An electron of incident momentum ki is scattered
elastically by the electric field of an atom of atomic number Z. The
potential due to the nucleus is of the form U0(r) = -Ze2/r.
This potential is screened by the atomic electron cloud. As a result, the
total potential energy of the incident electron is U(r) = (-Ze2/r)exp(-r/a),
where a is the radius of the atom. Let kf be the final
momentum of the electron and q = kf - ki
be the momentum transfer.
(a) Calculate the differential cross section, dσ/dΩ, in the Born approximation for the scattering of the electron by the atom (using the screened potential U(r)).
(b) Now calculate the differential cross section, dσ/dΩ, in the Born approximation for the scattering of the electron by the nucleus only (using the unscreened potential U0(r)).
(c) Plot the ratio of the two cross sections as a function of x = a|q| and briefly discuss the limits x --> 0 and x --> ∞.
Consider an electron of energy E0 and velocity v0 in the z-direction incident on an ionized Helium atom He+, with just one electron in its ground state. Compute the differential cross section dσ/dΩ for the incident electron to scatter into the solid angle dΩ about the spherical angles (θ,φ). Explain your assumptions and approximations.
exp(-iq∙r)U(r) = -e2∫d3r
exp(-iq∙r)∫d3r' ρt(r')/|r -
= -e2∫d3r' ρt(r')exp(-iq∙r')∫d3r exp(-iq∙(r - r')/|r - r'|
= -(4πe2/q2)∫d3r' ρt(r')exp(-iq∙r') = -(4πe2/q2)[2 - F(q)],
where F(q) = ∫d3r' ρ(r')exp(-iq∙r') is the atomic scattering form factor.
[To evaluate ∫d3r
r'| we have used
∫d3r e-λr exp(iq∙r)/r = 2π∫0∞dr r e-λr∫-11d(cosθ)eiqrcosθ
= 2π∫0∞dr e-λr(eiqr - e-iqr)/(iq) = (4π/q)Im(∫0∞dre-(λ-iq)r)
= (4π/q)Im(1/(λ - iq)) = 4π/(λ2 + q2).
Take the limit as λ --> 0.]
q2 = 4k2sin2(θ/2).
σkB(θ,φ) = [4m2e4/(16ħ4k4sin4(θ/2))][2 - F(q)]2
= (e4/(16E2sin4(θ/2))[2 - F(q)]2.
If we assume that F(q) = 1, i.e. that the incoming electron sees a screened nucleus of charge 1, we obtain the Rutherford scattering cross section.
Evaluate the differential scattering cross section for a particle of mass m in a repulsive potential U(r) = A/r2 in the Born approximation.
Details of the calculation:
In the Born approximation we have
σkB(θ,φ) = σkB(k,k') = [μ2/(4π2ħ4)]|∫d3r' exp(-iq∙r')U(r')|2,
where q = k' - k, k = μv0/ħ, k' = μv0/ħ (k'/k'), and μ is the reduced mass.
Here μ = m, k is the incident wave vector, and (k'/k') is a unit vector pointing in the direction (θ,φ).
With U(r) = A/r2 we have
∫d3r' exp(-iq∙r')U(r') = A∫∫∫dr'sinθ'dθ'dφ'exp(-iqr'cosθ')
= (4πA/q)∫0∞dr'sin(qr')/r' = 2π2A/q.
σkB(θ,φ) = [μ2/(4π2ħ4)][4π4A2/q2] = μ2π2A2/(ħ4q2).
With q = 2ksin(θ/2) we have σkB(θ,φ) = μ2π2A2/(4ħ4k2sin2(θ/2)).
Evaluate, in the Born Approximation, the differential cross section for the scattering of a particle of mass m by a delta-function potential U(r) = Bδ(r). Comment on the angular and velocity dependence. Find the total cross section.
Consider two distinguishable spin-½ particles interacting
via the interaction
Hint = gS1·S2δ3(r),
where g is a constant, Si are the spin operators of the two particles, and r is the separation of the two particles.
(a) Using the Born approximation, calculate the differential cross section in the center of mass frame for particles in the spin-singlet and spin-triplet states, assuming the particles are non-identical.
(b) What is the total cross section if the particles are unpolarized?
The differential scattering cross section is proportional to the Fourier transform of the potential.
We can derive the above expression for dσkB/dΩ using time-dependent perturbation theory and Fermi's golden rule. If H = H0 + Wexp(±iωt), then the transition probability per unit time to a group of states nearly equal in energy E = Ei + ħω is given by
w(i,βE) = (2π/ħ)ρ(β,E)|WEi|2δE-Ei,ħω, where WEi = <ΦE|W|Φi>.
Here we consider an initial state with momentum ħk and a final state with momentum ħk'. The box normalized wave functions are given as
Φi(r) = L-3/2exp(ik∙r), Φf(r) = L-3/2exp(ik'∙r),
where L is the dimension of the box.
We let W = U(r), ω = 0, and β = dΩ. Then
WEi = <ΦE|U|Φi> = L-3∫d3r' exp(iq∙r')U(r').
The energy eigenstates of a particle confined to a cubical box with periodic boundary conditions are
Φnx,ny,nz(x,y,z) = L-3exp(i2π(nxx + nyy + nzz)/L),
with nx, ny, nz = 0, ±1, ±2, ... .
We have kx = 2πnx/L, ky = 2πny/L, kz = 2πnz/L.
If k is large, then the number of states with wave vectors whose magnitudes lie between k and k + dk within an element of solid angle dΩ is
dN = k2dΩdk/(2π/L)3 = L3k2dΩdk/(2π)3.
dN/dk = L3k2dΩ/(2π)3.
The density of states is ρ(E,dΩ) = dN/dE = (dN/dk)(dk/dE).
With E = ħ2k2/(2μ) we have dN/dE = μħkdΩL3/(2πħ)3.
We now can insert WEi and ρ(E,dΩ) into the expression for the transition probability per unit time.
w(i,dΩE) = [μħkdΩ/(4π2ħ4L3)]|∫d3r' exp(-iq∙r')U(r')|2.
Here w(i,dΩE) is the probability per unit time that a particle scatters into the solid angle dΩ. The cross section is the transition rate per incident flux. The incident flux is the incident probability density times the velocity, |Φi(r)|2v = L-3v = L-3ħk/μ. Therefore we have
σkB(k,k')dΩ = [μ2/(4π2ħ4)]|∫d3r' exp(-iq∙r')U(r')|2dΩ.
This derivation makes it clear how to incorporate a spin-dependent potential into the Born approximation.
Consider a periodic scattering potential with translational invariance U(r + R) = U(r), where R is a constant vector. Show that in the Born approximation scattering occurs only in the directions defined by (ki - kf)·R = 2πn where ki is the initial wave vector, kf is the final wave vector and n is an integer.
U(r) = ∑-∞+∞C(km)exp(ikm∙r).
Here kn = n2π/R(R/R),
Δk = kn+1 - kn
= 2π/R, n = integer.
(Note: R/R denotes the unit vector in the direction of R.)
exp(-i(q - km)∙r') = 2π ∑-∞+∞C(km)δ(
km - q)
The integral is zero unless ( km - q) = 0, (kf - ki) = km, (kf - ki)·R = m2π.
Scattering occurs only in the directions defined by (kf - ki)·R = 2πm.