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

232 9 Point Defects

varying at distances that greatly exceed the average distance between the defects. Then we calculate the mean field of all images of the defects multiplying (9.5.15) by the number of defects in a specimen and disregarding surface effects

εimll =	4π	R3 nε1ll = πγ	4G	Ω0 n,	(9.5.17)
	3		3K

where n is the number of dilatation centers in a unit volume.

We see that although in an unbounded crystal the dilatation centers do not generate relative volume change in the elastic medium between defects, in a finite-dimension specimen there always exists a finite dilatation (Ω0 > 0) or compression (Ω0 < 0) proportional to the defect concentration. However, since the result (9.5.17) is independent of the specimen dimension R, it should also remain valid in the limit R → ∞ for n = constant.

9.5.1

Problem

1. Find the hydrostatic compression of an isotropic medium around an elastic dipole with a unit vector l of its axis.

	Solution.	Ω0	1
	ε kk ≡ div u(r) = −				, r = 0.
			lilk i k
		4π		r

10

Linear Crystal Defects

10.1 Dislocations

Dislocations are linear defects in a crystal near which the regular atomic arrangement is broken. In a theoretical treatment, dislocations in real crystals perform as important a role as electrons do in metals.

There are many microscopic models of dislocations. In the simplest model the dislocation is taken to be the edge of an “extra” half-plane present in the crystal lattice. In the conventional atomic scheme of this model where the trace of the half-plane coincides with the upper semiaxis Oy (Fig. 10.1), the edge of the extra half-plane on the z-axis, is called an edge dislocation. The regular crystal structure is then greatly distorted only in the near vicinity of the isolated line (the dislocation axis) and the region of irregular atomic arrangement has transverse dimensions of the order of a lattice constant. If we surround the dislocation with a tube of radius of the order of several interatomic distances, the crystal outside this tube may be regarded as ideal and subject only to elastic deformations (the crystal planes are connected to one another almost regularly) and inside the tube the atoms are considerably displaced relative to their equilibrium positions and form the dislocation core. In Fig. 10.1 the atoms of the dislocation core are distributed over the contour of the shaded pentagon.

Nevertheless, deformation even occurs far from the dislocation. The deformation at a distance from the dislocation axis may be seen by tracing a path in the plane xOy (Fig. 10.1) through the lattice sites along the closed contour around the dislocation core. We consider the displacement vector of each site from its position in an ideal lattice and find the total increment of this vector in the path. We go around the dislocation axis along the external contour starting from the upper left angle (atom 1) and see that the atomic displacement at the end of the path is nonzero and equal to one lattice period along the x-axis. This singularity of the dislocation deformation can be considered as the initial one when we define a dislocation in a 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

234 10 Linear Crystal Defects

Fig. 10.1 A scheme of atomic arrangement in the vicinity of an edge dislocation.

We denote the vector connecting atoms 1 and 2 by b. This vector is called the Burgers vector of a dislocation. The possible values of the Burgers vectors in an anisotropic solid are determined by its crystallographic structure and correspond, as a rule, to a small number of certain directions in a crystal. The dislocation lines are arranged arbitrarily, although their arrangement is limited by a set of definite crystallographic planes.

Let τ be the unit vector of a tangent to the dislocation line. For an edge dislocation τ b. Edge dislocations with opposite directions of b differ in that the “extra” crystal half-plane lies above and below the xOz plane (Fig. 10.2a). If dislocations such as 1 and 2 are observed in a crystal simultaneously, they are called opposite-sign dislocations (for instance, the first one may be called a positive edge dislocation). When opposite-sign dislocations merge, annihilation takes place, resulting in the elimination of two defects and in a “reunification” of the regular atomic plane.

Fig. 10.2 Annihilation of edge dislocations: (a) two dislocation of opposite signs; (b) reproduction of an atomic plane after the dislocations merge.

When τ b the corresponding dislocation is called a screw dislocation. The presence of a linear screw dislocation in a crystal converts the lattice planes into a helicoidal surface similar to a spiral staircase. Figure 10.3 shows a scheme of atomic-plane arrangement in the presence of a screw dislocation coinciding with the OO line.

10.2 Dislocations in Elasticity Theory 235

Fig. 10.3 Screw dislocation in a crystal.

If the dislocation line is not perpendicular and not parallel to the Burgers vector, it is called a segment of mixed type. Dislocation segments of an edge, screw and mixed type can arrange themselves continuously along a line forming a dislocation line. The dislocation line cannot end inside a crystal. It must either leave the crystal with each end at the crystal surface or (as is generally observed in real cases) form a closed dislocation loop. It is clear that the Burgers vector is constant along the dislocation line.

