong>and</strong>assume 4.1.2PhaseSpace<strong>and</strong>Emittance<br />
parameters,theemittance",thebetatronfunctionT<strong>and</strong>theTfunction;itsequationisgiven morecanonicallyconjugatebutthephasespacepropertiesexposedpreviouslyarestillapplicablein thetracespace.Forsakeofsimplicity,thephasespacedistributionisgenerallyarbitrarilybounded<br />
whereTisdenedasT=1+2T byellipsessincetheyhavethegoodpropertiestomapintoellipsesundercanonicaltransformation. Suchellipseisgenerallyreferredasthephasespaceellipse.Itcanbefullyspeciedwiththree by3: geometricemittance<strong>and</strong>correspondstotheareaoftheellipse: T.Inthispreviousequation,theemittanceisgenerallynamed Tx2+2Txx0+Tx02=" Zellipsedxdx0def =" (4.5)<br />
<strong>and</strong>beingthebeammatrix: ThebilinearformexpressedinEqn.(4.5),canberewritteninamatrixform!x!xTwith!x=(x;x0) def = TT!def TT = 1112 1222! (4.6)<br />
Despitethisdenitionofemittanceistheonegenerallyusedbyexperimentalist,itsuersfrommany problemespeciallyinpresenceofnon-gaussianphasespacedistributionorwhennonlineareects nonlinearprocessesgenerallyyieldnonlineardistortionsofphasespacewhichrenderthegeometric emittanceconceptdiculttoquantifyaphasespacewhichshowsagreatdealofstructure(e.g. arepresentinthetransportchannel(chromaticaberration,wakeeld,spacecharge,...).These (4.7)<br />
order,hx2i,hx02i,hxx0i,momentsofthephasespacedistribution.Thenwec<strong>and</strong>enearootmean Aconvenientwayistostatisticallycharacterizethephasespaceusingtherst,hxi,hx0i<strong>and</strong>second squareemittance[27]as: lamentation).<br />
thermsemittance.Also,mostofthetimeonerathernormalizedtheemittancewithrespecttothe momentum<strong>and</strong>denethenormalizedrmsemittanceas: Itisalsocommontondintheliteraturetheeectiveemittancewhichisthedenedas4times 3InthisSectiontheTwissparametersareindexedwiththesubscriptTtoavoidconfusionwithothervariables.<br />
~"x=hh(xhxi)2ih(x0hx0i)2ih(xhxi)(x0hx0i)i2i1=2 ~"nx=~"x (4.9) (4.8)<br />
Later,whereconfusioncannotoccur,wewillomitthissubscript.
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