On an Expression for the Total Cross Section

The well-known expression for the toral cross section inrerms of the imaginary part or'the coherent forward scattered amplitUde is. derived in a simple and general ¥fay from wave theory. 'The derivation also shows that there is a "shadow remnant" beyond the actual shadow, within· which the forward coherent scattering intensity is diminished, of approximate radius (r/k) 1/2, where r is the distance from the scattering center and k=2n/).. is the wave number. A qualitative understanding of the shadow remnant is obtain~ in terms of the uncertainty principle. ' It is well known that the total cross section (elastic. plus in~lastic and absorption, or coherent plus incoherent) in any scattering process is proportional' to the imaginary part of the elastic or coherent scattere,d amplitude in the forward direction}> The physical reason for this proportionality is that any- reduction in the incident beam, which represents a loss by scattering or absorption, must be produced by interference bet~een the incident plane wave and the coherent scattered wave in the forward direction, and hence·. must be a linear function of the forwar.sf scattered elastic amplitude. This suggests that' the theorem referred to above can be established by a straightforward consideration of, the incid~t and scattered waves. It is shown below that this can be done, and that the region in the forward direction over which substantial destructive interference occurs can easily be found. The relation of this interference. region to shadow formation is also discussed. Finally, the situation is discussed qualitatively from the point of view of the uncertainty principle. All of the calculations and results are applic;able, to the relativistic case. We write the total stationary wave function It as the sum of a coherent part Uc and an inc