The Born approximation

Problem:

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 --> ∞.

Solution:

Problem:

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.

Solution:

Problem:

Evaluate the differential scattering cross section for a particle of mass m in a repulsive potential U(r) = A/r2 in the Born approximation.

Solution:

Problem:

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. 

Solution:

Problem:

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?

Solution:

Problem:

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 (kikf)·R =  2πn where ki is the initial wave vector, kf is the final wave vector and n is an integer.

Solution: