Theory of the Diffuse Scattering of X-Rays by Crystals in the Region of theKCritical Absorption Wave-Length

The scattering of x-rays according to the classical theory from a bound electron of natural frequency ${\nu }_q$ is first considered. Then $Z$ electrons with various ${\nu }_q$'s are assembled into an atom and the scattering from the atom studied. The scattered rays are separated into coherent and incoherent parts by a study of the interference between the scattered rays from two similar atoms. From this $S_{\mathrm{coh}}$ and $S_{\mathrm{incoh}}$ are obtained for the atom. Kramers, Kallmann and Mark's idea of virtual oscillators is used so that $S_{\mathrm{coh}}$ and $S_{\mathrm{incoh}}$ are averaged for the distribution of oscillators. The ${\lambda }^3$ absorption law suggests either $2\nu _K^{}{}_{}{}^2\nu _q^{}{}_{}{}^{-3\mathrm{}}d{\nu }_q$ or $4\lambda _K^{}{}_{}{}^{-4\mathrm{}}\lambda _q^{}{}_{}{}^{3}d{\lambda }_q$ as the fraction of oscillators with frequencies between ${\nu }_q$ and ${\nu }_q+d{\nu }_q$ or with wave-lengths between ${\lambda }_q$ and ${\lambda }_q+d{\lambda }_q$. In determining the averages it makes a difference which distribution is used. The theory gives $S_{\mathrm{coh}}={(f-\Delta f)}^2\times \frac{(1-e^{-2M})}{Z}$, where $\Delta f$ is the atomic structure factor decrement and $e^{-2M}$ the Debye-Waller temperature factor. Also the theory gives $S_{\mathrm{incoh}}=1-\left(\frac{1}{Z}\right)\Sigma E_r^{}{}_{}{}^2+Y$, where $Y$ is negligible for $\lambda >{\lambda }_K$ but is quite large for $\lambda <{\lambda }_K$. Its size when $\lambda <{\lambda }_K$ suggests that $Y$ is the classical analog of the energy which goes into the $K$ fluorescent rays. Since these rays are removed by absorption in aluminum in experiments on diffuse scattering, $Y$ is omitted in $S_{\mathrm{incoh}}$. Finally $S=S_{\mathrm{coh}}+\frac{S_{\mathrm{incoh}}}{{\{1+(\frac{h}{\mathrm{mc}}\left) \mathrm{vers}\phi \right\}}^3}$. Hönl's theoretical value of $\Delta f=2.3\mathrm{}$ is compared with McNatt's experimental value 2.5 for Cu $K\alpha$ x-rays scattered by zinc.