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

12.4 Frequency of Local Vibrations in the Presence of a Two-Dimensional (Planar) Defect 293

One can see that the behavior of ωd as a function of κ depends on the signs of the

If W0 < 0 a dispersion relation for the localized wave is characterized by curve (1) in Fig. 12.4 for all κ. At W0 > 0, localized states can exist only for wave numbers 0 < κ < κ0 (curve (2) in Fig. 12.4). The value of κ0 for which the dispersion curve corresponding to the localized wave touches the top boundary of the band can be found from the equation

V0 (κ0 ) = 0.

Fig. 12.4 Dispersion lines of localized waves: (1) vibrations exist at all wave vectors; (2) the dispersion line ends on the boundary of the frequency band; (3) the dispersion line for localized vibrations of the type of a surface wave.

2. V0 < 0. Now (12.4.10) necessarily has a solution for ω < s1κ

12.4 Frequency of Local Vibrations in the Presence of a Two-Dimensional (Planar) Defect 295

surface vibrations it is natural to turn to elasticity theory making some assumptions on the properties of a planar defect that models the free crystal surface.

The vibrations localized near the planar defect are described as follows. The equations of elasticity theory for the bulk vibrations that occur with frequency ω have the form

and the elasticity modulus tensor of a crystal λiklm in an isotropic approximation is reduced to the two Lamé coefficients λ and G

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

If a solid has a defect coincident with the plane z = 0, the “perturbed” Lamé coefficients and the mass density are given by

λ = λ + Lhδ(z), G = G + Mhδ(z) ,

(12.4.24)

ρ = ρ + ∆ρhδ(z) = ρ [1 − (h − a)δ(z)] ,

where h is the “thickness” of the planar defect; L and M are characteristics of its elastic properties; a is the interatomic distance along the z-axis in an ideal crystal.

Substituting (12.4.24) into (12.4.23), we get a system of equations generalizing the scalar equation (12.4.7). It is clear that such a description of the planar defect has a literal meaning only for hk 1. Therefore, the singular functions describing the space perturbation localization in (12.4.24) must be treated with caution. Analyzing the vibrations near the linear defect, we have seen that using the δ-like perturbation in the long-wave approximation leads to introducing a finite upper limit of the integration over the wave vectors. Similarly, in the case of a perturbation of the type (12.4.24) caused by the planar defect, when we use the method of Fourier transformations along the z-axis, it is necessary to restrict the possible values of kz by the limiting value k0 1/h. In the case we are interested in it is determined by

The solution of a similar elasticity theory problem has shown that the frequency of waves localized near the defect splits off from the edge of the lowest-frequency band of vibrations (in the isotropic approximation, from the boundary of the transverse vibration frequency). This frequency is at a distance of δω κ3 from the band boundary, in agreement with (12.4.20).

This approach may also be used to study surface waves in an elastic half-space (Rayleigh waves), assuming that they are localized vibrations near the planar defect.

If this planar defect is a free crystal surface, than as a result of the perturbation of elastic moduli the connection between elastic half-spaces z > 0 and z < 0 vanishes.

296 12 Localization of Vibrations Near Extended Defects

Substituting (12.4.26) into (12.4.24), taking (12.4.25) into account, and using the described method of finding the possible frequencies of surface vibrations, one can show that these waves have the dispersion relation ω = ssκ, where ss coincides with the known velocity of Rayleigh waves (Kosevich and Khokhlov, 1970). We do not prove this statement here because the calculations are cumbersome.

13

Elastic Field of Dislocations in a Crystal

13.1

Equilibrium Equation for an Elastic Medium Containing Dislocations

Our initial definition of a dislocation

L

may be rewritten in a somewhat different form using the notation of a distortion tensor

uik = iuk

L

In dislocation theory the elastic distortion tensor may be regarded as an independent quantity describing the crystal deformation. Similarly to the strain tensor ε ik and the stress tensor σik it is a single-valued function of the coordinates even in the presence of a dislocation.

Let us write down the condition (13.1.2) in a differential form. For this purpose we transform the integral around the contour L into an integral over any surface S spanning this contour:

dxiuik = dSieilm l umk.

L

The constant vector b is represented in the form of an integral over the same surface by means of the 2D δ-function introduced in (10.3.4)

S

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

298 13 Elastic Field of Dislocations in a Crystal

As the contour L is arbitrary, the equality of these integrals means the equality of the integrand expressions:

This is just the desired differential form of (13.1.1). It is clear that on the dislocation line (ξ = 0), which is a line of singular points, a representation in the form of derivatives is meaningless.

If there are simultaneously many dislocations in a crystal with relatively small separations (but large compared to the lattice constant), their averaged consideration becomes important. This consideration is useful in problems where an exact field distribution between separate dislocations is of no interest and in which a theory operates with physical quantities averaged over small volume elements. It is clear that many dislocation lines should run through such “physically infinitesimal” volume elements.

The equation that expresses the main property of dislocation deformations is formulated by generalizing (13.1.4). We introduce the dislocation density tensor αik by requiring its integral on the surface spanned on any contour L to be equal to the sum of Burgers vectors b of all dislocation lines enveloped by this contour:

which in the case of a single dislocation shows that the Burgers vector is constant along the dislocation line.

