High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
whereiistheerrorontheithbeamsizemeasurement. Itiswellknown[40]thatthevalueof0thatminimizedthe2satises: 0B@PNi=1R211(i)(i) PNi=12R11(i)R12(i)(i) PNi=1R212(i)(i) 2(i) 1CA=0B@PNi=1R411(i) PNi=12R311(i)R12(i) 2(i)PNi=12R311(i)R12(i) 2(i)PNi=14R211(i)R212(i) 2(i)PNi=1R211(i)R212(i) 2(i)PNi=12R11(i)R312(i) 2(i) matricesequation.Itisworth(forsoftwareimplementation)tonotethatthecurvaturematrixhas The3x3matrixisnamedcurvaturematrix.Thesolutionfor0isobtainedinvertingtheprevious PNi=1R211(i)R212(i) c,b=C0 2(i)PNi=12R11(i)R212(i) 2(i) PNi=1R412(i) 2(i) 2(i)1CA0B@(0) (4.26) 11 12 221CA (0)<br />
thefollowingform: (0)<br />
whichyieldsaverytractableformforitsinverse,theerrormatrix: E=1 jCj0B@4(CED2)2(BECD)4(BDC2) 2(BECD)(AEC2)2(ADBC) 4(BDC2)2(ADBC)4(ACB2)1CA C=0B@A2BC 2B4C2B C2BE1CA (4.28) (4.27)<br />
matrixelementnamely: Theelementsoftheerrormatrixarethevariance<strong>and</strong>covariancenumbersonthecomputed(0) wherejCjdenotesthedeterminantofC: jCj=4(ACEAD2B2E+2BCDC3) 8>:11=pE11 11;12=E12 12=pE22 22=pE33 (4.29)<br />
Twissparametersatthelocation(0).Letbethecomputedparameteri.e.theemittanceorthe Twissparameters.Thentheerroronisgivenby: Usingthest<strong>and</strong>arderrorpropagationtheory[40]onecanestimatetheerrorsonthecomputed 11;22=E13 12;22=E23 (4.30)<br />
Thepartialderivativeintheerrorpropagationformulaearegivenby: ()2= +2@ @11@ @T @112(11)2+@ @12(11;12)2+2@ @ @11=T2,@~" @~" @122(12)2+@ @11@ @12=T,@~" @22(11;22)2+2@ @22=T2, @222(22)2+ @12@ @22(12;22)2(4.32)<br />
(4.31)<br />
@11=TT @T@11=2TT<br />
2~",@T 2~",@T @12=1+2T @12=TT ~",@T ~",@T @22=TT @22=2T 2~". 2~", (4.33)
Change language