{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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 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 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le 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 47 "Classical Differential Sc attering Cross Section" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 538 "We calculate the deflection function in classical m echanics which relates the polar scattering angle to the impact parame ter for a central potential. The calculation is based on the first int egral of the motion, i.e., rather than solving Newton's equation repea tedly in order to measure the relationship, we calculate the deflectio n function from an integral. For a numerical calculation (when the int egral cannot be found in closed form, as, e.g, for Rutherford scatteri ng) this integral needs to be calculated for each impact parameter." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 477 "In this worksheet the calculation is carried out for ion-atom scattering assu ming a simple screened Rutherford potential (Bohr potential). One of t he objectives is to verify that the differential cross section remains finite at forward angles, i.e., to demonstrate that the singularity i n the cross section (and in fact non-integrability) for scattering fro m the pure Coulomb potential is caused by the long-range nature, i.e., a lack of convergence at large impact parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart; wi th(plots): Digits:=11:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the na me changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 375 "First we define some relevant parameters: we choose Bohr units , in which the electron mass equals unity, and we consider proton-atom scattering for Z2=10 (neon atoms). The Bohr potential parameter was d etermined from experimental scattering data for neon atoms to be a=0.5 2 a.u. (S. Hagmann et al. Phys. Rev. A25, p.1918ff.). A neon atom has \+ a mass of about 20 proton masses." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "M1:=1836: M2:=20*1836: mu:=M1*M2/(M1+M2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG#\"&SA\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Z1:=1: Z2:=10: a:=0.52;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"#_!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "V:=r->Z1*Z2*exp(-r/a)/r;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"VGf*6#%\"rG6\"6$%)operatorG%&arrowGF(**%#Z1G\"\"\"%#Z2GF.-%$expG6#,$ *&9$F.%\"aG!\"\"F7F.F5F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The procedure follows expressions as given in H. Goldstein (3rd editi on), chapter 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 "First we define a procedure which computes the distance \+ of closest approach for given initial velocity and impact parameter. I t is based on the perihelion condition when the potential energy reach es its maximum along the trajectory for repulsive scattering. For give n impact velocity at infinity (" }{TEXT 19 2 "v0" }{TEXT -1 23 ") and \+ impact parameter " }{TEXT 19 1 "b" }{TEXT -1 50 " one defines the tota l relative scattering energy " }{TEXT 19 1 "E" }{TEXT -1 66 " at infin ity (zero potential), and the angular momentum magnitude " }{TEXT 19 1 "L" }{TEXT -1 35 ". The distance of closest approach " }{TEXT 19 4 " minR" }{TEXT -1 91 " results as a solution to the energy conservation \+ statement, which is a nonlinear equation." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "minR:=proc(v0,b) local E,L,peri; global mu,a,Z1,Z2; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "E:=mu*v0^2/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "L:=mu*v0*b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "peri:=E-L^2/(2*mu*r_min^2) - V(r_min);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fsolve(peri,r_min=0..infinity) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We pick the impact velocity at infinity in Bohr units. " }{TEXT 19 3 "v=1" }{TEXT -1 133 " would be intermediate valu e corresponding to a proton speed comparable to the classical orbit sp eed of a hydrogen 1s-state electron." }}{PARA 0 "" 0 "" {TEXT -1 82 "W e would like to explore fast and slow collisions ('less' and 'more' in teraction)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v0:=0.5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G$\"\"&!