Temperature Dependence of the Structure of Liquid Indium

Abstract

The x-ray diffraction pattern of liquid indium were measured at 170, 280, 390, 500, and 650\ifmmode^\circ\else\textdegree\fi{}C. For comparison, liquid mercury was measured at room temperature. All data were taken with a $\ensuremath{\theta}\ensuremath{-}\ensuremath{\theta}$ diffractometer from the open surface of the melt between values of $K=4\ensuremath{\pi}\frac{sin\ensuremath{\theta}}{\ensuremath{\lambda}}=1.5 \mathrm{and} 15$ ${\mathrm{\AA{}}}^{\ensuremath{-}1}$. Absolute intensity data ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$ were obtained by scaling the measured intensity of In to that of liquid mercury (coh=coherent, e.u. =electron units). The values of ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$ for $K>12$ did not show extensive modulation and were in good agreement with the square of the dispersion-corrected scattering factor $f$ of In. The interference function $I(K)$ was calculated by dividing the ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$ values by the theoretical ${f}^{2}$ values. Fourier transform of $I(K)$ yielded the radial distribution function $\mathrm{RDF}=4\ensuremath{\pi}{r}^{2}\ensuremath{\rho}(r)$ and pair probability function $g(r)=\frac{\ensuremath{\rho}(r)}{{\ensuremath{\rho}}_{0}}$, where ${\ensuremath{\rho}}_{0}$ is the atomic density. The RDF curve of Hg is completely free of ripples below $r<D$, where $D$ is the hard-sphere diameter, indicating that ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$ and ${f}^{2}$ were determined accurately. In the case of In, ripples were found below the first peak in the RDF. We conclude that these ripples are a consequence of the use of inappropriate ${f}^{2}$ values rather than errors in ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$, since Hg and In were measured under identical conditions. Fourier transform of the ripple-free RDF yielded an $I(K)$ curve which was about 10% higher in the region of the first peak. Dividing ${{I}_{\mathrm{e}.\mathrm{u}.}}^{\mathrm{coh}}$ by the corrected $I(K)$ leads to values of the scattering factor which are 5% lower in the range of $K=1.5 \mathrm{to} 8$ ${\mathrm{\AA{}}}^{\ensuremath{-}1}$ than the Dirac-Slater scattering factors. The distribution of the atoms in liquid In can be approximately described by the hard-sphere model with a packing density of 0.45 compared to 0.74 in the solid. This density corresponds to a hard-sphere diameter $D=2.86$ \AA{}, which is the first value of $r$ in the RDF where $\ensuremath{\rho}(r)={\ensuremath{\rho}}_{0}$. The interatomic distances ${r}_{1}$ taken as the position of the first peak maximum in the RDF and the coordination number CN decrease with increasing temperature. Both variations are a consequence of the excess or free volume created in the liquid. The electrical resistivity ${\ensuremath{\rho}}_{R}$ and thermoelectric power $Q$ of liquid In were calculated from the measured $\mathrm{I}(K)$ and the theoretical values of the Fourier transform $U(K)$ of the pseudopotential for different temperatures. The predicted values of ${\ensuremath{\rho}}_{R}$ are about 50% lower than those observed experimentally. The theory also under-estimates the temperature dependence of the resistivity by about a factor of 3.