Electronic charge distribution in silicon

Abstract

The structure factors ${F}_{h}$ of eight high-order [0.64\ensuremath{\le}(sin\ensuremath{\theta})/\ensuremath{\lambda}\ensuremath{\le}1.56 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$] reflections were measured for a number of wavelengths to an accuracy of a few thousandths of an electron per atom. Most of these have never been measured before to this level of accuracy. A monolithic double-crystal diffractometer of novel design was employed in measuring thin-crystal Laue-case rocking curves which exhibit fine structure strongly dependent on ${F}_{h}$. Computer fitting of the theoretical curve to the measured one yields ${F}_{h}$. An energy-dispersive mode of operation allowed simultaneous measurements of the rocking curves of a whole family of planes to be carried out. The ${F}_{h}$ values obtained, while in excellent agreement with previously measured and theoretical ones for (sin\ensuremath{\theta})/\ensuremath{\lambda}1 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$, are consistently and increasingly lower than the theoretical relativistic Hartree-Fock ${F}_{h}$ values for (sin\ensuremath{\theta})/\ensuremath{\lambda}>1 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$. This systematic trend most probably reflects the inadequacy of the Debye parameter B calculated from medium- and low-order ${F}_{h}$'s for high-order reflections. Assuming different vibrational amplitudes for intermediate- and inner-shell electrons, our data yield ${B}_{\mathrm{low}=0.4632\ifmmode\pm\else\textpm\fi{}0.0041}$ A${\mathrm{\r{}}}^{2}$ for reflections up to 880 [(sin\ensuremath{\theta})/\ensuremath{\lambda}=1.04 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$], in excellent agreement with previous measurements, and ${B}_{\mathrm{high}=0.5085\ifmmode\pm\else\textpm\fi{}0.0035}$ A${\mathrm{\r{}}}^{2}$ for reflections higher than 10 10 0 [(sin\ensuremath{\theta})/\ensuremath{\lambda}\ensuremath{\ge}1.30 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$]. We also find no anharmonic contribution to the temperature factor of ${F}_{h}$ within the limit of accuracy of the experiment, in good agreement with previous x-ray measurements and some neutron measurements, but in contradiction to some of the neutron forbidden-222-reflection results.