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "r_clap:=minR(v0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'r_clapG$\",#o#)[+?!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The distance of closest approach is slightly larger than \+ the impact parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "The scattering angle is given now by eq. (3.96) in Go ldstein (3rd ed). We need an integral from the distance of closest app roach to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 125 "It appears as if Maple can calculate the integral nume rically. It won't calculate the anti-derivative for the Bohr potential ." }}{PARA 0 "" 0 "" {TEXT -1 83 "We reduce the precision to which the integral is computed somewhat with respect to " }{TEXT 19 6 "Digits" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "theta:=p roc(v0,b) local E; global mu; E:=mu*v0^2/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "evalf(Pi)-2*evalf(Int(b/(r*sqrt(r^2*(1-V(r)/E)-b^2)), r=minR(v0,b)..infinity),Digits-1); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "theta(v0,0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +c7>^T!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "For a small impact p arameter and small impact velocity a large scattering angle is found. " }}{PARA 0 "" 0 "" {TEXT -1 187 "We now set up a loop over impact par ameter. We wish to explore small and large impact parameters, because \+ we are interested in a comparison with the unscreened Rutherford scatt ering case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "For small impact parameters (large deflection angles) we expec t our results to agree for both potentials, as the main deflection occ urs at the closest approach. For large " }{TEXT 19 1 "b" }{TEXT -1 88 " values the screened case leads to tiny deflection angles which becom e insignificant as " }{TEXT 19 1 "b" }{TEXT -1 87 " goes to infinity. \+ In the pure Coulomb potential there is always a deflection, even as " }{TEXT 19 1 "b" }{TEXT -1 66 " becomes infinite, which represents a pa thology (borderline case)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "db:=0.01; N:=200:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dbG$\"\"\" !\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "PP:=[seq([db*i,thet a(v0,db*i)],i=1..N)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "PP[ 200];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"$+#!\"#$\")wm38!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "P1:=loglogplot(PP,style=poin t,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "P2:=loglog plot(2*arccot(b*mu*v0^2/(Z1*Z2)),b=db..N*db,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(P1,P2,labels=[\"b\",\"theta \"],axes=boxed);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 570 266 266 {PLOTDATA 2 "6(-%'CURVESG6%7dw7$$!,+++++#!#5$\", OU'o'e$!#67$$!,V+q*)p\"F*$\",P'Q6(>#F-7$$!,`uyG_\"F*$\",.[\"\\**)*!#77 $$!,(3+%zR\"F*$!,c&*fBM#!#87$$!,d**H5I\"F*$!,VcJ5!))F87$$!,'\\([=A\"F* $!,T85nh\"F-7$$!,+'>!\\:\"F*$!,p;l6E#F-7$$!,I,5p4\"F*$!,%*)etLGF-7$$!, 1\\dd/\"F*$!,M)*[\"\\LF-7$$!,+++++\"F*$!,5ds#=QF-7$$!,%[J2'e*F-$!,Gm#G \\UF-7$$!,&Rv=3#*F-$!,%\\\"\\%[YF-7$$!,nZm0'))F-$!,u'pm?]F-7$$!,Lk>(Q& )F-$!,K=?)p`F-7$$!,%4u3R#)F-$!,&[h.*p&F-7$$!,K<+)ezF-$!,p8m3,'F-7$$!,g y5bp(F-$!,F)eT2jF-7$$!,*[\\FZuF-$!,)Q?W!f'F-7$$!,.*RY7sF-$!,m&QUhoF-7$ $!,K/+(*)pF-$!,\"Gyh@rF-7$$!,G02yx'F-$!,#o&*4stF-7$$!,;>tdd'F-$!,M['z8 wF-7$$!,+k@FQ'F-$!,w>8v%yF-7$$!,Fe()y>'F-$!,vN_R2)F-7$$!,J\"**f?gF-$!, ,^IPH)F-7$$!,/_m-&eF-$!,Ph\"R2&)F-7$$!,%eBO'o&F-$!,h'zT:()F-7$$!,lo>%G bF-$!,blQ#=*)F-7$$!,5-?gP&F-$!,1QPi6*F-7$$!,FX(yG_F-$!,*H$e(4$*F-7$$!, ;1$Q'3&F-$!,?Q6\"*\\*F-7$$!,o@+&[\\F-$!,()\\wXo*F-7$$!,6gg[\"[F-$!,eJ2 k')*F-7$$!,(H3@&o%F-$!,_U$[/5F*7$$!,lb>$fXF-$!,Bu1?