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

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 $\frac{1}{{\epsilon }^{\alpha }}$, where $\epsilon$ is the frequency measured from the threshold, and $\alpha$ an exponent involving the various phase shifts ${\delta }_l$ which describe scattering by the deep hole. $\alpha$ 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.