Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

We note that the equations of motion (6.1.18) for the operators ak and a†k are firstorder differential equations, whereas (1.6.6) for the normal coordinates are equations of the second order.

To determine the action of the operators ak and a†k on the occupation numbers, we use the matrix of the coordinate and the momentum of a 1D harmonic oscillator. If M is the mass, ω the frequency, and n is the state number or the energy level number (6.1.11) of a harmonic oscillator, the nonzero matrix elements of its coordinate X and the momentum Y are1

Using (6.1.20) in the case M = 1 and the linear relations (6.1.13), it is easily seen that the matrix elements of the operators ak and a†k are nonzero only for transitions in which the corresponding occupation numbers Nk (with the same value of k) change by unity.

We denote by

|. . . Nk . . . ≡| Nk = Ψ{N}k (Q)

the wave function of some crystal state that is the eigenfunction of the operator a†k ak and is characterized by a set of occupation numbers Nk. Then, only the following

ized wave functions (Q is a set of coordinates).

Thus, the operator ak transforms the function with the occupation number Nk into the function with the occupation number Nk − 1, i. e., reduces the occupation number by unity, and the operator a†k increases it by unity.

We denote by A the mean value of the operator A in a certain quantum state |Nk :

On the basis of (6.1.21) or (6.1.22) we can conclude how to calculate the quantummechanical average for the operator A that has the form of the sum of products of any number of the operators ak and a†k . If the number of operators ak in this product does not equal the number of operators a†k (with the same value of k), the mean value (6.1.23) vanishes automatically. In particular,

At the same time, (6.1.10) follows naturally from (6.1.21) and its generalization

is consistent with commutation relations (6.1.8).

6.2 Phonons

Let us consider the ground state of a crystal corresponding to the least vibration energy. The obvious definition of the ground state: Nk = 0 for all k follows from the form of energy levels (6.1.12) and the fact that Nk are non-negative numbers. The energy of the ground state

is called the zero-point energy and the vibrations with Nk = 0 are called zero lattice vibrations. Let |0 be the wave function of the ground state. Then, according to (6.1.22)

Thus, the wave function of the ground state (the state vector) is the eigenfunction not only of the binary operator a†k ak , but also of the operator ak . In the latter case it corresponds to a zero eigenvalue.

6.2 Phonons 171

Any excited state corresponds to a certain set of nonzero integers Nk. As follows from (6.1.22), the wave function of the excited state |Nk can be obtained by an

Nk-fold action of the operator a†k on the ground-state vector |0 . On writing the vibration energy (6.1.12) in the form

it becomes clear that a weakly excited (with small vibrations) crystal state is equivalent to an ideal gas of quasi-particles, the energy of each particle being h¯ ω(k), and their numbers in different states being given by the set Nk. The quasi-particles arise due to the quantization of vibrational collective excitations, which represent the elementary excitations of a vibrating crystal.

Using the de Broglie principle and the general statements of mechanics, the motion of an individual quasi-particle can be characterized by the velocity v = ∂ω/∂k and the quasi-momentum

We call the quantity (6.2.4) a quasi-momentum, as k is a quasi-wave vector and its specific properties in a periodic structure are automatically extended to the vector p.

The quasi-particles introduced in this way are called phonons and the operator a†k ak is the operator of the number of phonons. The names of the operators ak and a†k reflect the properties of (6.1.22): the operator ak decreases by unity the number of these phonons with quasi-wave vector k, and the operator a†k increases by unity the number of these phonons in the crystal are called the annihilation (or absorption) and the creation (emission) operators of a phonon.

If the creation and annihilation operators of particles obey the commutation relations (6.1.8), the corresponding particles are described by Bose statistics. In our case this assertion proves the fact represented by (6.1.10), implying that in a state with quasi-wave vector k there can be any number of phonons.

We note that in terms of thermodynamics a weakly excited state of the crystal is equivalent to an ideal gas of phonons, whose Hamiltonian has the form

The ground state of a crystal is a phonon vacuum and its physical properties are manifest in the existence of zero vibrations. The intensity of zero vibrations is characterized by the squared amplitude of each normal vibration that is determined by the same formula as for a 1D harmonic oscillator. Let us consider the square of the shift at the n-th site, i. e., the squared operator (6.1.15) and find its mean value in the ground state. Taking into account (6.1.24), (6.1.25) we obtain

172 6 Quantization of Crystal Vibrations

When a crystal is excited, phonons appear and their number depends on a specific choice of Nk, but phonon gas characteristics involve only the mean occupation numbers. We denote the phonon mean number in the corresponding state through fα(k) and call it the fα(k) phonon distribution function.

The function of the equilibrium boson gas distribution is known to be given by the Bose–Einstein distribution. Using this function one should remember that the total number of phonons characterizing the intensity of the lattice mechanical vibrations is not conserved and depends on the crystal excitation degree. Thus, the chemical potential of the phonon gas is zero and the mean thermodynamic values of the occupation numbers are determined by

where the brackets . . . denote averaging in the equilibrium thermodynamic state and the temperature T here and below is given in units of energy.

