Plane-wave Born approximation calculations have been carried out for the (p,d) reaction at intermediate beam energies with the purpose of elucidating the manner in which the various physical parameters of the problem determine the behaviour of the cross-sections. Analytical formulae for the differential cross section are derived for three choices of the bound state wave function: square well, Woods–Saxon, and truncated harmonic oscillator. The formulae can essentially be factored into two parts, one which oscillates rapidly with momentum transfer q(and is not physically important) and the other which varies smoothly with q and describes the 'envelope' of the calculated angular distribution. For the square well and truncated harmonic oscillator form factors, this envelope function varies as q−8 and q−10 respectively, but in the Woods–Saxon case it is given predominantly by an exponential q-dependence. To illustrate our main points, calculations are presented for the 208Pb(p,d) reaction, primarily at 200 MeV, as well as normalized fits to presently available data at 185 MeV on light and intermediate mass nuclei and at 700 MeV on 12C. The envelope dependence of the square well and Woods–Saxon form factors gives a reasonably good account of these data.
Correction of the near threshold behavior of electron collisional excitation cross-sections in the plane-wave Born approximation