Theory of soft-x-ray-absorption thresholds: AmorphousMgxSb1−xalloys and metallic Li, Na, Mg, and Al

Abstract

The trends as a function of free-electron density, $n={\left[\right(\frac{4}{3}\left)\pi r_s^3{a}^3\right]}^{-1\mathrm{}}$, in the soft-x-ray absorption spectra ${\epsilon }_2\left(\omega \right)$ of amorphous ${\mathrm{Mg}}_x{\mathrm{Sb}}_{1-x}$ alloys and metallic Li, Na, Mg, and Al are analyzed in terms of the many-electron theory of x-ray threshold anomalies: ${\epsilon }_2\left(\omega \right)=A_l^2{\left[\frac{(\hslash \omega -E_T)}{\xi }\right]}^{-{\alpha }_l}$. The threshold exponents ${\alpha }_l$ are calculated as functions of $r_s$ using the Nozières-deDominicis theory and a (Hulthén) screened-potential approximation to the electron-hole interaction. These calculated exponents are in marked disagreement with the empirical rule ${\alpha }_0\approx 0.068r_s$ for Na, Mg, and Al. The cutoff energy $\xi$ is extracted from ${\mathrm{Mg}}_x{\mathrm{Sb}}_{1-x}$ alloy data and, within experimental error, is independent of $x$. Thus $\xi$ cannot be identified with a Fermi energy. The value $\xi =0.24\mathrm{}\pm 0.1$ eV is too small to be identified as a conduction-band width, but could be identified with the exciton rydberg $R=13.6\left(\frac{m}{m_0{\epsilon }_0^2}\right)$ eV, where $m$ is the conduction-band mass and ${\epsilon }_0$ is the static dielectric constant of ${\mathrm{Mg}}_3$ ${\mathrm{Sb}}_2$. The analyses indicate the qualitative importance of final-state interactions in shaping the absorption thresholds, but improved, quantitatively accurate, theoretical understanding of the threshold exponents ${\alpha }_l$ and the characteristic energy $\xi$ is needed. Future experiments on x-ray edges of alloys such as ${\mathrm{Mg}}_x{\mathrm{Sb}}_{1-x}$ should be coupled with independent experimental determination of the variation of $r_s$ with composition $x$, in order to facilitate comparison of the many-electron theory with data.