1: n(k) dk. it is necessarily tied to the. out a possible mis-interpretation

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1 ADDEXDUM TO T R A Early August 6, 1968 Several SAC! pysicists ave pointed out a possible is-interpretation of te results of te tick target bresstralung TJI progra described in Since te output of te progra is given in ters of differential radiation probability per radiation lengt, it is necessarily tied to te forula for radiation lengt used in te progra, Tis forula is identical to tat used in te Alvarez progra and is based on te coplete screening approxiation of te Bete-Maxion equation Te result is tat te integral 1: n(k) dk Q is sligtly less tan one, even for te tin target case, - and sligtly dependent on bea energy Eo Te progra results are correct wen te nuber of electrons,into te target are onitored, and te radiation lengt used for te target aterial is cornpatable wit tat used in te progra If quantaeter easureents of te total energy in te poton bea are used, owever, ten care ust be taken to properly noralize te quantity kn(k) Tn, correct poton spectru can be obtained by dividing tis quantity by te above integral A deck is available for evaluating tis integral nuericillyo Te deviation fro one is of te order of one or two per cent for te tin target case, but can be considerably iger for tick target cases

2 T Marc 1966 not to be abstracted, quoted or A THICK TARGET BREMSSTRAHLUG PROGRAM of te autor R A Early A tick target bresstralung progra as been written in Fortran IV Te progra is based on a Balgol progra originally written by R Alvarez at HE PL Te tin target calculation uses te forulation of Bete-Maxion2 wic includes screening effects and a coulob correction ter Tis section is identical to te original Balgol progra Te tick target routine is different due to difficulties encountered near te ig energy tip in te original progra Te Fortran IV progra can be run on te SDS 9 coputer in te Central Lab and a binary tape (Magpak) is available A deck is available for use on te IBM 79 also Execution tie is about seconds per point on eiter acine Te tick target calculations allow for te cange in electron energy spectru wit distance into te target due to radiation losses, as well as poton losses due to pair production in te target Secondary potons due to bresstralung of pairs are not coputed Tin Target Calculation Te tin target routine is contained in subroutine BETHE Te differential radiation probability per unit radiation lengt is coputed fro te expression for te cross section equations in Ref 1 were: r = classical electron radius Z = atoic nuber k = poton energy E = electron energy E =Eo-k -1-

3 +1 and 2 are te screening functions of Bete and Heitler' and f(z) is te coulob correction ter of Bete and Maxion27 Te contribution fro electron-electron bresstralung is calculated fro57 6 q2 are tescreeningfunctionsforelectron-electronbresstralung Te screening functions 17 c $q1,~ and ~ zb,2 are coputed in subroutines Wereand FERMIandWEIAMB For Z greater tan one, subroutine screening functions for te Feri-Toas odel For FERMI coputes te Z equal to one, subroutine WELAMB coputes te screening functions for te Weeler-Lab odel Te functions are coputed fro expressions wic fit te curves in Ref 2 and 6 wit te screening paraeters y and E as arguents Were and E = 1 pk/(eoe Z2/) p = electron rest ass y = 1 pk/eoez 1/)

4 Electron Energy Spectru Te electron energy spectru is coputed fro te expressions in Ref 7 and is contained in subroutine EYGES Te forula used in te progra is were B = kn(k) for k=o (approxiately /) rr(eo, E, x) de is te probability of finding an electron of original energy Eo wose energy lies between E and E + de at distance x into te target x is in units of radiation lengts 8 For a =, tis expression is identical to tat of Bete-Heitler for w(y) dy were y = Qn(Eo/E) Fro Fig 1 of Ref 7 it appears tat a = 25 approxiates te energy spectru better and tis value as been used in te progra Te value of a ay be canged by altering one card in subroutine EYGES and anoter in subroutine WEX In practice, te actual value of a used as little effect for targets of te order of 1 radiation lengt 9 Te gaa function is coputed fro a polynoial approxiation were: al = a5 = a2 = a6 =, a = a7 = a = a8 = 5868 For x > 1 te recursion relations is used rp +X) = xr(x) For Bx less tan one, T(E,, E, x) becoes infinite as E approaces Eo Tis proble will be discussed in te section dealing wit tick target calculations Poton Losses Due to Pair Production Soe of te potons generated in te target will be lost due to pair production before leaving te target Te total cross section is coputed by te etod - -

5 suggested in Ref 11, page In tis etod, te total cross section for arbitrary Z and k is found by scaling fro a reference curve for a given Z Were Zo is te reference Z Te curve for air (Z = 72) on page 9 of Ref 11 is used as te reference curve An expression fitting tis curve is used for te first ter on te rigt of te above expression Tick Target Calculations It is assued tat te tin target expression is correct for an infinitesal tickness dx Te nuber of potons of energy between k and k + dk produced between x and x + dx by electrons of energy between E and E + de is PO n(k)dkdxde = - A a(e,k) rr(eo, E, x) de dkdx is te cross section in ern2 for production of a poton of energy between k and k + dk by an electron of energy E In order to account for losses due to pair production, a correction ter is included were TH is te target tickness in radiation lengts and (Z, k) is te pair total cross section for pair production Finally, Eo TH n(k)thick = (A) j 1 [KK(E 9 k)]th'j,'j n(eo' E, x) PR (k,x)dxde k+p 1 x is in radiation lengts and - noralizes kn(k) to one radiation TH lengt were wic is te differential radiation probability per radiation lengt Lf B