A simple form of the Hamiltonian (6.2.5) and the Bose-type distribution (6.2.7) allow one easily to construct the thermodynamics of a weakly excited crystal. If mechanical atomic vibrations exhaust all possible forms of internal motions in a crystal (6.2.3) determines the total crystal energy whose mean value coincides with the internal energy. If there are other forms of motion in a crystal (electron motion, change of spin magnetic moment or similar) (6.2.3) gives only a so-called lattice part of the crystal energy. In the latter case all results below listed results can be applied only when the phonon interaction with elementary excitations of the other types is very weak.

6.3

Quantum-Mechanical Definition of the Green Function

The initial definition of the Green function used in quantum theory differs at first sight from that accepted in Chapter 4. Nevertheless, it leads to the same properties of the Green function. We illustrate this by considering an example of the Green retardation function and restrict ourselves to a scalar model.

The quantum-mechanical expression for the Green retardation function is

where u(n, t) is the atomic displacement operator at time t; Θ(t) is the discontinuous unit Heaviside function

The presence of the function Θ(t) in (6.3.1) is connected with the singularity of the Green retarding function shown, e. g., in (4.6.5). It is necessary to recall the proce-

174 6 Quantization of Crystal Vibrations

Let us verify by means of a direct calculation that the Green function (6.3.1) actually determines a retarded solution to (6.3.7) with the Fourier components (4.6.5). We expand the displacements in (6.3.1) in normal vibrations, introducing the phonon creation and annihilation operators and taking into account their commutation relations as well as the time dependence (6.1.19). The following formula is then easily obtained

Furthermore, (6.3.8) for t = 0 justifies the rule (6.3.3).

Substituting (6.3.8) into (6.3.1) we write

The latter has the form of a spatial Fourier expansion of the Green function with Fourier components (4.6.5).

We emphasize the fact that (6.3.8) is valid for any vibrational crystal state. Thus, the averaging contained in the definition (6.3.1) can be carried out both in the sense of quantum mechanics and thermodynamics.

6.4

Displacement Correlator and the Mean Square of Atomic Displacement

The quantum definition of the Green function (6.3.1) involves averages such asu(n, t)u(n , 0) , determining the correlation of atomic displacements at different crystal lattice sites. Some physical properties of the crystal are generated by this correlator. A pair correlation function of displacements (or simply a pair correlator) will be referred to as the average

where both the spatial homogeneity of an unbounded crystal and time uniformity are taken into account.

Using (6.1.16), we go over from the displacements to the operators aα(k), a†α(k) and use the properties of their mean values and also of the polarization vectors

+ (Nα(k) + 1) e−iωαt+ikr(n) , where ωα = ωα(k) and e(α) = e(k, α) = e(−k, α).

6.4 Displacement Correlator and the Mean Square of Atomic Displacement 175

After thermodynamic averaging when N(k) is determined by (6.2.7), we obtain

(6.4.3) It is clear that the correlation function is complex. The availability term in (6.4.3) is of quantization origin. Indeed, in the classical limit h¯ → 0 when h¯ coth(h¯ ω/2T) →

(2T/ω), there remains

It follows from the definition (6.4.1) and the relation (6.4.2) that the mean square of atomic displacement at any site (independent, naturally, of the site number) equals

In a monatomic crystal lattice there is no index s, so that one can use the relation

To characterize the atomic displacements in the excited crystal states it is reasonable to consider the mean thermodynamic values of the displacement squares at nonzero absolute temperature T of the crystal. The averaging (6.4.5) over the phonon equilibrium distribution and performing an integration over frequencies gives the mean

The expression (6.4.6) is much simplified at high temperatures when the ratio T/¯h

A simple (linear) temperature dependence of the mean square of atomic displacement at high temperatures remains in a polyatomic lattice, but the expression for u2

178 6 Quantization of Crystal Vibrations

We substitute into (6.5.7) the explicit expression (6.5.4) and compare the exponent (6.5.7) with (6.4.5) for a scalar model. It is clear that

Although we derived (6.5.8) with respect to a scalar model, after obvious generalization it remains valid for any crystal lattice:

Thus, the quantity uj (n) has the Gaussian probability density

In a classical theory of the crystal lattice, this result was obtained by Debye (1914) and Wailer (1925); the quantum derivation was first made by Ott (1953).

The probability density of an atomic displacement from its crystal lattice site is strongly temperature dependent, since u2 is a function of temperature. The most remarkable fact here is that the atom exhibits an appreciable probability to be displaced from its equilibrium position even at T = 0 K due to the zero vibrations. Thus, although the energy of zero vibrations may not be manifest, the zero motion associated with it is accessible to observation.

6.6

Quantization of Elastic Deformation Field

In studying classical mechanics of a crystal lattice, it has been established that in the long-wave limit (ak 1) the equations of crystal motion transform into the dynamic equations of elasticity theory. Such a transition corresponds formally to the limit a → 0 and establishes a relation between the mechanics of a (crystal) discrete system and that of a continuous medium (continuum).

On the other hand, the quantum equations for crystal motion transform into the classical dynamic equations by passing to the limit h¯ → 0 (if α = const). We clarify whether the limiting transition α → 0 (if h¯ = const) is meaningful, as the lattice constant a and the Planck constant h¯ are considered in the crystal quantum theory as two independent parameters.