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Journal ArticleDOI

Additional WKB Inversion Relations for Bound‐State and Scattering Problems

15 May 1971-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 54, Iss: 10, pp 4174-4177

AbstractNew inversion relations for the diatom system (elastic scattering of two atoms by a single potential, or a bound diatomic molecule) are presented. They are exact within the WKB approximation for the relevant quantities and are closely related to the well‐known RKR expressions. Explicit formulas are given for bound‐state and scattering input.

Topics: WKB approximation (58%), Elastic scattering (54%), Scattering (53%), Bound state (52%), Diatomic molecule (51%)

Summary (1 min read)

II. Inversion from Eigenvalues

  • The principal disadvantage o~ Equation (9) is that one cannot solve these equations to obtain e)..'":plicit e>.:pressions for r< and r> (due to the logarithms).
  • For bound-state problems there is no apparent reason >vhy any pair of equa.tions should be easier to use than the usual RKR pair [Equations (10) and (11) ].
  • Equations (12) and (13) do give alternate inversionrelations, however, v.rhich may be of use in some situations.

b. Curve Crossing Probability

  • In the semiclassical treatment 1 7 of' electronic transitions in atom-atom collisions which take place via a crossing of the two curves, the transition probability is the product of a slm-rly varying factor and an oscillatory factor.
  • The phase of the oscillatory fae;tor is a phase integral; more specifically, it is the difference in phase integrals on the initial and final potential curves from their respective classical turning points to the crossing point.
  • If the E and £ dependence of this phase integral can be extracted from the differential and/or total cross section for the electronic transition, then the inversion formulae of Section II can be utilized to construct the two potentials involved.

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