Intensity and Shape of the X-Ray-Edge Singularity

Abstract

The singularity at the Fermi edge in the soft-x-ray spectra of metals is known to be described by a factor ${\left|\frac{{\xi }_0}{\epsilon }\right|}^{\alpha }$ multiplying the one-electron intensity. Using a separable potential, Nozières and deDominicis showed that $\alpha$ is a function of the Fermi-electron phase shifts. However, ${\xi }_0$, treated until now as a constant, has not yet been derived. We extend the above factor to other frequencies, writing it in the form $G\left(\epsilon \right){\left|\frac{\xi \left(\epsilon \right)}{\epsilon }\right|}^{\alpha }$. Using simplified diagrams we can introduce a realistic potential and orthogonalized plane waves and we determine $G\left(\epsilon \right)$ and $\xi \left(\epsilon \right)$ in a range of about 3 eV from the edge. The calculations are applied to $\mathrm{Na}{L}_{2,3}$, $\mathrm{Li}K$, and $\mathrm{Be}K$ bands. The factor $G\left(\epsilon \right)$, related to the open-line part of the problem, presents a singularity in the slope at $\epsilon =0\mathrm{}$. This fact, important in the $K$ bands, was not realized before. However, the edge singularity does not appear to be strong enough to explain the premature peak in the $K$-emission bands. The $p$-scattering resonance discussed by Allotey is probably dominant here.