Critical-Point Structure in Photoelectric Emission Energy Distributions

Abstract

Structure in photoelectric energy distributions is analyzed on the basis of a model which assumes that the electrons are created in the volume by direct transitions, and ignores lifetime broadening and the distortion of the distribution function due to the escape probability. In this model, the structure consists of square edges and logarithmically infinite peaks due to two-dimensional critical points where the optical energy surface is tangent to the electron energy surface. As the photon energy is varied the loci of these critical points trace out "critical lines" in $k$ space. A plot of the electron energy of the structure versus photon energy is called the "$E-\omega$ image" of the critical line. Simple properties of these lines and their images are derived and illustrated by explicit calculations for the band structure of silicon. The critical lines pass through the three-dimensional critical points of the electron energy function (ECP) and those of the optical energy function (OCP). An ECP and an OCP will coincide when required to by symmetry, and are then called symmetry critical points (SCP). ECP's are extrema of $E\left(\hslash \omega \right)$ along critical lines, while OCP's are extrema of $\hslash \omega \mathrm{}\left(E\right)\mathrm{}$. $E-\omega$ images will "kink" or intersect at SCP's. The "strength" of critical-point structure varies rapidly along critical lines and tends to infinity as an SCP is approached. These properties should be useful in inferring energy-band information from experimental plots of $E-\omega$ images. They will also be helpful in analyzing machine calculations of photoelectric energy distributions.