{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 61 "Classical deflection fun ction and differential cross section:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 130 "We start with a repulsive exponentia l potential, and look at energies such that the particles turn around \+ when scattering head-on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 115 "Explore what happens when the kinetic energy of t he scattering particle exceeds the maximum value of the potential." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Then chan ge the sign of V to explore the attractive exponential potential." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=14:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "V:=1.0*exp(-r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&$\"#5!\"\"\"\"\"-%$expG6#,$%\"rGF)F*F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Pick a wavenumber:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K:=1.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E0:=0.5*K^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E0 G$\"\"&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Pick an impact pa rameter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b0:=2.0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G$\"#?!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 21 "The angular momentum:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "L0:=K*b0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L0G$ \"#?!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Calculate the distan ce of closest approach: (no radial kinetic energy at this point) E = C F + V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r0:=fsolve(2*(E0-V )-L0^2/r^2,r=0.01..5*b0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r0G$\" /T8^-,^A!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pi:=evalf(Pi );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#piG$\"/)*e`EfTJ!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fI:=unapply(1/r^2/sqrt(2*(E0 -V)-L0^2/r^2),r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fIGf*6#%\"rG6 \"6$%)operatorG%&arrowGF(*&\"\"\"F-*&)9$\"\"#F-,($\"#5!\"\"F-*&$\"#?F5 F--%$expG6#,$F0F5F-F5*&$\"$+%!\"#F-F0F@F5#F-F1F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "theta0:=abs(-pi+2*L0*evalf(Int(fI,r0+1E-10..1E4,metho d=_Gquad,epsilon=1E-7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'theta0 G$\".1vGQ]\"R!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "theta0: =abs(-pi+2*L0*evalf(Int(fI,r0..infinity,epsilon=1E-10)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'theta0G$\".;k*H(3\"R!#8" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 22 "Plot of the integrand:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "#plots[logplot]((fI(r)),r=r0+1E-6..500);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Now we set up a b-loop:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "E0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "db:=2E-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dbG$\"\"#!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Np:=150;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NpG\"$]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "DbTh:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to Np \+ do: b0:=db*i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L0:=K*b0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "r0:=fsolve(2*(E0-V)-L0^2/r^2,r=0.01*b0.. 50*b0); if not type(r0,numeric) then print(\"no closest approach found \"); break fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "fI:=unapply(1/r^2 /sqrt(2*(E0-V)-L0^2/r^2),r); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "th eta0:=abs(-pi+2*L0*evalf(Int(fI,r0..infinity,epsilon=1E-8)));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "if type(theta0,numeric) then DbTh:= [op(DbTh),[b0,theta0]]: fi: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "i,nops(DbTh);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$^\"\"$]\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(DbTh,style=point); " }}{PARA 13 "" 1 "" {GLPLOT2D 582 332 332 {PLOTDATA 2 "6&-%'CURVESG6$ 7bt7$$\"3/+++++++?!#>$\"3=+t9y`0\"F @$\"36+)Rhq^CW#F-7$$\"39+++++++9F@$\"38+#)*f[`MM#F-7$$\"3-+++++++;F@$ \"3A+w$[#Qe]AF-7$$\"3#**************z\"F@$\"31+3Y#Q\\P;#F-7$$\"35+++++ ++?F@$\"3!**\\WTZCF3#F-7$$\"3-+++++++AF@$\"33+/Y1x>2?F-7$$\"3!******** 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