{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 211 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 305 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 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 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 207 "" 0 "" {TEXT 211 38 "DCS for a Lennard-Jones type potential" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We calculate the differential cross section for a Lennar d-Jones potential while testing the following approximations:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "1) Legend re polynomials replaced by Bessel [checked against the true result]" } }{PARA 0 "" 0 "" {TEXT -1 135 "2) true phase shifts replaced by high-e nergy approximation or by the semiclassical results (WKB) [not checked against the true results]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The idea is to test the partial-wave to impact p arameter conversion, and to understand the emergence of classical scat tering" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Eventually, one wants to test also:" }}{PARA 0 "" 0 "" {TEXT -1 92 "3 ) replacement of Bessel function by a difference of two exponentials ( large L*theta limit)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart; Digits:=14: with(plots): p i:=evalf(Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K:=60;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"#g" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 83 "A strong p otential (V0) and high wavenumber K are chosen for the following reaso ns:" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 111 "We want a strong pocket so that interferences from positive and negat ive classical deflections can be observed;" }}{PARA 0 "" 0 "" {TEXT 211 172 "We want these interferences at relatively high partial waves, so that a continuous-L approach (impact parameter picture) is possibl e (although we will work with discrete L)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "V0:=3000; V:=1/r^12-1/r^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G\"%+I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,& *&\"\"\"F'*$)%\"rG\"#7F'!\"\"F'*&F'F'*$)F*\"\"'F'F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 136 "The high-energy approximation will be n eeded for high L, since the SCL calculation goes bad there (for some u nresolved technical reason)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "HEM1p:=proc(K,Lc) local r0; global V0,V; r0:=(Lc+0.5)/K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "-2*V0/K*evalf(Int(eval(V*r,r=r0+s^2)/sqrt (s^2+2*r0),s=0..infinity,epsilon=1E-12));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "HEM 1p(K,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/-f9fVn@!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "HEM1p(K,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!/s'4'zq*4%!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "The result f or lower L looks highly suspicious. The HE approximation goes bad ther e (it is not a technical problem with evaluating the integral!)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "T he JWKB phase shifts can be calculated as:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "K:='K': L:='L': series(sqrt(K^2-((L+1/2)/r)^2),r=in finity)-K;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$*$)%\"KG\"\"#\"\"\"# F)F(F)*,\"\")!\"\"F%F*,&*&F(F)%\"LGF)F)F)F)F(F'!\"#%\"rGF1F-*,\"$G\"F- F%F*F.\"\"%F'!\"%F2F6F--%\"OG6#*&F)F)*$)F2\"\"'F)F-F)F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "asy_ex:=-(2*L+1)^2/8/(K*r^2)*(1+(2* L+1)^2/16/(K*r)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'asy_exG,$*, \"\")!