-\"F*7$$!,A*\\(pV%F -$!,*o)H#R5F*7$$!,$fF)zJ%F-$!,wPqh0\"F*7$$!,Q.k@?%F-$!,hSWG2\"F*7$$!,( HRN*3%F-$!,7ym#*3\"F*7$$!,n3+%zRF-$!,,?^a5\"F*7$$!,GVh@(QF-$!,vG597\"F *7$$!,g42vw$F-$!,mtbr8\"F*7$$!,UW:`m$F-$!,JR)p_6F*7$$!,^Btac$F-$!,xK[! 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The plot for " }{TEXT 19 2 "P2" }{TEXT -1 41 " is coded after eq. (3.101) in Goldstein." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 435 "It is interesting to obs erve that the truncation of the integral in the calculation of theta t o a finite upper limit (instead of infinity) can lead to serious error s at intermediate and larger impact parameters. It means that it is es sential in a numerical evaluation of the integral to map the entire in tegration range even for a short-range scattering potential. This is s omewhat unexpected, particularly if one has investigated the " }{TEXT 19 7 "theta-b" }{TEXT -1 130 " relationship using numerical solutions \+ to the differential equation for which a rather finite integration ran ge suffices usually." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 258 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 238 "Explore t he relationship between impact parameter and scattering angle for diff erent impact velocities while keeping all other parameters fixed. Does the b-value where the screened and unscreened results merge change wi th impact velocity?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "Now we want to demonst rate the behaviour of the differential cross section at small angles. \+ Does the screening of the potential prevent the cross section from blo wing up at small angles?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " PP[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"\"\"!\"#$\",UgbQG#!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We calculate the differential c ross section using Goldstein eq. (3.93), and take the inverse of " } {TEXT 19 9 "dtheta/db" }{TEXT -1 14 " to calculate " }{TEXT 19 9 "db/d theta" }{TEXT -1 15 ". We calculate " }{TEXT 19 13 "dsigma/dOmega" } {TEXT -1 69 " using a central finite-difference formula on the equispa ced grid of " }{TEXT 19 1 "b" }{TEXT -1 8 "-values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for i from 2 to N-1 do: dsdO[i]:=db*i/abs ((PP[i+1][2]-PP[i-1][2])/(2*db))/sin(PP[i][2]); od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "In order to graph it properly as a function of \+ the polar scattering angle we use the " }{TEXT 19 1 "b" }{TEXT -1 22 " -range as a parameter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P Pc:=[seq([PP[i][2],dsdO[i]],i=2..N-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P3:=loglogplot(PPc,style=point,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "P4:=loglogplot(1/4*(Z1*Z2/(m u*v0^2))^2*csc(theta/2)^4,theta=PP[1][2]..PP[N-50][2],color=blue):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "display(P4,P3,labels=[\"th eta\",\"ds/dO\"],axes=boxed,title=\"Bohr potential(red), Coulomb poten tial (blue) differential cross section: dsigma/dOmega\");" }}{PARA 13 "" 1 "" {GLPLOT2D 570 266 266 {PLOTDATA 2 "6)-%'CURVESG6$7gn7$$!,#=a*H R#!#5$\",h2'z#*oF*7$$!,vI6>D#F*$\",wrg%GjF*7$$!,*RXfX@F*$\",.3&>.fF*7$ $!,VW`-1#F*$\",*zC$=c&F*7$$!,kkf*))>F*$\",)>%fmF&F*7$$!,B.'3u=F*$\",MF qr\"[F*7$$!,')**GLy\"F*$\",_%)[TX%F*7$$!,xOTVk\"F*$\",U#f@)*QF*7$$!,:6 :#R:F*$\",z3MxZ$F*7$$!,8c$)QQ\"F*$\",=-rk&GF*7$$!,q14(p7F*$\",Y_e)*R#F *7$$!,$o)pG7\"F*$\",)fPq7=F*7$$!,4\"3L85F*$\",+C<[P\"F*7$$!,LW1$o$)!#6 $\",:%eq'p'Ffo7$$!,m$[n4rFfo$\",r'oBu;Ffo7$$!,x'))=RhFfo$!,YacA>#Ffo7$ $!,'p*y&*R&Ffo$!,=R$HL^Ffo7$$!,r#p*pu%Ffo$!,o-xCs(Ffo7$$!,k-P;;%Ffo$!, E'*pQ+\"F*7$$!,!G\\WZOFfo$!,&zOp17F*7$$!,W#*p_<$Ffo$!,/`uAR\"F*7$$!,PN b!)z#Ffo$!,m'\\'*R:F*7$$!,:Ob#4CFfo$!,de_:p\"F*7$$!,\\.950#Ffo$!,-^U0$ =F*7$$!,)H8nJF*7$$!,:!Rrg9Ffo$!,?S.y0#F*7$$!,ie&3f6Ffo$! 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The Bohr cross section does not blow up \+ as badly at small angles, yet it also seems to continue to rise. 