With such an interpretation of dislocations, the tensor uik becomes the primary quantity that describes the crystal deformation. The displacement vector u cannot then be introduced. Indeed, at uik = iuk the l.h.s. of (13.1.4) would identically become zero over the whole crystal volume.

Equations (13.1.6) or (13.1.4), together with

and Hooke’s law constitute a complete system of equilibrium equations of an elastic medium with dislocations.

13.2 Stress Field Action on Dislocation 299

In a scalar elastic field the role of a distortion tensor is played by the vector h that in a medium without vortices is determined by

where α is the density of vortices. For a separate vortex, α = τbδ(ξ).

An analog of the stress tensor is the vector σ = Gh that is determined by the elastic

13.2

Stress Field Action on Dislocation

Let us consider a dislocation loop D in a field of external (with respect to a dislocation) elastic stresses σik and find a force acting on the dislocation. We calculate the work done by external forces δR for an infinitely small movement of the loop. If this work is given as

D

where δX is the displacement element of a dislocation line, F will determine the force with which an elastic stress acts on a unit length of the dislocation.

Let the dislocation displacement generate a certain change in the displacement vector δu. The work of external stresses performed in a volume V with a dislocation loop is

where S∞ is the surface enveloping the volume V.

In the given case it is convenient to use transformations of the integral (13.2.2) based on Gauss’s theorem. Therefore the displacement u near a dislocation should be considered as a single-valued function of the coordinates. Then, the vector u will have a discontinuity along the surface SD spanned on the dislocation loop. To exclude discontinuity points of the vector u, we surround the loop D with a certain closed surface SD outside which (in the volume V ) the function u = u(r) is continuous.

13.2 Stress Field Action on Dislocation 301

If we disregard the inhomogeneity of material generated by a dislocation and that is unimportant in evaluating the interaction between the dislocation and an elastic field, the volume integral remaining on the r.h.s. of (13.2.3) is simplified. Indeed, when the cut banks come closer (h → 0) and the tube radius decreases (ρ → 0), the integral over the volume V , because of continuity of σik and ε ik, transforms into an integral over the whole volume

The surface integral in (13.2.3) can also be simplified by a corresponding limiting transition. We note that the integral over the surface of the tube St vanishes as ρ → 0, since the dislocation field has the following obvious property:

lim ρu(r) = 0.

ρ→0

On the cut banks still present in the integrals, the values of the continuous functions σik in the limit are the same and the limiting values of u are different in a constant value b. Thus, instead of (13.2.3), we get

SD

where σik is the stress deviator and ε ik is the strain deviator [(ε ik = ε ik − (1/3)δikε ll)]. The first two terms of the integral (13.2.3) due to the obvious identity in (13.2.5) follow from

In the last term on the r.h.s. in (13.2.5) the symbol of an infinitely small variation δ is taken outside the integral sign since the stress distribution σik is assumed to be given.

If the element of the dislocation line dl is displaced by δX the area of the surface element SD changes by

The relation (13.2.6) should be used in transforming the last integral in (13.2.5).

If the displacement δX is in the dislocation slip plane, (13.2.6) characterizes the changes in the crystal completely. If the displacement is perpendicular to the slip plane, additional conditions have to be taken into account that follow from the medium continuity. When the dislocation moves without breaking the medium continuity the local relative variation of the volume given by (10.3.4) exists. Using this formula, a formal replacement can be made to transform the second integral in (13.2.5)

302 13 Elastic Field of Dislocations in a Crystal

Taking into account (13.2.6), (13.2.7), we single out in the work δR the part connected with inelastic medium deformation that accompanies the dislocation migration:

The first integral in (13.2.8) equals the increase in elastic energy of a medium in a solid volume. The linear integral over the dislocation loop

will determine the work done for a dislocation displacement. Comparing (13.2.1), (13.2.9) it is clear that the force acting per unit length is

A formula such as (13.2.10) where the stress deviator σik is replaced by the stress tensor σik was first obtained by Peach and Koehler (1950). The necessity of taking into account an inelastic change in the medium volume in climbing of a dislocation, i. e., the necessity to replace σik by σik − (1/3)δikσll, was indicated by Weertman (1965).

As simple examples of using (13.2.10), we consider the forces acting on screw and edge dislocations. Let the Oz-axis be parallel to the dislocation line (τz = −1). In the case of a screw dislocation, bz = b and

In the case of an edge dislocation, we direct the Oz-axis along its Burgers vector (bx = b). Then,

It is interesting to determine the projection of the force (13.2.10) on the slip plane of the corresponding dislocation element. Let κ be a vector that is perpendicular to the dislocation line in the slip plane. Then, this projection (we denote it as f ) is equal

where n = [kτ] is the vector normal to the slip plane.

Since the vectors n and b are mutually perpendicular, by choosing two of the coordinate axes along these vectors, we see that the force f is determined by one of the components σik only. If the dislocation is a plane curve lying in its slip plane and is in a homogeneous elastic stress field then the force f is the same for all dislocation line elements, independent of their position on the slip plane.