Effect of Electron-Hole Scattering Resonance on X-Ray Emission Spectrum

Abstract

Soft x-ray emission from metals results when an electron in the conduction band makes a radiative transition to an inner hole of the atom. The object of this paper is to study the effect of the electron-hole scattering resonances on the soft-x-ray emission spectra of metals. A detailed theoretical calculation was done for the emission spectra of Li and Na. From this calculation, the qualitative features of the observed $K$ emission spectrum of Li can be explained if a $p$-scattering resonance exists in the band near the Fermi surface. The resonance is caused by the electron-hole scattering. A resonance violates the Friedel sum rule unless a localized state analogous to a "local moment" is invoked. The crystal was divided into Wigner-Seitz spheres. The normal Li ion was removed from the center of the sphere and replaced by a Li ion with a $1s$ hole. A Hartree-type potential was constructed for this Li ion. Multiple scattering of other ions outside the Wigner-Seitz sphere was treated in the optical model. Scattering wave functions for the electrons inside the Wigner-Seitz sphere were computed numerically and used to calculate the matrix elements. The emission spectrum of Li was computed for various effective masses. The ${\mathrm{L}}_{23}$ emission spectrum of Na was computed along similar lines. No localized state was found in the band. This result may explain why no anomaly exists for the ${\mathrm{L}}_{23}$ emission spectrum of Na. The spectrum of the unobserved ${\mathrm{L}}_1$ emission of Na has also been computed. Here, there exists a $p$-scattering resonance, as in the $K$ emission spectrum of Li. From the calculation, it may be concluded that a lack of localized state is the reason why the emission spectra of other substances are normal.