\"\",&*&\"\"#\"\"\"%\"LGF,F,F,F,F+%\"KGF(%\"rG!\"#,&F,F,**\"#;F( F)F+F.F0F/F0F,F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "etaSC :=proc(K,L) local aL,val,kL,r1,i,asy_ex; global pi; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "kL:=R->evalf(eval(sqrt(K^2-2*V0*V-((L+1/2)/r)^2) ,r=R)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "aL:=fsolve(kL(R)^2,R=1E -20..200)+1E-20; if not type(aL,numeric) then RETURN(\"No aL found\") \+ fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "asy_ex:=-(2*L+1)^2/8/(K*r^2 )*(1+(2*L+1)^2/16/(K*r)^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r1:= max(10,aL): for i from 1 to 10 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "if abs(eval(asy_ex,r=r1)-(kL(r1)-K))<1E-7 then break fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r1:=r1*2; od: if i=11 then RETURN(\"incre ase r1!\") fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "val:=evalf(Int(k L(r)-K,r=aL..r1,epsilon=1E-10)) + evalf(int(asy_ex,r=r1..infinity)) -K *aL+(L+1/2)*pi/2; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K:=60;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "etaSC(K,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"- !p\"4%)yC!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "etaSC(K,10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!/#>\"yYH$o$!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 94 "The full SCL phase shift is not as crazy as the HE approximation! Let's compare them at low L:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "L_HE:=[]: L_SC:=[]: for L from 1 t o 100 do: val:=etaSC(K,L): L_SC:=[op(L_SC),[L,val]]: val:=HEM1p(K,L): \+ L_HE:=[op(L_HE),[L,val]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([L_SC,L_HE],color=[red,blue],view=[0..100,-60..15]);" }} {PARA 13 "" 1 "" {GLPLOT2D 580 430 430 {PLOTDATA 2 "6&-%'CURVESG6$7`q7 $$\"\"\"\"\"!$!3M+Q'3))49+&!#;7$$\"\"#F*$!3=+13Ud'y%[F-7$$\"\"$F*$!3%* *pm:6!4'p%F-7$$\"\"%F*$!3#**>nS!R3YXF-7$$\"\"&F*$!3E+)z6M[yR%F-7$$\"\" 'F*$!3)***Q-`\\Q^UF-7$$\"\"(F*$!3&**\\Q.e&p1TF-7$$\"\")F*$!31+e,xByjRF -7$$\"\"*F*$!3<+&zI\"ykAQF-7$$\"#5F*$!3)**>>\"yYH$o$F-7$$\"#6F*$!3C+:> &3Eda$F-7$$\"#7F*$!35+a?va%*4MF-7$$\"#8F*$!3)***R!=icfF$F-7$$\"#9F*$!3 ***>knjjP9$F-7$$\"#:F*$!34+F$f(4P8IF-7$$\"#;F*$!37+>sXMy%)GF-7$$\"#F*$!3$)*R@W60*4DF-7$$ \"#?F*$!39+1,VMf)Q#F-7$$\"#@F*$!31+xmik6pAF-7$$\"#AF*$!35+$\\WD\"[^@F- 7$$\"#BF*$!3++0e8`pN?F-7$$\"#CF*$!3#**\\z:cm<#>F-7$$\"#DF*$!3:+GGfLq4= F-7$$\"#EF*$!3++;$3_9&*p\"F-7$$\"#FF*$!3#**zz'G$47f\"F-7$$\"#GF*$!3)** R(p\\vz%[\"F-7$$\"#HF*$!3-+]W_%*G!Q\"F-7$$\"#IF*$!3/+fqDepx7F-7$$\"#JF *$!3#***\\O)*z-x6F-7$$\"#KF*$!31+SugyHy5F-7$$\"#LF*$!3x+!z\\()y^\")*!# <7$$\"#MF*$!3g++.A;,n))Fhu7$$\"#NF*$!3e**poPShQzFhu7$$\"#OF*$!3$****z# [*H,.(Fhu7$$\"#PF*$!3!***poA,rThFhu7$$\"#QF*$!3%)**f$4g7NF&Fhu7$$\"#RF *$!3%)***)4[IqDWFhu7$$\"#SF*$!3++!Rd8b%)f$Fhu7$$\"#TF*$!37+!>M&4&>z#Fh u7$$\"#UF*$!36+q-`9Q1?Fhu7$$\"#VF*$!3!***4**Gp%>C\"Fhu7$$\"#WF*$!3')** *p\"y`d))\\!#=7$$\"#XF*$\"39++w#RhmA#F`y7$$\"#YF*$\"3(****pN'[#RA*F`y7 $$\"#ZF*$\"3&***Hw!>y+g\"Fhu7$$\"#[F*$\"3))**>wBvYbAFhu7$$\"#\\F*$\"31 +S5e:H))GFhu7$$\"#]F*$\"3=++#fJn#)\\$Fhu7$$\"#^F*$\"3O+5!)Ro4&3%Fhu7$$ \"#_F*$\"3))**>%*yhY[YFhu7$$\"#`F*$\"3?+Sc\"GW!)=&Fhu7$$\"#aF*$\"3.++u $y\"[.dFhu7$$\"#bF*$\"3c**\\*Qo4W>'Fhu7$$\"#cF*$\"3i**fD\"*yVgmFhu7$$ \"#dF*$\"3U+?kNO:,rFhu7$$\"#eF*$\"3O+5DD(>h^(Fhu7$$\"#fF*$\"3%***HjQE( [!zFhu7$$\"#gF*$\"3c+5FP/#pE)Fhu7$$\"#hF*$\"3%)**48'[S&R@BJ\"*Fhu7$$\"#!)F*$\"3o*** *GZb8*z)Fhu7$$\"#\")F*$\"3w++D,GDF%)Fhu7$$\"##)F*$\"3k++0cg`>!)Fhu7$$ \"#$)F*$\"3O++c/8#Ge(Fhu7$$\"#%)F*$\"3#*****p0!Gt7(Fhu7$$\"#&)F*$\"3K+ +y()\\6mmFhu7$$\"#')F*$\"3l***H,k7I@'Fhu7$$\"#()F*$\"3U++.u`Fhu7$$\"#*)F*$\"3%****4,3/$**\\Fhu7$$\"#!*F*$\"3 #****4*)\\;fl%Fhu7$$\"#\"*F*$\"3*)****f&\\#QUVFhu7$$\"##*F*$\"3')****3 s_OcSFhu7$$\"#$*F*$\"37++Wy9F&z$Fhu7$$\"#%*F*$\"39+++WzbcNFhu7$$\"#&*F *$\"32++A\\<(yL$Fhu7$$\"#'*F*$\"3))***4\\Z'4PJFhu7$$\"#(*F*$\"3%)***z1 8dB&HFhu7$$\"#)*F*$\"3?++]4R+#y#Fhu7$$\"#**F*$\"3*****f&3))eCEFhu7$$\" $+\"F*$\"3')*****o\"4%)yCFhu-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F`[m- F$6$7`q7$F($!38+'>x+$!3++?2FCVL]!\"&7$FC$!3W+#4?^/K,)!\"'7$FH$!31+ g#GZm-m\"Fj\\m7$FM$!3A+uKgsD!>%!\"(7$FR$!30+ZM(*exK7Fa]m7$FW$!3!)*>n4' zq*4%F_[m7$Ffn$!3(**zCEu5r]\"F_[m7$F[o$!3\")*f\")*QCoAg!\"*7$F`o$!31++ Puj$Ge#F^^m7$Feo$!33+Q$zFBn<\"F^^m7$Fjo$!3/+qA*>%[\\c!#57$F_p$!3***R#) =D<&RGFh^m7$Fdp$!3/+M*yrJg[\"Fh^m7$Fip$!3q*z\"G&*\\8h!)!#67$F^q$!3z*f \"4qdO:XFb_m7$Fcq$!3***>?r-,Kg#Fb_m7$Fhq$!3,+)[n3!QS:Fb_m7$F]r$!3&**** *\\x8fK$*!#77$Fbr$!31+'>35Irx&F_`m7$Fgr$!3=+G'z@4rk$F_`m7$F\\s$!3(***4 3]0?WBF_`m7$Fas$!3*)*pcGlD=`\"F_`m7$Ffs$!30+v!48'G;5F_`m7$F[t$!3*)*>PM Z@u$o!#87$F`t$!3E+!pn8Q(fYFbam7$Fet$!3;+1a\"zZN@$Fbam7$Fjt$!32+SERGbSA Fbam7$F_u$!31+FgRc%zd\"Fbam7$Fdu$!3,+D,6Qf@6Fbam7$Fju$!3t*fQ*))*[(R!)! 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We go for 1000 partial wave phase shifts calculated by the SC L method and the HE approximation which also happens to be a high-L hi gh-K approximation to the full quantum mechanical phase shift (approxi mates the Born approximation in the high-L regime)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "L:='L': PHE:=[]: PSCL:=[]: for L from 0 to 1000 do: val:=HEM1p( K,L): PHE:=[op(PHE),[L,val]]: val:=etaSC(K,L): PSCL:=[op(PSCL),[L,val] ]: od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 260 "First we graph the ph ase shift vs L in the interesting regime where in the classical deflec tion function cross-over from repulsive to attractive scattering and r ainbow scattering occurs. 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\"$$**F)$\"3()*********fJS#Fidy7$$\"$%**F)$\"3-+++++L\"R#Fidy7$$\"$&** F)$\"3%**********\\&zBFidy7$$\"$'**F)$\"31+++++'yO#Fidy7$$\"$(**F)$\"3 ++++++CcBFidy7$$\"$)**F)$\"33+++++pWBFidy7$$\"$***F)$\"3)**********4KL #Fidy7$$\"%+5F)$\"3,+++++z@BFidy-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-% &TITLEG6#Q&K=~606\"-%+AXESLABELSG6$Q!F`e^lFde^l-%%VIEWG6$;F]xF_jn;$!#7 !\"\"F`o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Compare the SCL and HE results at the highest L values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "N:=nops(PSCL); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"%,5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "PSCL[N],PHE[N];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$\"%+5$\"'z@B!#57$F$$\"/wV+S]%G#!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 133 "We should take a look at where they begin to deviate: ( for some reason the logplot fails in the mws version! 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^l-%&TITLEG6#Q&K=~606\"-%+AXESLABELSG6$Q!Fid^lF]e^l-%%VIEWG6$;FjzFfjl; $!#7!\"\"$F\\oFee^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 341 "Next, we look at the differ ential cross section (DCS) using both the Bessel approximation to the \+ Legendre polynomials and without. In both cases we use the combination of SCL and HE phase shifts stored in the table PSC1. We limit the sum mation over partial waves at 500. This has a small effect on the accur acy of the DCS, but saves us time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "P(2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\" \"#!\"\"*(\"\"$F%F&F'%\"xGF&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "S_B:=theta->1/K^2*abs(add((L1-0.5)*(exp(2*I*PSC1[L1,2])-1)*Be sselJ(0,(L1-1)*2*sin(0.5*theta)),L1=1..501))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_BGf*6#%&thetaG6\"6$%)operatorG%&arrowGF(*&%\"KG!