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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Space Charge over Emittance Ratio (no unit)<br />
10 2<br />
10<br />
0 200 400 600 800 1000 1200<br />
Distance from the Photocathode (cm)<br />
-4<br />
10 -3<br />
10 -2<br />
10 -1<br />
10 0<br />
10 1<br />
Rr tion. EnergyGain Theaccelerationinacceleratingcavitiesisprovidedbythelongitudinalcomponentoftheelectric Figure6.3:\spacechargeoveremittanceratioforthetransverse(Rr)<strong>and</strong>longitudinal(Rz)direc- eldofthefundamentalmode.Sucheldcanbewrittenapproximately:<br />
Rz betweentheparticle<strong>and</strong>theRF-wave.Becauseof<strong>their</strong>energyattherstcavityentrance,350keV, theelectronsarenotrelativistic<strong>and</strong>thereforeoneelectronisnotgoingtokeepthesamerelative phasewithrespecttotheRF-wave,sucheectisnamedphaseslippage.Let'sdenethephase E0isthepeakeld,zisthepositionwithrespecttothecavitycenter,<strong>and</strong>istheosetphase (z)as: Ez=E0cos(kz)cos(!t+)=E0 2(cos(!t+kz)+cos(!t++kz)) (6.4)<br />
TheEqns.(6.5)<strong>and</strong>(6.6)togetherformacoupleddierentialequationsystemthatcanbesolved Moreoverthenormalizedenergygainis: d(z) dz=eE0 (z)def =!tkz=kZz 2mc21(cos((z)+2kz)+cos((z))) 0 p211!dz+ (6.6) (6.5)<br />
numericallyusingst<strong>and</strong>ardtechnique.Figures6.4presentstheenergygaininthetwocavitywith
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