Kosevich A.M. The crystal lattice

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12.2 Quasi-Local Vibrations Near a Dislocation

283

The behavior of the function K

(x) at small and large values of its argument is well

known:

(x)=−log

, x  1, K

(x)=



−x

, x  1.

Thus, the vibration amplitude distant from the dislocation axis has a characteristic

exponential decrease conﬁrming that the vibrations are localized near the dislocation.

The limiting dependence K

(x) for x  1 shows that the vibration amplitude has a

logarithmic singularity as ρ → 0. The equation (12.1.19) derived in the long-wave

approximation is valid only for ρ  a; therefore, the extremely small distance ρ for

which (12.1.19) is valid may not be smaller than a.

12.2

Quasi-Local Vibrations Near a Dislocation

The equation for the eigenfrequen c ies (12.1.2) is equivalent to the condition

1 −U

(ε, k)=0, (12.2.1)

where G

(ε, k) is the Green function of the equation for 2D crystal vibrations (ε =

), with the spectrum of frequency squares (12.1.10) beginning with s

,wherek

is a ﬁxed parameter. Th is function has an obvious deﬁnition

(ε, k)=

(2π)



(ε, k) d

where G

(ε, k) is the Green function of an ideal crystal in a scalar model (4.5.10).

The dislocation localized waves have freque ncies for which Im G

(ε, k)=0.But

by examining quasi-local vibrations near a heavy impurity, we made it clear that in

the frequency range where Im G

= 0, resonance vibrational states may exist. In this

case the frequencies of these vibrations are determined by

1 −U

Re G

(ε, k)=0. (12.2.2)

Using (12.1.12), it is easy to see that

Re G

(ε, k)=

4πs

log



− ε



. (12.2.3)

It is clear that the r.h.s. of (12.2.3) tends to −∞ at the point ε = s

and determines

the function symmetrical with respect to this point (Fig. 12.1). Finding from the plot

a solution to (12.2.1), (12.2.2) we conclude that for U

< 0 there are simultaneously

solutions both to (12.2.1) for ε < s

(dislocation waves) and to (12.2.2) for ε > s

(quasi-local vibrations near the dislocation). But the latter have the physical meaning

284

12 Localization of Vibrations Near Extended Defects

Fig. 12.1 T he real (1) and imaginary (2) parts of the Green function as

functions of ε at ﬁxed k

Fig. 12.2 T he position of local ω

and quasi-local ω

frequencies of

dislocational vibrations with ﬁxed k

of isolated frequencies only in the case when the damping of a corresponding reso-

nance frequency (i. e., the imaginary part of this frequency) is small. We know that

the damping is determined by Im G

(ε, k)=πg

(2)

(ε, k),whereg

(2)

(ε, k) is a two-

dimensional vibration density. In the considered range of frequency squares, πg

(2)

practically constant (line 2 in Fig. 12.1) πg

(2)

=(a/s

)

∼ 1/ω

. Analyzing the

width o f the quasi-local vibration peak, we have

Γ = πg

(2)



dε

Re G



−1

= g

(2)



2πs





ε − s



∼ ε −s

, (12.2.4)

i. e., it has the order of magnitude of the square of a resonance frequency measured

from the edge of the spectrum s

. This means that the quasi-local frequency near a

dislocation is very weakly pronounced.

In conclusio n, we consider in brief a short qualitative characteristic of disloca-

tion localized vibrations of a real crystal lattice that has three polarizations of the

displacement vector with three branches of the dispersion law. In the isotropic ap-

proximation, there exist two branches of transverse vibrations with the dispersion law

12.3 Localization of Small Vibrations in the Elastic Field of a Screw Dislocation

285

= s

(κ

+ k

) and one branch of longitudinal vibrations with the dispersion law

= s

(κ

+ k

),sothats

> s

always. If the value of k is ﬁxed, there are two

bands of continuous frequency values of the bulk vibrations (Fig. 12.2) ω > sk.

