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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:" }}}{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 fo
r Z2=10 (neon atoms). The Bohr potential parameter was determined from
 experimental scattering data for neon atoms to be a=0.52 a.u. (S. Hag
mann et al. Phys. Rev. A25, p.1918ff.). A neon atom has a mass of abou
t 20 proton masses." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "M1:=
1836: M2:=20*1836: mu:=M1*M2/(M1+M2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "Z1:=1: Z2:=10: a:=0.52;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "V:=r->Z1*Z2*exp(-r/a)/r;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "The procedure follows expressions as given in H. Goldstei
n (3rd edition), 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 pa
rameter. It is based on the perihelion condition when the potential en
ergy reaches its maximum along the trajectory for repulsive scattering
. For given impact velocity at infinity (" }{TEXT 19 2 "v0" }{TEXT -1 
23 ") and impact parameter " }{TEXT 19 1 "b" }{TEXT -1 50 " one define
s the total relative scattering energy " }{TEXT 19 1 "E" }{TEXT -1 66 
" at infinity (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 c
onservation statement, which is a nonlinear equation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "minR:=proc(v0,b) local E,L,peri; gl
obal 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 inf
inity in Bohr units. " }{TEXT 19 3 "v=1" }{TEXT -1 133 " would be inte
rmediate value corresponding to a proton speed comparable to the class
ical orbit speed of a hydrogen 1s-state electron." }}{PARA 0 "" 0 "" 
{TEXT -1 82 "We would like to explore fast and slow collisions ('less'
 and 'more' interaction)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"v0:=0.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r_clap:=minR(v
0,2);" }}}{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 angl
e is given now by eq. (3.96) in Goldstein (3rd ed). We need an integra
l from the distance of closest approach to infinity. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "It appears as if Mapl
e can calculate the integral numerically. 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 r
espect to " }{TEXT 19 6 "Digits" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 51 "theta:=proc(v0,b) local E; global mu; E:=mu*v
0^2/2;" }}{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);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 89 "For a small impact parameter and small im
pact velocity a large scattering angle is found." }}{PARA 0 "" 0 "" 
{TEXT -1 187 "We now set up a loop over impact parameter. We wish to e
xplore small and large impact parameters, because we are interested in
 a comparison with the unscreened Rutherford scattering case." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "For smal
l impact parameters (large deflection angles) we expect our results to
 agree for both potentials, as the main deflection occurs at the close
st approach. For large " }{TEXT 19 1 "b" }{TEXT -1 88 " values the scr
eened case leads to tiny deflection angles which become insignificant \+
as " }{TEXT 19 1 "b" }{TEXT -1 87 " goes to infinity. In the pure Coul
omb potential there is always a deflection, even as " }{TEXT 19 1 "b" 
}{TEXT -1 66 " becomes infinite, which represents a pathology (borderl
ine case)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "db:=0.01; N:=
200:" }}}{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];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "P1:=loglogplot(PP,
style=point,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "
P2:=loglogplot(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 "" }}}{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 pa
rameters. The plot for " }{TEXT 19 2 "P2" }{TEXT -1 41 " is coded afte
r eq. (3.101) in Goldstein." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 435 "It is interesting to observe that the truncati
on of the integral in the calculation of theta to a finite upper limit
 (instead of infinity) can lead to serious errors at intermediate and \+
larger impact parameters. It means that it is essential in a numerical
 evaluation of the integral to map the entire integration range even f
or a short-range scattering potential. This is somewhat unexpected, pa
rticularly if one has investigated the " }{TEXT 19 7 "theta-b" }{TEXT 
-1 130 " relationship using numerical solutions to the differential eq
uation for which a rather finite integration range suffices usually." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Exerci
se 1:" }}{PARA 0 "" 0 "" {TEXT -1 238 "Explore the relationship betwee
n impact parameter and scattering angle for different impact velocitie
