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

662 General Analysis of Vibrations of Monatomic Lattices

ω = − |ωα (k)| describes the same crystal vibrational state as the wave with vector −k and frequency ω = − |ωα (−k)|. Consequently, in order to describe independent crystal states it suffices to consider the frequency of one sign that corresponds to all

possible k vectors inside a single unit cell of the reciprocal lattice. This allows us in what follows to discuss vibrations with positive frequencies only.

2.3

Normal Modes of Vibrations

We have seen that crystal eigenvibrations can be represented in the form of plane waves (2.2.3) whose frequencies are connected with a quasi-wave k by the dispersion law ω = ωα (k), α = 1, 2, 3. To distinguish between the displacements of different branches of the vibrations we explicitly write (2.2.3)

u(n, t) = e(k, α)ei[kr(n)−ω t].

(2.3.1)

Since the equations of motion (2.1.12) or (2.2.1) are homogeneous, their solutions are found up to a constant factor. With this in mind we determine e(k, α), as the unit vector called the polarization vector. The dependence of the vector e on k and α follows from equation like (2.2.4), which makes it possible to choose the vectors e real2 and possessing the property e(k, α) = e(−k, α). Various branches of vibrations correspond to different solutions of some eigenvalue problem, which is why the linear dependence of the eigenfunctions (2.3.1) requires the polarization vectors of vibrations of different branches to be orthogonal:

e(k, α)e(k, α ) = δα,α .

(2.3.2)

If the vector k is directed along a highly symmetrical direction (e. g., along a fourfold symmetry axis) there is one longitudinal vibration whose vector e is a simple classification of the possible types of wave polarization breaks down, only three polarization vectors remain mutually orthogonal (2.3.2). For some highly symmetrical directions in a crystal the vibration of the same branch corresponding to a certain value of the index α can be either transverse or longitudinal, depending on the direction of the vector k.

We choose the time dependence as e−iω t and consider the normalized solutions to (2.2.1) or (2.2.2) in the form

1
φkα (n) = √N e(k, α)eikr(n).	(2.3.3)

According to the properties of a quasi-wave vector we assume the vector k to be in one unit cell of a reciprocal lattice (or in the first Brillouin zone). Under this condition

2) The polarization vector can be chosen as real only in a monatomic lattice.

2.4 Analysis of the Dispersion Law 67

the eigenfunctions (1.3.3) possess the natural orthogonality properties
∑φkα (n)φk α (n) = δkk δα,α ,	(2.3.4)

where the asterisk denotes complex conjugation; δkk is the three-dimensional Kronecker symbol.

Thus, the crystal eigenvibrations (2.3.3) are numbered by (k, α). The eigenfunctions (2.3.3) are often called the normal modes of the vibrations.

There is no concept of polarization in a scalar model and the coordinate dependence of normal vibrations is written in the form

1			eikr(n),
φk (n) =	√			(2.3.5)
		N
which provides the normalization condition
∑φk (n)φk (n) = δkk .				(2.3.6)

2.4

Analysis of the Dispersion Law

To analyze the dispersion law we write (2.2.10) for a scalar model. Consider the vibrations with small k, i. e., those for which ak 1. We expand the cosine on the r.h.s. of (2.2.11) in powers of its argument and use the fact that the function α(n) decreases rapidly with increasing n. Then, in the main approximation

		1	3
	ω2 (k) = −		∑ (kaα )(kaβ ) ∑α(n)nα nβ .		(2.4.1)
		2m
			α,β=1	n

We denote k = kκ, by introducing the unit vector in reciprocal space κ and represent (2.4.1) as ω2 = s2 (κ)k2 , where

		1	3
	s2 (κ) = −		∑ (κaα )(κaβ ) ∑α(n)nα nβ .		(2.4.2)
		2m
			α,β=1	n

Thus, for small k we get the linear dispersion law of sound vibrations that is typical for an anisotropic continuum

ω = s(κ)k,

(2.4.3)

here s is the phase velocity of an acoustic wave. This result seems to be quite natural, so long as small k correspond to large wavelengths λ, and the condition ak 1 determines the requirement λ a determining the possibility of passing over from crystal-lattice mechanics to that of a continuum.

According to (2.2.11), for arbitrary k values, in particular for ak 1, the character of the dispersion law is determined mainly by the specific form of the matrix α(n). In

68 2 General Analysis of Vibrations of Monatomic Lattices

the general case one can assert that for the coefficients α(n) decreasing fast enough with increasing number n the function ω(k) is continuous, differentiable, and always bounded.

Thus, the following is typical of the dispersion law: the possible frequencies of crystal vibrations fill the band of a finite width (0, ωm) beyond which there are no vibrational frequencies. It is easy to evaluate the order of the maximum frequency value ωm, which is of the order of magnitude ω s/a 1013 s−1 (the typical sound velocity in a crystal s 105 cm/s).

There exists a very simple model of the spectrum of crystal vibrations that takes into account the availability of a maximum frequency and permits one easily to perform a lot of calculations explicit using the dispersion law. This is the so-called Debye model based on the assumption that the dispersion law is linear for all k, but is restricted in frequencies: ω = sk, ω < ωD. The frequency ωD ωm 1013 s−1 is called the

Debye frequency.

In the real situation near the upper edge of the band of possible frequencies, i. e., when ωm − ω ωm the frequency and quasi-wave vector are described by the following quadratic dependence

ω = ωm −	1	γij(ki − kmi	)(kj − kmj ),	(2.4.4)
	2
or by
ω2 = ωm2 − ωm γij(ki − kmi )(kj − kmj ).				(2.4.5)

Here the vector km is determined by the condition ω(km ) = ωm and the matrix of constant coefficients γik is defined positively. The terms linear in k − km do not enter in (2.4.4) as the frequency ωm is maximum by definition.

A dispersion law of the type (2.4.4) or (2.4.5) is known as a quadratic dispersion law.

Bearing in mind the results of the dispersion law of a scalar model, we go over to considering the general case when the frequency dependence on a quasi-wave vector is obtained by solving (2.2.12).

We first note that from (2.2.9) there follows the property of the dynamical matrix in k-representation: A(k) = A(−k).

Thus, the solution to (2.2.12) has the same property, namely, the dispersion law is

described by the function invariant relative to an inversion in reciprocal space
ω(k) = ω(−k).	(2.4.6)
For a scalar model this property follows directly from (2.2.10).

2.4 Analysis of the Dispersion Law 69

In the limiting case of long waves (ak 1) for the matrix A(k) there holds an expansion of the type (2.4.1) that follows directly from (2.2.9):

			1		3
	A(k) = −				∑ (kaα )(kaβ ) ∑A(n)nα nβ
			2
				α,β=1		n
			k2		3	(2.4.7)
	=	−			∑ (κaα)(κaβ ) ∑A(n)nα nβ .
			2
					α,β=1	n

All elements of the matrix A(k) are thus proportional to the square of the wave vector k2 . Therefore, the squares of frequencies, being the solution to (2.2.12), are also proportional to k2

ω2 = s2α (κ)k2 ,	α = 1, 2, 3,	(2.4.8)
and in the long-wave limit we get three sound dispersion laws
ω = sα (κ)k,	α = 1, 2, 3.	(2.4.9)

Three branches of vibrations for which (2.4.9) generalize the relation (2.4.3) correspond to the three different sound velocities sα (κ).

Consequently, at the point k = 0 there is a degeneration, i. e., several branches of vibrations coincide. Due to unambiguity of ω2 as the wave-vector function at the point k = 0, its expansion as a power series in ki is impossible. The relation (2.4.8) cannot generally be considered as an expansion of the function ω2 in powers of the wave-vector components. This is just the point in which the long-wave dispersion law of a three-dimensional crystal differs from the dispersion law (1.4.1) for a scalar model.

Fig. 2.1 Dispersion diagram with a point of degeneracy (kB, ω0 ) on the

Brillouin zone boundary.

The form of the dispersion law at ak 1 reflects the specific properties of a real crystal. Thus, we can make some general remarks concerning the dispersion law on Brillouin zone boundaries.

70 2 General Analysis of Vibrations of Monatomic Lattices

The normal component of the gradient in the k-space, k ω, vanishes on the Brillouin zone boundary, if at the corresponding point there is no degeneracy. This property of the dispersion law has a simple physical meaning. The gradient determines the group velocity of the wave (2.3.1):

v =	∂ω	= k ω.	(2.4.10)
	∂k

When the vector k ends on the Brillouin zone boundary the group-velocity component normal to it vanishes and the vibrational motion (2.3.1) acquires the character of a stationary wave with respect to this direction.

If at the point considered the degeneration occurs on the zone boundary, the dispersion law plots may approach the Brillouin zone boundary at an arbitrary angle (Fig. 2.1), the points k = ±kB give the zone boundary positions. The degeneracy point on the zone boundary corresponds to the frequency ω = ω0.

Finally, near the upper edge of frequencies for each branch of vibrations one can expect a quadratic dispersion law of the type (2.4.4) or (2.4.5).

2.5

Spectrum of Quasi-Wave Vector Values

Some vibrations (2.2.3), being independent states of motion of the whole crystal lattice, are characterized by different quasi-wave k values.

It is known that for physically nonequivalent crystal vibrations it suffices to consider the k values lying inside one unit cell of the reciprocal lattice. However, not all points inside the unit cell in k-space may correspond to independent crystal states. This follows from the fact that a set of points inside a unit cell composes a continuum but the set of independent vibrations coinciding with the set of degrees of freedom of the crystal lattice turns out to be countable even in the case of an infinite crystal.

For a crystal of finite dimensions the above-mentioned fact is obvious. Therefore, the general qualitative study of crystal vibrations should not be regarded as complete until the spectrum of possible k values has been determined.

When the form of the equations of motion is given, i. e., with the given force matrix A(k) in (2.2.2), certain boundary conditions should be formulated to define the spectrum of eigenvalues. However, it seems that a specific form of reasonable boundary conditions has little influence on the spectrum of k values in a crystal consisting of a great number of atoms. Proceeding from this assumption we choose the boundary condition so that it simplifies the solution of the problem as much as possible. Such a condition is the cyclicity requirement according to which

u(rn ) = u(rn + N1 a1 ) = u(rn + N2 a2 ) = u(rn + N3 a3 ).

(2.5.1)

In formulating (2.5.1) it is assumed that a crystal has a form of a parallelepiped with edges Nα aα , α = 1, 2, 3, i. e., it contains N = N1 N2 N3 atoms.

2.5 Spectrum of Quasi-Wave Vector Values 71

The cyclicity conditions (2.5.1) are called the Born–Karman conditions. In a onedimensional case the Born–Karman conditions admit a very simple interpretation. We can close up a linear periodic chain of N1 points into a ring, after which the N1 + n atom actually coincides with the n-th atom. But in a three-dimensional case a similar attempt to interpret the conditions (2.5.1) does not produce a clear representation.

In studying the bulk dynamical properties of the crystal, we will always proceed from the boundary conditions (2.5.1).

Imposing the requirement (2.5.1) on (2.2.3), we obtain

kaα =	2π	pα, α = 1, 2, 3,	(2.5.2)
	Nα

where pα are integers. To consider the k values lying in one unit cell of the reciprocal lattice, we assume pα to belong to a set pα = 0, 1, 2, . . . , Nα. In a cubic lattice the formula (1.5.2) will be simplified if we put Lα = Nα a and direct the coordinate axes along the four-fold symmetry axes:

kx =	2π	p1, ky =	2π	p2, kz =	2π	p3.	(2.5.3)
	L1		L2		L3

Finally, the set pα is generally taken to be symmetrical with respect to the number-

	ing axis	Nα + 1
	pα = 0, ±1, ±2, . . . , ±		, α = 1, 2, 3.
		2

It follows from (2.5.2) or (2.5.3) that the discrete values of the k vector components are divided by the intervals ∆k 1/L that decrease with increasing the linear crystal dimensions. Therefore, when all linear crystal dimensions are macroscopic the spectrum of k values can be regarded as quasi-continuous. The last property of the k spectrum was used to analyze the dispersion law considering the frequency as a continuous function of quasi-wave vector.

Proceeding further from the quasi-continuity of the spectrum of k values we change the summation over the discrete values of a quasi-wave vector for the integration. Taking (2.5.2), (2.5.3) into account it is easy to obtain the rule governing this transition

to the integration	V
∑ f (k) =			f (k) d3 k,	(2.5.4)
	(2π)	3
k

where the integration is carried out over the volume of a single unit cell in k-space (or the Brillouin zone).

We note that if we put f (k) ≡ 1 in (1.5.4), we obtain the simple relation

∑ =	V		d3 k =	V	= N,
	(2π)	3
k				V0

implying that the number of independent k vector values in one unit cell equals the number of unit cells (the number of atoms in a monatomic lattice).

722 General Analysis of Vibrations of Monatomic Lattices

2.6

Normal Coordinates of Crystal Vibrations

We have seen that the crystal eigenvibrations can be represented in the form of plane

monochromatic waves (2.3.1) where the frequency ω is related to the quasi-wave vector k by the dispersion law ω = ω(k).

It is clear that the harmonic waves (2.3.1) do not describe the most general motion of atoms in a crystal. But the general solution of the equations of motion (2.1.12) can certainly be expressed through a sum of all possible waves such as (2.3.1). In particular, an arbitrary coordinate dependence of the displacement of a vibrating crystal can be realized by an appropriate set of normal modes (2.3.3).

We shall now expand the crystal vibrations into normal modes for a scalar model, disregarding the polarization vectors and the presence of several branches of the dispersion law. The generalization to a real scheme of three-dimensional lattice vibrations involves no difficulties. It will be carried out after performing all the necessary calculations.

Thus, we represent an arbitrary motion of atoms of the crystal lattice as a superposition of normal vibrations (2.3.5):

1
u(n, t) = ∑Qk (n)ψk (n) =	√		φk (n).	(2.6.1)
		m
k

The quantities Qk (t) ≡ Q(k) are called complex normal coordinates of lattice vibrations3. Since the atom displacements (2.6.1) are described by a real function the normal coordinates should have an obvious property

Qk (t) = Q−k(t).

(2.6.2)

Therefore, (2.6.1) is equivalent to

u(n) =	1		∑k	Q	eikr(n) + Q e−ikr(n)	,	(2.6.3)
	2√mN			k	k

showing that the displacements u(n) are real.

Now express the mechanical energy of a vibration crystal through normal coordinates. For the kinetic energy K we have

	m			du(n) 2			˙ ˙
K =	2		∑	dt		= ∑ Qk Qk ∑φk (n)φk (n)				(2.6.4)
	1		n			1	k,k	2	n
			˙	˙			˙
=	2		∑Qk Q−k		=	2	∑ Q(k)		.
			k				k

√

3) The factor m in the definition of normal coordinates reflects the specific feature of (2.2.1) and is introduced to describe the dynamics of a polyatomic crystal lattice (Section 3.2) in a convenient way.

+ ω2 (k)Q(k) = 0.

2.6 Normal Coordinates of Crystal Vibrations 73

Performing transformations in (2.6.4) we used the definition (2.3.4) as well as the properties (2.3.6) and (2.6.2).

Let us transform the potential energy of small crystal vibrations, depending on squared displacement, as follows in terms of normal coordinates

	U =	1	∑ α(n − n )u(n)u(n )
		2
			n,n
	=	1		∑ Q(k)Q(k ) ∑ α(n − n )φk (n)φk (n)
		2m
		1		k,k	n,n	1
	=			∑ Q(k)Q(−k) ∑α(n)e−ikr(n) =			∑ω2 (k) |Qk |2 .
		2m				2
				k,k	n		k

The last in the chain of transformations was performed by making use of (2.2.10) that determines the dispersion law.

Thus, the energy and, hence, the Lagrangian function of crystal vibrations are reduced to a sum of terms that refer to separate normal coordinates. In particular, the Lagrangian function has the form

	L = K	−	U =	1	∑	Q˙	Q˙	k −	ω2 (k)Q Q	k	.
				2
							k		k
					k

The equation of motion for every normal coordinate follows from (2.6.5): d2 Q(k)

dt2

We introduce the generalized momentum conjugate to Qk

Pk =	∂L	= Q˙ (k),
	˙
	∂Q(k)

and obtain the Hamiltonian function for crystal vibrations:

H = 12 ∑ |P(k)|2 + ω2 (k) |Q(k)|2 .

k

(2.6.5)

(2.6.6)

(2.6.7)

Changing to a final formula of the type of (2.6.7) is trivial in general: it suffices to take into account that the normal modes (2.3.3) refer to certain branches of vibrations

and, therefore, the coordinates Q and moments P
Q = Qα (k), P = Pα (k).
Then,
H = 21 ∑ |Pα (k)|2 + ωα2 (k) |Qα (k)|2 .	(2.6.8)
k,α

Thus, the independent oscillations are numbered by a pair of indices (k, α) and their number equals that of the vibrational degrees of freedom of a monatomic lattice, i. e., 3N (three branches of vibrations and N physically inequivalent k values for each branch).

742 General Analysis of Vibrations of Monatomic Lattices

2.7

The Crystal as a Violation of Space Symmetry

The motion of a crystal lattice in which each atom vibrates around its equilibrium position can be expanded in terms of motions of independent oscillators, i. e., normal vibrations. The crystal energy (or its Hamiltonian function) is separated into the terms corresponding to individual normal modes.

Separation of independent motions that may be superpositioned to compose any complex motion of a system of many particles (atoms) is known as the procedure of introducing the collective excitations and relevant collective coordinates (or variables). For small crystal vibrations, i. e., mechanically weakly excited states of a crystal body, the collective excitations are represented by normal modes and collective coordinates by normal coordinates.

The dispersion law of collective vibrations of a monatomic lattice has the universal

property: the frequencies of all three branches of vibrations vanish at k → 0. The extremely long-wave vibrations (k = 0, λ = ∞) are equivalent to the displacement of

a lattice as a whole and this property is a direct result of the crystal-energy invariance with respect to its translational motion as a whole. In proving the relation (2.1.6) we proceeded from the fact that due to space homogeneity the internal state of a body is

independent of the position of its center of masses.

However, this property (i. e., the condition ω(k) → 0 as k → 0) of the frequency spectrum of crystal eigenvibrations can be explained in another way. Since the space where a crystal exists is homogeneous, the movement from one point of free space to another by an arbitrary vector, including an infinitely small one, is equal to the transformation into an equivalent state. For this reason the energy of a system of interacting atoms does not change for arbitrary translations of the whole system. The symmetry connected with the Lagrangian function (or Hamiltonian function) invariance relative to transformations of a continuous group of translations is inherent to any system of particles.

However, in the crystal ground state the atoms form a space lattice whose symmetry is lower than the initial one: the physical characteristics of an equilibrium crystal are invariant under a discrete group of translations, since they are described by some periodic functions reflecting the lattice periodicity.

When the symmetry of the ground state of a system is lower than that of the corresponding Lagrangian function, the initial symmetry is broken spontaneously.

If the properties of the ground state of a system with a large number of degrees of freedom break its symmetry with respect to transformations of a certain continuous group then the collective excitations whose frequencies tend to zero at k → 0 arise in the system (Goldstone, 1961). These excitations seem to strive to re-establish the broken symmetry of the system. The number of branches of such Goldstone excitations is determined by the number of broken independent elements of a continuous symmetry group of the Lagrangian function of the system (by the number of “disappeared” generators of the initial symmetry group).

2.8	Macroscopic Equations for the Displacements Field	75

The ground state in a crystal breaks the symmetry relative to continuous translations in three independent directions, which is “generated” by three components of the momentum. The role of the three branches of the collective excitations generated by a spontaneous symmetry breaking is played by the three branches of harmonic crystal vibrations. Thus, the crystal eigenvibrations are Goldstone excitations and, for this reason, their dispersion laws should possess the properties discussed (ω(k) → 0 as k → 0).

2.8

Long-Wave Approximation and Macroscopic Equations for the Displacements Field

We know that the dispersion law for long-wave vibrations (ak 1) coincides with the dispersion law of sound vibrations in a continuous medium. Now we show how the equations of motion of a crystal are simplified for long-wave vibrations, i. e., in what manner the limiting transition from the equations of crystal lattice mechanics to those of a continuous solid is made. It is clear that as one of the results of such a limiting transition we should obtain the known equations of elasticity theory.

Generally, the equation of motion of a homogeneous three-dimensional crystal lattice may be written as

m	d2 ui(n)	= − ∑αik(n − n )uk (n ),	(2.8.1)
	dt2
		n

and the matrix A(n) obeys the requirement

A(n) = A(−n), ∑A(n) = 0.

Consider those solutions to (2.8.1) that describe displacement fields weakly varying in space. Let λ be the characteristic wavelength of the relevant displacement with λ a. The difference in atom displacements in neighboring unit cells δun is then very small compared to the displacement δu(n) u. Since the natural discrete step in the lattice is small due to the condition a λ, to analyze the displacements weakly varying in space we use some simplifications. First, we assume that the atom coordinate r = x(n) takes a continuous series of values and then the displacement u(n) is a continuous function of r. We next denote the new function by the same letter and write (2.8.1) as

m	d2 ui(r)	= − ∑αik(n − n )uk (r ),	(2.8.2)
	dt2
		n

where r = x(n); r = x(n ).