{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 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 66 "Classical deflection func tion and differential cross section (DCS)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "We are using reduced units where \+ m=1, the length scale=1, and a potential energy scale=1 (very much lik e hbar=m_e=e=1 Bohr atomic units)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart; Digits:=14: with(plots): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pi:=evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#piG$\"/)*e`EfTJ!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "try some potentials: (comment out the one you don't want)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 83 "V:=-100.0*exp(-r)/r; # screened Rutherford/C oulomb potential (attractive/repulsive)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*($\"%+5!\"\"\"\"\"-%$expG6#,$%\"rGF)F*F/F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "V:=300*(1/r^12-1/r^6); # a L ennard-Jones potential" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,&*&\" $+$\"\"\"%\"rG!#7F(*&F'F(F)!\"'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Pick a wavenumber:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K:=30;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E0:=0.5*K^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E0G$\"%+X!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Pick an impact parameter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b0:=2.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G$\"#?!\"\"" }}} {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 distance of closest approach: (no radial kinetic ene rgy at this point) E = CF + 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$\"/^%o#e]*)>!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fI:=unapply(1/r^2/sqrt(2*(E0-V)-L0^2/r^2),r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fIGf*6#%\"rG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-*&)9$\"\"#F-,*$\"%+!*!\"\"F-*&\"$+'F-F0!#7F5*&F7F-F0!\"'F -*&$\"'++O!\"#F-F0F>F5#F-F1F5F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "the large-r behavior of the integrand (while assuming a finite -range potential) will help to perform the integral to infinity and ta ke out an unknown. We can use the leading order (L-independent) or try to do better." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "L:='L': s eries(1/r^2/sqrt(k^2-L^2/r^2),r=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"\"F%*&*$)%\"kG\"\"#F%#F%F*)%\"rGF*F%!\"\"F%*,F* F.F'#F.F*%\"LGF*F)!\"#F-!\"%F%-%\"OG6#*&F%F%*$)F-\"\"'F%F.F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "asy_ex:=1/K/r^2*(1+L0^2/2/(K *r)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'asy_exG,$*(\"#I!\"\"%\"r G!\"#,&\"\"\"F,*&$\"/++++++?!#8F,F)F*F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "logplot([fI(r),asy_ex],r=r0+1E-3..15,color=[red,bl ue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 754 236 236 {PLOTDATA 2 "6'-%'CURV ESG6$7hn7$$\"3#**4Xo#e]!*>!#<$!/q#\\ge]r&!#97$$\"3IJky(Qn$**>F*$!/y!Hr [42\"!#87$$\"3#HwF([*G#3?F*$!/ii,Wi77F37$$\"3`%4p'404vO/#F*$!/'[qPXIX\"F3 7$$\"3;_dP9$)Rh?F*$!/:d\\cJ>:F37$$\"3;y59eX%o4#F*$!/vd4U%eh\"F37$$\"3j /k!>!3HK@F*$!/NK6e9(o\"F37$$\"3mcqV*G$=.AF*$!/+=F>#Hz\"F37$$\"394x'pxv SF#F*$!