Determine the total cross section for the scattering of slow particles (ka<1) by a potential V(r)=Cd(r-a).
What must V0a2 be for a 3-dimensional square well potential in order that the scattering cross section be zero at zero bombarding energy (Ramsauer-Townsend effect)? Find the leading term in the expression for the differential cross section for small bombarding energies.
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 V0(r)=-Ze2/r. This potential is screened by the atomic electron cloud. As a result, the total potential energy of the incident electron is V(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=ki-kf be the momentum transfer.
(a) Calculate the differential cross section, d
s/dW, in the Born approximation for the scattering of the electron by the atom (using the screened potential V(r).)(b) Now calculate the differential cross section, d
s/dW, in the Born approximation for the scattering of the electron by the nucleus only (using the unscreened potential V0(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®¥.From measurements of the differential cross section for scattering of electrons off protons (in atomic hydrogen) it was found that the proton had a charge density given by
r(r)=aexp(-br) where a and b are constants.(a) Find
a and b such that the proton charge equals e, the charge on the electron.(b) Show that the protons mean square radius <r2> is 12/
b2.(c) Assuming a reasonable value for <r2>1/2
calculate a in esu/cm3.Evaluate, in the Born Approximation, the differential cross section for the scattering of a particle of mass m by a delta-function potential V(r)=Bd(r). Comment on the angular and velocity dependence. Find the total cross section.