Bonding-Electron Distributions, Anharmonicity, and the Temperature Dependence of the Forbidden Si(442) Reflection

Abstract

The forbidden Si(442) reflection was detected and its temperature dependence measured between 20 and 500°C using $\mathrm{Cu}K\alpha$ x-ray radiation. The reflection is extremely weak at room temperature with an integrated intensity of 4.6×${10}^{-11\mathrm{}}$ and a structure factor $F\left(442\right)=0.035\mathrm{}\pm 0.002$. $F\left(442\right)\mathrm{}$ decreases to zero at 250°C and then increases to almost twice its room-temperature value at 500°C. This temperature dependence is consistent with contributions to the structure factor from both the antisymmetric bond charge distribution $F_{\mathrm{bond}}$, and anharmonic thermal motion of the atoms $F_{\mathrm{anh}}$. It was shown that $F_{\mathrm{bond}}$ is of opposite sign and larger in magnitude than $F_{\mathrm{anh}}$ at room temperature. The anharmonic contribution increases with temperature and cancels $F_{\mathrm{bond}}$ at about 250°C producing zero intensity. The combination of these measurements and a recent neutron determination of $F_{\mathrm{anh}}$ yielded $F_{\mathrm{bond}}=0.076\mathrm{}\pm 0.006$ at 25°C. This structure factor is the Fourier transform of the valence-electron distribution and like the (222) structure factor, provides a sensitive check on the accuracy of the wave functions determined from band calculations. The values of $F_{\mathrm{bond}}$ show a tendency to be temperature independent. However, rather large experimental errors due to the very low intensities prevented conclusions about the temperature dependence of $F_{\mathrm{bond}}$.