With the corresponding sign of the perturbation U

, the frequency lying near the

boundary of the corresponding band may split off from the lower edge of each of the

bands. One of these frequencies (the lowest one, ω

) corresponds to the vibration

localized near the dislocation. It arises near the edge of the transverse vibration band

and the localized vibrations have the form of transverse waves running along the dis-

location. The dislocation axis participates in these vibrations, bending and vibratin g

like a spanned string. As the quantity s

k − ω

is exponentially small, the bending

waves have a velocity that practically d oes not differ from that of s

. The character

of vibrations allows us to formulate the elastic string model oftenusedindifferent

applications of a dynamic theory of dislocations. In this model a dislocation line is

considered as a h eavy string vibrating in a slip plane. The ratio of linear tension to

dislocation mass is such that the dispersion law of string-bending vibrations coincides

practically with the dispersion law of transverse sound waves in a crystal ω = s

The frequency “split off” from the boundary of the longitudinal vibrations spectrum

(the frequency ω

in Fig. 12.2) could be considered as discrete only if the interaction

between different branches of vibrations is disregarded. But the linear defect violates

the independence of different types of vibrations so that they are “mixed together”.

Since the frequency ω

is in the region of a continuous spectrum of transverse vibra-

tions, it gets broadened and the corresponding vibration is transformed into a quasi-

local one.

Finally, even in the case of an independent branch of vibrations, the quasi-local

vibrations discussed in detail above are possible. These vibrations in Fig. 12.2 corre-

spond to the frequencies ω

and ω

. The quasi-local peak width at the frequency ω

has been evaluated, and the peak width at ω

cannot be smaller. Hence, only the fre-

quencies of bending vibrations of a dislocation as a spanned string are actually singled

out.

12.3

Localization of Small Vibrations in the Elastic Field of a Screw Dislocation

The elastic vibrations near the dislocation as a source of static stresses in a crystal

can be included, using a simple anharmonic approximation, into the initial state of a

vibrating crystal and small vibrations on the background of a distorted lattice can be

considered.

We represent the vibrating crystal displacement in the form

(ξ, z, t)=u

(ξ)+u(ξ, z, t) ,

where u

(ξ) is a static ﬁeld of the screw dislocation coinciding with the axis Oz

286

12 Localization of Vibrations Near Extended Defects

((10.2.8) is convenient for this case)

(ξ)=

bθ

2π

arctan





, (12.3.1)

u(ξ, z, t) are small dynamic displacements relative to a stationary crystal with a dislo-

cation.

Apart from a static deformation, the equations of motion of a vibrating crystal with

a dislo cation will include anharmonicities (cubic and fourth-order ones). Then, re-

stricting ourselves to the approximation linear in dynamic displacements, we g et in a

scalar model the following equation for the displacement u:

∂

∂t

− µ

∂

∂x

− (λ + 2µ)

∂

∂z

= A

∂u

∂x

∂

∂x

∂z

+ B



∂u

∂x



∂

∂x

+ C



∂u

∂x



∂

∂z

(12.3.2)

where λ and µ are the renormalized second-order elastic moduli taking into account

anharmonicities (µ > 0, λ + 2µ > 0); A is the third-order elastic modulus; B and

C are the fourth-order elastic moduli of a nonlinear elasticity theory (from general

considerations, A ∼ B ∼ C ∼ µ).

Equation (12.3.2) should be supplemented with a certain boundary condition on the

surface of a dislocation tube r = r

to describe the phonon reﬂection from the dislo-

cation axis. We use further the fact that no phonons penetrate into the region of the

dislocation core. But we try, using the same approximation in which (12.3.2) is writ-

ten to take correctly into account in the boundary conditions the symmetry of lattice

distortions along the screw dislocation axis. If f (θ, z) is some function characterizing

these distortions on a dislocation tube, it should have screw symmetry

f (θ + θ

, z)= f



θ, z −

bθ

2π



, f (θ, z + b)=f (θ, z), (12.3.3)

that takes into account static displacements (12.3.1) about the screw dislocation.

In view of (12.3.3), we write the solution (12.3.2) as

u = χ(r)e

ik(z+bθ)+imθ−iωt

, (12.3.4)

where m = 0, ±1, ±2,....

It is typical that the solution (12.3.4) satisfying the screw symmetry (12.3.3) is

nonperiodic with respect to the angle θ, and, therefore, is not a single-valued function

of the coordinate x(ξ, z). We remember that u(x) is the atomic displacement relative

to their equilibrium positions in a crystal with a dislocation, and x(ξ, z) are the atomic

positions (coordinates) in a nondeformed crystal. The atom coordinates in a crystal

with a dislocation, i. e., in a statically deformed med ium, are different from those in

12.3 Localization of Small Vibrations in the Elastic Field of a Screw Dislocation

287

the initially nondeformed medium, n amely, the coordinates ξ(x, y) are unchanged and

the third coordinate (along the dislocation axis) is Z = z + u

(θ)=z +(bθ/2π).