A crystal lattice with dislocations will sometimes be called a dislocated lattice (or a dislocated crystal).

10.2

Dislocations in Elasticity Theory

The main property of a dislocation implies that when a circuit is made around the dislocation line the total increment of the elastic displacement vector is nonzero and equal to the Burgers vector. Thus, a dislocation in a crystal will be said to be a specific line D having the following general property: after a circuit around the closed contour L enclosing the line D (Fig. 10.4) the elastic displacement vector u changes by a certain finite increment b equal (in value and direction) to one of the lattice periods. This property is written as

L dui =	∂ui	dlk = −bi,	(10.2.1)
	L ∂xk

assuming that the direction of the circuit is related by the corkscrew rule with a chosen direction of the tangent vector τ to the dislocation line. The dislocation line is in this case a line of singular points of the fields of strains and stresses1.

1) We do not consider the dislocation line as a local inhomogeneity of a crystal.

236 10 Linear Crystal Defects

Fig. 10.4 Mutual orientations of the vector n and τ.

The majority of the essential physical properties of dislocations is not connected with microscopic models and can be described phenomenologically in the framework of elasticity theory using a similar definition.

From a mathematical point view, the condition (10.2.1) means that in the presence of a dislocation the displacement vector is a many-valued function of the coordinates that receives an increment in a passage around the dislocation line. In this case there is no physical ambiguity: the increment b denotes an additional displacement of crystal atoms by one lattice period that, due to the translational invariance, does not change its state. In particular, the stress tensor σik characterizing the elastic crystal state is a single-valued and continuous function of the coordinates.

In a crystal with a separate dislocation, instead of many-valued function u(r), we can regard the displacement vector u as a single-valued function that undergoes a fixed jump b on some arbitrary chosen surface SD spanning the dislocation loop D:

δu ≡ u+ − u− = b,

(10.2.2)

where u+ and u− are the u(r) values from the upper and lower sides of the surface SD , respectively. The “upward” direction (positive) is determined by the direction of a normal n to the surface SD (this direction is connected by the corkscrew rule with the τ-vector direction, Fig. 10.4). If the jump δu is the same at all points of the SD surface the distortion tensor uik is a continuous and differentiable function on this surface.

Using (10.2.2) we can formally give another definition of a dislocation, namely, by defining it as the line D on which the surface SD with given jump (10.2.2) of the vector u(r) is spanned. In some cases this definition is more convenient than the initial one, e. g., using it we can easily find the field around the dislocation. If we calculate the strain tensor for a crystal with a dislocation, i. e., in the presence of the jump (10.2.2)

	on the SD surface, it will have on this surface a δ-like singularity
	(S) =	1	(ni bk + nk bi )δ(ζ ),	(10.2.3)
		2
	ik

where ζ is the coordinate along the normal n. The value ζ = 0 corresponds to the SD surface.

10.2 Dislocations in Elasticity Theory 237

As there is no physical singularity in the space near a dislocation the stress tensor σik as noted above should be a continuous function everywhere. Meanwhile, the stress

(S) = (S)

tensor σik λiklm lm having a singularity on the surface SD is formally associated with the strain tensor (10.2.3). To eliminate this stress tensor, it is necessary to define

the function’s body forces distributed over the surface SD with a specially chosen density f (S):

fi(S) = − k σki(S) = −λiklm k lm(S).

(10.2.4)

Thus, the problem of finding a many-valued function u(r) is equivalent to that of finding a single-valued but discontinuous function in the presence of body forces (10.2.3), (10.2.4). Substituting (10.2.4) into (9.5.1) and performing the integration we obtain

ui (r) = −λiklmbm nl k Gij(r − r ) dS .	(10.2.5)
SD

In principle, (10.2.5) allows one to obtain elastic displacements in a crystal when the form of the dislocation loop is arbitrary. The general formula (10.2.5), however, is complicated and the calculation of a displacement field even with simple dislocation line shapes is quite cumbersome. In the case of a straight-line dislocation, when we deal with the plane problem of elasticity theory, it is simpler to solve an equilibrium equation under the condition (10.2.1).

Studying the lattice dynamics, we used a scalar model to simplify the calculations. To the same aim we clarify to what the dislocation-type linear defect corresponds in a scalar elastic field model.

Let b characterize the linear defect intensity of a scalar field u and the defect itself is defined by

du =	∂u	dxk = −b,	(10.2.6)
	∂xk

LL

which plays the role of a boundary condition for the field equation (2.9.21). In a static case, (2.9.21) reduces to the Laplace equation

∆u = 0.

