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Journal of Mathematical Physics

Cross sections for screened potentials

Mark D. Semon, John R. Taylor

Published Aug 1, 1976• 11 citations
Publisher

Abstract

We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.

DOI
10.1063/1.523085

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We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.
An analysis of potential scattering in terms of a Gaussian wavepacket is given. It is used to derive rigorously a bound alpha on the ratio of the modulus of the difference between the exact and Born transition amplitudes for any given momentum transfer to the modulus of the Born amplitude in the forward direction. Bounds on alpha are given in the case of a square-well potential, a screened Coulomb potential and an exponential potential; the last two are applied to elastic electron-helium scattering.
The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux-across-surfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V(x) decays like |x|−γ at infinity with γ>n∈N then the generalized eigenfunctions of the corresponding Hamiltonian −1/2Δ+V are n−2 times continuously differentiable with respect to the momentum variable.
Quantum Rutherford scattering and scattering of classical waves off Coulomb-like potentials have similar formal structures and can be studied using the same mathematical techniques. In both contexts, the long-range nature of the interaction leads to a divergent total cross-section, which has been interpreted and regularized in various ways in the past literature. We review in detail the origin of this divergence, in both real and multipole spaces, and show that it arises from incorrectly using approximations out of their domain of validity. We also stress that although classical and quantum Rutherford scattering share the same formalism, the natures of the associated physical observables differ. We comment on the role of interference: while interference can be safely neglected in a quantum context (due to the fact that the observable quantity is a flux, and the incoming flux is collimated), in a classical context one expects to see a superposition of transmitted and scattered waves in a broad region downstream of the target, hence a cross-section is not connected to any physically observable quantity.
A development of the scattering amplitude in potential scattering which is based upon replacing a smooth potential by a sequence of potential steps is given. The Schrödinger equation is solved exactly in each interval over which the potential is constant and by matching boundary conditions an expression relating the coefficients of two linearly independent solutions at small radius to those at large radius is found. From this the scattering amplitude is calculated. The limits of validity are found to be dependent upon the value of the expression (r/2)ddrln(2mℏ2[E−V]). Limiting cases of the amplitude are shown to be a modified WKB approximation and the Born approximation. A particular application to the Yukawa potential is discussed. The discussion is restricted to attractive potentials.