THE EGS5 CODE SYSTEM

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so thatP r{ ˆx < x } = G(h(x)) = F (x) .This is the basis of the so called “direct method” of sampling ˆx in which we setF (x) = ζand solve for x. The x-values so chosen will have the distribution F .Another method involves evaluating a function of several variables using uniform random variablesfor arguments. Thus we letˆx = h(ˆζ 1 , . . . , ˆζn ) , (2.16)thenP r{ ˆx < x} = ∫ h( ⃗ ζ)
The result of this algorithm is that ˆx will have the density function f given byn∑f(x) = α i f i (x) g i (x) . (2.22)i=1Thus, if we have a function f which can be put in this form for which the f i can be sampledeasily and the g i evaluated relatively easily, then we have a good method of sampling from f. Itcan be shown that the mean number of tries to accept a value is ∑ α i . If all the g i = 1, we havethe pure “composition” method; and if n = 1, we have the pure “rejection” method. For short wewill call this the “mixed” method. We shall sometimes use the mixed method for some of the f ialso. In these cases we will use the notation (for example)and so on.f 2 (x) =n 2 ∑i=1α 2j f 2j (x) g 2j (x) . (2.23)Finally, we consider the problem of sampling from a joint density function f(x 1 , x 2 , . . . , x n ).Define the marginal density functions∫g m (x 1 , x 2 , . . . , x m ) = f(x 1 , x 2 , . . . , x n ) dx m+1 dx m+2 . . . dx n∫= g m+1 (x 1 , x 2 , . . . , x m , x m+1 ) dx m+1 .We see that g n = f. Now consider h m given byWe see that(2.24)h m (x m |x 1 , x 2 , . . . , x m−1 ) ≡ g m (x 1 , x 2 , . . . , x m )/g m−1 (x 1 , x 2 , . . . , x m−1 ) . (2.25)∫h m (x m |x 1 , x 2 , . . . , x m−1 ) dx m = 1 (2.26)from the definition of the g’s. We see that h m is the conditional density function for x m given thespecified values for x 1 , x 2 , . . . , x m−1 . It can be easily seen that f can be factored into a productof the h’s; namely,f(x 1 , x 2 , . . . , x n ) = h 1 (x 1 ) h 2 (x 2 |x 1 ) h 3 (x 3 |x 1 , x 2 ) . . . (2.27). . . h n (x n |x 1 , x 2 , . . . , x n−1 ) .The procedure then is to get a sample value x 1 using density function h 1 . Then use this value x 1 todetermine a density function h 2 (x 2 |x 1 ) from which to sample x 2 . Similarly, the previously sampledx 1 , x 2 , . . . , x m determine the density function h m+1 (x m+1 |x 1 , x 2 , . . . , x m ) for x m+1 . This schemeis continued until all x i have been sampled. Of course, if the ˆx i are independent random variables,the conditional densities will just be the marginal densities and the variables can be sampled inany order.There are other methods analogous to the one-dimensional sampling methods which can beused for sampling joint distributions. The reader is referred to the references cited above for moredetails.25
Page 1 and 2: THE EGS5 CODE SYSTEM 1Hideo Hirayam
Page 6 and 7: 4.1 UCCYL - Cylinder-Slab Geometry
Page 8 and 9: E CONTENTS OF THE EGS5 DISTRIBUTION
Page 10 and 11: 2.17 Convergence of energy depositi
Page 12 and 13: List of Tables2.1 Symbols used in E
Page 14 and 15: C.2 Goudsmit-Saunderson-related sub
Page 16: With the release of the EGS4 versio
Page 19 and 20: “shower book”.For various reaso
Page 21 and 22: 1.2.2 EGS1About this time Nelson be
Page 23 and 24: MSCAT.These versions of EGS, PEGS,
Page 25 and 26: ten for energies less than 100 MeV
Page 27 and 28: - Molière multiple scattering (i.e
Page 29 and 30: EGS5. The primary advantages of thi
Page 31 and 32: ICRU37-compliant using the NIST dat
Page 33 and 34: high Z materials. Del Guerra et al.
Page 35 and 36: One of these six cross sections is
Page 37 and 38: at low energies. The latter, couple
Page 39: If E 1 and E 2 are expressions invo
Page 43 and 44: 2.4 Particle Transport SimulationTh
Page 45 and 46: from appropriate distribution funct
Page 47 and 48: parameters which may be needed. The
Page 49 and 50: arrived at their values in a very m
Page 51 and 52: Math FORTRAN ProgramTable 2.1 (cont
Page 53 and 54: Figure 2.2: Feynman diagrams for br
Page 55 and 56: defined byδ ij = 1 if i = j,0 othe
Page 57 and 58: Davies, Bethe and Maximon[49] (e.g.