If we consider (12.3.4) as a function of atomic coordinates X(ξ, Z) in a deformed

medium, then it is a single-valued function. Ambiguity of the expression (12.3.4) for

the displacement u as a function x(ξ, z) is a direct result of the fact that (12.3. 2) is

written in the coordinate system connected with the initial undeformed medium.

Before writing the equation for χ(r) that follows from (12.3.2) we note that the

spatial derivatives in (12.3.2) have different orders. It follows from (12.1.14) that the

frequency square of a local vibration ω

differs insigniﬁcantly from (s

k )

.Butthen

∂

∂x

∼ κ

⊥

u ∼

−ω

k )

∂

∂z



∂

∂z

where κ

⊥

is the inverse radius of the vibration localization near the dislocation axis.

Thus, it is meaningless to retain on the r.h.s. of (12.3.2) the term with a fourth-order

elastic modulus B. Taking this into account, we get the following ordinary differential

equation

dχ

−



⊥



χ = 0, (12.3.5)

where

ν =



m +

2π



+ m



2π





2π



A + C

, (12.3.6)

and the parameter κ

⊥

is determined by a relation such as (12.1.20)

⊥

(k)=

−ω

, ρs

= λ + 2µ , ρs

= µ . (12.3.7)

Equation (12.3.5) is a Schrödinger equation for the radial part of the wave function

in cylindrical coordinates where the Planck constant should be equal to unity, and the

particle mass to 1/2. It describes a particle located in the potential ﬁeld U = ν/ r

and having the energy E = −κ

⊥

If m = 0, it follows from (12.3.6) that in the long-wave approximation (bk  1)

ν ≈ m

> 0. (12.3.8)

But under the condition (12.3.8) and with a repulsive potential on the dislocation axis

(12.3.5) has no discrete spectrum corresponding to ﬁnite (localized) states. Thus, in

the long-wave approximation for m = 0 there are no localized vibrations near the

dislocation.

The situation is quite different in the case m = 0 when

ν =



2π





1 +



, M = A + C.

For M > −µ, as before, ν > 0 and there are no localized vibrations near the disper-

sion, but if M < −µ,thenν < 0 and the situation changes. The (12.3.5) describes

288

12 Localization of Vibrations Near Extended Defects

the particle motion in a 2D potential well, and there is an inﬁnite number of discrete

levels with the point of condensation at E = 0. These levels can be calculated quasi-

classically.

Equation (12.3.5) corresponds to the Hamiltonian

H = p

+ p

+ U(r) ≡ p

−

(here τ

= −ν). Thus, the Bohr quasi-classical quantization condition is



dr ≡





+ Edr = π



N +



, (12.3.9)

where N is a natural number (N  1); r

= τ

/ |E| =(τ/κ

⊥

)

We perform a trivial integration in (12.3.9) and retain logarithmically large terms

only (ln(r

)  1)

τ ln





= π



N +



, λ =

⊥

. (12.3.10)

The dispersion law for localized vibrations follows from (12.3.6), (12.3.7), (12.3.9)

= s

−|ν|





exp







−

2π

(2N + 1)

b |k|



|M|

−1







, (12.3.11)

where the pre-exponential factor is determined to an order of magnitude only.

Let us note two important peculiarities of the dispersion law (12.3.11). First, the

second term on its right-hand side depends less on k than in (12.1.14). Second, the

eigenfrequencies of localized vibrations form an inﬁnite geometric progression that

describes the concentration of local frequencies toward the edge of the continuous

spectrum. The latter should be taken into account in evaluating the statistical weight

of local vibrations that determines their contribution to the low-temperature thermo-

dynamic properties of a crystal.

If the signs of the perturbations on the dislocation axis and the r atio of M to µ

are such that there are vibrations with frequencies (12.1.14), (12.1.11), then these

frequencies are mixing. But these vibrations “do not interfere” with one another under

the condition that the frequency (12.1.14) is lower than the ﬁrst of the frequencies

determined by (12.3.11) (with a given k).

