Intensities of X-Ray Reflections from Bismuth Crystals Between 25° and 530° Abs.

The temperature variation of the "integrated intensities" of reflections of $\mathrm{Mo} K\alpha$ radiation from Bi crystals on (111) are reported over a range from near the boiling point of hydrogen to the melting point of the crystal. A modified Bragg spectrometer with stationary ionization chamber was used. The cryostat in which the crystal is mounted for observations at low temperatures is described as well as the thermostatic arrangement used for high temperatures. The observations were taken for the second, third and fifth order. The relative values of the integrated intensities ($\frac{J_T}{J_N}$) were found to follow the Debye-Waller relation: $J_T=J_0·e^{-2M·{sin}^{2}v}$ for $\Theta \gtrless T$ within the range of the experimental error. Also the ${sin}^{2}v$-relation was found to hold well for the three orders used, thus the reduction $\ln \frac{\left(\frac{J_T}{J_0}\right)}{{sin}^{2}v}=f\left(T\right)\mathrm{}$ could serve to determine the Debye factor in first approximation. As $f\left(T\right)\mathrm{}$ is almost linear, $\frac{M^{\prime }}{T}$ (the slope of the line) was determined to be 1.15.${10}^{-2\mathrm{}}$. By an approximation method the characteristic temperatures of Bi were calculated without ($\Theta$) and with (${\Theta }^{*}$) consideration of the zeropoint energy. The values obtained are: $\Theta ={92.5}^{\circ }$ and ${\Theta }^{*}={95.9}^{\circ }$ (abs.). Although the scattering of the observations is of nearly the same order of magnitude as the difference between the Debye function with and without zero-point energy, the deviations from the former are smaller and our measurements thus support the assumption of the zero point energy.