1. Lecture 9 The field of sound waves. Thermodynamics of crystal lattice.
Phonons and second sound.
The Debye model.
The Debye temperature.
Specific heat of the solid in the Debye model.

2. In the same way that the energy of the electromagnetic radiation is quantized in the form of photons so the energy of the elastic waves, or sound waves, inside a solid medium can be considered to be quantized in the form of phonons.
The energy of a phonon of frequency  is again and, because the phonons have integral angular momentum, the assembly of phonons in the solid may again treated as a boson gas.
To illustrate this point, we may consider the Hamiltonian of a classical solid composed of N atoms whose positions in space specified by the coordinates (x1, x2,.…., x3N).

3. and the potential energy by
In the state of the lowest energy, the values of these coordinates may be denoted by
Denoting the displacements of the atoms from their equilibrium positions by the variables i (i=1,2,…3N), the kinetic energy of the system in the configuration (xi) is given by
(9.1)
and the potential energy by

4. (9.2)
The main term in this expansion represents the (minimum) energy of the solid when all the N atoms are at rest at their mean positions ; this energy may be denoted by the symbol 0.
The next set of terms in the expansion is identically equal to zero, because the function (xi) has its minimum value at (xi)=( ) and hence all its derivatives must vanish there. The second-order terms of the expansion represent the harmonic component of the atomic vibrations.

5. If we assume that the overall amplitudes of the atomic vibrations are not very large
we may retain only the harmonic terms of the expansion and neglect all the successive ones; we are then working in the so-called harmonic approximation. Note that the inharmonic components are important at phase transition (from one crystal symmetry to another and solid-liquid phases)
(9.3)
where
(9.4)

6. We now introduce a linear transformation, from the coordinates i to the so-called normal coordinates qi, and choose the transformation matrix in such a way that the new expression for the Hamiltonian does not contain the cross terms, i.e.
(9.5)
where i (i=1,2,..3N) are the characteristic frequencies of the so-called normal modes of the system and are determined essentially by the quantities ij or, in turn, by the nature of the potential energy function (xi).
The expression (9.5) suggests that the energy of the solid, over and above the (minimum) value 0, may be considered as arising from a set of 3N one-dimensional, non interacting, harmonic oscillators, whose characteristic frequencies i are determined by the nature of the interatomic interactions in the system.

7. Classically, each of the 3N normal modes of vibration corresponds to a wave of distortion of the lattice points, i.e a sound wave.
Quantum-mechanically, these modes give rise to quanta, called phonons, in very much the same way as the vibrational modes of the electromagnetic field give rise to photons.
However, there is one important difference, i.e. while the number of normal modes in the case of electromagnetic field is indefinite, the number of normal modes (or the number of phonon energy levels) in the case of a solid is fixed by the number of lattice sites in it.
This introduces certain differences in the thermodynamic behavior of the sound filed in contrast to thermodynamic behavior of the radiation field; however, at low temperatures, when the high-frequency modes of the solid are not very likely to be excited, these differences become rather insignificant and we obtain a striking similarity between the two sets of results.

8. The thermodynamics of the solid can now be studied along the lines of a system of harmonic oscillators. First of all, we note that the quantum-mechanical eigenvalues of the Hamiltonian (9.5) would be
(9.6)
where the numbers ni denote the “states of excitation” of the various oscillators (or, equally well the occupation numbers of the various phonon levels). The internal energy of the system is then given by
(9.7)
The expression within the curly brackets gives the energy of the solid at absolute zero. The term 0 is necessarily negative and larger in magnitude than the total zero-point energy,
of the oscillators

9. together they determine the binding energy of the lattice
together they determine the binding energy of the lattice. The last term in the formula represents the temperature dependent part of energy, which determines the specific heat of the solid:
(9.8)
To proceed further, we must have knowledge of the frequency spectrum of the solid. To acquire this knowledge from first principles is not an easy task.
Accordingly, one either obtains this spectrum through experiment or else makes certain plausible assumptions about it. Einstein, who was the first to apply quantum concept to the theory of solids (1907), assumed, for simplicity, that the frequencies i are all equal in value! Denoting this (common) value by E, the specific heat of the solid is given by
(9.9)

10. where E(x) is so-called Einstein function:
(9.10)
with
(9.11)
At sufficiently high temperatures, when T>>E and hence x<<1, the Einstein result tends towards the classical one, viz. CV=3Nk.
At sufficiently low temperatures, when T<<E and hence x>>1, the specific heat falls at an exponentially fast rate and tends to zero as T0.
The dashed curve in Fig. 9.1 depicts the variation of the specific heat with temperature as, given by the Einstein formula (9.9)

11. T3-law
1.0
0.5
T/E
CV/3Nk
Fig.9.1 The specific heat of a solid, according to the Einstein model (dashed line), and according to the Debye model (solid line). The circles denote the experimental results for copper.

12. The theoretical rate of fall, however, turns out to be rather too fast in comparison with the observed rate. Nevertheless, Einstein’s approach to the problem did at least provide a theoretical basis for understanding the observed departure of the specific heat of solids from the classical law of Dulong and Petit, whereby CV=3R5.96 calories per oK of the substance.
Debye (1912) on the other hand, allowed a continuous spectrum of frequencies, cut off at an upper limit D such that the total number of normal modes of vibration is equal to 3N, that is
(9.12)
where g()d denotes the number of normal modes of vibration whose frequency lies in the range (,+d).

13. For g(), Debye adopted the Rayleugh expression (8.49),
modified so as to suit the problem under study. Writing cL for the velocity of propagation of the longitudinal modes and cT for the velocity of propagation of the transverse modes eqn.(9.12) becomes
(9.13)
whence we obtain for the cut-off frequency
(9.14)

14. Accordingly, the Debye spectrum may be written as
(9.15)
Before we proceed further to calculate the specific heat of solids on the basis of the Debye spectrum, two remarks appear in order.
First, the Debye spectrum is only an idealization of the actual situation obtaining in a solid; it may be compared with a typical spectrum.
While for low-frequency modes ( the so called acoustical modes) the Debye approximation is reasonably valid, there are serious discrepancies in the case of high-frequency modes ( the so-called optical modes).
At any rate, for “averaged” quantities, such as the specific heat, the finer details of the spectrum are not very important. In fact, Debye approximation serves the purpose reasonably well; things indeed improve if we take account of the various peaks in the spectrum by including in our result a number of “suitably weighted” Einstein terms.

15. Second, the longitudinal and the transverse modes of the solid should have their own cut-off frequencies, D,L and D,T say, rather than having a common cut-off at D, for the simple reason that, of the 3N normal modes of the lattice, N are longitudinal and 2N transverse. Accordingly, we should have, instead of (9.13),
(9.16)
We note that the two cuts-offs D,L and D,T correspond to a common wavelength
which is comparable to the
mean interatomic distance in the solid. This is quite reasonable because, for wavelengths shorter than min, it would be rather meaningless to speak of a wave of atomic displacements.
In the Debye approximation, formula (9.8) for the specific heat of the solid becomes

16. (9.17)
where D(x0) is the so called Debye function:
(9.18)
with
(9.19)
where D being the so-called Debye temperature of the solid. Integrating by parts, the expression for the Debye function becomes
(9.20)

17. For T>>D, which means x0 <<1, the function D(x0) may be expressed as a power series in x0:
(9.21)
Thus, as T , CV  3Nk; moreover, according to this theory, the classical result should be applicable to within ½ percent so long as T>3D. For T<<D, which means x0 >>1, the function D(x0) may be written as
(9.22)
whence
(9.23)

18. Thus, at low temperatures the specific heat of the solid follows the Debye T3-law:
(9.24)
Thus, while
in the limit T we recover the well-known classical behavior (CV=const),
in the limit T0 we obtain the typical phonon behavior (CVT3).
It is clear from eqn. (9.24) that a measurement of the low-temperature specific heat of a solid should enable us not only to check the validity of the T3-law but also to obtain an empirical value of the Debye temperature D.

19. The value of D can also be obtained by computing the cut-off frequency D from a knowledge of the parameters N/V, cL and cT ; see formulae (9.14) and (9.19).
(9.14)
(9.19)
The closeness of these estimates is another evidence in favor of Debye’s theory. Once D is known, the whole of temperature range can be covered theoretically by making use of the tabulated values of the function D(x0). A typical case is shown in Fig We note that not only was T3-law obeyed at low temperatures, the argument between theory and experiment was good throughout the range of observations.

20. T3-law
1.0
0.5
T/E
CV/3Nk
Fig.9.1 The specific heat of a solid, according to the Einstein model (dashed line), and according to the Debye model (solid line). The circles denote the experimental results for copper.

22. Back to Dispersion Curves
We know we can measure the phonon dispersion curves - the dependence of the phonon frequencies upon the wavevector q.
To calculate the heat capacity, we begin by summing over all the energies of all the possible phonon modes, multiplied by the Planck Distribution.
number of polarizations is 3 for three dimensions
Planck distribution gives us the number of phonons at a given temperature and energy
h bar omega – energy of a phonon of polarization p and wavevector q
Planck
Distribution
sum over all
wavevectors
sum over all
polarizations

23. Density of States g() What’s this good for?
For photons, there are an infinite number of modes for each energy, but not so far solids. We are working with a discrete system – within which the atomic spacing puts limits.
number of modes
unit frequency
g()

24. Density of States: One Dimension
determined by the dispersion relation
Each normal vibrational mode of polarization p has the form of a standing wave.
If the ends are fixed, what modes, or wavelengths, are allowed?

25. Density of States: One Dimension
To calculate the density of states, use
number of modes
unit frequency
g()
There is one mode per interval  q =  / L with allowed values...
So, the number of modes per unit range of q is L / .

26. Density of States: One Dimension
There is one mode for each mobile atom.
To generalize this, go back to the definition...the number of modes is the product of the density of states and the frequency unit.

27. Density of States: One Dimension
monatomic lattice
diatomic lattice
Knowing the dispersion curve we can calculate the group velocity, d/dq.
Near the zone boundaries, the group velocity goes to zero and the density of states goes to infinity. This is called a singularity.

28. Periodic Boundary Conditions
No fixed atoms – just require that u(sa) = u(sa + L).
This is the periodic condition.
The solution for the displacements is
The allowed q values are then,
This method results in the same number of modes: one per mobile atom, but now we have both positive and negative values of q and q = 2/L.

29. Density of States: 3 Dimensions
Let’s say we have a cube with sides of length L.
Apply the periodic boundary condition for N3 primitive cells:

30. Density of States: 3 Dimensionsqz
There is one allowed value of q per volume (2/L3) in q space or
allowed values of q per unit volume of q space, for each polarization, and for each branch.
The total number of modes for each polarization with wavevector less than q is qy qx
In q-space, we are looking at all possible modes from q = 0 to maximum value of q. Since we are looking in three dimensions, this is a sphere. We are talking about an infinite cubic lattice within which there is this sphere.
How many q values are possible within this sphere?