Comparison and Appraisal of Approximation Formulas for Total Elastic Molecular Scattering Cross Sections

Abstract

The Massey—Mohr (MM), Schiff (S), and Landau—Lifshitz (LL) approximations for the total elastic cross section (Q) are intercompared. All can be shown to follow from the same assumption, (i.e., the classical small-angle deflection function, thence the Jeffreys—Born phases via the semiclassical equivalence relationship), sufficing to determine the velocity dependence of Q. Thus, for V=±C(s)/rs, Q(s)=p(s)[C(s)/hv ]2/(s—1) The coefficient p(s) is the same for the S and LL approximations; the ratio pSLL(s)/pMM(s)≥1, (<1.075); it is 1.0709 and 1.0458 for s=6 and 12, respectively. A numerical calculation for a repulsive (s=12) interaction shows that the SLL formula reproduces the partial-wave calculated Q to within ½%. A graphical presentation suggests the generality of this result; it also indicates the source of bias in the MM approximation. For a ``realistic'' intermolecular potential, (restricting consideration to collisions in the ``thermal'' energy range), the influence of the repulsion is only to produce undulations in Q(v); the correct value of C(6) may be obtained by velocity averaging the ``apparent'' CSLL(6).