B1 Kanzaki Forces

In a perfect crystal, the equilibrium positions of all atoms are such that they exert no net forces on each other. However, when the crystal is subject to either external forces or internal forces originating from crystal defects, mutual interaction forces arise. For example, the atoms surrounding a vacancy move to new positions in response to the missing interaction forces from the atom that would normally occupy the vacant site. One may imagine that these missing forces are applied to atoms in a perfect crystal and given such magnitudes and directions that they pro­duce the same strain and stress fields as exist in a crystal with a real vacancy. These fictitious forces that are applied to a perfect crystal are known by the name of their inventor,26 as the Kanzaki forces.

For a localized defect in an elastic medium, the elastic strain or stress field can be generated with a finite number of point forces fa) acting at the lattice sites r0 + R(a), where e center of

the defect region and R(a) is the lattice vector from this center to the adjacent atom on which the force is to be applied. In harmonic crystal lattices, or equiva­lently, in solids that deform according to linear elas­ticity theory, the displacement field created by all these point forces is then given by

Z

rn (r) = £ Gj (r, r’ + R(a>)ff [B1]

a

where Gj is either the lattice Green’s function when the solid is described by a harmonic crystal lattice or the elastic Green’s function when it is described as an elastic continuum.

For distances |r — r0| >> |R(a)|, we can expand the elastic Green’s function into a Taylor series around the point r0. Using the notation

д r n

Gj ’k = ~dXk Gj [B2]

one obtains

i (r) = Gj (r: r)Y f/ + Gjk (r’ r) 53 Rka)f +2G„,k-,(r, r’)J2rS“)r!“)/;m + — m

The set of z forces must of course be self- equilibrating so as to not impose any net force or net force moment on the solid medium. Hence,

z

Y fya) = 0 [B4]

a=l

and

z

Y J R^ — fk“)Rj“)) = 0 [B5]

a=1 ‘ ‘

As a consequence of eqn [B3], the first term in the multipole expansion (B3) vanishes. The next term contains the first moment of the forces, which is called the dipole tensor of the defect, and is denoted by

z

Pjk = f (“)Rka) = Pkj [B6]

a=1

In an infinite medium, the elastic Green’s function has the form

G» = gj(o) ij |r — r’

where o is the solid angle of the unit vector parallel to (r-r’). It is then permissible to change the differ­entiations with respect to r to differentiations with respect to r, taking into account changes in sign. For example,

j = — G»k [B12]

The multipole expansion for the displacement field of a point defect can then finally be written as

U(r) = — G»k(r — r’)Pjk + (r — r’)Pjki

— 3TG;^kim(r — r’) Pjkim +••• [B13]

Using eqn [B11], we find that the first term falls off as 1/r2, the second as 1/r3, etc.

B2 Volume Change from Kanzaki Forces

and as indicated, it is a symmetric tensor by virtue of eqn [B5]. The second moment of the forces z Pjki = f;a)Rka)Rf) [B7] a=1
and the total volume change is then
1
o„ (r)d3r
is called the quadrupole tensor, and the third moment	[B15]
3K
We now transform this volume integral by using the following identity:
[B8]
Pa

the octupole tensor.

In terms of these multipole tensors, the displace­ment field of the defect region can be written in the series expansion

U (r) = Gj k (r, r’) Pjk + Gj ki’ (r, r’) Pjki + [B9]

If the crystal lattice defect has certain symmetries, some terms in this multipole expansion may not be present. For example, if the defect possesses an inver­sion symmetry, then for each force there exists an equal but opposite force at a position equal to but opposite to the position vector belonging to the mir­ror force. For such a defect,

Pa> = (-1)nPjki……… [B10]

n indices

and all multipole tensors of odd rank vanish.

s ii simdmi (simxi );m sim, mxi (simxi );m +fixi [B16]

Here, f(r) represents the distribution of internal forces that enter into the equilibrium equations that the stresses must satisfy, namely

s im, m + fi 0

With the formula [B15], the total volume change can be written as the sum of two terms. The first one contains as its integrant the divergence of the vector field simxj, and it can therefore be converted with the Gauss theorem into a surface integral. As a result

DV ък{ $xisimnmdS +

The surface integration is over the surface tractions simnm, where n is the surface normal vector. If no

external loads are applied on the surface of the solid, then the surface tractions vanish, and the first term in [B16] is zero. When the internal body forces are Kanzaki forces, then

f(r) = £f(a)d(r — R(a)) x,- = R

and using (B6) one obtains

DV = ^ (P11 + P22 + P33) [B19]

It is immediately obvious from this derivation that when more than one point defect is present in the solid, the volume change is simply the sum of indi­vidual traces of their dipole tensors divided by three times the bulk modulus of the solid.