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{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 "" 
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thetaFg^p-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(FiinFdf
p" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "
Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Here we see how the \+
numerical evaluation of the integral agrees with the Rutherford result
 for small impact parameters. 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 
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{PARA 0 "" 0 "" {TEXT -1 82 "For the range of theta-values shown the R
utherford cross section follows a simple " }{TEXT 19 9 "1/theta^4" }
{TEXT -1 29 " behaviour, as we are in the " }{TEXT 19 16 "sin(theta)=t
heta" }{TEXT -1 191 " regime. The Bohr cross section does not blow up \+
as badly at small angles, yet it also seems to continue to rise. To fi
gure out what is really going on there, we need to consider much large
r " }{TEXT 19 1 "b" }{TEXT -1 52 "-values, i.e., much smaller polar sc
attering angles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 63 "We repeat the loop for the calculation with the Bohr pote
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "There appears to be a problem wi
th noise at the larger-" }{TEXT 19 1 "b" }{TEXT -1 34 " end, we should
 probably increase " }{TEXT 19 6 "Digits" }{TEXT -1 88 ". 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]
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style=point,color=red,labels=[\"theta\",\"ds/dtheta\"],axes=boxed,titl
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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
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{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 }
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