\" #-%$absG6#-%$addG6$*(,&%#L1G\"\"\"$\"\"&!\"\"F;F8,&-%$expG6#*&^#\"\"#F 8&%%PSC1G6$F7FBF8F8F8F;F8-%(BesselJG6$\"\"!,$*(FBF8,&F7F8F8F;F8-%$sinG 6#*&F9F89$F8F8F8F8/F7;F8\"$,&FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "S_P:=theta->1/K^2*abs(add((L1-0.5)*(exp(2*I*PSC1[L1,2 ])-1)*P(L1-1,cos(theta)),L1=1..501))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_PGf*6#%&thetaG6\"6$%)operatorG%&arrowGF(*&%\"KG!\"#-%$absG6 #-%$addG6$*(,&%#L1G\"\"\"$\"\"&!\"\"F;F8,&-%$expG6#*&^#\"\"#F8&%%PSC1G 6$F7FBF8F8F8F;F8-%\"PG6$,&F7F8F8F;-%$cosG6#9$F8/F7;F8\"$,&FBF(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S_B(1E-3),S_P(1E-3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/Or,.+=^!\"*$\"/v]_QGYg\"$L$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 139 "The Legendre polynomial calculation sufferes badly from numerical precision problems (cancellations of significant digits), a s shown below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits:=1 00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "val:=[S_B(1E-3),S_P(1E-3)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(val,5);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$$\"&!=^\"\"!$\"&$*p#\"$q\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits:=500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"$+&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "va l:=[S_B(1E-3),S_P(1E-3)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(val,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"&!=^\"\"!$\" &x6&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 437 "The summation with the Bessel functions is al so susceptible to errors. This is one of the reasons why replacing the sum over L with an integral over b (or continuous L) may actually res ult in more accurate results. Maple's numerical implementation of Bess elJ is also not very useful (it requires high settings of Digits in so me of the code further below), so we replace it by code adapted from N umerical Recipes (Press, Teukolsky et al)." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 84 "_q0:=[-.1562499995E-1,.1430488765E-3,-.6911147651E- 5,.7621095161E-6,-.934945152E-7]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "_p0:=[1.0,-.1098628627E-2,.2734510407E-4,-.2073370639 E-5,.2093887211E-6]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "_r0 :=[57568490574.0E0,-13362590354.0E0,651619640.7E0,-11214424.18E0,77392 .33017E0,-184.9052456E0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "_s0:=[57568490411.0E0,1029532985.0E0,9494680.718E0,59272.64853E0,2 67.8532712E0,1.0E0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "_r1:=[72362614232.0E0,-7895 059235.0E0,242396853.1E0,-2972611.439E0,15704.48260E0,-30.16036606E0]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "_s1:=[144725228442.0E0, 2300535178.0E0,18583304.74E0,99447.43394E0,376.9991397E0,1.0E0]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "_p1:=[1.0E0,.183105E-2,-.351 6396496E-4,.2457520174E-5,-.240337019E-6]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "_q1:=[.04687499995E0,-.2002690873E-3,.8449199096E-5 ,-.88228987E-6,.105787412E-6]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "J0:=proc(x) loca l y,ax,z,xx; global q0,p0,r0,s0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if abs(x)<8. then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "y:=x**2;\n" } {MPLTEXT 1 0 132 "RETURN((_r0[1]+y*(_r0[2]+y*(_r0[3]+y*(_r0[4]+y*(_r0[ 5]+y*_r0[6])))))/(_s0[1]+y*(_s0[2]+y*(_s0[3]+y*(_s0[4]+y*(_s0[5]+y*_s0 [6]))))));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ax:=abs(x);\n" }{MPLTEXT 1 0 10 "z:=8./ax;\n" } {MPLTEXT 1 0 8 "y:=z^2;\n" }{MPLTEXT 1 0 19 "xx:=ax-.785398164;\n" } {MPLTEXT 1 0 151 "RETURN(sqrt(.636619772/ax)*(cos(xx)*(_p0[1]+y*(_p0[2 ]+y*(_p0[3]+y*(_p0[4]+y*_p0[5]))))-z*sin(xx)*(_q0[1]+y*(_q0[2]+y*(_q0[ 3]+y*(_q0[4]+y*_q0[5]))))));\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fi : end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=14:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "J0(4.01);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!/!fBo.Z'R!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "BesselJ(0,4.01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!/3P4PqkR !#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "S_B:=theta->1/K^2*a bs(add((L1-0.5)*(exp(2*I*PSC1[L1,2])-1)*J0((L1-1)*2*sin(0.5*theta)),L1 =1..501))^2;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ S_BGf*6#%&thetaG6\"6$%)operatorG%&arrowGF(*&%\"KG!\"#-%$absG6#-%$addG6 $*(,&%#L1G\"\"\"$\"\"&!\"\"F;F8,&-%$expG6#*&^#\"\"#F8&%%PSC1G6$F7FBF8F 8F8F;F8-%#J0G6#,$*(FBF8,&F7F8F8F;F8-%$sinG6#*&F9F89$F8F8F8F8/F7;F8\"$, &FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "SLsc1:=[]: for iT from 1 to 100 do: th:=iT*1E-4: SLsc1:=[op(SLsc1),[th,S_B(th)]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=30;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "SLs c2:=[]: for iT from 1 to 100 do: th:=iT*1E-4: SLsc2:=[op(SLsc2),[th,S_ B(th)]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "PL1:=plot(SLsc1,color=red): PL2:=pl ot(SLsc2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pl ots[display](PL1,PL2);" }}{PARA 13 "" 1 "" {GLPLOT2D 579 307 307 {PLOTDATA 2 "6&-%'CURVESG6$7`q7$$\"\"\"!\"%$\"/9u(4)Hn^!\"*7$$\"\"#F*$ \"/Di3czl^F-7$$\"\"$F*$\"/DFiEHj^F-7$$\"\"%F*$\"/#4,1\"zf^F-7$$\"\"&F* $\"/'=/K$Hb^F-7$$\"\"'F*$\"/1nuE!)\\^F-7$$\"\"(F*$\"/]yiIKV^F-7$$\"\") F*$\"/JQF\"fe8&F-7$$\"\"*F*$\"/\"3q?;u7&F-7$$\"#5F*$\"/]OG.+=^F-7$$\"# 6F*$\"/>c(>=w5&F-7$$\"#7F*$\"/GS!>xi4&F-7$$\"#8F*$\"/1tT`)R3&F-7$$\"#9 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m-%&STYLEG6#%%LINEG-%&TITLEG6#Q&K=~606\"-%+AXESLABELSG6$Q!F[\\bmF_\\bm -%%VIEWG6$;Fac^l$\"\"'!\"\";$F`c^lFf\\bm\"%+5" 1 6 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 130 "Destructi ve interference at orders 2,3,... are pronounced at theta=0.05, 0.08, \+ 0.12, 0.15, 0.19, 0.22, 0.26, 0.30, 0.34, 0.38,0.42" }}{PARA 0 "" 0 " " {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "V0,K;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+I \"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 87 "Fraunhofer minima: theta_m = arcsin(m/k/a) a = slit width, k = 1/wavelength, m=order)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#a1:=evalf(solve(arcsin(1/K/a)=0.03,a),3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a2:=evalf(solve(arcsin(2/K/a )=0.05,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G$\"$n'!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a3:=evalf(solve(arcsin(3/K/a )=0.08,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a3G$\"$E'!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a4:=evalf(solve(arcsin(4/K/a )=0.12,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a4G$\"$d&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a5:=evalf(solve(arcsin(5/K/a )=0.15,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a5G$\"$e&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a6:=evalf(solve(arcsin(6/K/a )=0.19,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a6G$\"$H&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a7:=evalf(solve(arcsin(7/K/a )=0.22,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a7G$\"$N&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a8:=evalf(solve(arcsin(8/K/a )=0.26,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a8G$\"$>&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a9:=evalf(solve(arcsin(9/K/a )=0.30,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a9G$\"$3&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a10:=evalf(solve(arcsin(10/K /a)=0.34,a),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a10G$\"$+&!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 265 "We are finding some consisten cy with the Fraunhofer single-slit diffraction formula and a slit widt h of about 0.5x for K=60. Does this mean that the potential which is a ttractive for larger L (b-values) and repulsive for smaller ones acts \+ a little like an aperture?" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 142 "Note: we should not oversimplify and believe \+ that we are seeing Fraunhofer diffraction as that also makes a predict ion about the peak heights." }}{PARA 0 "" 0 "" {TEXT 211 153 "Fraunhof er diffraction is derived in scalar diffraction theory (which is an ap proximation for the theory of light). How applicable is it to matter w aves?" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 124 "However, we can (for the sake of argument) look at the impact par ameter b(L0) at which the deflection function changes sign." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "L0:=71;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L0G\"# r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b0:=evalf((L0+0.5)/K,3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G$\"$>\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "We are interested in exploring the L-theta relationship. Pick \+ an angle theta, and watch how the sum to the DCS accumulates as a func tion of a cutoff in the partial wave number (labeled by L). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "We begin with a small scattering angle (theta=0.01 \+ rad) and observe whether only large impact parameters (or angular mome nta L) make a substantial contribution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=14;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Dig itsG\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "th:=1.0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#thG$\"#5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sacc:=[]: for L1 from 1 to 601 do: L:=L1-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "dat[L]:=L*(exp(2*I*PSC1[L1,2])-1) *J0(L*2*sin(0.5*th));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v_n:=abs(a dd(dat[i],i=1..L1))^2/K^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Sacc: =[op(Sacc),[L,v_n]]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "PL1:=pl ot(Sacc,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "dis play(PL1,title=cat(\"K= \",convert(K,string),\" , theta= \",convert( th,string)));" }}{PARA 13 "" 1 "" {GLPLOT2D 580 296 296 {PLOTDATA 2 "6 &-%'CURVESG6$7e`m7$$\"\"!F)$\"/6\\n[=:U!#=7$$\"\"\"F)$\"/A\\NF0HO!#<7$ $\"\"#F)$\"/h>&>eYO$F27$$\"\"$F)$\"/-c/OoN7!#;7$$\"\"%F)$\"/;QeY_M7F=7 $$\"\"&F)$\"/kY()[ApTF27$$\"\"'F)$\"/AfBH.tA JF=7$$\"#6F)$\"/9[gTb4MF=7$$\"#7F)$\"/K'>%Gf&f#F,7$$\"#8F)$\"/'[_uBBK& F,7$$\"#9F)$\"/^<`\"4Oi#F=7$$\"#:F)$\"/:e/,J(=#F=7$$\"#;F)$\"/JC'=!)G_ (F27$$\"#F)$\"/;>TCdm:F =7$$\"#?F)$\"/vaqco%Q%F=7$$\"#@F)$\"/$G#[G+,YF=7$$\"#AF)$\"/(z;g)3<&*F 27$$\"#BF)$\"/D-q)Qlc$F=7$$\"#CF)$\"/%4H(z]6^F=7$$\"#DF)$\"/c8N!RBV(F, 7$$\"#EF)$\"/#4W.aGO\"F=7$$\"#FF)$\"/*['>TF=7$$\"#LF)$\"/Ci\\Mn\"f#F27$$\"#MF)$\"/Ea:KmkC F=7$$\"#NF)$\"/e1>@ALUF=7$$\"#OF)$\"/L!3&p%z!