12.4

Frequency of Local Vibrations in the Presence of a Two-Dimensional (Planar)

Defect

A two-dimensional defect of the crystal breaks the lattice regularity along a certain

surface inside a solid. If the character of the crystal structure distortions is the same

12.4 Frequency of Local Vibrations in the Presence of a Two-Dimensional (Planar) Defect

289

along the indicated surface (more frequently, it is a plane), the break in the rigorous

lattice periodicity is called a stacking fault (Fig. 12.3). The possible types of planar

stacking fault are completely determined by the crystallography of a g iven lattice.

Fig. 12.3 Planar stacking fault that is perpendicular to the z-axis.

We consider an unlimited, extended planar defect, assuming it to be coincid ent with

aplanez = 0 p arallel to some crystallographic plane. We restrict ourselves to the case

of a symmetric lattice and assume that the z-axis is a four-fold symmetry axis of an

ideal cry stal. In general, the symmetry group of the plan e defect is smaller than th e

corresponding group of atomic planes for an unbounded crystal lattice. Thus, there is

no four-fold symmetry axis perpendicular to the defect plane in Fig. 12.3.

We show ﬁrst that the equations of vibrations localized near such a defect coincide

with the equations of vibrations of a certain 1D crystal. The stacking fault does not,

generally, change the atomic mass; thus, it may seem that its inﬂuence on lattice vi-

brations is described only by the corresponding force matrix perturbations (just such

perturbations are usually taken into consideration in qualitatively discussing the prob-

lem). However, if the interatomic distances in the stacking fault differ from those in a

nondefect crystal lattice (h = a in Fig. 12.3) a local change in the mass density takes

place. We intend to account for this fact.

The planar defect running through the whole crystal does not break the lattice ho-

mogeneity in a plane perpendicular to the z-axis. Then the p erturbation potential U

∗

due to the change in the mass den sity along the defect layer can be written in the form

analogous to (11.1.7)

∗

(n, n



)=U



, U

= −ω

∆ρ

where ∆ρ is the change in the mass density. Assuming such a form of the perturbation

we unite two atomic layers (or crystallographic planes) on both sides of the plane

z = 0 in Fig. 12.3 into one planar defect.

290

12 Localization of Vibrations Near Extended Defects

The perturbation of the force matrix U(n, n



) in this crystal should be dependent

on the differences n

− n



and n

− n



and obey the requirement

∑

U(n

, n

; n

, n



∑

U(n

, n

; n



, n

)=0. (12.4.1)

Therefore, an equation of crystal vibrations including both types of perturbations is

(n) −

∑



α(n − n



)u(n



)

∑



U(n

− n



, n

− n



; n



)u(n



)+U

u(n

, n

,0)δ

(12.4.2)

Since the crystal is structurally uniform in the x0 y plane, it is convenient to switch

in the equations of motion from a site representation in this plane to a two-dimensional

k-represen tation, retaining the site rep resentation along the the z-axis. We represent

the amplitude of vibrations in the form

u(n)=χ

ia(k

)

, κ =(k

, k

and rewrite (12.3.2) in the κ-representation

) −

∑



− n



)χ



∑



, n



)χ



)+U

(0)δ

(12.4.3)

where

∑



α(n)e

−ia(k

)

, n



∑



U(n

, n

; n

, n



−ia(k

)

It is very important that the quantity κ appears in (12.4.3) as a parameter. If we

assume this parameter to be ﬁxed and are interested only in the dependence of the

vibration amplitude on z, then omitting the index κ in all the quantities in (12.4.3), we

obtain an equation for 1D crystal vibrations

χ(n) −

∑



Λ(n − n



)χ(n



∑



V(n, n



)χ(n



) , (12.4.4)

where n = n

and the matrix V(n, n



) is equal

V(n, n



)=U(n, n



)+U



The matrix Λ(n) has the following property

∑

Λ(n)=ω

(κ,0), where the function

(κ, k

) d etermines the dispersion relation of an ideal crystal.

12.4 Frequency of Local Vibrations in the Presence of a Two-Dimensional (Planar) Defect

291

In order to take into account a band structure of the dispersion relation and have

at the same time a possibility to obtain simple analytical results of calcu lations we

analyze the dynamics of the planar defect in a strongly anisotropic crystal (see Sec-

tion 2.10). Let the crystal has possess a primitive body-centered tetragonal lattice.

