Fundamentals of the Physics of Solids / 12-The Quantum Theory of Lattice Vibrations

408 12 The Quantum Theory of Lattice Vibrations

for ω > ω0. By evaluating the Jacobian for the new sets of variables, we have

for ω > ω0. In either case the integrand vanishes unless r = |ω − ω0|1/2. Assuming once again that the quadratic approximation for the dispersion curve is valid in a finite neighborhood of the saddle point, integration is performed in the region where

Positive and negative values of x3 contribute equally, therefore it is sufficient to calculate the integral over positive values. To include only points within the sphere of radius R, the following restriction must be imposed on the variable θ:

(12.2.50) for ω > ω0. Because of the delta functions these integrals are also elementary. For frequencies close to the saddle point ω0, i.e., for |ω − ω0| R2, the final result is

The density of states has a kink at ω0, with a square-root singularity on one side.

412 12 The Quantum Theory of Lattice Vibrations

therefore Tm = 0. The finite-temperature instability of the symmetry-breaking state is due to the excessively high number of soft Goldstone modes – acoustic phonons in our case – that are present even at very low temperatures and that disrupt the crystalline order.

Let us now calculate the mean square displacement in a three-dimensional crystal at finite temperature. Using the density of states (12.2.19) determined for the Debye model (with p = 1 for the monatomic basis),

Instead of the Debye frequency ωD the Debye temperature ΘD defined via

is commonly used. Changing to the variable t = β ω,

When only the first-order temperature correction to the zero-point vibrations is taken into account at low temperatures (T ΘD),

Above the Debye temperature the integral can be evaluated using the expansion valid for t < 1,

It seems reasonable to assume that the crystal melts when the root-mean- square displacement becomes comparable to the lattice constant a – that is, at the melting point Tm

where α should be less than, but on the order of unity. Using the volume v of the primitive cell instead of the lattice constant,

Experimental data are in good agreement with the above relation if α ≈ 1/4. This is the Lindemann criterion for melting,7 even though it was formulated somewhat di erently before the advent of the Debye model.

12.3.2 The Specific Heat of the Phonon Gas

We shall now determine the contribution of atomic vibrations to the specific heat of solids. The previous expression for the thermal energy, (12.1.12), is expected to be modified by zero-point vibrations as

(12.3.15) To demonstrate this, we shall make use of a statistical mechanical formula for the internal energy of a system with discrete energy levels Ei:

It is a simple matter to evaluate the sum on the right hand side, the partition function

i

Writing the energy of the phonon system as a sum of individual modes,

where ni(q, λ) can be any nonnegative integer. Then

Z = ? e−βEi (q,λ)

q,λ i

?

=e−β ωλ(q)/2 + e−3β ωλ (q)/2 + e−5β ωλ(q)/2 + . . .

7 F. A. Lindemann, 1910.

414 12 The Quantum Theory of Lattice Vibrations

From this formula the above expression for the internal energy is indeed recovered. Alternatively, the internal energy of the phonon gas can be expressed in terms of its Helmholtz free energy

which is equivalent to (12.3.16). In terms of the density of states, the Helmholtz free energy and the internal energy read

Let us return to form (12.3.15) of the internal energy. At high temperatures the condition β ωλ(q) = ωλ(q)/kBT 1 holds for all frequencies of the spectrum, and so it is su cient to keep the first two terms in the series expansion of coth x given in (3.2.83):

When the specific heat is calculated from this expression, the leading term gives the classical Dulong–Petit value. Its first correction would be of order 1/T 2, however this is suppressed by other contributions that are neglected in the harmonic approximation.

The frequency of acoustic phonons usually ranges from zero to order 1012 Hz. As the phonon density of states is proportional to ω2, acoustic phonons of frequency 1012 Hz can be considered as typical. Even for optical phonons the frequency is at most 1013 Hz – so typical phonon energies are between 10 and 100 meV. This also means that the Dulong–Petit law is expected to be valid only at temperatures well above 100 K. At lower temperatures energy and specific heat can be determined only numerically from the above expressions, and only when the spectrum is known. The energy can be

q,λ

Expressing the sum over the phonon quantum numbers by the density of states,

From the previous expressions for the energy, the specific heat is

At low temperatures, where practically only low-energy acoustic phonons (which belong to the linear regime of the dispersion curve) are excited, the Debye model provides a good approximation. Using the Debye model density of states in (12.3.26),

Besides the factor p, which specifies the number of atoms in the basis, all information about the material characteristics of the crystal is contained in the parameter ωD. Using the Debye temperature ΘD instead,

0

416 12 The Quantum Theory of Lattice Vibrations

Introducing the Debye function through the definition

A simple integration by parts shows that this is equivalent to the expression

When the condition T ΘD is met, the energy and the specific heat contain the asymptotic form of the Debye function valid for large values of x. According to (C.2.14),

In the low-temperature region the temperature dependence of the specific heat is usually well approximated by the cubic term. When CV /T is plotted against T 2, as in Fig. 12.8, a straight line can be fit fairly well to the measured data.

While a pure T 3 temperature dependence is observed in ionic crystals that are electrical insulators, the finite intercept for metals indicates that their specific heat contains an additional term which is linear in T . We shall see in Chapter 16 (Volume 2) that this is due to the electrons that are not bound to the ion cores and move almost freely.

On the other hand, it follows from (C.2.13), the expansion of the Debye function valid for small values of x, that at high temperatures the Dulong– Petit value is recovered from (12.3.35). Since (12.3.34) and (12.3.35) give good approximations for the specific heat both in the lowand high-temperature regions, the numerical evaluation of the integral in the Debye function may give an interpolation formula that is valid in the intermediate region as well. As indicated in Fig. 12.9, experimental data for the temperature dependence of specific heat show remarkable agreement with this interpolation formula.