6 I, T(E,, E, x) becoes infinite as E approaces Eo However, te integral wic is te probability of an electron aving energy greater tan E, reains finite Tis is andled in te following way Te integral over E is done nuerically by Gaussian integration over te lower 95% of te integration interval Te upper 5% of te integration interval is done by applying te law of te ean for b integrals If we ave an integral like f f(x) g(x) dx and if f(x) and g(x) are a continuous for a < x < b, and g(x) does not cange sign, ten: b f'f(x) g(x) dx = f([) [ g(x) dx were a < [ <b J a J a (b - a) is sall and te functions PR(k, x) and sligtly over te interval so tat we ave b TH /- i, Were Te integral W(Eo,E,x) is known for te case a = (Ref 8, page 79) and can be coputed by a series approxiation -5-

7 Were Yo = ln(eo/e) and T(l+Bx) is coputed by te polynoial expression discussed previously Te largest possible value of Yo in te progra is ln(l/ 95) = 9 so te series converges rapidly Te series is carried out until te absolute value of te last ter in te bracket is less tan of te su of te previous ters For a > we ave (l+a)bx is just a constant (E/Eo)a is very nearly one since a is sall ( 25) and at te lower liit of te integration (E/Eo) is always greater tan 95 Te following approxiation is ade Wic is just anoter application of te law of te ean for integrals Integration Routine Te integrals over E and x integration Te abscissas and are done nuerically by 16-point Gaussian weigt factors for up to 96 points can be found in Ref 1 Gaussian integration was cosen because te integrand is sarply peaked in te region E - E, due to n(eo, E,x) and routines suc as Sipson's Rule require a greater nuber of points for te sae accuracy A 16-point Gaussian integration is exact (except for round off errors) if te function to be integrated can be fit by a polynoial of degree 1 Te procedure in te progra is to first find te 16 values of x and E over teir respective intervals Te integration eac of te E values eld constant Ten, te integration of x is carried out first wit is carried out over E Te x integration is perfored first because n(eo, E, x) is less peaked for larger x Tis elps te E integration Te single integral over x to te double integral result For target ticknesses wic is perfored next is also done by Gaussian integration and added tion lengts, te contribution of te double integral is very sall of te order of 1 radia- Te second execution tie is due to te large nuber of points required (256) for te double integration Te nuber of points in te x interval could be reduced for te case of very tin targets wit no loss in accuracy -6 -

8 Input Paraeters Te progra input paraeters are read fro data cards as follows: Paraeter Data Card Colun EO - incident electron energy - MeV 1-1 Ek - poton energy - MeV 11-2 Z - target atoic nuber 21- AWT target weigt atoic 1- RHO - target density 1-5 TH - target tickness - real lengts 51-6 Te Bete-Maxion forulation applies only for te extree relativistic case, so Eo and Eo-Ek sould be large copared to p Ek ust be greater tan zero and less tan EkMAX = EO -,51976 If eiter condition is violated, a flag is printed out and te next data card is read Any value of Z can be used Any tickness TH can be used, eiter radiation lengt Output Paraeters less tan or greater tan one Te output forat is sown in Fig 1 Te output paraeters are: EO, Ek, Z, AWT, TH - sae as input paraeters B - k n(k)thi for k = Rad Lengt - Radiation lengt of target in c kn(k)thi - Tin Target differential radiation probability per radiation lengt kn(k)thi CK - Tick target differential radiation probability per radiation lengt THICK/THI - Ratio of k (k) THICK to kn(k)thi AVG Loss Due to Pairs - Approxiate average fractional loss of potons due to pair production in target, coputed fro PR(k,X) for X = TH/2 Progra Results Figures 2 troug 6 illustrate soe of te progra results Figure 2 is a plot of te tin target kn(k) for electron energies of 2 to 2 BeV In Figure, te tin target spectru is plotted vs Eo - k in te ig energy tip region Tick target effects are sown in Figs, 5, and 6 Figure sows te siilarity between THICK/THI and te integrated electron energy spectru W(Eo,E,x) Tis is wat one would expect, since for target tickness like -7-

9 radiation lengts, te tick target effects are ainly due to te altered electron energy spectru Losses due to pair production are sall for suc tin targets Te ratio of kn(k)thick / kn(k)thi is plotted in Figs 5 and 6 Figure 5 sows te tip region and Fig 6 sows te entire spectru Tese curves depend ainly on E/Eo so tat tey can be applied for any Eo fro 2 to 2 BeV wit an error of only a few percent -8-

10 REFERECES R A Alvarez, HEPL-228, 1961 H A Bete and L C Maxion, Pys Rev - 9, 768 (195) H Davies, H A Bete, and L C Maxion, Pys Rev - 9, 788 (195) H A Bete and W Heitler, Proc Roy SOC 816, 8 (19) H W Koc and J W Motz, Rev Mod Pys - 1, 92 (1959) J Josep and F Rorlic, Rev Mod Pys -, 5 (1958) L Eyges, Pys Rev - 76, 26 (199) W Heitler, Te Quantu Teory of Radiation, rd ed (Claredon Press, Oxford, 195) C Hastings, Jr, Approxiations for Digital Coputers (Princeton University Press, Princeton, 1955) M Abraowitz and I A Stegun, Handbook of Mateatical Functions, (ational Bureau of Standards Applied Mateatics Series 55, 196) E Segre, Experiental uclear Pysics Vol I, (Jon Wiley and Sons, Inc ew York, 195)

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