s while keeping all other parameters fixed. Does the b-value where the
 screened and unscreened results merge change with impact velocity?" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 186 "Now we want to demonstrate the behaviour
 of the differential cross section at small angles. Does the screening
 of the potential prevent the cross section from blowing up at small a
ngles?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "PP[1];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 96 "We calculate the differential cross secti
on using Goldstein eq. (3.93), and take the inverse of " }{TEXT 19 9 "
dtheta/db" }{TEXT -1 14 " to calculate " }{TEXT 19 9 "db/dtheta" }
{TEXT -1 15 ". We calculate " }{TEXT 19 13 "dsigma/dOmega" }{TEXT -1 
69 " using a central finite-difference formula on the equispaced 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 "PPc:=[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/(mu*v0^2))^2*csc(th
eta/2)^4,theta=PP[1][2]..PP[N-50][2],color=blue):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 147 "display(P4,P3,labels=[\"theta\",\"ds/dO\"],
axes=boxed,title=\"Bohr potential(red), Coulomb potential (blue) diffe
rential cross section: dsigma/dOmega\");" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 82 "For the range of theta-values shown the Rutherford cross \+
section follows a simple " }{TEXT 19 9 "1/theta^4" }{TEXT -1 29 " beha
viour, as we are in the " }{TEXT 19 16 "sin(theta)=theta" }{TEXT -1 
191 " regime. The Bohr cross section does not blow up as badly at smal
l angles, yet it also seems to continue to rise. To figure out what is
 really going on there, we need to consider much larger " }{TEXT 19 1 
"b" }{TEXT -1 52 "-values, i.e., much smaller polar scattering angles.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "We re
peat the loop for the calculation with the Bohr potential:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "db:=0.04; N:=200:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "PP:=[seq([db*i,theta(v0,db*i)],i=1..N)]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "loglogplot(PP,style=po
int,color=red,labels=[\"b\",\"theta\"],axes=boxed,title=\"Bohr potenti
al: theta(b)\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "There appears
 to be a problem with noise at the larger-" }{TEXT 19 1 "b" }{TEXT -1 
34 " end, we should probably increase " }{TEXT 19 6 "Digits" }{TEXT 
-1 88 ". However, when we try that the calculation of the scattering a
ngle gets stuck, because " }{TEXT 19 10 "evalf(Int)" }{TEXT -1 75 " fa
ils 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/d
theta in order to address 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]*s
in(PP[i][2])],i=2..N-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
128 "loglogplot(PPc,style=point,color=red,labels=[\"theta\",\"ds/dthet
a\"],axes=boxed,title=\"Bohr potential differential cross section\");
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "We should be aware of the fa
ct that the cross section can't be calculated by inverting the derivat
ive of theta'(b) when the deflection angle is so small that the absolu
te error in the integral exceeds the actual value. The data below demo
nstrate the failure of the procedure:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "seq(dsdO[i],i=N-50..N-1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "Apparently the results for " }{TEXT 19 13 "dsigma/dtheta
" }{TEXT -1 110 " turn around at the smallest theta-values shown, whic
h indicates the differential cross section is integrable." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "In general, we ha
ve a hard time to verify in finite-precision arithmetic that the cross
 section is bounded. For some choices of scattering parameters the cal
culation fails before a maximum in the differential cross section is r
eached." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 
11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 135 "Verify the large-scat
tering angle regime: how well does the numerical Bohr calculation agre
e with the analytical Rutherford result for " }{TEXT 19 13 "dsigma/dOm
ega" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 259 14 "Mini-projects:" }}{PARA 0 "" 0 "" {TEXT -1 88 "Explore
 some other repulsive central potential that has a finite range. Inves
tigate the " }{TEXT 19 7 "b-theta" }{TEXT -1 60 " relationship and the
 differential scattering cross section." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 145 "Investigate potentials with a power
-law fall-off that decay faster than the Coulomb potential and explore
 the forward differential cross section." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Investigate attractive finite-rang
e 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 "29 0 0" 62 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }