Calculations of Surface Thermal Expansion

Using a method previously described, we have calculated the coefficients of thermal expansion for the first two interplanar spacings near the (111) and (100) surfaces of Ar, Kr, and Xe. The bulk thermal expansion, which is obtained as a by-product in the calculation, is found to be in good agreement with experimental measurements at all temperatures up to the melting point (largely because of a cancellation of errors at higher temperatures). This fact provides some confidence in the method and in the results for the surface thermal expansion. At high temperatures, the results for the surface thermal expansion are in agreement with the prediction of an approximate model which we gave earlier, $\frac{{\alpha }_{\mathrm{surface}}}{{\alpha }_{\mathrm{bulk}}}\approx \left(\frac{3}{4}\right)\frac{{〈u_z^{}{}_{}{}^2〉}_{\mathrm{surface}}}{{〈u_z^{}{}_{}{}^2〉}_{\mathrm{bulk}}}$. At low temperatures, $\frac{{\alpha }_{\mathrm{surface}}}{{\alpha }_{\mathrm{bulk}}}$ passes through a rather high peak [with a value of greater than 6 for the (100) surface] because of dispersion of the surface modes. We are not able to give a conclusive explanation of the large apparent discrepancy between our calculations and the experimental observations of Ignatiev and Rhodin, which if taken at face value indicate that $\frac{{\alpha }_{\mathrm{surface}}}{{\alpha }_{\mathrm{bulk}}}$ is greater than twice our result of about 1.9 for the (111) surface of Xe between 55 and 75 °K. However, it is possible that factors other than thermal expansion influence the shifts in the Bragg peaks which are observed experimentally, as has been found to be the case in other attempts to measure surface thermal expansion. A nonkinematical calculation of temperature effects in low-energy-electron diffraction from Xe(111) would be of interest in this regard, and also in regard to apparent discrepancies between theoretical and experimental "effective Debye temperatures" at the lowest energies. Experimental observation of the strong peak in $\frac{{\alpha }_{\mathrm{surface}}}{{\alpha }_{\mathrm{bulk}}}$ would also be of interest; this peak occurs at roughly 6% of the bulk Debye temperature and should therefore be observable in metals or other materials at cryogenic temperatures.