Kosevich A.M. The crystal lattice

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13.5 Dislocation Field in a Sample of Finite Dimensions

313

to the cylinder equilibrium equations should involve, apart from (13.3.3), additional

stresses compensating the twisting moment (13.5.1), i. e., creating an average moment

= −M

. These stresses (and also the corresponding displacements) are easily

obtained from a rod torsion theory. It is known that for a rod twisted under the action

of the moment M

, there arises a displacement vector component u

providing a

torsion angle that is homogeneous along the rod length:

dθ

∂

∂z





∂u

∂z

= constant,

where C =(1/2)πGR

is the torsional rigidity of a rod.

Taking this into account, it is easy to write an equilibrium distribution of additional

displacements and stresses in a cylinder with a screw dislocation along its axis

= −

br z

πR

, δσ

zϕ

= −

2π

r. (13.5.2)

The second term on the r.h.s. of (13.5.2) shows the role of ﬁnite dimensions of the

specimen in generating the elastic stresses around the dislocation. With increasing

cylinder radius R, the contribution of the last term decreases for r  R. The torsion

angle also decreases:

dθ

= −

πR

However, the total torsion of the cylindrical specimen

δθ = −

πR

, (13.5.3)

for L  R may be essential. Torsional deformation is observed in long and thin

thread-like crystals with a screw dislocation (whiskers).

Now consider a superstructure formed by a system of a large number of screw dislo-

cations oriented along the axis of the cylinder of radius R and we will be interested in

the macroscopic properties of such a superstructure. Then, in the main approximation

the distribution of the dislocations can be assumed continuous, characterized by the

density n

= 1/S

,whereS

is the average area o f the xy plane per dislocation.

The dislocation creates a stress (13.3.3) in an unbounded media (denote it as σ

But because the stress ﬁeld of the screw dislocations is similar to the electric ﬁeld of

linear charges, the stresses at a distance r from the axis of the cylinder are created by

all the dislocations intersecting an area S = πr

around the axis of the cylinder and

are equal to the stresses around one dislocation lying along the axis of the cylinder

(x = y = 0) and carrying the total “charge” (the Burgers vector bS/S

)ofallthose

dislocations:

≡ σ

Gbn

r. (13.5.4)

These stresses, ﬁrst, create a force actin g on a d islocation lying a distance r from

the axis of the cylinder:

= bσ

r . (13.5.5)

314

13 Elastic Field of Dislocations in a Crystal

Second, the dislocation ﬁeld (13.5.4) creates the following moment of torque on the

end of the cylinder:



rσ

dS = 2π



dr =

GbR

. (13.5.6)

One sees that (13.5.6) differs from (13.5.1) with the multiplier (1/2)πR

. Conse-

quently, in order to calculate strains and stresses produced by the dislocation system

in the cylinder it is sufﬁcient to multiply (13.5.2) by this multiplier:

= −

rz, δσ

= −

Gbn

r . (13.5.7)

The stresses described by the second relation in (13.5.7) can be looked upon as a

certain external ﬁeld in relation to the dislocation system. Comparing (13.5.7) with

(13.5.4) and (13.5.5) one can see that the interaction force of a given dislocation with

remaining continuously distributed dislocations is exactly compensated by these “ex-

ternal stresses”. This means that the expected repulsion of discrete dislocations calcu-

lated according to formula (13.5.5) with the use of (13.3.3) is eliminated on average

when the boundary conditions and symmetry of the problem with a continuous distri-

bution of dislocations are correctly taken into account. In other words, the equilibrium

state of such a dislocation system is stabilized by the twisting of the sample.

Another more trivial example of the inﬂuence of boundary conditions for a free

crystal surface on the distribution of the elastic ﬁelds around the dislocation is the

problem of a screw dislocation parallel to the plane of a free surface of an isotropic

medium. Let the plane zOy coincide with a su rface of the solid, and the dislocation

parallel to the z-axis have the coordinates x = x

, y = 0. The stress ﬁeld leaving

the free medium surface is described by the sum of dislocation ﬁelds and its mirror

reﬂection in the plane yO z as if they were in an unbounded medium

= −

2π



(x − x

)

+ y

−

(x + x

)

+ y



2π



x − x

(x − x

)

+ y

−

x + x

(x + x

)

+ y



(13.5.8)

The dependence of stresses (13.5.8) on the distance x

is typical for a ny linear dis-

location near the crystal surface and is very important in discussing various boundary

problems.

13.6

Long-Range Order in a Dislocated Crystal

According to (13.3.5), far from the dislocation an unexpected dependence of u

on r

∼ b ln r) is observed. This dependence is quite natural, if we consider the dis-

13.6 Long-Range Order in a Dislocated Crystal

315

placement as a strain (or stress) ﬁeld “potential” generated by a linear source. The

vector u has a simple physical meaning in a crystal with an isolated dislocation. It de-

termines the displacement of an atom in a dislocated crystal lattice with respect to its

equilibrium position in the same lattice without a dislocation. Thus, the displacements

of atoms extremely distant from the linear dislocation axis increase logarithmically

with increasing crystal dimension.

Although the relative displacements of the neighboring atoms, g iven by the defor-

mation tensor proportional to 1/r, are vanishingly small as r → ∞ , it is necessary

to specify more accurately the n otion of crystal order in a dislocated lattice since the

displacement vector increases at large distances from the dislocation. Since disloca-

tions can generate arbitrarily large atomic displacements without breaking the crystal

structure, it is unn ecessary to require in describing the crystalline ordering in a real

crystal that the same periodic spatial atomic network is preserved in the whole space.

To determine the crystal order, we consider the scheme of a deformed crystal

(Fig. 7.7) used to explain the effect of lattice deformation on the phonon spectrum.

A system of atoms has no space periodicity, and the “unit cells” in its different parts

differ in form and dimension, but it is nevertheless considered as a deformed crystal.

To make this system ordered, we introduce a curvilinear net describing at each point

of the space a quite deﬁnite crystal structure. Going along this net, it is possible to es-

tablish a relation between a local short-range order and that in any p art of the crystal.

The point defects do not break the general structure of the net.

In a crystal that has several dislo cations (Fig. 13.6), in addition to local crystal net

distortions, there are “topological” faults of the spatial lattice that signiﬁcantly deform

the long-range order.

Fig. 13.6 A dislocated crystal with topological long-range order.

The increase in the density of the opposite-sign dislocations makes the whole struc-

ture amorphous without breaking the short-range order. Thus, an amorphous solid can

be considered as the limiting state of a crystal with a large numbe r of dislocations such

316

13 Elastic Field of Dislocations in a Crystal

that the average distance between them is of a comparable order of magnitude as the

lattice period.

However, we are interested in the crystal state, whose existence implies that the

long-range order distortions induced by dislocations are small. These distortions can

be displayed during the following process of observation. Basing our treatment on the

local short-range order in a crystal, we pass around a macroscopic contour that would

be closed in an ideal crystal. For instance, if we pass an external contour in a counter-

clockwise direction (Fig. 13.6) starting from the point a, it will not be closed and we

will arrive at th e po int e noncoincident with the contour origin. The impossibility of

closing a similar contour characterizes the fault in the long-range order in the crystal.

The situation when continuous “short-range ordering” in passing around a closed

contour cannot result in a common short-range order at all contour points has been

called (in a theory of crystal singularities) frustration.

In the given case the degree of “frustration” in the structure produced is character-

ized by the vector (ae), which is equal to the total Burgers vector B of all dislocations

enveloped by tracing around a contour. Thus, a quantitative measure of breaking the

long-range order in a certain part of a dislocated crystal is the total Burgers vector of

dislocations coupled with a “closed” macroscopic contour that envelopes the crystal

part concerned.

We denote by L the length of the contour abcde (Fig. 13.6). For the system under

consideration, the in e quality B  L is satisﬁed. If a similar inequality is satisﬁed for

any macroscopic contour in a dislocated crystal then the long-range order fault can be

assumed to be unessential.

For the atomic system concerned, in determining the crystal order we replace the

literal translational symmetry of an ordered system by a set of the following structural

properties.

1. The system has a short-range order given entirely by the crystallographic struc-

ture of substance. To formulate this p roperty, it is necessary to exclude the

vicinities of point defects or of certain speciﬁc lines–dislocation cores.

2. During a continuous motion from a spatial point to the neighboring one, it is

possible to establish a correspondence between equivalent atoms and crystallo-

graphic directions in the unit cells positioned close to one another.

3. The total Burgers vector coupled with any closed macroscopic contour inside

the system remains small compared to the length of this contour. A closed

contour is constructed by using the short-range order at each of its intermediate

points, and the Burgers vector coupled with it equals the displacement of an

atom at the end of the contour relative to an atom at its beginning.

It is natural to consider the atomic system characterized by these pr operties as a

crystal with long-range order. A real crystal has long-range order only in this sense.

13.6 Long-Range Order in a Dislocated Crystal

317

We note that the suggested deﬁnition of crystal long-range order does not give the

information about the smallness of atomic displacements with respect to their equi-

librium positions in the initial crystal lattice. Sometimes, the long-range order deter-

mined in this way is called topological order.

The vanishing of topological long-range order means the destruction o f a crystal

structure. When the absence of long-range order in a solid substance is not accompa-

nied by the loss of shear elasticity, we assume that a solid is in an amorphous state. If

the long-rang e order vanishes with increasing tem perature simultaneously with shear

elasticity (the substance starts leaking), we can speak of crystal melting. Unfortu-

nately, there is no systematic theory of 3D crystal melting.

The situation is opposite in a 2D case. In a 2D crystal, a dislocation is a point

defect, whose 2D displacement vector ﬁeld u(u

, u

) is not different from that of

a linear edge dislocation in a 3D crystal (13.3.5). The energy of this defect can be

obtained by multiplying the linear dislocation energy per un it length by the lattice

constant a:

E = ε

, ε

aGb

4π(1 −ν)

. (13.6.1)

Since the energy ε

is comparable with the atomic energy of a 2D crystal and the de-

pendence on crystal dimension (13.6.1) is very weak, a thermo-ﬂuctuational initiation

of the dislocation can be assumed in a 2D crystal.

With given temperature T, the thermodynamic equilibrium condition is determined

by the minimum of the free energy of the solid F. We consider the change in F associ-

ated with the appearance of one dislocation and note that the dislocation contribution

to the conﬁguration entropy in a 2D crystal is

δS = ln

. (13.6.2)

Then

δF = E − TδS =(ε

−2T) ln

. (13.6.3)

It is clear that for T < (1/2)ε

the appearance of a dislocation increases the free

energy of the system and, thus, the dislocatio n initiated ﬂuctuationally will inevitably

escape from the crystal. In total thermodynamic equilibrium there are no isolated free

dislocations.

For T > (1/2)ε

, the free energy of the system decreases with the appearance of

a dislocation and thus the pro cess of their initiation in a 2D crystal becomes advanta-

geous thermodynamically. Hence, with increasing temperature at T = T

=(1/2)ε

there occurs a phase transition of a 2 D crystal into a state with an arbitrary number of

free dislocations (Berezinsky, 1970, Kosterlits, Taules, 1973). As a result, the initial

long-range order in a 2D crystal is broken.

It is easy to follow what mechanism is responsible for breaking the 2D crystal long-

range order. At low (T < T

) but ﬁnite temperatures a certain equilibrium density

318

13 Elastic Field of Dislocations in a Crystal

of dislocation dipoles exists in a crystal. This is the case because the energy of an

opposite-sign dislocation pair, in contrast to (13.6.2) is independent of crystal dimen-

sion. Indeed, the dip ole en ergy can be written as E

= 2E + E

int

, where the inter-

action energy of a dislocation pair E

int

is determined by (13.6.7) shown in solving

Problem 2. In this formula it is necessary to put b

= b = −b

. Thus,

= 2ε



+ sin



+ 2U

, (13.6.4)

where the angle ϕ d e termines the dipole orientation and U

is the dislocation core

energy.