Assume that neighboring atomic layers perpendicular to the four-fold symmetry axis

have weak interlayer bonds and the stacking fault is parallel to such layers. The disper-

sion relation for a crystal of this type is given in the nearest-neighbors approximation

by (2.10.2) that determines the vibration band of the strongly anisotropic crystal.

Under the condition ω

 ω

the band width (for any value of k) is determined

by the weak interlayer interaction and is proportional to ω

. The maximum frequency

of the vibrations is determined by the strong interlayer interaction and is proportional

to ω

. As a result the band is very narrow and strongly elongated. This speciﬁc form

of the frequency distribution allows us to propose a simple description of the crystal

dynamics at low enough frequencies ω  ω

admitting ω ∼ ω

For the frequencies mentioned above we can use the long-wave approximation in re-

spect to κ-dependence, and the dispersion relation (2.10.2) acquires the form (2.10.4):

= ω

(κ, k

) ≡ s

+ ω

sin

; κ

= k

+ k

), s

, (12.4.5)

where b is the interatomic distance along the z-axis and s

is the sound velocity in the

basal plane.

If the value of κ is ﬁxed then (12.4.5) can be considered as a dispersion relation

for 1D crystal vibrations of the optical type. Remembering the results of Section 11.1

we expect to ﬁnd discrete frequencies of localized 1D modes outside the continuous

spectrum of an ideal crystal, namely the discrete frequencies squared should lie either

under ω

(κ) or above ω

(κ)+ω

. Assume the perturbation to be small enough

and refer to Section 1.1 where it was shown that in such a case (1) possible discrete

frequencies are located close to the edges of the continuous spectrum; (2) the planar

defect can be assumed to be concentrated in the plane z = 0. The later means that the

Kronecker delta δ

in the r.h.p. of (12.4.3) can be substituted with the Dirac delta-

function and the total perturbation matrix V

should be assumed to have the following

form

V(n, n



)=b

δ(z)δ(z



) , V

=[U

(κ)+U

(ω)] . (12.4.6)

Of course, (12.4.6) can be used only for calculations of the localized frequencies close

to boundaries of the continuous spectrum.

Due to the condition (12.4. 1) and the broken symmetry o f the problem, the function

(κ) has the following general expansion in powers of κ : U

(κ)=−W

αβ

(α, β = 1, 2). We substitute (12.4.6) into (12.4.4)

χ(z) −

∑



Λ(n − n



)χ(z



)=bV

χ(0 )δ(z) , (12.4.7)

292

12 Localization of Vibrations Near Extended Defects

and use the ordinary Fourier expansion

χ(z)=

2π



ikz

dk , χ

∑

χ(z)e

−ikz

. (12.4.8)

The equation for the Fourier component is then reduced to the relation



− ω

(κ, k)



= V

χ(0 ) . (12.4.9)

In the approximation taken for calculations, the dispersion law of the crystal concerned

is given by (12.4.5).

From (12.4.8), (12.4.9) we just ﬁnd the dependence of the vibration amplitude on

the coordinate z,

χ(z)=

2π

χ(0 )



−

cos kz dk

−ω

(κ, k)

. (12.4.10)

Setting z = 0 in (12.4.10), we ﬁnd an equation for the possible frequencies of such

vibrations

2π



−

−s

−ω

sin

= 1. (12.4.11)

Before calculations of the integral in (12.4.11) look on the sketch of the vibration

spectrum in Fig. 12.4. In Fig. 12.4. the hatched region corresponds to a band in the

continuous spectrum of vibrations of an ideal crystal at small κ. The bottom ω

low

(κ)

and top ω

(κ) boundaries of this band are given by the expressions

low

(κ)=s

κ, ω

(κ)=ω

+ s

. (12.4.12)

Notations proposed in (12.4.12) allow us to give the simple representation of the cal-

culation in (12.4.11):



− ω

low

(κ)



− ω

(κ) . (12.4.13)

Analyzing (12.4.12) and (12.4.13) we obtain localized states and their dispersion

relations of the following types.

1. V

> 0. In this case a frequency of the localized wave lies above the band of the

continuous spectrum



ω > ω

(κ)



=[ω

− ω

low

(κ)][ω

−ω

(κ)] . (12.4.14)

Considering small perturbations we suppose |V

|ω

. Under such a condition

(12.4.14) has the following solution

= ω

(κ)+



∆ρ

+ W

αβ



. (12.4.15)