Temperature-Dependence of the Forbidden (222) Reflection in Silicon

X-ray crystallography has now been used for over a hundred years to study the atomic and molecular structure of crystals. It has been responsible for 28 Nobel Prizes, including the determination of the structure of DNA and the discovery of quasicrystals. It is based on Bragg scattering, which consists of constructive interference when x-rays with particular wavelengths and propagation directions reflect on certain crystal planes. 

In the case of silicon, which has the diamond structure, the two atoms in the primitive cell reduce the symmetry and the (222) reflection (allowed by lattice translational symmetry), is nominally forbidden. However, there is not total interference because of tetrahedral rather than spherical symmetry. Such asymmetry arises from anharmonic vibrations and the bond charge distribution. Therefore, the weakly allowed (222) X-ray reflection in silicon is useful for studying bond charge. Previous theories have been somewhat ad hoc, not dealing fully with electron-phonon induced valence charge density thermal shifts. Our formulation of this shift uses full second-order electron-phonon perturbation theory. This work serves as a stepping stone to study the change of density and wavefunctions with temperature in the pyroelectric effect (change of polarization with temperature), which has a wide range of applications.