## Abstract

The thermodynamic properties of H$_2$O ice and D$_2$O ice have been analysed in terms of the effective harmonic spectrum of lattice vibrations. In this approximation the vibrations are treated as harmonic, but with frequencies which are both volume and temperature dependent. Analysis of the heat capacity then gives properties of the effective spectrum at 0 $^\circ$K, and the average temperature dependence of the spectrum at constant volume. It has been shown that the parts of the spectrum associated with translational and librational motion of the molecules are distract, and their contributions to the thermodynamic properties may be separated. The moments $\overset{\bar}\nu^{-2}$, $\overset{\bar}\nu^{-1}$, $\overset{\bar}\nu$~, $\overset{\bar}\nu^2$, $\overset{\bar}\nu^4}$ and $\overset{\bar}\nu^6$ together with the geometric mean frequency vg, have been derived for the separate parts of the spectrum. The reduced librational moments, ($\overset{\bar}\nu^n_L$)$^{1/n}$~ are independent of ~ within the experimental uncertainty. This shows that the spectrum of librational frequencies consists of a fairly sharp and symmetrical band. From the very low temperature data the coefficients of the terms $\nu^2$, $\nu^4$,and $\nu^6$ in the low frequency expansion of the translational spectrum have been derived, together with the position and approximate weight of the first peak in the spectrum. The ratios of the moments for H$_2$O and D$_2$O show that the intermolecular forces are stronger for $D_2$O than for H$_2$O. From the thermal expansion data, by analysing the Gruneisen function $\gamma$(T, V) = $\beta$~V/$\chi_s$C~$_p$, values for some of the parameters $\gamma$(~n) = -(1/~n)dln$\overset{\bar}\nu^n$~/dln V, which describe the volume dependence of the moments, have been obtained for the separate parts of the spectrum. These are useful in estimating the volume dependence of various crystal properties. The properties of the effective harmonic spectrum and its average total temperature dependence (including thermal expansion effects) are in good agreement with the results of a variety of spectroscopic experiments. The moments derived from the thermodynamic data have been used to calculate the mean square amplitudes of vibration of O, H and D atoms and the results are m good agreement with the results of X-ray and neutron scattering experiments.