Specific Heat of an Ordered and a Disordered Crystal

The specific heat at constant volume, $C_v$, for a two-component cubic crystal was calculated as a function of temperature assuming two extreme conditions to exist—that the crystal was (1) ordered and (2) totally disordered. These results were computed using 18 moments of the frequency spectrum which have been computed for both states of order. The parameters used in this calculation correspond to Debye temperatures ${\Theta }_D\left(\mathrm{order}\right)=143\mathrm{}$°K and ${\Theta }_D\left(\mathrm{disorder}\right)=167\mathrm{}$°K. The results indicate that measurable differences do exist in the specific heat for these two states of order above 20°K; in particular $C_v$ (disorder) exceeds $C_v$ (order) by 3.4% at 40°K and by 18% at 20°K. Three methods of determining $C_v$—the $T^3$ law for order only, Thirring's high-temperature expansion, and Padé approximants relating to Thirring's expansion for order and disorder—are discussed and used in our calculations. Some evidence is shown that the assumption of independence of forces for nearest neighbors is valid for Au${\mathrm{Cu}}_3$.