Theory of Phonon-Aided Optical Absorption in the Alkali Metals. I. Sodium

Using the formalism of Nettel, the optical absorption $\sigma \left(\omega \right)$ due to all single-phonon exchange processes has been calculated for sodium from the near-infrared to the near-ultraviolet region, i.e., $\hslash \omega =0.5-4.0\mathrm{}$ eV, and at various temperatures. An initial study was made to determine the correct form of wave function to be used in calculating the absorbing power. The Butcher nearly-free-electron theory was reformulated in terms of orthogonalized plane waves (OPW). This investigation led to the conclusions that (i) a single OPW is not adequate to use as a wave function in the calculation of the absorbing power, (ii) the core-induced absorption does not contribute significantly to the interband absorption, and (iii) a nearly-free-electron wave function is adequate to use as a first approximation to calculate $\sigma \left(\omega \right)$ for the alkali metals. Consequently, a nearly-free-electron wave function was used in the subsequent calculations of the optical absorption due to single-phonon exchange processes. The phonon spectrum for sodium was calculated throughout the Brillouin zone using the pseudopotential method of Schneider and Stoll. In the infrared and intermediate regions, the results of this first-principles calculation are in excellent agreement with the recent experimental data of Smith, but do not agree with the experimental results of Mayer and Hietel; i.e., (i) there is no evidence in the computed curves for the existence of the anomalous resonance absorption, nor (ii) does this theory obtain the temperature behavior of the anomaly exhibited in the data of Mayer and Hietel. In the interband region the results are also in good qualitative agreement with the data of Smith. The temperature dependence of the computed interband absorption (which enters intrinsically via the phonon occupation operator) is again in disagreement with the data of Mayer and Hietel.