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Problem 1:Determine the total cross section for the scattering of slow particles (ka<1) by a potential V(r)=Cd(r-a). |
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Problem 2: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. |
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Problem 3: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®¥. |
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Problem 4: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. |
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Problem 5: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. |