On the extraction of optical sums from experimental generalized oscillator strengths and the experimental determination of the x-ray incoherent scattering factory by electron scattering

The connection between an experimentally measured Bethe surface and certain optical sum rules is discussed. The simplest procedure for constructing optical sums involves summing (integrating) the generalized oscillator strength over energy loss at a fixed scattering angle. This involves an approximation and an error formula is developed to estimate the magnitude of the correction involved. The approximation is shown to be extremely accurate in the limit of high incident energy and large momentum transfer. The sum S(?1,K) which is simply related to the X-ray incoherent scattering factor is found to have a small first order correction in the case of electron scattering from He with 25 keV electrons even at angles as small as 1°.