Temperature dependence of elastic constants

8.1. Normal temperature dependence, caused by anharmonicity Wachtman et al. (1961) (see also Durand 1936) noted empirically that the temperature dependence of Young's modulus E of several oxides could be well fitted to

E(T) = [1 - bTexp(-T0/T)]E(0). (4.24)

An expansion of the exponential term for T 3> TQ gives, to leading order,

E(T)^[l-b(T-T₀)]E(0). (4.25)

Ch. 4. What values do the elastic constants take?

35

30

Q_ O 25

c? 20

15

0 200 400 600 800 1000 T (K)

Fig. 4.5. Temperature dependence of C44 in Al (solid curve), with data from Every and McCurdy (1992). The dashed line is an extrapolation to low T of the low-order anharmonic correction. Its intercept with the vertical axis gives the value of C44 in a

"harmonic" system. The measured C44 is lower, because of the zero-point vibrations.

This temperature dependence is observed for many systems other than oxides, and for elastic constants other than E (see, for instance, fig.

4.5). Anderson (1966a) has shown how eq. (4.24) can be understood from the quasi-harmonic model of lattice vibrations. We shall partly follow that work but also make contact with the treatment in Chapter 8, of anharmonic lattice vibrations. The elastic constants can be related to the long-wavelength limit of phonons. Anharmonicity shifts a phonon frequency co(q, s) by an amount Aco(q, s). In an Einstein model, we may write for shifts Aco relative to co at T = 0, (eq. (8.27)),

Aft>(q,.s) = fc(q, s)

co(q,s) e x p ^ / D - l * ⁽ -^{2 6 )} Here q is the wave vector of the phonon, s is a mode index (longitudi-nal or transverse modes etc.), fc(q, s) is a dimensionless proportio(longitudi-nality constant and 6>_E is an Einstein temperature characteristic of the entire phonon spectrum. Let Y be an elastic constant. A dimensionally correct relation between Y and co is (Y/p)^l/2 = co/\q\. The mass density p varies as l/V and |q| as V~^1/3, where V is the sample volume (per mole etc.). Within our simple model, the thermal expansion V(T)-V(0) has the same temperature dependence as Aco/co. (In fact, Aco is to a large extent due to the thermal expansion.) Thus, an expression of the type (co/\q\)p^l/2 has a temperature-dependent shift which varies with T

Temperature dependence of elastic constants 59

as [exp(6E/T) — I ]^{- 1}- In our model the same temperature dependence enters all (small) shifts in the elastic constants. We summarise this by the relation

Y V 0 e / e x p ( 0 E / r ) - r

where ay is a dimensionless proportionality factor varying with the elastic parameter Y under consideration. Comparing the high tem-perature expansions Texp(—T0/T) = T — T0 + T^/IT — ••• and 9E[exp(9E/T) - I ] "¹ = T - 9E/2 + (1/12)0²/T + • • •, we note that the two leading terms are identical if T0 = 0E/2. This gives a theoretical justification for the empirical rule (eq. (4.25)) at intermediate and high temperatures, and for the fact that T₀ was observed to be about 1/3 of the Debye temperature. Since the Debye temperature is less than 500 K for most solids, we also understand why the linear temperature depen-dence expressed by the series expansion (eq. (4.25)) is such a good approximation at ambient and higher temperatures. (Close to the melt-ing temperature, high-order anharmonic effects usually give a stronger temperature dependence than linear in 7\ cf. Chapter 8, §6.)

Our shifts A&>(q, s) were taken relative to T = 0 K. In fact, there is a shift even at T = 0, due to the anharmonicity related to the zero-point vibrations. The intercept of the linear portion of ctj (T) with the Q; -axis at T - 0 K gives the "harmonic" ctj (see Appendix E and fig. 4.5).

The temperature-dependent part of the energy U (or enthalpy / / , Helmholtz energy F, Gibbs energy G) of an insulator varies as T⁴ at low T (since Cp and Cv ~ T³). From the fundamental relations of the elastic coefficients expressed as derivatives of [/,//, F or G (Chapter 3, §3), it follows that dY/dT ~ T³ at low T. The Einstein model used above to account for the temperature-dependent factor in the empirical relation (eq. (4.24)) gives too rapid (exponential) a temperature depen-dence at low T. However, the absolute magnitude of the shift A Y at these temperatures is so small that this discrepancy is of little practical importance. In metals, £/, / / , F and G vary as T² due to the excitation of electron states, but this term is important only at such low T that it is of no interest in the present context.

B c

C/>

c o .o o iS to

temperature

Fig. 4.6. A schematic illustration of an anomalous temperature dependence of an elastic coefficient, caused by features in the electron band structure close to the Fermi level.

The smooth dashed curve shows the extrapolation to low T that would result if an-harmonic effects is the only cause of a temperature dependence. The hatched parts correspond to the additional electronic contribution. It may increase or decrease the

elastic coefficient.

8.2. Anomalous temperature dependence, caused by electronic structure

In some cases, one or several of the elastic coefficients c exhibit a marked temperature dependence also at relatively low temperatures, often below room temperature. Figure 4.6 shows schematically such a behaviour. The smooth decrease with increasing temperature, shown as a dashed curve for low T and a solid curve for high 7¹, is the normal variation of c due to anharmonic effects. Superimposed is a variation illustrated by the shaded area, leading to a low-temperature dependence of c on T as given by the solid curve. Although the "anomalous" contri-bution at low T is not yet fully understood, it is naturally explained by features in the electron band structure in the immediate vicinity of the Fermi level, which can be reached by thermal excitations. The tempera-ture dependence then has its roots in Fermi-Dirac statistical factors for the electrons. The "normal" variation in the elastic constants at high 7\

related to anharmonicity, has its roots in the increased vibrational dis-placement, i.e. in the Bose-Einstein factors for the phonons. (The most important result of anharmonicity is to cause thermal expansion, which affects the elastic constants through their volume dependence.) There is a close connection between an anomalous temperature dependence of c^ as discussed here, and a dependence of q7 on the composition of certain alloys (see §11).

Dependence on lattice structure and order 61

9. Dependence on lattice structure and order