International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.3, pp. 8991
Section 1.3.5. Pressure dependence and temperature dependence of the elastic constants^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France, and ^{b}Laboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France 
In a solid, the elastic constants are temperature and pressure dependent. As examples, the temperature dependence of the elastic stiffnesses of an aluminium single crystal within its stability domain (the melting point is 933 K) and the pressure dependence of the elastic stiffnesses of the ternary compound KZnF_{3} within its stability domain (the crystal becomes unstable for a hydrostatic pressure of about 20 GPa) are shown in Figs. 1.3.5.1 and 1.3.5.2, respectively.

Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Every & McCurdy, 1992). 

Pressure dependence of the elastic stiffness of a KZnF_{3} crystal. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980). Copyright (1980) IEEE. 
We can observe the following trends, which are general for stable crystals:
These observations can be quantitatively justified on the basis of an equation of state of a solid: where represents the stress tensor, the strain tensor, X the position of the elementary elements of the solid and Θ the temperature.
Different equations of state of solids have been proposed. They correspond to different degrees of approximation that can only be discussed and understood in a microscopic theory of lattice dynamics. The different steps in the development of lattice dynamics, the Einstein model, the Debye model and the Grüneisen model, will be presented in Section 2.1.2.7 . Concerning the temperature and the pressure dependences of the elastic constants, we may notice that rather sophisticated models are needed to describe correctly the general trends mentioned above:
Table 1.3.5.1 gives typical values of for some cubic crystals considered within their stability domain. In column 6, the `elastic Debye temperature' of the crystal, , has been calculated according to the formula where h is the Planck constant, is the Boltzmann constant, v is an average velocity (see for instance De Launay, 1956) and n is the number of atoms per unit volume.

It is interesting to compare , the `elastic Debye temperature', with , the `calorimetric Debye temperature'. The definition of will be given in Section 2.1.2.7 . It results from the attempt at founding a universal description for the thermal properties of solids when the temperature is expressed as a reduced temperature, ; is obtained from calorimetric measurements at low temperature. It is worth noting that accurate values of lowtemperature elastic constants and lowtemperature calorimetric measurements lead to an excellent agreement between and [better than 2 or 3 K (De Launay, 1956)]. This agreement demonstrates the validity of the Debye model in the vicinity of 0 K. From Table 1.3.5.1, we can observe that for ionic crystals is, in general, greater than . This remark is not valid for covalent and metallic crystals. Typical orders of magnitude are given in Table 1.3.5.2. These statements concern only general trends valid for stable crystals.

In the case of temperatureinduced phase transitions, some elastic constants are softened in the vicinity and sometimes far from the critical temperature. As an example, Fig. 1.3.5.3 shows the temperature dependence of in RbCdF_{3}, CsCdF_{3} and TlCdF_{3} single crystals. RbCdF_{3} and TlCdF_{3} undergo structural phase transitions at 124 and 191 K, respectively, while CsCdF_{3} remains stable in this temperature range. The softening of when the temperature decreases starts more than 100 K before the critical temperature, . In contrast, Fig. 1.3.5.4 shows the temperature dependence of in KNiF_{3}, a crystal that undergoes a para–antiferromagnetic phase transition at 246 K; the coupling between the elastic and the magnetic energy is weak, consequently decreases abruptly only a few degrees before the critical temperature. We can generalize this observation and state that the softening of an elastic constant occurs over a large domain of temperature when this constant is the order parameter or is strongly coupled to the order parameter of the transformation; for instance, in the cooperative Jahn–Teller phase transition in DyVO_{4}, is the soft acoustic phonon mode leading to the phase transition and this parameter anticipates the phase transition 300 K before it occurs (Fig. 1.3.5.5).
As mentioned above, anharmonic potentials are needed to explain the stress dependence of the elastic constants of a crystal. Thus, if the strainenergy density is developed in a polynomial in terms of the strain, only the first and the second elastic constants are used in linear elasticity (harmonic potentials), whereas higherorder elastic constants are also needed for nonlinear elasticity (anharmonic potentials).
Concerning the pressure dependence of the elastic constants (nonlinear elastic effect), considerable attention has been paid to their experimental determination since they are a unique source of significant information in many fields:
References
De Launay, J. (1956). The theory of specific heats and lattice vibrations. Solid State Physics, Vol. 2, edited by F. Seitz & D. Turnbull, pp. 219–303. New York: Academic Press.Fischer, M. (1982). Third and fourthorder elastic constants of fluoperovskites CsCdF_{3}, TlCdF_{3}, RbCdF_{3}, RbCaF_{3}. J. Phys. Chem. Solids, 43, 673–682.