Atomic Structure and Vibrations in Zinc Crystals. VI. Determination of Electron Asymmetry and the Two Principal Characteristic Temperatures

The results reported in papers by Brindley, Miller and the two previous papers of this issue of the Physical Review are discussed. The results show that the true atomic structure factor $f$ (the effect of atomic vibrations having been removed) is a function of the orientation angle $\psi$ out to $\frac{\left(sin \frac{1}{2}\phi \right)}{\lambda }=0.6\mathrm{}$. It is shown how the $f$ values may be obtained for two values of $\psi$ from diffuse scattering measurements alone at room temperature. The $f$ curves for $\psi ={14}^{\circ } \mathrm{and} {90}^{\circ }$ coincide for $\frac{\left(sin \frac{1}{2}\phi \right)}{\lambda }$ between 0.6 and 1.0. At 0.6 the curves fork. Correcting Miller's $F$ values ($\lambda =0.71\mathrm{}\mathrm{A}$) for atomic vibrations and Brindley's $F$ values ($\lambda =1.54\mathrm{}\mathrm{A}$) for atomic vibrations and dispersion, $f$ points are obtained which mostly fall between or on the $f$ curves for $\psi ={14}^{\circ }$ (this is practically the same as the $f$ curve for $\psi =0^{\circ }$) and $\psi ={90}^{\circ }$. It is shown that the difference between the two $f$ curves is due to a reasonable asymmetry in the spatial distribution function of the valence electrons. The $\Delta F$'s between Miller's and Brindley's $F$ values are corrected for atomic vibrations, and the resultant $\Delta f$'s are all equal within experimental error. The average $\Delta f=2.35\mathrm{}$ in good agreement with the quantum mechanical $\Delta f=2.36\mathrm{}$. The characteristic temperatures associated with vibrations parallel and perpendicular to the principal axis have been found. The characteristic temperature for parallel vibrations decreases while that for perpendicular vibrations remains almost constant with rise of temperature in agreement with Zener's theory. The average characteristic temperature for vibrations in all directions is just slightly greater than the characteristic temperature for specific heat in agreement with Zener's theory. The decrease of the characteristic temperature with temperature appears to begin when the root mean square displacement exceeds about 0.13A.