Since the energy (13.6.4) is bounded, there exists a ﬁnite density of dislocation

dipoles of the given dimension r, proportional to the Boltzmann factor exp(−βE

where β =(1/T). We calcu late the mean square of the dislocation dipole dimension:

r

 =



−βE

rdrdϕ









−βE

rdrdϕ







−1

(13.6.5)



3−2βε









1−2βε







−1

1 − βε

2 − βε

4−2βε

− a

4−2βε

2−2βε

− a

2−2βε

Since we are interested in the limit R → ∞ (R  a) the mean square (13.6.5) will

have different forms for 4 < 2βε

when T < T

, and with 4 > 2βε

when T > T

In the ﬁrst case T < T

, in the limit R → ∞ we have

r

 =

1 − βε

2 − βε

− T





. (13.6.6)

Thus, the average dimension of equilibrium d islocation dipoles in an ordered 2D

crystal is limited, but increases with increasing temperatu re and goes to inﬁnity at the

phase transition point T = T

In the second case (T > T

), in the limit R → ∞ we get r

 = ∞. Hence, it

follows that when T > T

the dislocation dipoles are broken down, on average, and a

phase with free 2 D dislocations is occurs. A more d etailed description of the prope r-

ties of the resulting phase requires taking into account the dislocation interaction and

a knowledge of how to treat disordered systems.

13.6 Long-Range Order in a Dislocated Crystal

319

13.6.1

Problems

1. Calculate the shear stresses generated by an inﬁnitely extended dislocation wall

(Fig. 10.8c) at large distances.

Hint. Write down the total ﬁeld of all dislocations

(x, y)=bMx

∞

∑

n=−∞

− (y −nh)

+(y − nh)

]

and make use of the Poisson summation formula (9.4.3).

Solution.

= 4π

−2πx/h

cos



2π



, x  h.

2. Find the interaction energy of two linear edge dislocations lying in the parallel glide

planes.

Hint. Make use of the fact that the d esired energy is equal to the work done to remove

one dislocation in the glide plane from inﬁnity (from crystal surface) to the line of its

location in the ﬁeld of a ﬁxed second dislocation. Thus, the interaction energy per unit

length of the dislocation equals



r cos ϕ

dx at R → ∞,

where F

is determined by an equation such as (13.3.10).

Solution.

= −b



+ sin



. (13.6.7)

3. Describe the distribution of straight dislocations near the obstacle under the action

of a homogeneous compressive stress.

Hint. Make use of (13.3.13), setting σ

= σ

=constant, a

= 0,andﬁnda

= −L

from the condition



−L

ρ(x) dx = N.

Solution.

ρ(x)=

πbM



L + x

|x|

, L =

2NbM

(−L < x < 0).

Dislocation Dynamics

14.1

Elastic Field of Moving Dislocations

Let us now elucidate what form the total system of equations of elasticity theory takes

that determines strains and stresses in a crystal when d islocations perform a given

motion.

Equation (13.1.6) is independent of whether the dislocations are at rest or in motion.

However, in the dynamic case the distortion tensor should change with time and this

will be determined by the character of dislocation motion.

If during the deformation of a medium the dislocations remain stationary the fol-

lowing equality is valid

∂v

∂x

∂u

∂t

, (14.1.1)

where v = v(r, t) is the velocity of displacement of a medium element with a coordi-

nate r at time t.

If the dislocations move and their density changes with time, (14.1.1) is incompati-

ble with (13.1.6). Thus, we replace it with

∂v

∂x

∂u

∂t

− j

, (14.1.2)

where the tensor j

should be chosen so that these equations are compatible. The

compatibility condition of (13.1.6), (13.1.2) has the form

∂α

∂t

+ e

ilm

∇

∂j

∂t

= 0. (14.1.3)

To associate the tensor j

with dislocation motion, we note that the condition

(14.1.3) can be regarded as the differential form of the conservation law of the Burg-

ers vector in a medium. Indeed, we integrate both sides of (14.1.3) over the surface

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

322

14 Dislocation Dynamics

spanned on a certain closed line L and introduce the total Burgers vector b of the

dislocations enveloped by the contour L. Using Stokes’s theorem, we obtain

= −



It is clear from th is equality that the integral on its r.h.s. determines the Burgers

vector “ﬂowing” per unit time through the contour L, i. e., carried over by dislocations

that cross the line L. Thus, it is natural to consider j

as the dislocation ﬂux density

tensor, and (14.1.3) as an equation for the continuity of dislocation ﬂow.