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However, whe n we try that the calculation of the scattering angle gets stuck, beca use " }{TEXT 19 10 "evalf(Int)" }{TEXT -1 75 " fails to return a value (the integrator can't reach the desired accuracy)." }}{PARA 0 "" 0 " " {TEXT -1 88 "In the graph below we show dsigma/dtheta in order to ad dress the integrability question:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for i from 2 to N-1 do: dsdO[i]:=db*i/abs((PP[i+1][2] -PP[i-1][2])/(2*db))/sin(PP[i][2]); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PPc:=[seq([PP[i][2],6.283*dsdO[i]*sin(PP[i][2])],i=2. .N-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "loglogplot(PPc, style=point,color=red,labels=[\"theta\",\"ds/dtheta\"],axes=boxed,titl e=\"Bohr potential differential cross section\");" }}{PARA 13 "" 1 "" {GLPLOT2D 642 216 216 {PLOTDATA 2 "6*-%'CURVESG6#7`w7$$!,%*)etLG!#6$!, $H7s47!#57$$!,%\\\"\\%[YF*$!,QDRA^'F*7$$!,p8m3,'F*$!,Rl:Wn#F*7$$!,\"Gy h@rF*$\",s%*)*=d#!#77$$!,vN_R2)F*$\",&*olpj#F*7$$!,blQ#=*)F*$\",wdckk% 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$\",%4DCazF-Fhjn/$\"\")FjhnQ&1e+08F_in/$\",d**H5I)F-Fhjn/$\",ZD@rZ)F-F hjn/$\",8**f?g)F-Fhjn/$\",V+q*)p)F-Fhjn/$\",/D^\"y()F-Fhjn/$\",+/)4X)) F-Fhjn/$\",q)**3.*)F-Fhjn/$\",%4DCa*)F-Fhjn/$\"\"*FjhnQ&1e+09F_in/$\", d**H5I*F-Fhjn/$\",ZD@rZ*F-Fhjn/$\",8**f?g*F-Fhjn/$\",V+q*)p*F-Fhjn/$\" ,/D^\"y(*F-Fhjn/$\",+/)4X)*F-Fhjn/$\",q)**3.**F-Fhjn/$\",%4DCa**F-Fhjn /$\"#5FjhnQ&1e+10F_in/$\",'**H5I5FcjnFhjn/$\",b77x/\"FcjnFhjn/$\",\"** f?g5FcjnFhjn/$\",/+(*)p5FcjnFhjn/$\",]7:y2\"FcjnFhjn/$\",S!)4X3\"FcjnF hjn/$\",()**3.4\"FcjnFhjn/$\",4DCa4\"FcjnFhjn" 1 5 0 1 10 0 2 6 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "We should be aware of the fact that the cross section ca n't be calculated by inverting the derivative of theta'(b) when the de flection angle is so small that the absolute error in the integral exc eeds the actual value. The data below demonstrate the failure of the p rocedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(dsdO[i],i= N-50..N-1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6T$\",]$G+NG\"\"$$\",**>a xL$F%$\",_[Y!4QF%$\",WRK()y%F%$\",D(oJhdF%$\",iR%4VmF%$\",&ytdF6$\",\")=FUH'F6$\",c5p0\"pF6$\",P037=*F6$\",>25uB\"\"\"&$\", 8F&3T8FS$\",R([=i9FS$\",Hys19#FS$\",kD5kD#FS$\",zHQ1^#FS$\",nmmmm#FS$ \",k/(4@IFS$\",\\'['['[FS$\",;\"oPY_FS$\",+++vo&FS$\",0$)*Q.iFS$\",[\" [\"[\"oFS$\",#3/-^vFS$\",4444p\"\"\"'$\",++++q\"Fjo$\",5kD5k*FS$\",=TH NA#Fjo$\",w6%HNAFjo$\",'eF[MEFjo$\",@'eF[EFjo$\",nmmm@$Fjo$\"\"\"%)inf inityG$\",++++D$Fjo$\",&*y:j7%FjoFip$\",E0@%oTFjo$\",dG9do&Fjo" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Apparently the results for " } {TEXT 19 13 "dsigma/dtheta" }{TEXT -1 110 " turn around at the smalles t theta-values shown, which indicates the differential cross section i s integrable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "In general, we have a hard time to verify in finite-prec ision arithmetic that the cross section is bounded. For some choices o f scattering parameters the calculation fails before a maximum in the \+ differential cross section is reached." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 135 "Verify the large-scattering angle regime: how well does \+ the numerical Bohr calculation agree with the analytical Rutherford re sult for " }{TEXT 19 13 "dsigma/dOmega" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 14 "Mini-projects:" }} {PARA 0 "" 0 "" {TEXT -1 88 "Explore some other repulsive central pote ntial that has a finite range. Investigate the " }{TEXT 19 7 "b-theta " }{TEXT -1 60 " relationship and the differential scattering cross se ction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "Investigate potentials with a power-law fall-off that decay faste r than the Coulomb potential and explore the forward differential cros s section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Investigate attractive finite-range scattering potentials " } {TEXT 19 4 "V(r)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "42" 0 } {VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }