Coulomb Corrections to Inelastic Electron Scattering

The problem of making an improvement in the first-Born-approximation treatment of inelastic electron scattering involving Coulomb multipole transitions is approached by expressing the scattering cross section in terms of integrals over the nuclear-transition charge-density form factor with a kernel which contains all the electron physics. The distorted-wave Born approximation is used, and the interaction with the transverse electromagnetic field is neglected. Various approximations to the electron wave functions may be used to construct the kernel, leading to varying degrees of accuracy in the results. The kernel is computed in second Born approximation, using the radius approximation to allow for the finite nuclear size. An analytic expression is obtained for all multipole orders, and explicit results are given for the multipolarity $J=0, 1, \mathrm{and} 2$. Also an analytic expression for all multipole orders for the kernel in second Born approximation using the Yukawa static charge distribution is given. Integration of the radius approximation kernel with some simple transition form factors is carried out for $J=0, 1, \mathrm{and} 2$ and analytic results are given. An application of these results for correction of nuclear transition rate estimates from electron-scattering data is presented. Analysis of the corrections in the limit of zero momentum transfer shows that the ratio of the cross section to the Mott cross section does not go to zero as ${\left(\mathrm{momentum}\mathrm{transfer}\right)}^{2J}$ for $J>\mathrm{}3\mathrm{}$ in second Born approximation. This is contrary to expectation. This work provides a simple means of getting some correction to the first Born approximation without the necessity of a digital computer.