Wigner-Wilkins neutron/nucleus scattering kernel quantum-mechanically derived

TL;DR

This paper provides a detailed quantum-mechanical derivation of the Wigner-Wilkins neutron/nucleus scattering kernel, demonstrating its equivalence to the classical version and clarifying its foundational basis in neutron thermalization.

Contribution

The paper offers a self-contained, explicit quantum derivation of the Wigner-Wilkins kernel, emphasizing its equivalence to the classical kernel and addressing gaps in previous literature.

Findings

Quantum and classical Wigner-Wilkins kernels are identical.

The derivation uses Fermi pseudopotential and first-order Born approximation.

Explicit calculation of the final integral confirms the classical result.

Abstract

We undertake herein to derive the Wigner-Wilkins [W-W] neutron/nucleus scattering kernel, a foundation stone in neutron thermalization theory, on the basis of a self-contained calculation in quantum mechanics. Indeed, a quantum-mechanical derivation of the W-W kernel is available in the literature, cited below, but it is, in our opinion, robbed of conviction by being couched in terms of an excessive generality. Here, by contrast, we proceed along a self-contained route relying on the Fermi pseudopotential and a first-order term in a time-dependent Born approximation series. Our calculations are fully explicit at every step and, in particular, we tackle in its every detail a final integration whose result is merely stated in the available literature. Furthermore, and perhaps the most important point of all, we demonstrate that the quantum-mechanical W-W kernel outcome is identical down to the last iota with its classical antecedent, classical not only by virtue of historical precedence but also by being based on classical Newtonian mechanics.