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

2.11 “Bending” Waves in a Strongly Anisotropic Crystal 87

Fig. 2.4 Dispersion laws of a strongly anisotropic layered crystal:

(a) general scheme; (b) experimental curves for graphite (Nicklow et al. 1972). In the upper part of the curves, the dependence on the direction of k in the basal plane is indicated.

wavelength vibrations in the form

into account.

3

Vibrations of Polyatomic Lattices

3.1

Optical Vibrations

A polyatomic crystal lattice is different from a monatomic one in that its unit cell contains more than one atom. In other words, the number of mechanical degrees of freedom per unit cell of a polyatomic lattice is necessarily more than three.

The last point introduces not only quantitative but also qualitative changes in the spectrum of crystal vibrations. The specific features of polyatomic lattice vibrations will first be studied for a very simple model.

Let us consider the vibrations of an atomic crystal with a densely packed hexagonal lattice or a crystal with the NaCl structure (Fig. 0.2). The unit cell of the lattice of such crystals has two atoms, i. e., six degrees of freedom. It is convenient to introduce the vector of displacement of the center of mass of a pair of atoms, u(n), i. e., the center of mass of a unit cell (three degrees of freedom), and the vector of a relative displacement of atoms in a pair, ξ (n) (three degrees of freedom).

In terms of these displacements, we can write explicitly the energy (or the Lagrangian function) of a vibrating crystal. However, we shall not write this value directly in the general form, but confine ourselves to a scalar model. Now this model cannot be called “scalar”, since we have to operate with two scalar functions u(n) and ξ (n), where ξ (n) mean, e. g., changes in the distance between the atoms of a chosen pair. Thus we should actually transform a scalar model to a two-component one.

Assuming that the displacements of the centers of mass of the unit cells u(n) and the relative displacements of atom pairs ξ (n) are small, we represent the potential energy of crystal vibrations in the form corresponding to a harmonic approximation

U = 12 ∑α1 (n − n )u(n)u(n ) + 12 ∑α2 (n − n )ξ (n)ξ (n ) (3.1.1) + ∑β(n − n )u(n)ξ (n ) ,

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

3.1 Optical Vibrations 91

The compatibility condition homogeneous equations (3.1.7) for the two unknowns u0 and ξ0

considered as an equation for finding the squares of the possible vibration frequencies at fixed k transforms to the second-degree algebraic equation, then becomes

Thus, each value of the quasi-wave vector in our model of a polyatomic lattice corresponds to two frequencies ω = ωα (k), α = 1, 2, i. e., the dispersion law has two branches of the dependencies ω on k. The typical distinctions between these two branches can be shown on considering their limiting properties.

The most essential differences are observed at small k (ak 1). In this case, as we have already seen in sect.1, expansion of the function A1 (k) in powers of the quasi-wave vector starts with the quadratic terms

A1 (k) = Ms20 k2 .

Due to the above-mentioned properties of the matrix α2 (n) the expansion of the function A2 (k) starts with a zero term and does not contain the first power of k

Finally, to simplify the analysis, we assume that the matrix β(n) is symmetric β(n) = β(−n). Then, using (3.1.4), we have

Hence, |B(k)|2 (ak)4 at ak 1 and, in the main approximation with respect to the small parameter ak (3.1.9) is split into two independent ones

92 3 Vibrations of Polyatomic Lattices

The dispersion law (3.1.11) gives the above-discussed dispersion of sound vibrations. Indeed, substitute (3.1.11) into (3.1.7) and perform the limit transition k → 0. Since A2 (0) = 0 we have ξ0 = 0. Thus, under long-wave vibrations with the dispersion law (3.1.11), the unit cell centers of mass vibrate with the relative po-

sition of atoms in a pair remaining unchanged. Therefore, using (3.1.6), we get u(n) = u0 eiω t, ξ (n) = 0.

A feature of the dispersion law (3.1.12) is that the corresponding vibrations with an infinitely large wavelength have the finite frequency ω0. It follows from (3.1.7) that at k = 0 this vibration is

Under such crystal vibrations the centers of mass of the unit cells are at rest and the motion in the lattice is reduced to relative vibrations inside the unit cells. The presence of vibrations such as (3.1.13) distinguishes a diatomic crystal lattice from a monatomic one.

For arbitrary k, the form of the dispersion law is strongly dependent on the dynamical matrix properties. In simple models it is generally observed that ω1 (k) < ω2 (k). The dependence ω = ωα (k) along a certain “good” direction in the reciprocal lattice has the form of a plot in Fig. 3.1, where b is the period of a reciprocal lattice in the chosen direction. The low-frequency branch of the dispersion law (ω < ωm) describes the acoustic vibrations, and the high-frequency one (ω1 < ω < ω2) the optical vibrations of a crystal. Thus, the polyatomic crystal lattice, apart from acoustic vibrations (A) also has optical vibrations (O).

The generally accepted name for high-frequency branches of the vibrations is explained by the fact that in many crystals they are optically observed. In the NaCl ion crystal, the unit cell contains two different ions whose relative displacement changes the dipole moment of the unit cell. Consequently, the vibrations connected with relative ion displacements interact intensively with an electromagnetic field and may, thus, be studied by optical methods.

When ω1 (k) < ω2 (k) for a fixed direction of k, the spectrum of optical vibration frequencies is separated by a finite gap from the spectrum of acoustic vibration frequencies. However, it is possible that for some k the condition ω1 (k) > ω2 (k) is fulfilled. Then, in k-space there are points of degeneracy where the acoustic and optical vibration frequencies coincide and the plots of two branches are tangential or intersect. The simplest plot of the dispersion law for a diatomic lattice is given in Fig. 3.1 ( 12 b indicates the Brillouin zone boundary).

Regardless of the complicated form of the dispersion law for the optical branch, the corresponding frequencies always lie in a band of finite width and its ends are generally the extremum points for the function ω = ω2 (k). The latter means that the dispersion law near the ends of the optical band is quadratic.

The characteristic optical vibration frequencies have the same order of magnitude as the limiting frequency of acoustic vibrations, i. e., ω0 ω1 ωm 1013 s−1.

3.1 Optical Vibrations 93

Fig. 3.1 Acoustic and optical branches of the dispersion law.

But in the crystal due to some specific physical reasons the frequencies of some optical vibrations may increase markedly. In particular, this may occur in so-called molecular crystals.

Let us consider a crystal lattice of two-atom molecules positioned at its sites. The optical branch of vibrations of such a crystal describes intramolecular motions weakly dependent on intermolecular interactions. These motions are characterized by frequencies close to the eigenfrequencies of a free molecule. Then, ω0 ω1 ωm, ω0 − ω1 ω0 and the optical branch of vibrations has a narrow high-lying frequency range.

Let ξ (n) be a coordinate of intramolecular vibrations at the point n. Then the assumption of a strong atomic interaction inside a molecule is reduced to the conditions

Under these conditions the equations for intramolecular vibrations with frequencies ω ω0 have the form

and reduce to the equations for one scalar quantity ξ (n).

Thus, a scalar model may also be used for qualitative analysis of the optical lattice vibrations. For this, it suffices to replace the requirement (3.1.4) for the matrix α1 (n) with the condition

1

µ ∑α2 (n) = ω02 = 0, (3.1.15)

determining the long-wavelength limit for the frequency of optical vibrations.

943 Vibrations of Polyatomic Lattices

3.2

General Analysis of Vibrations of Polyatomic Lattice

We now consider the complete equations of three-dimensional vibrations of a crystal

whose unit cell contains q atoms. We number different atoms in one unit cell by the index s and denote by us(n) the displacement vector of the s-th atom in a unit cell with number-vector n, e. g., the (n, s atom). The Lagrangian function of small crystal

vibrations is

and a summation over all indices is assumed. The equations of motion are obtained from (3.2.1) by the usual method

The invariance of (3.2.1) relative to the motion of a crystal as a whole (uks (n) = u0k ) leads to the following constraints imposed on the force matrix elements

The relations (3.2.4) follow directly from (3.2.3). For stationary vibrations with the frequency ω, we have

in which

Aijss (k) = ∑αijss (n)e−ikr(n). n

Transform (3.2.5), (3.2.6) to a canonical form of the equations of some eigenvalue problem. For this we introduce instead of the displacement vectors, the new variables

vs(n) = √msus(n),