=F=7$$\"#PF)$\"/h)R:ctx$F =7$$\"#QF)$\"/+;H(R74$F27$$\"#RF)$\"/6L2cdN&)F=7$$\"#SF)$\"/>$*3\"=9@& F=7$$\"#TF)$\"/<`Z7I]SF=7$$\"#UF)$\"/aAE[AV9!#:7$$\"#VF)$\"/\">8!Q#*=F F[y7$$\"#WF)$\"/:;ezg5FF[y7$$\"#XF)$\"/,9I!3#\\>F[y7$$\"#YF)$\"/X!H*H, WFF[y7$$\"#ZF)$\"/+,w;KsIF[y7$$\"#[F)$\"/`F'*zISMF[y7$$\"#\\F)$\"/PA(G I\\t\"F[y7$$\"#]F)$\"/q$z<,\"*[#F=7$$\"#^F)$\"/oI0+8([\"F[y7$$\"#_F)$ \"/e+4_)z&zF[y7$$\"#`F)$\"/Jr7+R-7!#97$$\"#aF)$\"/)e`\")*z&>\"Fc\\l7$$ \"#bF)$\"/US95D]7Fc\\l7$$\"#cF)$\"/wZu7w97Fc\\l7$$\"#dF)$\"/36d!=0<\"F c\\l7$$\"#eF)$\"/BYuV+[>Fc\\l7$$\"#fF)$\"/yT&f\"3sQFc\\l7$$\"#gF)$\"/s 1$)R)Gp%Fc\\l7$$\"#hF)$\"/J,b[2KSFc\\l7$$\"#iF)$\"/s1n1mlNFc\\l7$$\"#j F)$\"/$yNGRld$Fc\\l7$$\"#kF)$\"/+N#HW<_$Fc\\l7$$\"#lF)$\"/6(fMWMX$Fc\\ l7$$\"#mF)$\"/*)*z#y\"*RPFc\\l7$$\"#nF)$\"/1mo*R2t$Fc\\l7$$\"#oF)$\"/. 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"" }}{PARA 0 "" 0 "" {TEXT 211 8 "V0=2000:" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }{TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 8 "V0=3000:" }} {PARA 0 "" 0 "" {TEXT 211 0 "" }{TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }{TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 0 "" } {TEXT 211 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 0 "" }{TEXT 211 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 0 "" }{TEXT 211 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 380 "The derivatives of these graphs (with respect to L) could be comp ared indirectly with what is shown in Fig. 5.1 in Child. The question \+ is: does the final value (once all relevant L are included) indeed rec eive its main contribution from L-values close to L0(theta)? What if t here are two such L0 values, or three (positive Theta, and two occurre nces of negative deflection Theta)?" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 42 "For theta=0.1 we could have L0=64, 69 \+ , 95" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 187 "We observe small contributions for L<80, and a really big one aro und L=100 with ample interference. The outermost contributions dominat e due to the weighting with impact parameter (or L)." }}{PARA 0 "" 0 " " {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 242 "From the comparison \+ of graphs for neighboring angles (theta=0.1/0.11/0.13) which cover a ' wiggle' in the DCS plot we can trace the origin of these wiggles: for \+ some angles the individual L contributions add up more coherently than for others." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 211 11 "Assignment:" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 337 "Summarize your observations \+ as to the interference phenomenon in the DCS. Compare the findings fro m this worksheet with those from M.S. Child; the latter are based on s emiclassical methods and consider interference phenomena from particul ar classical paths, and an Airy function approach to turn them into pr oper(?) quantum calculations." }}{PARA 0 "" 0 "" {TEXT 211 0 "" }} {PARA 0 "" 0 "" {TEXT 305 26 "Further independent study:" }{TEXT 211 441 " how do the interference patterns change with energy (wavenumber \+ K)? Can one understand the changes in the 'aperture' parameter a? It c ertainly is energy dependent in this case. One can derive the total cr oss section from the optical theorem (using the imaginary part of the \+ scattering amplitude at theta=0). This total cross section provides a \+ 'target size' that depends on energy. Is the aperture parameter a rela ted to this 'target size'?" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}{PARA 0 "" 0 "" {TEXT 211 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 0 "" }}}} {MARK "34 0 0" 132 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }