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

242 10 Linear Crystal Defects

on the surface S, then where b is a fixed vector identical at all points on the surface S, the distortion tensor uik remains a continuous and differentiable function of the coordinate everywhere except for a closed line on which the surface S is spanned.

However, the requirement of continuity for the distortion tensor is, to some extent, excessive as the physical state of the elastic body depends only on stresses and elastic strains proportional to them. Thus, for studying the elastic fields generated by discontinuities δu on surfaces not distinguished by their physical properties in the body volume we restrict ourselves to the requirement of unambiguity and continuity of the strain tensor ik. It then turns out that (10.2.2) does not present a general form of the change of the vector u on the surface S at which the strain tensor ε ik conserves continuity and differentiability as a function of the coordinates. The most general form of the discontinuity at the surface S is

δu = u+ − u− = b + [Ωr],

(10.4.1)

where b and Ω are constant vectors (b is a translation vector and Ω is a rigid rotation vector); r is the point radius vector on the surface S. The vector Ω in a crystal should coincide with one of the elements of crystal rotational symmetry.

Under the condition (10.4.1) an antisymmetric part of the distortion tensor is broken on the surface. On reminding ourselves of the definition of the rotation vector under the deformation (1.9.4) we see that the rotation vector ω undergoes a step at the surface:

δω ≡ ω+ − ω− = Ω.

(10.4.2)

The jumps (10.4.1), (10.4.2) generate an elastic field singularity in the crystal concentrated along the line D on which the surface S is spanned.

If Ω = 0, then (10.4.1) is transformed into (10.2.2), and the translation vector b coincides with the Burgers dislocation vector.

If b = 0, but Ω = 0, the singularity arising in a solid is called a disclination. The disclination vector Ω is sometimes called the disclination power, and sometimes the Frank vector. We shall use the latter name.

It follows from (10.4.2) that in the presence of a disclination the Frank vector describes a relative rigid rotation of two parts of a solid positioned on both sides of the surface S. It is clear that for an unambiguous definition of δu in (10.2.1), the space position of the vector Ω, i. e., the position of the disclination axis should be fixed. The displacement of the disclination rotation axis by the vector R amounts to adding an ordinary dislocation with Burgers vector b = [ΩR]. Therefore, the definition of a disclination in (10.4.1) becomes unambiguous if we rewrite it as

δui(r) = eilmΩl (xm − xm0 ),

(10.4.3)

where r0 is the radius vector of the point through which the rotation axis given by the vector Ω runs. It follows from a direct calculation that (10.4.2) is a consequence of (10.4.3).

10.4 Disclinations 243

As the disclination is a linear singularity of the elastic deformation field, it may be defined in a form that does not use the notion of an arbitrary surface S, i. e., in a form analogous to the definition of a dislocation (10.2.1). Indeed, we introduce a continuous and differentiable function ω(r) (the medium element rotation at point r as a result of an elastic deformation of a solid). The disclination will then be said to be a specific line D with the following property: in passing around any closed contour L enclosing the line D, the elastic rotation vector ω gets a certain finite increment Ω. This property is written as

dωi =	∂ωi	dxk = −Ωi.	(10.4.4)
	∂xk

L

The Frank vector Ω unambiguously determines the properties of the disclination D only when the point through which the disclination rotation axes runs is specified.

Let τ be a unit vector tangent to the line of the disclination. If τ Ω, the disclination is called a wedging or a slope disclination. If τ Ω we have a twisting disclination.

A disclination in a crystal is most vividly exemplified by a 60◦ wedging disclination in a hexagonal crystal when this defect is parallel to the six-fold symmetry axis. Analyzing the atomic arrangement in a plane perpendicular to the axis of this disclination in a nondefective crystal (Fig. 10.7a) and also their distortion in the presence of a positive (Fig. 10.7b) and a negative (Fig. 10.7c) wedging disclination, we note the following peculiarities. Choosing the sign of a wedging disclination, unlike choosing it for an edge dislocation, has an absolute character: the atomic displacements in the vicinity of a positive disclination is inverse to the atomic displacements in the vicinity of a negative disclination. In the first case, crystal stretching is observed along the contour that encloses the disclination, and in the second case, we have crystal compression.

Another important peculiarity of the wedge disclination observed is the change in the crystal lattice symmetry in the vicinity of a disclination. Indeed, for a 60◦ positive wedging disclination there arises a five-fold symmetry axis coincident with the vector Ω (Fig. 10.7b), and a 60◦ negative disclination generates pseudosymmetry with a seven-fold symmetry axis (Fig. 10.7c). In a perfect crystal such rotational symmetry is impossible.

The two types of linear defects considered here (dislocations and disclinations) are in fact two independent forms of the same family of peculiarities inherent to the deformation of continuous media that are called Volterra dislocations. The dislocations in a crystal are translational Volterra dislocations, and disclinations are rotational Volterra dislocations. Generally, the Volterra dislocation may have a mixed character, i. e., simultaneously represent a translational dislocation and a disclination.

We now turn to finding an elastic field around a separate disclination. Note that a simple tool for calculating this field can be obtained on the basis of (10.4.3). If we consider the disclination as a line limiting the surface S with given rotation vector

244 10 Linear Crystal Defects

Fig. 10.7 A 60◦ wedge dislocation parallel to the six-fold symmetry axis: (a) ideal structure in a basal plane; (b) structure with a positive disclination; (c) structure with a negative disclination.

jump (10.4.2), then it corresponds formally to a dislocation with “Burgers vector” distributed over the surface S

bi(S) = eilmΩl (xm − xm0 ).

(10.4.5)

It is true that in contrast to the Burgers vector of an ordinary dislocation distribution (10.4.5) is dependent on the coordinate on the surface S. However, this does not prevent us from using (10.2.5), without taking the Burgers vector outside the integral sign. Hence, the problem of finding the displacement vector u(r) around a separate disclination loop is reduced to calculation of the integral

u

(r) =

−

λ

e

mpq

Ω

p

(x

x0 )

k

G

(r

−

r ) dS .

(10.4.6)

i

jklm

q −

q

ij

l

S

The expression (10.4.6) allows us to determine the elastic deformations of a crystal with an arbitrary disclination loop.

10.5

Disclinations and Dislocations

In some cases a simple isolated disclination can easily be represented by a planar “pileup” of continuously distributed dislocations. Equation (10.4.5) allows us to clarify how the “Burgers vector” of dislocations is distributed along the surface S, which is necessary for these dislocations to be equivalent to a disclination with Frank vector Ω.

The above point is illustrated by the relationship between a wedge disclination and a so-called dislocation wall. By “wall” we mean a large number of identical parallel linear edge dislocations distributed in the same plane perpendicular to their Burgers vectors (Fig. 10.8a). The dislocations are in parallel slip planes, the distances between which in the simplest case are the same, i. e., equal to h.

The geometrical meaning of a crystal deformation generated by this system of dislocations is easily understood. The presence of a wall leads to a misorientation of

10.5 Disclinations and Dislocations 245

the two parts of a crystal that are divided by the system of dislocations concerned (Fig. 10.8a). If h is the distance between the dislocations (in a macroscopic theory it is necessary that h b), the misorientation angle between the two parts of a crystal is ψ = b/h. Thus, the dislocation wall is a model of the boundary between two blocks or subgrains with small misorientation. If the boundary consists of edge dislocations, then the axis around which the neighboring subgrains are inclined is in the plane dividing them. Such a low-angle grain boundary is called an inclination boundary.

Let a half-infinite inclination boundary end at a straight line A (Fig. 10.8b), the dislocations being distributed in it continuously with the Burgers vector density b/h. If we now imagine a closed contour enclosing the line A and intersecting the inclination boundary at the point y, then such a contour encloses dislocations with the total Burgers vector

by

Bx = h , (10.5.1) where the coordinate y is measured from the line A.

Fig. 10.8 Dislocation wall and a wedging disclination: (a) low-angle grain boundary, (b) dislocation wall ends at the line A, (c) representation of the dislocation wall by a disclination dipole.

Comparing (10.5.1) and (10.4.6), we can conclude that when the dislocation wall (Fig. 10.8b) is treated macroscopically then it is equivalent to a negative wedge dislocation with Frank vector Ωz = −b/h and rotation axis coincident with the line A. Naturally, the Frank vector in this case is equal to the misorientation angle between the parts of a crystal.

In this situation the rotation by an angle Ω should not be an element of the group symmetry of the ideal crystal, but the boundary between two misoriented parts of a crystal is then taken as a plane stacking fault.

If the inclination boundary ends at a certain line B (Fig. 10.8c), the latter should be coincident with the rotation axis of the second disclination. Its Frank vector Ω then satisfies the requirement that the rotation through the angle Ω + Ω be an element of

the group symmetry of the crystal lattice. In particular, the most interesting case is

Ω = −Ω.

246 10 Linear Crystal Defects

A pair of disclinations of the same type but with opposite sign and parallel rotation axes forms a disclination dipole. In the example concerned (Fig. 10.8c) when ΩB = −ΩA, the wedging disclinations positioned along the lines A and B form a dipole. If the distance between the lines A and B is equal to L, the deformation caused by the dipole described can be thought of as a peculiar wedging-out of a crystal: i. e., a semi-infinite planar plate of thickness l = ΩL whose edge has a triangular cross section with angle Ω at the vertex is set into the crystal cut.

At distances greatly exceeding L, the dipole of wedging disclinations is regarded as an edge dislocation with Burgers vector bx = l = ΩL and the “core” enclosing the whole region of an inclination boundary of length L.

10.5.1

Problems

1. Obtain an expression for the displacement vector around the dislocation in an isotropic medium, isolating explicitly the contour integration over the dislocation loop.

Hint. Use (10.2.5) and (9.5.10) for the Green tensor in an isotropic medium and also the Stokes theorem.

	Solution.
	u = b	O	+	[bτ] dl	+ grad ψ;	ψ =	λ + µ	[bR]τ	dl	,
		4π		R			4π(λ + 2µ)		R

where O is the solid angle subtended at the observer by the dislocation loop.

11

Localization of Vibrations

11.1

Localization of Vibrations near an Isolated Isotope Defect

We begin studying the influence of defects on lattice vibrations by using a scalar model, treating independently one branch of crystal vibrations only. An equation for stationary vibrations of frequency ω in this model is

1
ω2u(n) −			∑α(n − n )u(n ) = 0.					(11.1.1)
	m
				n
The dispersion law for an ideal lattice follows from (11.1.1)
ω2 = ω02(k) ≡						1	∑α(n) cos kr(n).	(11.1.2)
						m
Equation (11.1.1) describes acoustic crystal vibrations for which
ω02 =				1	∑α(n) = 0,			(11.1.3)
				m

and where the possible vibration frequencies are in a finite range (0, ωm).

If we remove the requirement of (11.1.3), then (11.1.1) can be used to analyze qualitatively the optical lattice vibrations (Chapter 3) upon determination of the extremely

long-wave frequency of vibrations by
ω02(0) = m1 ∑α(n) = 0.	(11.1.4)

The eigenfrequencies of (11.1.1) will then be within a certain interval (ω1, ω2), where

ω1 > 0.

The simplest point defect arises in a monatomic species when one of the lattice sites is occupied by an isotope of the atom making up the crystal. Since the isotope atom differs from the host atom in mass only, it is natural to assume that the crystal

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

248 11 Localization of Vibrations

perturbation does not change the elastic bond parameters. Let the isotope be situated at the origin (n = 0) and have a mass M different from the mass of the host atom m. With such a defect we get, instead of (11.1.1),

ω2 Mu(0) − ∑α(n )u(n ) = 0,								n = 0;
			n						(11.1.5)
ω2mu(n)	− ∑			α(n		n )u(n ) = 0, n = 0.
					−
			n
Equations (11.1.5) can be written more compactly as
mω2u(n) − ∑α(n − n )u(n ) = (m − M)ω2δn0,									(11.1.6)
	n
by introducing the 3D Kronecker delta δnn .
We denote							∆m
∆m = M − m,						U0 = −		ω2,	(11.1.7)
							m
and rewrite (11.1.6) in a form typical for such problems
ω2u(n) −		1	∑α(n − n )u(n ) = U0ω2δn0.						(11.1.8)
		m
			n
We write a formal solution to (11.1.8) as
		u(n) = U				G0(n)u(0),			(11.1.9)
					0	ε

where Gε0 is the Green function for ideal lattice vibrations, ε = ω2; u(0) is a constant multiplier still to be defined.

If we reject the scalar model and proceed to the general case of a simple lattice, it is easy to write a formula analogous to (11.1.9). With an isotope defect at the site n = 0, the displacement vector of any atom in the crystal is

ui (n) = U Gik(n)uk (0).	(11.1.10)
0 ε

It is also easy to generalize (11.1.9), (11.1.10) for the case of a polyatomic lattice. However, in order not to complicate the formula, we return to the scalar model.

Setting n = 0 in (11.1.9), we find that (11.1.9) is consistent only when

1 − U0 Gε0 = 0.

(11.1.11)

The expression (4.5.12) for the Green function is substituted into (11.1.11):

	U		1
1 −	0	∑k		= 0.	(11.1.12)
	N		ε − ω2(k)

11.1 Localization of Vibrations near an Isolated Isotope Defect 249

Finally, after a transition from summation to integration over quasi-wave vectors and then changing to integration over frequencies:

	1 − U0	g0	(z) dz	= 0.	(11.1.13)
		ε − z

Here, g0 (ε) is the ideal lattice vibration density.

Before we proceed to analyze (11.1.13), we derive a corresponding general equation, taking into account the different polarizations of vibrations when it is necessary to use (11.1.10).

We set n = 0 in (11.1.10) and note that the resulting homogeneous system of three

algebraic equations for three independent ui (0) is solvable if
Det δik − U0Gεik(0) = 0.	(11.1.14)

An explicit expression for the Green tensor (4.5.14) does not allow us to reduce (11.1.14) to the relation containing only the ideal lattice vibration density g0(ε) and that is independent of the polarization vectors. In a cubic crystal or in the isotropic approximation, however, we have

Gεik(0) = 13 δikGεll (0),

and (11.1.14) is reduced to three identical equations of the type (11.1.13).

As (11.1.13) is a condition for solvability of the corresponding equation of motion, it is an equation to determine the squares of frequencies ε at which the atomic displacement around an isotope has the form of (11.1.9). In a theory of crystal vibrations with a point defect, equations such as (11.1.11)–(11.1.13) were first obtained by Lifshits (1947).

We start to analyze (11.1.13) for the case of acoustic vibrations when the unperturbed crystal frequencies are in the interval (0, ωm). It is clear that in this case (11.1.13) is meaningful only for ε > εm = ωm2 , but the denominator in the integral (11.1.13) is then always positive and (11.1.13) can only have a solution for U0 < 0, i. e., for a light isotope (M < m). In a 3D crystal, however, the availability of the necessary sign of the perturbation does not guarantee the existence of a solution to (11.1.13). This is easily seen when we graphically analyze (11.1.13). Introducing the

notation
F(ε) = ε	g0 (z) dz			,
		ε − z
and taking (11.1.7) into account, we rewrite (11.1.13) as
F(ε) = −		m
			.	(11.1.15)
		∆m