In particular, it is obvious that in the case of an isolated dislocation loop, the tensor

has the form

= e

ilm

δ(ξ) , (14.1.4)

where V is the dislocation line velocity at a given point. The vector of the ﬂux j

through an element dl of the contour is then proportional to dl [τV]=V [dlτ],i.e.,to

the projection velocity V onto the direction perpendicular both to dl and τ. It follows

from geometrical considerations that only this projection results in the dislocation

crossing the element d l.

If the dislocation distribution is described by the continuous functions α

(14.1.4)

is generalized by

= e

ilm ∑

where the index s d e notes the densities of dislocations of various types (for in stan ce,

dislocations of different sign in the case of a system of parallel edge dislocations) and

the vector V

is equal to the average velocity of dislocations of the type s at a given

point in the crystal.

The tensor j

has an independent meaning and is the principal characteristic of

dislocation motion.

Finally, we obtain a total system of differential equations describing the elastic

ﬁelds in a crystal with moving dislocations. This system consists of (13.1.6), (14.1.2)

and the equations of motion of a continuous medium:

∂v

∂t

= ∇

, σ

= λ

iklm

, e

ilm

∇

= −α

∇

−

∂u

∂t

= −j

(14.1.5)

In these equations the tensors α

and j

are the g iven functions of the coordinates

(and time) characterizing the dislocation distribution and motion.

The compatibility conditions for (14.1.5) are the conservation laws

∇

= 0,

∂α

∂t

+ e

ilm

∇

= 0. (14.1.6)

14.1 Elastic F ield of Moving Dislocations

323

Using the deﬁnition for the tensor of the dislocation ﬂow density and the system of

(14.1.5), the dynamics of dislocations in an elastic medium can be developed (Kose-

vich, 1962; Mura, 1963).

The relation between the trace of the tensor j

≡ j

) and the equation for con-

tinuity of a continuous medium is of special interest. The convolution j

is involved

in the equation obtained from (14.1.5):

div v −

∂ε

∂t

= −j

. (14.1.7)

It is easy to explain the physical meaning of (14.1.7). Indeed, the convolution ε

is a relative elastic change in the medium element volume obviously related with a

corresponding relative change in its density (ρ):

= −

δρ

. (14.1.8)

Substituting (14.1.8) into (14.1.7) and using the linearity of the theory, we get the

relation

∂ρ

∂t

+ div ρv = −ρj

. (14.1.9)

If, with a moving dislocation, the medium elements move without breaking continuity,

the l.h.s. of (14.1.9) vanishes due to the continuity equation and (14.1.10)

≡ j

= 0. (14.1.10)

For linear dislocations (14.1.10) has a simple interpretation. Indeed, in the case of

an isolated linear dislocation, the convolution j

is proportional to [bτ] V, i. e., propor-

tional to the projection of the dislocation velocity onto the directional perpendicular

to the vectors τ and b, or in other words, onto the direction perpendicular to the dis-

location glide plane. Thus, (14.1.10) implies that with medium continuity preserved

the vector of the dislocation velocity V lies in the glide plane, so that the mechanical

motion of a dislocation can o nly take place in this plane.

If dislocation motion is accompanied by the formation of some discontinuities, for

instance, by a macroscopic cluster of vacancies along some part of the dislocation line,

the l.h.s. of (14.1.9) is nonzero and equals the velocity of a relative inelastic increase

or decrease o f the mass of some elementary volume of the medium.

The action of this mechanism amounts to a macroscopic clustering of vacancies

or interstitial atoms along the dislocation line. We denote by q(r) a relative increase

of the speciﬁc volume of a medium at the point r per unit time. According to the

formulae (14.1.4), (14.1.9), the motion of an isolated dislocation should then generate

the following value of q:

q(r)=V [bτ] δ(ξ), (14.1.11)

when ξ is a 2 D radius vector measured from the dislocation axis.