Debye spectrum

Only a few years after the Einstein model, Debye (1912) introduced his famous model. It is widely used, and sometimes misused. We therefore give a detailed treatment, and comment in several later chapters on its virtues and shortcomings. Here it suffices to note that concepts such as Debye temperatures can be generalised and given useful and precise meanings that go far beyond the simple original ideas described here.

In the long-wavelength limit, i.e. for small q, we can write

Q)(q,s) = C ( q , s ) | q | . (6.8) C(q, s) is a directional-dependent velocity, defined for the three s-values corresponding to the acoustic branches. The simplest Debye model (Debye 1912) assumes a constant value C(q, s) = CS0Und,D for all (q, s) with &>(q, s) linear in |q| for all wave numbers |q| < qo.

The maximum frequency (the Debye frequency) in the model is cov = C_Sound,D#D. (In this book C_S0Und,D denotes the sound velocity in the De-bye model, and CD denotes the heat capacity in the DeDe-bye model.) From the general expression (6.5) we get the Debye density of states

FD(co) = 3V 4nq² (2n)³N Csound,D

3Vco²

q=co/C_sound,D ⁿ ^^sound,D

(6.9)

Debye spectrum 85

The value of the Debye wave number gD is fixed by the normalisation condition

/ FD(a))dco = 3. (6.10)

Jo

This yields

qD = (6n²N/V)^l/3 = (6jr²/^a)^{1 / 3}. (6.11) In applications to thermophysical properties, it is convenient to

intro-duce the Debye temperature 0_D as a measure of the maximum frequency

&>D. The two parameters are related by

hcoD = kB0D. (6.12)

In a real solid C(q, s) is anisotropic, and different for the longitudinal and the transverse acoustic branches. We write C(q, s) = CS(0, (/)) and dS = q² dQ = [co²/C²(0, (p)]dQ where (0, 0) are angular coordinates for q and dfi = sin 9 dO d0. Equation (6.5) becomes

3

V C

^ W y . f 1 dQ

S = \ ^A

This agrees with eq. (6.9) if we define the Debye sound velocity CSOund D by

3 _ A f 1 dQ

cl^-j-JcW^)^- ⁽⁶ ' ¹⁴⁾

If C_s is isotropic, but different, for the longitudinal (L) and the two degenerate transverse (T) branches, one has

3 1 2

- = —,+—• (6-15) c⁵ c⁵ c⁵

^ sound, D ^ L ^ T

Ch. 6. The phonon spectrum

The Debye temperature #D can now be expressed as h (6n²N\^ip

?D =

V

,2.XT „ \ V 3

^ sound, D

~ fe {-^^)

-

Here r is the number of atoms in a molecule (r = 1 for an element, 2 for NaCl, 5 for AI2O3), M is the mass of a mole of the material, Np, is Avogadro's number, and p is the mass density of the material. Since

&>D = Csound,D?D = CSOund,D(67r²A7 V)1/3 w e c a n a

l^{s o w r}i^{t e} the Debye density of states (eq. (6.9)) as

F_D(<») = -f^ = — . (6.17)

The Debye model assumes that o>(q, s) = C_s(9, 0)|q| is linear in |q| for all wave numbers. The linearity always holds in the small-|q| limit, for the acoustic phonon branches of any solid, but with increasing |q| there will be deviations from this simple relation. With the inclusion of only the first correction term, we can write

o>(q, s) = ci(0, 0, s)\q\ + c₂(0, 0, *)|q|². (6.18) Then one may prove that F(oo) has the form

F{cS) = aX(D² + a2co⁴ + • • •, (6.19)

that is, F{QJ>) only contains even powers of co, for small co. For larger co there are of course drastic deviations from such a power law, as is illustrated later in this chapter.

Finally some common misconceptions should be clarified. The cut-off frequency co_D is the highest frequency in the Debye model spectrum.