Page 59 and 60: cross section given in Equation 2.4
Page 61 and 62: and use this as the variable to be
Page 63 and 64: we haveNow defineδ ′ = ∆ C ∆
Page 65 and 66: This agrees with formula (10) of Bu
Page 67 and 68: That is, using Equations 2.122 and
Page 69 and 70: d˘Σ P air, Run−timedE=[23 − 1
Page 71 and 72: Angular distribution formulasThe fo
Page 73 and 74: at either x = 0, x = (πE 0 ) 2 (i.
Page 75 and 76: The following table, derived from t
Page 77 and 78: Figure 2.3: Feynman diagrams for tw
Page 79 and 80: whereX 0 = radiation length (cm),n
Page 81 and 82: whereα ′ 1 =α ′ 2 =k 0′k 0
Page 83 and 84: + 1 E 2′ − 1 E 1′− C 2 ln E
Page 85 and 86: PEGS functions BHABDM, BHABRM, and
Page 87 and 88: To find the limits of E, we first c
Page 89 and 90: Figure 2.5: Feynman diagram for sin
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Ī adj = average adjusted mean ioni
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Table 2.2 (cont.)Z Symbol Atomic De
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Table 2.3 (cont.)LABEL a m s x 0 x
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(c) If 10.5 ≤ −C < 11.0 then x
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obtain the real scattering angle. E
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ρ = material mass density (g/cm 3
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Hence,so thatln [1.13 + 3.76(αZ i
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g 3 (θ) =θ4 ()λf (0) (θ) + f (1
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Actually, b = 0 does not correspond
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Assume that an electron starts off
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At small path lengths t, a very lar
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xFinalDirectionφΘ∆xtInitialDire
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shown that this version of the rand
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and as we noted earlier, they are d
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FinalDirectionxEnergy HingesφΘt(
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Transport Steps,∆ E = E x ESTEPEt
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tranport step 1DEINITIAL1 DERESID1t
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= ∆E( ∣ ∣∣∣ dE−1 ∣ )
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Since the CSDA range is uniquely de
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short steps accurate, but slowstep
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Table 2.4: Materials used in refere
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Average Lateral Displacement (cm)0.
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100 MeV Electrons0.010.001LiCLWAlST
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actions involving photons with ener
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Cu 40 keVCounts (/keV/sr/source)10
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Table 2.6: Data sources for general
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2.16.2 Photoelectron Angular Distri
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where r 0 is the classical electron
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second term on the right-hand side
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ZθkYOφe0Xk0Figure 2.23: Photon sc
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Note the similarities and differenc
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XZωk0Oe0YFigure 2.25: Direction of
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W = Atomic, molecular and mixture w
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In order to use EGS5 to answer the
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write(6,100)100 FORMAT(’ PEGS5-ca
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! plate is 1 mm thick!-------------
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implicit noneinclude ’include/egs
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0.989 MeV kinetic energyBrem photon
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! locally (in fact EDEP = particles
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inmax=max(binmax,ebin(j))end dowrit
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0.40 0.0058 *0.60 0.0054 *0.80 0.00
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!----------------------------------
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endifif(loop.lt.3) thenwrite(6,120)
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180 FORMAT(/’ Knock-on electrons
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common/score/escore(3), iscore(3)re
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Brem photons can be created and any
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in any combination of 31 well speci
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end doend do! nmed and dunit defaul
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! ------------------------------clo
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if (iarg.eq.17) then! A Compton sca
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! the general purpose geometry subr
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eturnend!--------------------------
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open(UNIT= 6,FILE=’egs5job.out’
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write(6,130)130 format(/’ Start t
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icol=* int(dlog10(ebin(j)*10000.0/f
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0.0300 0.0000*0.0320 0.0001*0.0340
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0.0780 0.0014 *0.0800 0.0012 *0.082
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The main purpose of this section, h
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4.1.3 Leading Particle BiasingThe s
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• Sum the weighted energy deposit
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4z6Vac115Pb65Air749 1012AirAir8Vac
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Figure 4.3: UCBEND simulation at 3.
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necessary geometry input. The follo
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Appendix AEGS5 FLOW DIAGRAMSHideo H
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subroutineannihVersion051219-1435ia
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¡eq1anormr = 1./sqrt(anorm2)sineta
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1br = max(br,0.D0)ekse2 = br*ekines
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12 3br = br*pesg = eie*bryesesg.lt.
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¤ne¤nesubroutinecollis(lelec,irl,
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¦ne¦necallausgab(iarg)6iq(np).eq.
