High Brightness Electron Beam Diagnostics and their ... - CASA

w.r.t.theRF-wave.ThisfactisaconsequenceofphaseslippagebetweentheRFwaveandthe cavity#4,oneneedstoinjectthebunchwith'40deg. bunchwhichisnotyetrelativistic.Infacttoobtainthemaximumpossibleenergyattheexitof anelectronbeamofinitialenergyof350keV.Itisnotablyseenthatmaximumenergygainprovided bytherstcavity(cavity#4)isnotobtainedbyinjectingthebunchwitharelativephase=0
Reduced Energy Gain (no unit)
2
Cav #4
10
-0.25 -0.125 0.0 0.125 0.25
Distance (m)
0
10
5
2
1
=10 deg.
=-10 deg.
=0 deg.
10
-0.25 -0.125 0.0 0.125 0.25
Distance (m)
0
10
5
2
1
10 0
10
5
2
1
cavity. Radio-FrequencyinducedFocusing Figure6.4:Reducedenergygain,,alongtherst(cavity#4)andsecond(cavity#3)cryounit
=-16 deg.
Thefundamentalmodelongitudinalacceleratingeldalsoinduces,byvirtueofMaxwell'sequations,
=0 deg.
Thisradialeldtogethercombinedwiththeequationofmotion@tp=eEryieldsatransverse equationofmotion,e.g.forthehorizontalplane: aradialelectriccomponentthatwrites: d2x(z) dz2+0(z) Er(z)=r(z) (z)dx(z) dz+d2(z) 2d dzEz dz2x(z)=0 (6.8) (6.7)
Thisequationcanbesolvednumerically,butsomeapproximatedsolutionhavebeenderivedby Chambers[58,57],forpure-modeacceleratingcavities,andareinverygoodagreementwiththe
2
Cav #3
2