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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 260 42 "Scattering from a spheric
al potential well" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 334 "Our interest is in assembling the scattering amplitude f
or a spherical potential barrier or well. We calculate the phase shift
s as a function of energy using the matching condition, and assemble t
he scattering amplitude. We use a simple unit system (h-bar=m=1), and \+
measure length as a multiple of the scale set by the potential well." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(orthopoly);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "R:=1; U0:=9/10; # (U0>0
 = barrier, U0<0 = well)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "
k:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kappa:=sqrt(k^2-U0
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Now we need the two types o
f Bessel functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "jn:=
(n,x)->simplify(sqrt(Pi/2/x)*BesselJ(n+1/2,x));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "assume(r>0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "jn(2,r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 
"nn:=(n,x)->simplify((-1)^(n+1)*sqrt(Pi/2/x)*BesselJ(-n-1/2,x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nn(2,r);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "ul:=(l,x)->x*jn(l,x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "vl:=(l,x)->x*nn(l,x);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 121 "The regular and irregular solutions to the radial equ
ations in terms of the Ricatti-Bessel and Ricatti-Neumann functions." 
}}{PARA 0 "" 0 "" {TEXT -1 26 "We need their derivatives:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ulp:=unapply('simplify(diff(ul(l,x)
,x))',l,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ulp(2,r);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "vlp:=unapply('simplify(diff
(vl(l,x),x))',l,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "vlp(2
,r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Now we are ready for the \+
matching condition which determines the phaseshift." }}{PARA 0 "" 0 "
" {TEXT -1 80 "The derivative functions defined above do not work with
 composite arguments yet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "vlp(1,k*r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "We need to sp
ecify a simple argument, and then substitute. Our hope was that unappl
y would take care of that, but, of course, with the " }{TEXT 262 1 "l
" }{TEXT -1 97 "-value unspecified nothing can be done. Therefore, we \+
define the required function for specified " }{TEXT 261 1 "l" }{TEXT 
-1 58 " inside the loop. The trick is to force the evaluation of " }
{TEXT 19 3 "ulp" }{TEXT -1 5 " and " }{TEXT 19 3 "vlp" }{TEXT -1 85 " \+
before unapplying the answer (this does not work with the simple mappi
ng construct)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Lmax:=10;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for l from 0 to Lmax do
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rho:=kappa*R; Ulp:=unapply(sim
plify(ulp(l,r)),r): Vlp:=unapply(simplify(vlp(l,r)),r):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 103 "taneta[l]:=evalf((Ulp(k*R)-Ulp(rho)*ul(l,k*R)
/ul(l,rho))/(-Vlp(k*R)+Ulp(rho)*vl(l,k*R)/ul(l,rho))); od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "print(taneta);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 52 "For very small phase shift (which happens for larg
e " }{TEXT 263 1 "l" }{TEXT -1 22 " at finite wavenumber " }{TEXT 264 
1 "k" }{TEXT -1 69 ") the number can be undefined due to the lack of n
umerical precision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 394 "For scattering from repulsive potential barriers the a
bove matching condition works only when the particle can penetrate the
 barrier as the solution is assumed to be of a simple scattering type \+
there (regular scattering solution). For scattering energies below the
 barrier height the particle would be tunneling into the potential bar
rier, and an exponentially dying function would be required." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sigma:=4*Pi/k^2*add((2*l+1)*
sin(arctan(taneta[l]))^2,l=0..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "The partial-wave contributions:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "4*Pi/k^2*[seq((2*l+1)*sin(arctan(taneta[l]))^2,l=0..L
max)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "for Lm from 0 to
 Lmax do: dsdO[Lm]:=1/4/k^2*evalc(abs(add((2*l+1)*(exp(2*I*arctan(tane
ta[l]))-1)*P(l,cos(theta)),l=0..Lm))^2); od:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 77 "plot([seq(dsdO[Lm],Lm=0..4)],theta=0..Pi,color=[
red,blue,green,brown,black]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 11 
"Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 127 "Explore the convergence \+
of the partial-wave expansion of the differential cross section for di
fferent values of the wavenumber " }{TEXT 257 1 "k" }{TEXT -1 38 " (va
riation of the scattering energy)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 258 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 
110 "Compare the differential cross section at fixed energy for repuls
ive and attractive potential well scattering." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 3:" }}{PARA 0 "
" 0 "" {TEXT -1 402 "Explore the dependence of the differential cross \+
section at fixed energy on the size of the potential well (or barrier)
. Comment in particular on the amount of backscattering. Observe what \+
happens in scattering from barriers of growing size when the scatterin
g energy is just sufficient for barrier penetration without tunneling.
 What is the meaning of the structures in the differential cross secti
on?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
265 17 "Energy dependence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 225 "Let us calculate the total scattering cross secti
on as a function of energy in order to demonstrate the convergence of \+
the partial wave expansion at low energies. We need to automate the ca
lculation of the matching condition." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "R:=2; U0:=-4; # (U0>0 = barrier, U0<0 = well)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "Lmax:=10;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "for ik from 1 to 10 do: k:=ik/10; kappa:=sqrt(k^2-U0)
; Ev[ik]:=k^2/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for l from 0 to
 Lmax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rho:=kappa*R; Ulp:=una
pply(simplify(ulp(l,r)),r): Vlp:=unapply(simplify(vlp(l,r)),r):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "taneta[ik,l]:=evalf((Ulp(k*R)-Ulp(
rho)*ul(l,k*R)/ul(l,rho))/(-Vlp(k*R)+Ulp(rho)*vl(l,k*R)/ul(l,rho)));" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sigma[ik,l]:=4*Pi/k^2*(2*l+1)*sin
(arctan(taneta[ik,l]))^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "Lmax:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "P1:=pl
ot([seq([Ev[ik],log10(add(sigma[ik,l],l=0..Lmax))],ik=1..10)],style=po
int,color=red): display(P1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "Lmax:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "P2:=plot([seq([Ev
[ik],log10(add(sigma[ik,l],l=0..Lmax))],ik=1..10)],style=point,color=b
lue): display(P1,P2);" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Lmax:=2;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "P3:=plot([seq([Ev[ik],log10(add(sigma[ik,l],l=0..Lma
x))],ik=1..10)],style=point,color=green): display(P1,P2,P3);" }{TEXT 
-1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"Lmax:=3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "P4:=plot([seq([Ev[ik]
,log10(add(sigma[ik,l],l=0..Lmax))],ik=1..10)],style=point,color=black
): display(P1,P2,P3,P4);" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 216 "We see that at low scattering energies t
he cross section is dominated by isotropic (s-wave) scattering. As the
 energy increases the higher partial waves begin to contribute more to
 the total scattering cross section." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
266 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 278 "Explore the energ
y dependence of the total scattering cross section for attractive pote
ntial barriers of increasing depth and size. What is the interpretatio
n of the minimum in the total cross section at low energies (which app
ears for potential wells of sufficient depth/size)?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 2 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }