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

= λiklm k l um,

76 2 General Analysis of Vibrations of Monatomic Lattices

Expanding the function u(r ) as a series near the point r(x1 , x2 , x3 ) retaining only the second-order terms in |xk(n) − xk(n )|:

uk(r ) = uk (r) + (xl − xl ) l uk(r) + 12 (xl − xl )(xm − xm) l muk (r). (2.8.3)

Here we introduced the notation iφ ≡ ∂φ/∂xi ; it will often be used later.

We now make use of the fast decay of coefficients α(n) with increasing n and substitute the expansion (2.8.3) into (1.8.2), taking no account of the terms with higher space derivatives of the displacements:

	∂2 ui
m	∂t2 = cikuk + cikl l uk + ciklm l muk.	(2.8.4)

The constant coefficients on the r.h.s. of (1.8.4) are defined by the force matrix elements as:

	cik =	cki =				∑ αik(n) = 0;
					− n
	cikl =	∑αik(n − n )[xl (n) − xl (n )] = − ∑αik(n)xl (n) = 0;
		n				n
						(2.8.5)
	ciklm =	−	1	∑αik(n − n )[xl (n) − xl (n )][xm(n) − xm(n )]
			2
			1	n
	=	−		∑αik(n)xl (n)xm(n).
			2
				n

Thus, to describe the long-wave (slowly varying in space) crystal displacements we have the following system of second-order differential equations in partial derivatives

	∂2 ui
m	∂t2 = ciklm l muk.	(2.8.6)

Equation (2.8.6) coincides in its notation with the dynamical equation of elasticity theory

∂2 ui

ρ ∂t2 (2.8.7)

where ρ = m/V0 is the mean mass density in a crystal (V0 is the unit cell volume). The coefficients on the r.h.s. of (2.8.7) that give the crystal elastic moduli tensor

have, however, known symmetry with respect to a permutation of the first and second pairs of indices.

Comparing (2.8.6), (2.8.7) we see that for these equations to be the same, the following equality should hold:

1	(λiklm + λilkm) =	1	cimkl.	(2.8.8)
2
		V0

2.9 The Theory of Elasticity 77

The relation (2.8.8) will not be inconsistent only in the presence of the symmetry

ciklm = clmik.

(2.8.9)

The property (2.8.9) does not follow immediately from the definition (2.8.5) and imposes additional constraints on the force matrix elements of a crystal. We make use of (2.8.5) and write these constraints as

∑αik(n)xl (n)xm(n) = ∑αlm(n)xi (n)xk(n).

(2.8.10)

It is clear that the conditions (2.8.10) are actually the result of the invariance of crystal energy relative to a hard rotation of the type (2.1.7).

By imposing the constraints (2.8.10) on the matrix of atomic force constants we provide for the symmetry (2.8.9). This allows us to establish a relation between the tensors λiklm and ciklm, by solving the relation (2.8.8) for the elastic modulus tensor:

λiklm =	1	cimkl + ckmil − clmki .	(2.8.11)
	V
	0

The equality (2.8.11) determines the crystal moduli through the atom force constants, i. e., gives an exact relationship between macroscopic mechanical monocrystal characteristics and microscopic crystal lattice properties.

2.9

The Theory of Elasticity

The transformation from crystal lattice equations (2.8.1) to those of elasticity theory (2.8.7) is accomplished by changing the model of the substance construction. We go over from a discrete structure to a continuum, i. e., the lattice is replaced by a continuous medium. This radical change from a microscopic description of a crystal to a macroscopic one entails new concepts, terms and relations.

For a macroscopic (or continuum) description of crystal deformation, the concept of a displacement vector u as a function of coordinates r(x, y, z) and time t: u = u(r, t) is normally used. Using space derivatives of the displacement vector, the strain tensor (i, k = 1, 2, 3) is written as

ik =	1		∂ui	+	∂uk	+	∂ul ∂ul			.	(2.9.1)
	2		∂xk		∂xi		∂xi ∂xk

The latter is the main geometrical characteristic of the deformed state of a medium. The tensor ik defined by (2.9.1) is sometimes called the finite strain tensor, and the

part that is linear in displacements,

ik =	1		∂ui	+	∂uk	≡	1	( iuk + k ui) ,	(2.9.2)
	2		∂xk		∂xi		2

78 2 General Analysis of Vibrations of Monatomic Lattices

is the small strain tensor. The linear elasticity theory that we will be concerned with is based on the definition of the strain tensor (2.9.2).

Six different elements of the strain tensor (2.9.2) cannot be absolutely independent since all of them are generated by differentiating three components of displacement vectors. Indeed the strain tensor components ik are related by differential relations known as the Saint Venant compatibility conditions:

eilmekpn l p mn = 0.

(2.9.3)

All the components of the tensor of homogeneous deformations (independent of the coordinates) can, however, be arbitrary.

In addition to the strain tensor (2.9.2), often the distortion tensor uik = i uk whose symmetrical part determines the tensor ik introduced. The antisymmetric part of the distortion tensor gives the vector ω of the local crystal lattice rotation due to deformation:

	ω =	1	curl u.	(2.9.4)
		2
	The sum of diagonal elements of the tensor ik, i. e., the value of ekk			equals the

relative increase in the volume element under deformation. Consequently, the total change in the volume as a result of deformation ∆V can be written as

∆V = kk dV.

(2.9.5)

The time derivative of the vector u determines the velocity of displacements v = ∂u/∂t. If a crystal is deformed but retains its continuity (no breaks, cracks, cavities, etc.) the displacement velocity satisfies the continuity equation

	∂ρ	+ div ρv = 0,	(2.9.6)
	∂t

where ρ is the crystal density (the mass of a unit volume).

The forces of internal stresses arising under crystal deformation are characterized by the symmetric stress tensor σik; the force that acts on unit area is

Fi = σikn0k ,

where n0 is the unit vector normal to the area. If the area concerned is chosen on an external body surface then F equals the force created by external loads.

In the case of hydrostatic crystal compression under the pressure p the tensor σik reads

	σik = −pδik.			(2.9.7)
	On the basis of (2.9.7)	1
	p0 = −		σkk	(2.9.8)
		3

2.9 The Theory of Elasticity 79

is called the mean hydrostatic pressure when the stress tensor does not coincide with (2.9.7) and describes a more complex crystal state. When the tensor σik is different from (2.9.7) displacement stresses are present, which are usually characterized by the deviator tensor:

σik	= σik −	1	δki σll = σik + δik p0.	(2.9.9)
		3

If the crystal deformation is purely elastic, the stresses are related linearly to strainsik by the generalized Hooke’s law:

σik = λiklm lm,

(2.9.10)

where λiklm is the tensor of crystal elasticity moduli. For a cubic crystal, there are three independent elastic moduli (or stiffness constants):

λ1 = λ1111 = λ2222 = λ3333, λ2 = λ1122 = λ1133 = λ2233;
G = λ1212 = λ1313 = λ2323.	(2.9.11)

In an isotropic approximation these three moduli are related through λ1 − λ2 − 2G = 0. Therefore, the tensor λiklm for an isotropic medium reduces to two independent moduli that can be represented, for example, by the Lamé coefficients λ = λ2 and G:

λiklm = λδikδlm + G(δil δkm + δimδkl ).

(2.9.12)

The coefficient G in (2.9.11), (2.9.12) often denoted by µ is called the shear modulus and relates the nondiagonal (“oblique”) elements of the σik and ik tensors in an isotropic medium and in a cubic crystal

σik = 2G ik,

i = k.

(2.9.13)

Note that there is an obvious relation between the mean hydrostatic pressure p0 and the relative compression of an isotropic medium or a cubic crystal. From (2.9.10)– (2.9.12), we have

σll = 3K ll,

(2.9.14)

where K is the modulus of hydrostatic compression that, in a cubic crystal, is found from

	3K = λ1 + 2λ2,			(2.9.15)
	and, in an isotropic medium, from
	K = λ +	2	G.	(2.9.16)
		3

To determine the deformed and stressed crystal states in the presence of bulk forces, it is necessary to solve the following equation for an elastic medium

ρ	∂2 ui	= kσki + fi,	(2.9.17)
	∂t2

80 2 General Analysis of Vibrations of Monatomic Lattices

where the vector f describes the density of bulk forces acting on a crystal (the mean force applied to a crystal unit volume), and the tensor σik is related to the strains through Hooke’s law (2.9.10).

The dynamics of a free elastic field ( f = 0) is described by

ρ

∂2 ui

− λiklm k l um = 0.

(2.9.18)

∂t2

Equation (2.9.18) corresponds to the Lagrangian function

L =

1

∂u

2

1

∂uk ∂um

ρ

−

dV.

(2.9.19)

2

∂t

2

∂xi ∂xl

Equation (2.9.18) and the corresponding Lagrangian function, even in an isotropic case, are rather complicated. One of the difficulties in solving (2.9.18) for the three components of the displacement vector u is the following. Equation (2.9.18) is similar to a wave equation. In transforming to normal vibrations we can reduce it to three wave equations. But the latter describe the waves propagating with different velocities. Even in an isotropic approximation the elastic field has two different characteristic wave velocities (the velocities of longitudinal and transverse waves). This very much complicates the solution of dynamic problems.

To simplify the equations reflecting the main physical properties of an elastic medium, we formulate the analog of the scalar model for an elastic continuum, i. e., we introduce a scalar elastic field. We take as a generalized field coordinate the scalar value u(r, t) and assume the Lagrangian function of this field to be

L =	1			∂u		2	1		∂u	2
		ρ				−		G		dV.	(2.9.20)
	2			∂t			2		∂xi
The equation for the field motion stemming from (2.9.20) is
		ρ	∂2 u			− G∆u = 0,					(2.9.21)
			∂t2

where ∆ is the Laplace operator (∆ = 2k ). This is an ordinary equation of the waves propagation with the acoustic dispersion law:

ω2 = s2 k2 , s = G/ρ.

(2.9.22)

The main disadvantage of (2.9.21) as a model equation for crystal dynamics is its scalar character, which does not allow one to describe transverse elastic vibrations.

2.10

Vibrations of a Strongly Anisotropic Crystal (Scalar Model)

We write the dispersion law for monatomic lattice vibrations with interatomic nearestneighbor interactions in an explicit form, using a scalar model that enables us to find

2.10 Vibrations of a Strongly Anisotropic Crystal (Scalar Model) 81

the dependence of vibration frequencies on the quasi-wave vector through the simplest elementary functions. It turns out that for crystal directions where it is possible to distinguish longitudinal and transverse vibrations a scalar model describes the crystal longitudinal vibrations well. This can be explained as follows. There is no polarization of displacements in a scalar model and the only vector characteristic of a normal vibration is vector k. Therefore, the atomic displacements described by such a model can be associated only with the quasi-wave vector direction.

Going over to the formulation of a concrete problem, we simplify a model to describe in detail some interesting physical properties of a vibrating crystal, in particular, the vibrations of strongly anisotropic crystal lattices.

As an example of such an anisotropic model we consider a tetragonal lattice with different interactions of the nearest atoms in the basal plane (xOy) and along the fourfold axis (z). Choosing naturally the translation vectors, we denote |a1 | = |a2 | = a, |a3 | = b. The neighboring atom interaction in the basal plane will be described by the force matrix element α1 and the interaction along the axis z by the element α2 . On the basis of the relation (2.1.10), we have

α0 + 4α1 + 2α2 = 0.

(2.10.1)

Since α0 = α(0) > 0, it follows from (2.10.1) that 2α1 + α2 < 0. We assume that α1 < 0 and α2 < 0.

The dispersion law (2.2.10) for the lattice concerned is written as

2	2		2 akx		2 aky		2	2	akz
ω	(k) = ω1	sin			+ sin		+ ω2 sin			,	(2.10.2)
			2						2
					2

where ω12 = −4α1 /m and ω22 = −4α2 /m.

We assume the atomic interaction in the basal plane to be much stronger than that

along the four-fold axis:
ω1 ω2.	(2.10.3)

This assumption transforms a tetragonal lattice into a crystal lattice with a layered structure, whose separate atom layers are interrelated weakly. The formula (2.10.2) for such a crystal determines an extremely anisotropic dispersion relation, which is well shown in the low-frequency part of the vibration spectrum.

Consider the frequencies ω ω1 (e. g., ω ≤ ω2) for which the formula (2.10.2) is much simplified

ω	2	= s	2	k	2	+ ω	2	sin	2 akz		;	k	2	= k	2	+ k	2	,	s	2	=	1	a	2	ω	2	,	(2.10.4)
			1				2			2					x		y			1		4				1
where s1		has a meaning of a sound velocity in the basal plane.

In view of the condition (2.10.3), we keep the second term in the r.h.s. of (2.10.4) unchanged, in so far as the assumption ω ω1 does not imply that bkz is small. Thus, we take into account that at comparatively low frequencies the quasi-wave vector component along the z-axis may be large.

82 2 General Analysis of Vibrations of Monatomic Lattices

Within the long-wave limit when bkz 1, (2.10.4) gives the dispersion law of sound vibrations in an anisotropic medium

	ω2 = s2 k2	+ s2 k2	;	s2	=	1	b2	ω2	,	(2.10.5)
						2
	1	2 z		2				2

where s2 is the sound velocity along the z-axis.

If the lattice parameters a, b are little different (have the same order of magnitude), the sound velocity in the basal plane of a “layered” crystal will be much larger than that in a perpendicular direction (s2 s1 ).

Equation (2.10.4) is also simplified in the case when the vibration frequencies have

the range
ω2 ω ω1 .	(2.10.6)

For such frequencies the second term in (2.10.4) should not be taken into account, and the dispersion relation reduces to

ω = s1 k ≡ s1 k2x + k2y,

(2.10.7)

which coincides with the dispersion relation for sound vibrations in a two-dimensional elastic medium. Hence, under the conditions (2.10.6) the frequency of vibrations of a three-dimensional “layered” crystal is independent of the wave-vector component along the direction perpendicular to its “layers”.

Along with a “layered” crystal, one can consider a crystal model with a “chain” structure where one-dimensional chains of atoms weakly interact one with another. In order to obtain this model it suffices to assume

ω1 ω2.

(2.10.8)

Then the results stated above are easily transformed by changing the numbers 1 and 2 and also the components ki and kz.

In particular, at frequencies ω ≤ ω1 the dispersion relation (2.10.2) reduces to

	2	= ω	1		2	akx		2 aky		2 2
ω				sin			+ sin			+ s2 kz.
						2		2

In the long-wave limit (ak 1) we come again to (2.10.5), and in the frequency range ω1 ω2 the dispersion law of vibrations of such a crystal coincides with the dispersion law of elastic vibrations of a one-dimensional system

ω= s2 kz,

i.e., the vibration frequency is independent of the wave-vector projection onto the plane perpendicular to the direction of a strong interaction between atoms.

2.11 “Bending” Waves in a Strongly Anisotropic Crystal 83

2.11

“Bending” Waves in a Strongly Anisotropic Crystal

We consider a crystal with a simple hexagonal lattice in which the atoms interact in different ways in the basal plane xOy and along the six-fold axis Oz. We assume the crystal structure to be layered and the atom interaction in the plane xOy to be much larger than the atom interaction in neighboring basal planes. In describing the vibration of such “layered” crystal one can proceed from the model that takes exact account of the strong interaction between all atoms lying in the basal plane, and the weak interaction of neighboring atomic layers is taken into account in the nearestneighbor approximation along the six-fold axis.

A crystal with a chain structure may be considered simultaneously. A crystal with such a structure consists of weakly interacting parallel linear chains. In the model proposed, this corresponds to the fact that the atomic interaction along the six-fold axis is much stronger that the interaction between neighboring chains (or the nearest neighbors in the plane xOy).

An example of a chemical element that has three possible crystalline forms (approximately isotropic, layered and chain) is carbon. It exists in the form of diamond (an extremely hard crystal with a three-dimensional lattice), in the form of graphite (layered crystal) and in the form of carbene (a synthetic polymer chain structure).

For definiteness the following arguments are given for a layered crystal and intended for the model formulated above. The latter makes it possible to qualitatively describe the acoustic vibrations in graphite – a layered hexagonal crystal with very weak interactions between the layers4. The atomic forces between the neighboring layers in graphite are almost two orders less that the nearest-neighbor interaction forces within the layer.

Let a and b be interatomic distances in the xOy plane and along the Oz-axis, respectively. The vector n1 represents a set of two-dimensional number vectors connecting any one of the atoms with all remaining atoms in the same basal plane, n3 is the unit vector of the Oz-axis. Then nonzero elements of the matrix αik(n) in our model are represented by αik(n1 ) and αik(n2 ). Making use of the obvious force matrix symmetry in a hexagonal crystal, we write the elements αik(n3 ) responsible for the weak atomic layer interaction as follows

	αik(n	3	) =	α	δ , i, k = 1, 2,
					1	ik	(2.11.1)
	αzz(n3 ) =			α2 ,		αxz(n3 ) = αyz(n3 ) = 0.

Concerning the spectrum of acoustic vibrations of graphite we note that the parameter α2 is generally larger than α1, and α2 is determined mainly by central forces

|α1 | |α2 | ,

(2.11.2)

(for graphite α2 ≈ 10α1 ≈ 0.6104 dyn/cm).

4) Graphite has a complex lattice with atoms positioned in a separate basal plane as shown in Fig. 2.2.

84 2 General Analysis of Vibrations of Monatomic Lattices

To characterize the strong interaction in the basal plane, we introduce the notation α3 = αzz(n0 ), where n0 is the unit vector directed from an atom to any one of its six nearest neighbors in the plane xOy (Fig. 2.2). It may also be assumed that

αik(n0 ) αzz(0) α3, i, k = 1, 2. (2.11.3)

For graphite α3 105 dyn/cm.

Now the assumption of a layered crystal structure can be formulated in the form of

a quantitative ratio establishing a hierarchy of interatomic interactions
|α1 | |α2 | |α3 | .	(2.11.4)

We note an important property of anisotropic crystal vibrations whose displacement vector u is perpendicular to the strong interaction layers (perpendicular to the xOy plane). For a very weak layer interaction, these vibrations should resemble the bending waves in the noninteracting layers5, so that they may tentatively be referred to as “bending” vibrations. Simultaneously, assuming strong anisotropy of interatomic interactions (2.9.4) and the same order of the lattice constant values (a b), it is impossible in describing the “bending” vibrations to include only the nearest-neighbor interaction in the basal plane. Noting that the character of the bending vibrations is primarily determined by the force matrix elements αzz(n1 ), we take into account the conditions (2.8.10) imposed on the A(n) matrix elements, putting i, k = x or i, k = y and l, m = z:

2α1 b2 = ∑αzz(n1 )x2 (n1 ) = ∑αzz(n1 )y2 (n1 ).		(2.11.5)
n1	n1

Fig. 2.2 Choice of the nearest neighbors in the basis plane of an hexagonal crystal

By keeping in (2.11.5) the summation over the vectors n0 only, we get the equality

2α1 b2 = 3α3 a2 ,

(2.11.6)

5) The need to take into account the bending wave type of vibrations in layered crystals with a weak interlayer interaction was first indicated by Lifshits (1952).

2.11 “Bending” Waves in a Strongly Anisotropic Crystal 85

Fig. 2.3 Second difference in atom displacements that determines the bend energy.

which is impossible when the requirements a b and |α1 | I |α3 | are satisfied simultaneously.

Thus, such a model for a layered crystal with interaction between nearest neighbors only is in fact intrinsically inconsistent. To describe a crystal lattice with a characteristic layered structure having “bending” waves it is necessary, while keeping the relations such as (2.11.4), to take into account more distant interatomic interactions in the basal plane. The physical meaning of this assertion is easily understood if we consider the limiting case of noninteracting layers possessing bend rigidity. Analyzing one atomic layer allows one to conclude that the interaction of atoms displaced along the Oz-axis (Fig. 2.3) is determined by the difference of relative pair displacements of at least three atoms rather than by the relative displacement of two neighboring atoms.

The atomic interaction energy under bending vibrations of the plane layer depends on

δun = (1/2)[(un − nn−1 ) − (un+1 − un)] = un − (1/2)(un−1 + un+1 ).

We now turn to (2.11.5) and note that it does not contradict the assumption of a strong anisotropy of a crystal. This is due to the fact that the elements of the matrix αzz(n1 ) describing the interaction not only between nearest neighbors in the plane xOy can be quantities of the same order of magnitude and have opposite signs. The

signs of each of them satisfy the condition
∑αzz(n) ≡ αzz(0) + ∑ αzz(n1 ) + 2αzz(n3 ) = 0.			(2.11.7)
n		n=0
Taking into account the inequality (2.11.4), we conclude from (2.11.7) that
		∑ αzz(n1 ) < 0.	(2.11.8)
		n=0
To find the dispersion law of the vibrations, we calculate the tensor functions
Aij(k) = ∑αij(n)e−ikr(n) ≡ ∑ αij(n) [cos kr(n) − 1] .			(2.11.9)
	n	n=0
Assume that		αzz(n1 ) = αyz(n1 ) = 0.
			(2.11.10)