/I>lB#G(=F37$$\"3'zS[c7UuR#F*$!/,&pA'e!)>F37$$\"3E2\"HVZ33_#F*$ !/hFc'>h1#F37$$\"3S$=/f$oafEF*$!/KyB/mY@F37$$\"3_f#zu>&G)z#F*$!/@-F&\\ i@#F37$$\"3cprY;7gxIF*$!/mD*H'oMBF37$$\"3iNP<&p*ebLF*$!/+,!*\\^LCF37$$ \"3**Rp$**f?Lh$F*$!/smHDM#)GF37$$\"30cJ9x\\'e_&F*$!/!oSUQ9$HF37$$\"3sXwYsD^8PjF*$!/*H9gE\"eIF37$$\"3 3I#=]A/be'F*$!/>a%p)G$4$F37$$\"3uyR0KlTuoF*$!/`&*>)*QKJF37$$\"3v\"4(3y G:IrF*$!/kfqa_lJF37$$\"3[Ler\"=O2T(F*$!/%e!HNU+KF37$$\"3%)[<%*Qy\"zn(F *$!/5f!*4NKKF37$$\"3ux01pQpczF*$!/A^IZTkKF37$$\"3r?y;*G(p7#)F*$!/`4)*y \"GH$F37$$\"35YUhi:$))[)F*$!/uo'Q;CK$F37$$\"3/_8gmulv()F*$!/Eoi))4_LF3 7$$\"3\\9')>f-MD!*F*$!/%3F![6xLF37$$\"3y\"G([:X+&H*F*$!/79#3FLS$F37$$ \"3'*Q***)ySft&*F*$!/F;'f\"eHMF37$$\"3)o\"4i:'Rh%)*F*$!/WR\\Q]aMF37$$ \"30')GkVT)4,\"!#;$!/Zv\\6%zZ$F37$$\"3TU1&[tj-/\"Fdx$!/^ek'HK]$F37$$\" 35RK\")\\Fdm5Fdx$!/REC%H`_$F37$$\"3)fhK)QKm%4\"Fdx$!/HFc7Fdx$!/h^]ZrpOF37$$\"3[PA//5rq$F37$$\"3+p_Dy'4#Q8Fdx$!/%>Q Nk_s$F37$$\"3+w$oy(RNj8Fdx$!/eZ3XhTPF37$$\"3G_xv#=s@R\"Fdx$!/a'[ty*fPF 37$$\"3([7b(Gv%zT\"Fdx$!/VsQz2wPF37$$\"3mH*3OUHaW\"Fdx$!/jZ_t\"Hz$F37$ $\"3%4z%f.Jtr9Fdx$!/q6(yJ(3QF37$$\"#:\"\"!$!/rbj'3a#QF3-%'COLOURG6&%$R GBG$\"*++++\"!\")$F_^lF_^lFi^l-F$6$7Y7$F($!/!z@$[d(*=F37$FI$!/)G&oJ$z$ >F37$FS$!/LiKp[w>F37$FX$!/@-='yL,#F37$Fgn$!/Os'yN([?F37$F\\o$!/Jtv$Rp5 #F37$Fao$!/Xks9Oh@F37$Ffo$!/mu]=f=AF37$F[p$!/G[jz6sAF37$$\"3J9K(p?Vz$H F*$!/I#f()*pABF37$F`p$!/P+KbMqBF37$Fep$!/\\p@PldCF37$Fjp$!/G.j^.JDF37$ F_q$!/vMQ#e1g#F37$Fdq$!/e&pDIpm#F37$Fiq$!/AR'\\d!GFF37$F^r$!/#Q$zm_'y# F37$Fcr$!/W/IC![$GF37$Fhr$!/r'QExf)GF37$F]s$!//Mm!pV$HF37$Fbs$!/E2k9[y HF37$Fgs$!/4FCFh;IF37$F\\t$!/rLB**zfIF37$Fat$!/Z4$F37$Fft$!/&HkN \"fLJF37$F[u$!/\"*et8cmJF37$F`u$!/pxA$48?$F37$Feu$!/]yMF37$Fhx$!/&)y?\\X.NF37$F]y$!/lEQJ`DNF37$Fby$!/WkjN\\[ NF37$Fgy$!/`!30!yoNF37$F\\z$!/jn8?U!f$F37$Faz$!/U9%*zH5OF37$Ffz$!/0bQH fIOF37$F[[l$!/7)*\\OF37$F`[l$!/KbQ-#)pOF37$Fe[l$!/G,*p.&)o$F37$Fj[l $!/3go*)>2PF37$F_\\l$!/!Q8?Y`s$F37$Fd\\l$!/%zAY!pTPF37$Fi\\l$!/tur&[+w $F37$F^]l$!/9T=G9wPF37$Fc]l$!/p&>UxHz$F37$Fh]l$!/jTjwy3QF37$F]^l$!/$\\ PVga#QF3-Fc^l6&Fe^lFi^lFi^lFf^l-%+AXESLABELSG6$%\"rGQ!6\"-%%VIEWG6$;$ \"/^%o#e]!*>F3F]^l%(DEFAULTG-%*AXESTICKSG6$Fbjl7O/$!\"%F_^lQ&.1e-3F[jl /$!/gL/+(*)p$F3Fjil/$!/.GX(yG_$F3Fjil/$!/?n3+%zR$F3Fjil/$!/Sm&**H5I$F3 Fjil/$!/jh\\([=A$F3Fjil/$!/d)*f>!\\:$F3Fjil/$!/\"3I,5p4$F3Fjil/$!/2c! \\dd/$F3Fjil/$!\"$F_^lQ&.1e-2F[jl/$!/gL/+(*)p#F3Fjil/$!/.GX(yG_#F3Fjil /$!/?n3+%zR#F3Fjil/$!/Sm&**H5I#F3Fjil/$!/jh\\([=A#F3Fjil/$!/d)*f>!\\:# F3Fjil/$!/\"3I,5p4#F3Fjil/$!/2c!\\dd/#F3Fjil/$!\"#F_^lQ&.1e-1F[jl/$!/g L/+(*)p\"F3Fjil/$!/.GX(yG_\"F3Fjil/$!/?n3+%zR\"F3Fjil/$!/Sm&**H5I\"F3F jil/$!/jh\\([=A\"F3Fjil/$!/d)*f>!\\:\"F3Fjil/$!/\"3I,5p4\"F3Fjil/$!/2c !\\dd/\"F3Fjil/$!\"\"F_^lQ#.1F[jl/$!/,OV+q*)pF-Fjil/$!/L!GX(yG_F-Fjil/ $!/-s'3+%zRF-Fjil/$!/(Rm&**H5IF-Fjil/$!/K;'\\([=AF-Fjil/$!/t&)*f>!\\:F -Fjil/$!.13I,5p*F-Fjil/$!.ng0\\dd%F-Fjil/Fi^lQ#1.F[jl/$\"/*Rm&**H5IF-F jil/$\"/n>ZD@rZF-Fjil/$\"/)zK\"**f?gF-Fjil/$\"/.OV+q*)pF-Fjil/$\"/o$Q] 7:y(F-Fjil/$\"/F9+/)4X)F-Fjil/$\"/%>*p)**3.*F-Fjil/$\"/LR%4DCa*F-Fjil " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r1:=10: for i fr om 1 to 20 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if abs(evalf(fI(r 1))-eval(asy_ex,r=r1))<1E-9 then break else fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r1:=2*r1; od: r1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "corr:=int(asy_ex,r= r1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%corrG$\"/cbbb!oL) !#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "theta:=abs(pi-2*(pi -L0*(evalf(Int(1/r^2/sqrt(2*(E0-V)-L0^2/r^2),r=r0+1E-10..r1,method=_Gq uad,epsilon=1E-7))+corr)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thet aG$\"-G3+E5J!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 285 "This is not s o nice: we have handled the upper-boundary problem, but we should also do something about the lower boundary where the integrand has a 1/sqr t singularity. The results are somewhat sensitive to the cut-off. The \+ cut-off of 1E-10 can only be reduced when Digits is increased!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "We first re-map the variable to y=r-r0 so that the integration range starts at zero, and then we substitute y=t^2 to remove the singularity." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "theta:=abs(pi-2*(pi-L0*(eva lf(Int(2*t/(r0+t^2)^2/sqrt((K^2-eval(2*V,r=r0+t^2))-L0^2/(r0+t^2)^2),t =0..sqrt(r1-r0),method=_Gquad,epsilon=1E-8))+corr)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG$\"-w=oH7J!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This computes reasonably quickly!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "Our definition o f theta follows Fetter-Walecka, Eq. (5.15) which combines attractive a nd repulsive scattering. This precludes orbiting trajectories, which, \+ however, may be of interest. In QM the scattering angle is always betw een 0 and pi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "This angle is negative, \+ (attractive scattering), although in spherical polars 0 " 0 "" {MPLTEXT 1 0 5 "E0,K;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%+X!\"\"\"#I" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "db:=1E-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #dbG$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Np:=400; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NpG\"$+%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "# ,method=_Gquad" }}}{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 0 123 "r0:=fsolve(2*(E0-V)-L0^2/r^ 2,r=0.01*b0..500*b0); if not type(r0,numeric) then print(\"no closest \+ approach found\"); break fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "asy _ex:=1/K/r^2*(1+L0^2/2/(K*r)^2); r1:=10.: for j from 1 to 50 do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "if abs(eval(1/r^2/sqrt(K^2-2*V-L0^2 /r^2),r=r1)-eval(asy_ex,r=r1))<1E-8 then break else fi:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "r1:=2*r1; od: corr:=int(asy_ex,r=r1..infinity) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "theta:=abs(pi-2*(pi-L0*(evalf (Int(2*t/(r0+t^2)^2/sqrt(K^2-eval(2*V,r=r0+t^2)-L0^2/(r0+t^2)^2),t=0.. sqrt(r1-r0),method=_Gquad,epsilon=1E-8))+corr))); " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "if type(theta,numeric) then DbTh:=[op(DbTh),[b0,the ta]]: fi: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "i,nops(DbT h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$,%\"$+%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "logplot(DbTh,style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 672 416 416 {PLOTDATA 2 "6%-%'CURVESG6#7\\dl7$$\"\"\"!\"# $\"/I4u<^R\\!