Since g0(z) √εm − z for z → εm the function F(ε) is finite and positive at the point ε = εm (F(εm) = Fm > 0) and, in the small range ε − εm εm, has a negative

250	11 Localization of Vibrations
	derivative		g0 (z) dz
		F (ε) −εm
				.
			(ε − z)2
	For ε	εm due to the chosen normalization of the vibration density F(ε) ≈ 1,
	so that a plot of the function F(ε) has the shape of a curve (Fig. 11.1) and a plot
	of the r.h.s. of (11.1.15) is a horizontal line.		Thus, the solution ε > εm exists if

|δm/m| Fm > 1. If this condition is satisfied, the solution to (11.1.13) gives a discrete frequency ωd lying outside the continuous spectrum band (Fig. 11.1, εd = ωd2).

Fig. 11.1 A graphical method of finding the frequency of local vibrations.

Thus, in order for a vibration to appear at a discrete frequency lying near the acoustic band, an isotope should have the mass M satisfying the condition M < m(1 − 1/Fm) < m.

Consider the case when the solutions to (11.1.13) determine the discrete frequencies near the optical band of an ideal lattice. Let the continuous spectrum of an ideal crystal

occupy the interval (ε1, ε2), where ε1 = ω12 and ε2				= ω22 and the defect intensity is
characterized by the parameter U = εW0 . Then, instead of (11.1.15), we write
F(ε) =	1		,	(11.1.16)
	−
		W0
assuming that the function F(ε) is defined for ε				< ε1 and ε > ε2 (Fig. 11.2,
F1 = F(ε1 ) < 0; F2 = F(ε2 ) > 0).	Simple analysis leads to the conclusion

that the existence of discrete solutions to (11.1.16) is provided only by the defects whose parameter W0 satisfies certain conditions. If W0 > 0, it suffices to require

that W0		> 1/ |F|.				If W0 < 0, the absolute value of the parameter W0 is within
2	<	|	0	|	< 1.	d	is to the left of the point ε	1
1/F		W				In the first case the solution ε

(Fig. 11.2), and in the second case to the right of the point ε2.

Thus, depending on the sign of the parameter W0, discrete frequencies can arise either to the left of the continuous spectrum band (ωd < ω1 at W0 > 0) or to the

11.1 Localization of Vibrations near an Isolated Isotope Defect 251

Fig. 11.2 Determination of the position of local frequencies for different signs of ∆m.

right of it (ωd > ω2 at W0 < 0). Crystal vibrations with the frequencies described are called local vibrations, and the frequencies ωd – local frequencies. This name is attributed to the fact that the amplitude of the corresponding vibration is only nonzero in a certain vicinity near the point defect. Let us analyze the letter of the cases considered, assuming that W0 > 0 and the local frequency ωd is to the left of the continuous spectrum band (ωd < ω1). The local vibration amplitude is given by (11.1.9), implying its coordinate dependence is completely determined by the behavior of the ideal crystal Green function. We rewrite the Green function as

1

eikr(n)

V

eikr(n)d3 k

Gε0(n) =

∑

=

0

.

(11.1.17)

N

ε − ω

2

(k)

(2π)

3

ε − ω

2

(k)

k

It is known that the behavior of such integrals at large distances (r

a) is mainly

determined by the form of the integrand at small values of k (ak l). In other words, the major contribution to the integral (11.1.17) at large r comes from the integration in the region of small wave vectors where the dispersion law of an ideal crystal is quadratic in k. To avoid possible complications we write down the quadratic dispersion law for the isotropic model in the form

ω2(k) = ω12 + γ2 k2,

γ2 (ω22 − ω12)a2.

We take into account this dispersion law by changing ε

1

= ω2

and performing the

integration required in (11.1.17)

1

eikr(n)d3k

∞

k2 dk

π

=

2π

eikr cos θ dθ

ε1 − ε + γ2k2

0

ε1

− ε +

γ2 k2

0

=

2π2

r

√

exp

−

ε1 − ε .

γ2r

γ