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subroutinecomptVersion051219-1435ic
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3456icprof(medium).eq.3noyescallran
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subroutinecounters_outVersion051227
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1234567noii.ne.jjyesnoledgb(ii,medi
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©ne1 2 3neispl = (2*neispl + 1)/3f
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subroutineelectr(ircode)ielectr = i
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7 8 9 10 11 12 13detot = e(np)-ecut
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2324 25 26 27 282930ustep.gt.dnear(
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4142 43 44 45 4647 48 49ecut(irnew)
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5859 60 61 6263 64tmscat.eq.0.0noye
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73 7475 76noedep.lt.e(np)yescallran
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subroutinehardx(charge,kEnergy,keIn
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1 2 3 4 5iz = izziz.eq.0noxsi = zer
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13 14 15 16 17sint.ne.0.yesrdev = m
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19im=1im=im+1im >nmednoyesnoiprofm(
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2223write(kmpo,1610)read(kmpi,1260)
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26 27 28iprofm(im).ne.1noyeswrite(6
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30 31 32 33 34esig0(i,im) = esig0(i
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36 37 38noiedgfl(ii).ne.0.or.iauger
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nokaug.eq.6calllshell(3)kaug.eq.7ka
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subroutinekxrayVersion051219-1435ik
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123dfl3aug(5,iz).eq.0.nonauger = na
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12 3 4rnnow.le.omegal2(iz) + f23(iz
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1 2dflx3(6,iz).eq.0.nonxray=nxray+1
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1impacr(ir(np)).eq.1.and.iedgfl(ir(
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12fject = (ktot - k1grd(iprt,ik1))
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56789thr = 1./eta"Central correctio
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1 2 3delta = delcm(medium)*del"Reje
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89101112galpha.ge.0.0yesnoximid = 0
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subroutinephotoVersion051219-1435"C
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4 5 6 7 8rnnow.le.pbran(i)noyesiz =
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121314beta = sqrt((eelec - RM)*(eel
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subroutinephotonVersion051219-1435i
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678910idisc.gt.0yesnoedep = 0.iarg
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1718192021iausfl(iarg+1)ne0nocallpa
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2728iausfl(iarg+1)ne0noircode = 2np
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subroutinerk1Version060313-0945open
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4 5 6j=1j=j+1j>neke-1noyesj.eq.1noj
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18 19 20 21 22 23elkeold = elkek1ol
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1 2"end of file; go to 13"read(17,*
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4 5"go to 30"noabs(k1mine-k1grd(1,1
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7 8read(17,'(72a1)') bufferread(17,
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subroutine shower(iqi,ei,xi,yi,zi,u
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subroutineuphi(ientry,lvl)Version05
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subroutinerandomset(rndum)Version05
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subroutinerluxinitVersion051219-143
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1i=1i=i+1i>24noyesseeds(i) = real(i
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subroutinerluxinVersion051219-1435w
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Appendix BEGS5 USER MANUALHideo Hir
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Table B.1: Variable descriptions fo
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Page 333 and 334:
Page 335 and 336:
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Table B.12: Variable descriptions f
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Optional parameter modificationsThe
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the call to PEGS5 may be skipped if
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egions. Execution of EGS5 is termin
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of the transport in the walls of el
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call rluxinitafter specifying LUXLE
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END OF FILE ON UNIT 12PROGRAM STOPP
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do i=1,ncasesuf(1)=ufivf(1)=vfiwf(1
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crossed, then USTEP should be set t
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subroutine howfarimplicit noneinclu
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Table B.18: IARG values program sta
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As an example of how to write an AU
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!**********************************
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nreg=3do i=2,nregecut(i)=100.0pcut(
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nlines=0nwrite=15!-----------------
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if (nlines.lt.nwrite) thenwrite(6,1
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Appendix CPEGS USER MANUALHideo Hir
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is entered. On each pass through th
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(from previous figure)(to previous
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(from previous figure)||+ ---------
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+------+|BREMTR|+------+|V+------+|
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+------+|PAIRTR|+------+|V+------+|
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NameDCSLOADDCSSTORDCSTABELASTINOELI
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NameAFFACTAINTPALKEALKEIALINALINIAD
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Table C.5: Functions in PEGS, part
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+------+ +------+ +------+ +------+
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Table C.8: ELEM option input data l
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Table C.10: MIXT option input data
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Table C.12: PWLF option input data
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Table C.17: PLTN option input data
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ICPROF is set to 3, the user must c
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ColumnLine 123456789112345678921234
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interiors of the intervals. If FEXA
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C.3.6The TEST OptionThe TEST option
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C.3.9The HPLT OptionThe Histogram P
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Appendix DEGS5 INSTALLATION GUIDEHi
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egs5 directory (preferably using th
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6. The user is then asked to key-in
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* User code tutor1.f has been compi
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Appendix ECONTENTS OF THE EGS5DISTR
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All of the actual FORTRAN source co
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aprime.data Data for empirical brem
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ismuth krypton silverboron lanthanu
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The tutorial problems and advanced
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Bibliography[1] R. G. Alsmiller Jr.
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[28] A. F. Bielajew. HOWFAR and HOW
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[59] K. Flöttmann. Investigations
Page 431 and 432:
[92] H. Kolbenstvedt. Simple theory
Page 433 and 434:
[123] Y. Namito, H. Hirayama, A. Ta
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[156] Y. A. Shreider, editor. The M
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Index“shower book”, 37AE, 28, 3
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Klein-Nishina formula, 62Landau dis
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