Explicit anharmonicity

7LiH/⁷LiD 1

0 200 400 600 T(K)

Fig. 8.2. Effect of isotope mass on the lattice parameter of lithium hydrides.

4. Explicit anharmonicity

The frequency shifts discussed above vanish if the crystal volume, or rather all e,-, remain constant. However, there are also effects that we will refer to as explicitly anharmonic. They arise from higher-than-quadratic terms when the potential energy <E> is expanded in powers of displacements u from the equilibrium positions of the atoms, (eq.

(8.1)). (Due to the zero-point motion they are, to some extent, present also at T = 0.) To see the structure of the associated frequency shifts, it is illuminating first to study an anharmonic one-dimensional oscillator.

Let the mass M move in a potential

V(x) = (l/2)Mco²x² + V3x³ + V4x\ (8.20) where V3 and V4 are in some sense (decreasingly) small. When V3 =

V4 = 0, the energy eigenvalues are En == hco(n + 1/2). Quantum mechanical perturbation theory, applied to the state n and at T = 0, gives energy shifts AEn. In conventional notation, they can be written

, / E*n^f ^n

We have {n | V3X³ \n) = 0 , since in this term the integrand is an odd func-tion of x and the integrafunc-tion is over negative and positive x. In the last term of eq. (8.21) we only keep the lowest-order part of the numerator, i.e. (n\V$x*\n^f). Thus, the term V4*⁴ contributes to the first order (in the first term on the righthand side), while V3*³ contributes to second

I.UUO

O

* 1.006

1.004H

£ 1.002

1 nt\c\

Explicit anharmonicity 145

order in the perturbation expansion. It is necessary to keep both these terms, even though V4X⁴ was assumed to be much smaller than V3X³ in the relevant range of x-values, because the larger term enters only in a higher-order perturbation contribution. We shall use subscript 2 for the quasiharmonic shift (because only terms quadratic in the atomic displacements are kept), while frequency shifts originating from u³ and u⁴ are denoted A 3 and A4. (Other authors may use other conventions for the subscripts of A.) Neglecting damping of the phonons, we now get

o)(q, s) = o>₀(q, s) + A₂(q, s) + A₃(q, s) + A₄(q, s). (8.22) When one adds the explicitly anharmonic shifts A3 and A4 to the qua-siharmonic model, but still neglects damping, one sometimes calls it the pseudoharmonic approximation. (It should be mentioned that some authors refer to eq. (8.22) as the quasiharmonic approximation, but normally "quasi-harmonic" refers only to &>o + A2.)

The shifts A2(q, s), A3(q, s) and A4(q, s) can be written in con-densed form as (e.g. Maradudin and Fein 1962, Cowley 1963, 1968, 1970, Cowley and Cowley 1965, Wallace 1972)

A2(q, s) = (2/h) J^ Va(qs, -qs)ea, (8.23)

a

A₃(q,s) = -(lS/h²) J2 |V(q5,q₁5₁,q₂₅₂)|²_JR(0,l,2),

qi-*l,q2*2

(8.24)

A4(q,s) = (l2/h)^2v(.qs,-qs,qisu-qis\)

qisi

x[2n(l) + l], (8.25) with

Table 8.1

The relative frequency shifts — Atot/^harm(0) and — A&>(Vo)/<^harm(0) ^{n e a r} Tfus-Based on data from Fernandez Guillermet and Grimvall (1991a) (for Mo, W) and Rosen

and Grimvall (1983)

In the Bose-Einstein function w(i), as well as in the interaction function V(/,...), in the quantities A(/) and &>,-, and in the summations, the index i = 0 , 1 , 2 is short for (q, s), (qi, s\) and (q2, £2), respectively. In A₂, the thermal strain sa is to be calculated as in eq. (14.17). Va(0), V(0, 1) and V(0, 1, 2) are short for the Fourier transforms of the interatomic potential. The principal value (in the mathematical sense) should be taken in the sum over the singular terms in eq. (8.26). (In a non-primitive lattice there is an additional term in A4.)

One can show that A3 is always negative. A4 may have either sign but often cancels much of A3. Usually, A2 is much larger than A3 + A4, but see also the elements K, Mo and W in table 8.1. The temperature dependence of A3 and A4 comes from the Bose-Einstein factors n. At high temperatures the factors (n + 1/2), and therefore also A3 and A4, are linear in T (Appendix E). When 7 = 0 there remains the term 1/2 in (n + 1/2), i.e. the contribution from zero-point vibrations. Hence, A3

and A4 are not zero at T = 0 K.

Table 8.1 gives the quantities A_tot/&>harm(0) and Act)(Vb)/^harm(0) near the melting temperature 7>us for some metals. Here &>harm(0) is approximated by the frequency corresponding to the Debye temperature 0D derived from the experimental entropy at 300 K, and corrected for the

Explicit anharmonicity 147

350

g 300 250

200

0 0.2 0.4 0.6 0.8 1.0

T / T

f u s

Fig. 8.3. The entropy Debye temperature 0^(T) of tungsten, at constant pressure and at fixed volume, as derived from thermodynamic data.

electronic contribution to S. The total frequency shift Atot is obtained from 0&(T « T_fus) - e£(T = 300K), and Ao)(V₀) is the frequency shift that results when the thermal data have been reduced to fixed volume

Vo. Within low-order perturbation theory, A_tot = A2 + A3 + A4 and A&>(Vo) = A3 + A4, but close to Tfus there are higher-order anharmonic contributions to Atot and Aco(Vo). As shown in table 8.1, the thermal expansion accounts for almost all of the frequency shifts in Cu, Zn, Al and Pb, i.e. A&>(V0)/&>harm(0) ~ 0. However, in Mo and W there are considerable frequency shifts A(o(Vo) ^al^{s o} when the solid is held at a fixed volume Vo (see also fig. 8.3). In K, the shift Ato(Vb) is moderate in absolute magnitude, but is not much smaller than A2. A closely related consequence is that the vibrational part of the heat capacity Cy at high T is not approximately given by the classical value 3&B per atom. This fact is worth noting, since it is often assumed that Cy is close to 3kB

per atom at high T. Consider, for instance, the three compounds AI2O3, MgO and Mg2Si04. These solids are insulators, and therefore Cy has a contribution from lattice vibrations only. Cy of AI2O3 and MgO ap-proach the Dulong-Petit limit but Cy of Mg2SiC>4 steadily increases above this value as T increases. Wallace (1997) got results similar to those of table 8.1 in an analysis of 25 elements.

Example: temperature dependence of frequency shifts in an Einstein model. The summations in eqs. (8.24) and (8.25) cannot be carried out in a closed form, and one is left with a numerical calculation. However, a simple expression results if we replace co₂ and &>₃ in the Bose-Einstein factors by the same frequency, &>E. If we also take an Einstein

represen-i 1 1 r

j L

tation of the strain sa (cf. eq. (14.17)), including that due to zero-point vibrations, we get

A2(q, s) = k2(q, s)co0(q, s) ( — — + - ) , (8.27)

\exp(ha)_E/k_BT) - 1 2 / with the same expressions for A3 and A4 if £₂(q, s) in eq. (8.27) is re-placed by other functions, k^(q, s) and ^ ( q , s), respectively. Although the temperature dependence of A2, A3 and A4 is the same as in an Einstein model for the thermal energy, our description is not that of an ordinary Einstein model since the (dimensionless) quantities fc,-(q, s) (i = l, 2, 3) may vary with the mode (q, s).

5. Thermodynamic functions in anharmonic systems