(10.2.7)

If we introduce the vector h = grad u as a characteristic of the field state then the fixed circulation of this vector along any closed contour enclosing the defect line will be determined by (10.2.6). A similar defect is a vortex of the field h. Thus, a dislocation in an elastic field is an analog of a vortex of some scalar field.

It follows from (10.2.6), (10.2.7) that the dislocation field in a scalar model coincides with the vortex field in a liquid up to the notations.

In particular, a rectangular vortex perpendicular to the plane xOy and a dislocation

in a scalar model have potential field
u =	b	θ,	tan θ =	y	.	(10.2.8)
	2π
				x

238

10 Linear Crystal Defects

The field (10.2.8) generates the vector field h

hx =

b y

,

hy = −

b x

,

r2 = x2 + y2.

(10.2.9)

2π

r

2π

r

Since a screw dislocation with Burgers vector b parallel to the Oz-axis in an isotropic medium is a singularity of the scalar field, uz and its displacement field are ux = uy = 0, uz = u, where the function u is given by (10.2.8).

10.3

Glide and Climb of a Dislocation

The definition of a dislocation (10.2.2) is a formal tool allowing us to solve some static elasticity problems in a medium with dislocations. If we associate the vector u(r) having a discontinuity (10.2.2) with real atomic displacements in a crystal and try to reproduce the real process of dislocation generation (via relative displacements of atomic layers on both sides of the surface SD by the value b), we run into certain difficulties of a physical character. Indeed, when the condition (10.2.2) was formulated we supposed that crystal continuity is conserved along the surface SD . In particular, the interatomic distances remain unchanged (up to elastic deformations). However, when (10.2.2) is understood formally it is clear that crystal continuity is violated. In fact, when the cut boundaries are displaced by b the crystal volume changes inelastically

δV = nbδS,

(10.3.1)

per each element δS of a discontinuity surface. Therefore, the condition (10.2.2) implies that we “eject” the material where the atomic layers overlap under displacement and fill in the remaining “gaps” with additional material. However, a crystal has no mechanisms of automatic removal or supply of material in a solid. Thus, a purely mechanical way of dislocation generation through displacement of the atomic layers along an arbitrary surface SD without discontinuities appearing in physical quantities is impossible.

However, it follows from (10.3.1) that in a crystal there exists a specified surface Ssl at each point of which nb = 0 and the displacement described is shear-like with no effect on the crystal continuity. It is clear that it is a cylindrical surface whose elements are parallel to the vector b and its directrix is a dislocation loop. It is called a slip surface of the dislocation concerned and is an envelope of the family of slip planes of all dislocation line elements. By the slip plane of a dislocation element we understand a surface tangent to the corresponding element of the dislocation line and this plane is determined by the vectors τ and b. Possible systems of slip planes in an anisotropic medium are determined by the structure of a corresponding crystal lattice.

A slip plane is singled out physically because a dislocation-induced shift is possible along it (the interatomic distances in the vicinity of a slip plane surface remain

10.3 Glide and Climb of a Dislocation 239

unchanged after a shift). A comparatively easy mechanical displacement of the dislocation is possible in this plane. The latter follows directly from a microscopic picture of the dislocation defect and is demonstrated by means of a scheme with an extra halfplane (Fig. 10.5). Let an edge dislocation be generated by a shift b along a slip plane whose trace in Fig. 10.5 coincides with a crystallographic direction AB. We consider two atomic configurations near the dislocation core when an extra atomic half-plane is in the position MM (atoms are marked with black circles) and also when the role of an extra crystal half-plane is played by the atomic layer occupying the position N N (atoms are shown with light circles).

Fig. 10.5 A scheme of atomic-layer rearrangement with the edge dislocation gliding.

Although the transition from the first atomic configuration to the second one is connected with the dislocation migrating one interatomic distance to the right in the slip plane, the displacements of individual atoms are small compared to the value of b. This means that such collective atomic rearrangements that provide a dislocation migration may be realized under the action of comparatively small forces. If we compare such forces with macroscopic loads, it turns out that the corresponding shear stresses σs necessary to initiate dislocation motion are less by a factor of 102 − 104 than the shear modulus of a monocrystal G. The smallness of the parameter σs/G is a crucial physical factor that allows us to use linear elasticity theory to describe the mechanical processes accompanied by the motion of dislocations. Thus, a dislocation may move comparatively easily in its own slip plane. This motion of a dislocation is generally called sliding or gliding, or just glide. Finally, it is often called a conservative motion.

A simple mathematical model allows us to understand certain features of the dislocation dynamics and to explain qualitatively the high mobility of the dislocations.

Let us imagine that a chain of atoms (Fig. 1.11) is an edge series of one half of a plane crystal (y > 0, Fig. 10.6) displaced in a certain way with respect to another half of a crystal (y < 0, Fig. 10.6). Then the influence of the nondisplaced half of a crystal on the atoms distributed along the x-axis can be qualitatively described using the energy (1.6.4).

240 10 Linear Crystal Defects

Crystallographic atomic rows with an extra half of the atomic row (Fig. 10.6) above the x-axis represent a model of the dislocation in a crystal. Thus, the scheme described may be an analog of the problem of dislocations in a two-dimensional and threedimensional crystal. Such an interpretation of this model was performed by its authors (Frenkel and Kontorova, 1938).

Fig. 10.6 Edge dislocation in the Frenkel–Kontorova model.

We remind ourselves of an important result obtained in analyzing the crowdion dynamics (Section 9.4). In the continuum approximation (a crystal is considered as a continuous elastic medium) the crowdion energy is independent of the coordinate of its center, and thus the crowdion may move freely along a 1D crystal. A dislocation has the same property.

However, if we take into account the discreteness of the crystal, a resistance force arises that must be overcome to start the motion of either a crowdion or of a dislocation, i. e., a starting stress σs appears.

We estimate it from (9.4.7) determining the crystal energy dependent on a crowdion coordinate. The derivative of this energy with respect to the crowdion center coordinate x0 yields the force

f = −16πα0 a exp −	π2 l0	sin 2π	x0	,
	a		a

where l0 is the crowdion width; a is the interatomic distance (l0 a). This force exists by virtue of the lattice discreteness (a = 0) and is exponentially small with respect to the parameter l0 /a determining the degree of macroscopicity of a crowdion core.

Extending the results obtained in a 1D model to the case of a linear dislocation in a 3D crystal, it is possible to expect that they are valid as qualitative statements. The latter makes it possible to explain the smallness of a starting stress of the dislocation by the macroscopicity of its core and understand why glide is comparatively easy.

Quite a different physical nature is inherent to the real migration of a dislocation in a direction perpendicular to the slip plane. We consider an arbitrarily small displacement δX of an element of the dislocation loop length dl, assuming δX to have

10.4 Disclinations 241

a component normal to the slip surface. Such a migration of a dislocation element results in the surface area SD increasing by the crystal volume, a value that may be characterized by an axial vector δS = [δX τ]dl.

As a result, the	crystal volume exhibits an inelastic local	increase that, from
(10.3.1), is equal to
	δV = bδS = b[δX τ] dl = −[bτ]δX dl.	(10.3.2)

If δV = 0, then the deficiency (δV > 0) or excess (δV < 0) of material cannot be balanced in the ideal crystal volume in a mechanical way (if continuity is conserved). However, in a real crystal a slow-acting mechanism of condensation or rarefaction of a substance exists, requiring no macroscopic breaks of continuity. We are referring to the processes of formation and diffusion migration of point defects: atoms in the interstices (the material condensed) and vacant sites (the material rarefied). Thus, an inelastic increase in the volume (10.3.2) on the dislocation axis should be compensated by an equal decrease in the crystal volume due to the formation of a corresponding number of point defects in the vicinity of a dislocation core. As each crystal site is associated with unit cell volume V0 , the quantity (10.3.2) should be associated with the number |δV| /V0 arising from vacancies or vanishing interstitial atoms. However, since point defects of both types may vanish or be generated, the change in their number is related to the displacement of the dislocation line element by

δN = [bτ] δX dl, (10.3.3)

V0

where δN is the difference in numbers of interstitial atoms and vacancies produced. The point defects for which (10.3.3) is written appear or vanish just near the dislo-

cation core. Therefore, in macroscopically describing the dislocation motion the total variation in crystal volume can be concentrated at the dislocation line. Thus, the migration of a dislocation in a direction perpendicular to the slip plane is accompanied by a local increase in crystal volume with a relative value given by

δ 0	= δX [bτ]δ(ξ),	(10.3.4)
kk

where δ(ξ) is a 2D δ-function; ξ is a 2D radius vector in the plane perpendicular to the vector τ at a given point of the dislocation loop with its origin at the dislocation axis.

Dislocation migration with a velocity limited by the diffusion processes that provide changes in the volume (10.3.4) is called climb or nonconservative motion of a dislocation.

10.4 Disclinations

The definition of a dislocation based on (10.2.2) reflects an important property of deformation in a continuous medium. If the function u(r) exhibits a jump (10.2.2)