#97$$\"\"#F*$\"/*H#H^G2\\F-7$$\"\"$F*$\"/Un@G\"[([F-7$$\" \"%F*$\"/apC()3U[F-7$$\"\"&F*$\"/$Q)[l54[F-7$$\"\"'F*$\"/e%\\$)fex%F-7 $$\"\"(F*$\"/\"\\o%>MUZF-7$$\"\")F*$\"/trjga3ZF-7$$\"\"*F*$\"/J.s^YuYF -7$$\"#5F*$\"/-=d?4SYF-7$$\"#6F*$\"/[h%H>ag%F-7$$\"#7F*$\"/sfS#R/d%F-7 $$\"#8F*$\"/DXAS9NXF-7$$\"#9F*$\"/:LGb_*\\%F-7$$\"#:F*$\"/aW'RvNY%F-7$ $\"#;F*$\"/*GP+&GFWF-7$$\"#F*$\"/\"\\c&=G;VF-7$$\"#?F*$\"/W&>`Q&yUF-7$$\"#@F*$\"/X(=Z2/C%F- 7$$\"#AF*$\"/#R+@y=?%F-7$$\"#BF*$\"/X<@*RH;%F-7$$\"#CF*$\"/$eES\"eBTF- 7$$\"#DF*$\"/$eg0\"z$3%F-7$$\"#EF*$\"/ednobVSF-7$$\"#FF*$\"/jK(QmG+%F- 7$$\"#GF*$\"/eN2nqhRF-7$$\"#HF*$\"/*>$QW1?RF-7$$\"#IF*$\"/;r%oDz(QF-7$ $\"#JF*$\"/8`=gFNQF-7$$\"#KF*$\"/&*G^/5#z$F-7$$\"#LF*$\"/B?/MQ[PF-7$$ \"#MF*$\"/jWw'3Tq$F-7$$\"#NF*$\"/rJ6%f#fOF-7$$\"#OF*$\"/n7g!=Qh$F-7$$ \"#PF*$\"/hlVjwnNF-7$$\"#QF*$\"//'4@&3@NF-7$$\"#RF*$\"/wJ&zaPZ$F-7$$\" #SF*$\"/L1nVvDMF-7$$\"#TF*$\"/42$GiqP$F-7$$\"#UF*$\"/AcKflFLF-7$$\"#VF *$\"/m&*y;^xKF-7$$\"#WF*$\"/3Q(z/mA$F-7$$\"#XF*$\"/?Z2%4\\<$F-7$$\"#YF *$\"/(H5S)RAJF-7$$\"#ZF*$\"/c1kL/pIF-7$$\"#[F*$\"/1p#\\9[,$F-7$$\"#\\F *$\"//G-0ofHF-7$$\"#]F*$\"/hGH&3O!HF-7$$\"#^F*$\"/F\"\\-kl%GF-7$$\"#_F *$\"/#H*R1^)y#F-7$$\"#`F*$\"/Ol*45%HFF-7$$\"#aF*$\"/.:o?ApEF-7$$\"#bF* $\"/$y/+/zg#F-7$$\"#cF*$\"/Cw\")4TXDF-7$$\"#dF*$\"/ph^bp\"[#F-7$$\"#eF *$\"/#oB^2nT#F-7$$\"#fF*$\"/aB>PR]BF-7$$\"#gF*$\"/'3/&yp#G#F-7$$\"#hF* $\"/F-7$$\"#lF*$\"/`'yqf7#>F-7$$\"#mF*$\"/gk'\\vQ%=F-7$$\"#nF*$\" /bpJ7gk2*z9\"F-7$$\"#vF*$\"/<*pj%e\\5F-7$$\"#wF*$ \"/'R/\")R9[*!#:7$$\"#xF*$\"/Oq\")f%[V)F_cl7$$\"#yF*$\"/BYI.1atF_cl7$$ \"#zF*$\"/k1!)f*oB'F_cl7$$\"#!)F*$\"/O()>:'43&F_cl7$$\"#\")F*$\"/Xca`j $)QF_cl7$$\"##)F*$\"/xe\\f.UEF_cl7$$\"#$)F*$\"/)3m;()HN\"F_cl7$$\"#%)F *$\"/a!\\HoxH\"!#<7$$\"#&)F*$!/5Rcn)=Q\"F_cl7$$\"#')F*$!/%\\)H)Rf$GF_c l7$$\"#()F*$!/xQk=-aVF_cl7$$\"#))F*$!/$HmMg:%fF_cl7$$\"#*)F*$!/s*Q+nYg (F_cl7$$\"#!*F*$!/@ASdD]$*F_cl7$$\"#\"*F*$!/sC1(='=6F-7$$\"##*F*$!/GW^ Y978F-7$$\"#$*F*$!/'efsOm^\"F-7$$\"#%*F*$!/$o!p5HLF-7$$\"#'*F*$!/*HC**=*3AF-7$$\"#(*F*$!/X:6)p9Z#F-7$$\"#)*F*$!/51D S\\`FF-7$$\"#**F*$!/\"[$)4>y0$F-7$$\"$+\"F*$!/$*Q[/#zQ$F-7$$\"$,\"F*$! 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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 " For our choice of potential and energy we cover a large range of scatt ering angles for b ranging from 0 to 3 units." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "For the screened Coulomb potential we expect Rutherford scattering at the smallest impact para meters. A simple power law relates b and theta in this case (cf textbo ok solutions)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "For a Lennard-Jones type potential we should look at the \+ classical deflection function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "DbChi:=[]:" }}{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 0 123 "r0:=fsolve(2*(E0-V)-L0^2/r^2,r=0.0 1*b0..500*b0); if not type(r0,numeric) then print(\"no closest approac h found\"); break fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "asy_ex:=1/ K/r^2*(1+L0^2/2/(K*r)^2); r1:=10.: for j from 1 to 50 do:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 90 "if abs(eval(1/r^2/sqrt(K^2-2*V-L0^2/r^2),r=r 1)-eval(asy_ex,r=r1))<1E-8 then break else fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "r1:=2*r1; od: corr:=int(asy_ex,r=r1..infinity);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "chi:=(-pi+2*(pi-L0*(evalf(Int(2*t/ 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Child , Semiclassical Mechanics with Mol ecular Applications (Oxford Science, 1991; QD96 M65 C48). Alternativel y look at Fig. 5.3 in M.S. Child: Molecular Collision Theory, p.65, Ac ademic Press 1974." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "There is an accumulation point for negative deflection an gles:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "small (angular momentum L or) impact parameters b: strong repulsion f rom the 1/r^\{12\} core dominates the scattering;" }}{PARA 0 "" 0 "" {TEXT -1 56 "there is a critical b-value when the deflection is zero; " }}{PARA 0 "" 0 "" {TEXT -1 164 "then there is a minimum (stationary \+ point) in the deflection angle - this is called the rainbow angle, bec ause many trajectories result in very similar deflections." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#DbTh;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "This p art computes the DCS for simple, monotonic potentials (such as the scr eened Coulomb/Rutherford potential)." }}{PARA 0 "" 0 "" {TEXT -1 70 "F or the Lennard-Jones potential this will produce strange results!!!!! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "To ca lculate the differential cross section we need |db/dth|" }}{PARA 0 "" 0 "" {TEXT -1 54 "One way to do it, is to calculate dth/db and invert \+ it" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dfdThL:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to nops(DbTh)-1 do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "b:=0.5*(DbTh[i,1]+DbTh[i+1,1]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "dThdb:=(DbTh[i+1,2]-DbTh[i,2])/(DbT h[i+1,1]-DbTh[i,1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L0:=K*b;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "r0:=fsolve(2*(E0-V)-L0^2/r^2,r=1E- 10..500*b); if not type(r0,numeric) then print(\"no closest approach f ound\"); break fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "asy_ex:=1/K/r ^2*(1+L0^2/2/(K*r)^2); r1:=10.: for j from 1 to 20 do:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 91 "if abs(eval(1/r^2/sqrt(2*(E0-V)-L0^2/r^2),r=r1 )-eval(asy_ex,r=r1))<1E-8 then break else fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "r1:=2*r1; od: corr:=int(asy_ex,r=r1..infinity);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "theta:=abs(pi-2*(pi-L0*(evalf(Int( 2*t/(r0+t^2)^2/sqrt(K^2-eval(2*V,r=r0+t^2)-L0^2/(r0+t^2)^2),t=0..sqrt( r1-r0),method=_Gquad,epsilon=1E-8))+corr)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dfdTh:=b/sin(theta)/abs(dThdb);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "if type(dfdTh,numeric) then dfdThL:=[op(dfdThL),[thet a,(dfdTh)]]: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dfdThL[300];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\",3yY(pE!#8$\"/=S\\5A3@!\")" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "logplot([seq(dfdThL[i],i=1..300)],style=point, color=blue,symbolsize=25);" }}{PARA 13 "" 1 "" {GLPLOT2D 584 356 356 {PLOTDATA 2 "6'-%'CURVESG6#7h]l7$$\"/(Gci6q5$!#8$!/0c;qJas!#97$$\"/wj9 [&R3$F*$!/fh.%>OD(F-7$$\"/f_8Q*31$F*$!/C=RBd_sF-7$$\"/K0Vp#y.$F*$!/WlZ ]<^sF-7$$\"/?y?Dv9IF*$!/!RD\\E%\\sF-7$$\"/z:g)o;*HF*$!/(eTPDtC(F-7$$\" 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