But this does not mean that &>D is the highest frequency in the actual spectrum of a solid (cf. fig. 6.5). The cut-off frequency follows from the normalisation condition (6.4), and the connection to the actual spectrum is through the sound velocities, i.e. through the low-frequency part of F(co). Since the sound velocities are determined by the elastic proper-ties, one sometimes refers to the low co part as the elastic limit. It may

Debye spectrum 87

0 5 10 15 20 v (THz)

Fig. 6.5. The phonon density of states F(y) for TiC, based on neutron scattering experiments, and two Debye model representations of F(v)\ co = 2ixv.

also be called the long-wavelength part, because the wavelength of an elastic wave is k(q, s) = 27t/\q\ = 2nC(q, s)/co(q, s). Sound velocities vary as p~^{1 / 2} where p is the mass density, while the highest frequency in a solid with different masses tends to vary as M~^1/2 where M is the lightest atomic mass. In solids with two or more kinds of atoms, and with large mass ratios, the Debye cut-off frequency &>D, therefore, may be significantly lower than the highest frequency.

Another confusion may arise in the description of the low tempera-ture heat capacity of solids characterised by a phonon density of states that has clearly separated acoustic and optical parts, as in the example of TiC below. The Debye model heat capacity at very low T can be written C_D = Nk_B(l2n⁴/5)(T/e_D)³ (eq. (7.29)). Since only the low frequency phonons (the acoustic branches) contribute significantly to the heat capacity at low 7\ one could get a very good fit to experiments by modelling only that part. But one must now be careful with the normalisation condition for F(co). Comparison of the expression (7.29) for the heat capacity C_D with eq. (6.16) for the Debye temperature

#D yields N = NAr, when f F(co)da) = 3. If only the acoustic part of F(co) is considered, we should take r = 1 in eq. (6.16). The two approaches yield identical CD at low T since r cancels in N/6^, but

the proper r ( > l ) must be used when the model is considered at higher temperatures.

Example: F{co) and a Debye model for TiC. Figure 6.5 shows the phonon density of states F(co) obtained from neutron scattering experi-ments (Pintschovius et al. 1978) and two Debye model representations.

The Debye models have the same ^-dependence at low co, and therefore equally well account for the low temperature heat capacity. However, the one with the lower cut-off &>D = 27r VD assumes one atom per primi-tive cell, while the model with the higher cut-off correctly assumes two atoms per primitive cell in TiC. Only the latter model gives a reasonable description of the heat capacity at high temperatures. Note that &>D is not exactly equal to the highest frequency co in the true F(co).

Example: Debye temperature from approximate sound velocities. From a knowledge of the elastic coefficients ctj, one obtains the velocities C_s(0, 0) as eigenvalues of a secular equation (5.5), and then C_SOUnd,D and #D after a numerical integration over angles 0 and 0 (eq. (6.14)).

A much simpler, but approximate, method is to estimate the bulk mod-ulus K and the shear modmod-ulus G of a polycrystalline material by the Voigt-Reuss-Hill (VRH) approximation (Chapter 18, §3.3), then ob-tain the longitudinal sound velocity from pC[ = K + (4/3)G and the transverse sound velocity from pC\ — G, and apply eqs. (6.15) and (6.16) to get 0D- Anderson (1963, 1965) investigated how well a calculation of CS0Und,D and 6fo by the latter method approximates the exact 0D. For AI2O3 (trigonal lattice) he obtained CS0Und,D = 7190 m/s ("exact" numerical calculation) and C_SOUnd,D = 7093 m/s (Voigt-Reuss-Hill approximation), while for CaCC>3 (orthorhombic lattice) CS0Und,D

= 3942 m/s ("exact") and Csound,D = 3991 m/s (VRH). Calculations on a large number of other systems showed that the typical error in

#D is less than 2% when the approximate method is used. This also means that an accurate value of #D may be obtained from the measured longitudinal and transverse sound velocities of (statistically isotropic) polycrystalline materials, without recourse to the elastic constants Q;

of a single crystal. When the single crystal is isotropic, #D is given exactly by the VRH expression because then Ky = KR = AVRH and GY = GR = ^GVRH- Using the quantity ^AVRH = (Gv - GR)/(GV + GR) as a measure of elastic anisotropy in a single crystal, Anderson (1963) found that when ^AVRH < 0.2, the error in the predicted ^#D is ^ 2 % . In

Frequency moment representations of F(co) 89 Table 6.1

Vibrational properties described by a single parameter Physical property Zero-point velocity (T = 0) Heat capacity, high T

the example in Chapter 5 (§4) it is argued that #D can be well estimated if only G is known.

5. Frequency moment representations of F(co)