Singularities in the X-Ray Absorption and Emission of Metals. III. One-Body Theory Exact Solution

TLDR

The singularities of x‑ray absorption or emission in metals are studied by a new “one‑body” method that describes the scattering of conduction electrons by the transient potential due to a deep hole. The authors express the transition rate as a product of a one‑electron transient Green’s function L and a deep‑level Green’s function G, which satisfy simple Dyson equations solvable asymptotically via Muskhelishvili’s method. The x‑ray transition rate behaves as ε^(-α), with α determined by phase shifts δ_l; α can be >0 (infinite threshold) or 0 (zero threshold), and the results relate to the Friedel sum rule.

Abstract

The singularities of x-ray absorption or emission in metals are studied by a new "one-body" method, which describes the scattering of conduction electrons by the transient potential due to the deep hole. Using the linked-cluster theorem, the net transition rate in the time representation is expressed as the product of two factors: a one-electron transient Green's function $L$, and the deep-level Green's function $\mathcal{G}$. These factors obey simple Dyson equations, which can be solved asymptotically by using Muskhelishvili's method. The x-ray transition rate is found to behave as $\frac{1}{{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}}$, where $\ensuremath{\epsilon}$ is the frequency measured from the threshold, and $\ensuremath{\alpha}$ an exponent involving the various phase shifts ${\ensuremath{\delta}}_{l}$ which describe scattering by the deep hole. $\ensuremath{\alpha}$ may be >0 (infinite threshold) or 0 (zero threshold). The experimental implications of these results and their relation to the Friedel sum rule are briefly discussed.