On the evaluation of equivalent Debye temperatures and related problems

Abstract

The heat capacity, at constant (0 degrees K) volume, of face-centred cubic crystals is evaluated by a modified Houston (1948) approximation. There are four principal changes. (a) Instead of three directions in the reciprocal lattice, we use five, namely, the (100), (110), (111), (210) and (211) directions. (b) We replace the first Brillouin zone by a sphere of equal volume. In this way we avoid the difficulty, at low and medium temperatures, of the appearance of the characteristic normalization factor in Houston's method. At the same time we overcome the difficulty that the latter is only applicable to the integration of functions defined over a sphere. (c) We do not concern ourselves with the frequency spectrum. This is due to a change from integration over frequency to integration over wave-number in the expression for C$_{v}$. The reason for the change is that in this way we obtain integrands, in the five directions, that contain no singularities, vary smoothly, and are convenient for numerical evaluation on a digital computer. (d) At higher temperatures we give an expression for C$_{v}$ in terms of the first three even moments that we prove is in error by less than 0$\cdot $1%. We are able to ensure this accuracy by fitting an even polynomial to the Einstein function over the range 0 $\leq \hslash \omega $/kT $\leq 3$ to an accuracy better than 0$\cdot $1%. Where they overlap the $\Theta $(T) curve obtained using the three modifications described earlier lies about 1% below that obtained by the polynomial method. A correction is then applied to eliminate this discrepancy by allowing for the error introduced by replacing the first Brillouin zone by a sphere. The possible error in the final $\Theta $(T) curves is shown to be well below 1% for the whole temperature range. We show that the use of more than five directions is not required by comparing the results for five directions with those for six; (221) is the extra direction. The results differ by much less than 1%. A comparison is also made with earlier work using the conventional three- and five-direction Houston approximations and with the results of Leighton (1948) and Salter (1956). The change to wave-vector integration provides a very convenient technique for passing directly from the dispersion relations, which can be obtained experimentally without reference to a particular model, to various quantities containing averages over the normal mode distribution. The polynomial method differs from that introduced by Thirring (1913) who expanded the Einstein function in a Maclaurin series. We discuss both methods and find that Thirring's is more suitable for $\Theta $, while the polynomial method is more suitable for C$_{v}$. Our procedure is applied, by way of illustration, to five representative face-centred cubic metals, Al, Ag, Au, Cu and Pb. We use the nearest-neighbour non-central force theory of Begbie & Born (1947) and the nearest- and next-nearest-neighbour central force theory previously used by Leighton (1948). At each temperature we use the force constants derived from the elastic constants, measured experimentally at that temperature, by the method of long waves. To this extent, at least, we do take account of anharmonic effects. It is found, as was to be expected, that neither theory can quantitatively account for the measured specific heats of the metals studied. Our method may be used in the discussion of related problems such as the thermal expansion of crystals and the thermal conductivity of insulators.