Kosevich A.M. The crystal lattice

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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 multiplyin g (9.5 . 15) by th e

number of defects in a specimen and disregarding surface effects

ε

 =

4π

nε

= πγ

Ω

n, (9.5.17)

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 (Ω

> 0) or compression (Ω

< 0)

proportional to the defect concentration. However, since the result (9.5.17) is inde-

pendent 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.

≡ div u(r)=−

Ω

4π

∇

, r = 0.

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 th e 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 o f each site from its position in an ideal

lattice and ﬁnd the total increment of this vector in the path. We go around the dislo-

cation 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

 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 Burg-

ers 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” crys-

tal half-plane lies above an d 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 dislo-

cations (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: (

) two dislocation of op-

posite signs; (

) reproduction of an atomic plane after the dislocations

merge.

When τ  b the corresponding dislocation is called a screw dislocation. The pres-

ence of a linear screw dislocation in a crystal converts the lattice planes into a heli-

coidal surface similar to a spiral staircase. Figure 10.3 shows a scheme of atomic-plane

arrang ement 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 alon g the d islocation

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 o f 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



∂u

∂x

= −b

, (10.2.1)

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 dislo cation line is in this

case a line of singular points of the ﬁelds of strains and stresses

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 σ

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 S

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

, respectively. The “upward” direction (positive) is determined by the direction of a

normal n to the surface S

(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 S

surface

the distortion tensor u

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 S

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 S

surface, it will have on this surface a δ-like singularity



(S)

+ n

)δ(ζ), (10.2.3)

where ζ is the coordinate along the normal n.Thevalueζ = 0 corresponds to the S

surface.

10.2 Dislocations in Elasticity Theory

237

As there is no physical singularity in the space near a dislocation the stress tensor

as noted above should be a continuous function everywhere. Meanwhile, the stress

tensor σ

(S)

= λ

iklm



(S)

having a singularity on the surface S

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 S

with a specially chosen

density f

(S)

= −∇

(S)

= −λ

iklm

∇



(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

(r)=−λ

iklm



∇

(r − r



) dS



. (10.2.5)

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-lin e dislocation, when we

deal with the plane problem of elasticity theory, it is simpler to solve a n 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

∂x

= −b, (10.2.6)

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 c haracteristic 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 coin-

cides with the vortex ﬁeld in a liquid up to the notations.

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

in a scalar model have potential ﬁeld

u =

2π

θ,tanθ =

. (10.2.8)

238

10 Linear Crystal Defects

The ﬁeld (10.2.8) generates the vector ﬁeld h

2π

, h

= −

2π

, r

= x

+ y

. (10.2.9)

Since a screw dislocation with Burgers vector b parallel to the Oz-axis in an

isotropic medium is a singularity of the scalar ﬁeld, u

and its displacement ﬁeld

are u

= u

= 0, u

= 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 S

by the value b), we run into certain difﬁ-

culties of a physical character. Indeed, when the condition (10.2.2) was formulated we

supposed that crystal continuity is conserved along the surface S

. In particular, the

interatomic distances remain unchang e d (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) im-

plies 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 S

without discontinuities appearing in physical quantities

is impossible.

However, it follows from (10.3.1) that in a crystal there exists a speciﬁed surface

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 concerne d and is an envelope of the family of slip

planes of all dislo cation 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 possi-

ble 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 dislo-

cation 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 half-

plane (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 th e 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 p layed by the atomic layer occupying the position NN



(atoms are shown with light circles).

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

cation gliding.

Although the transition from the ﬁrst atomic conﬁguration to the second one is con-

nected with the d islocation 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 mi-

gration 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

necessary to initiate dislo cation motion are less by a factor of 10

− 10

than the

shear modulus of a monocrystal G. The smallness of the parameter σ

/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,orjustglide. Finally, it is often called a conservative motion.

A simple mathematical model allows us to understand certain features of the dislo-

cation 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 three-

dimensional 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 m ay 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 dislo ca-

tion, i. e., a starting stress σ

appears.

We estimate it from (9.4.7) determining the crystal energy dependent on a crow-

dion coordinate. The derivative of this energy with respect to the crowdion center

coordinate x

yields the force

f = −16 πα

a exp



−



sin



2π



where l

is the crowdion width; a is the interatomic distance (l

 a). This force

exists by virtue of the lattice discreteness (a  = 0) and is exponentially small with

respect to the parameter l

/a determining the degree of macroscopicity of a crowdion

core.

Extending the results obtained in a 1D model to the case o f a linear dislocation in

a 3D crystal, it is p ossible 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 con sid er an arbitrarily small dis-

placement 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 su rface area S

increasing by the crystal volume, a value that may be

characterized by an axial vector δS =[δXτ]dl.

As a result, the crystal volume exhib its an inelastic local increase that, from

(10.3.1), is equal to

δV = bδS = b[δXτ] dl = −[bτ]δX dl . (10.3.2)

If δV = 0, then the deﬁciency (δV > 0)orexcess(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 d islocation core. As each crystal site is

associated with unit cell volume V

, the quantity (10.3.2) should be associated with

the number |δV|/V

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)

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 mi-

gration 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

= δX[bτ]δ(ξ), (10.3.4)

where δ(ξ) is a 2D δ-function; ξ is a 2D radius vector in the p lane perpendicu lar to

the vector τ at a given point of the dislocation loop with its origin at the dislocation

axis.

Dislocation migration w ith a velocity limited by th e diffusion processes that pro-

vide 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)