Geology Reference
In-Depth Information
is to be regarded as belonging to other dislocations (R
q
1
=
2
where q is the
dislocation density). Within the region of the long-range stress field, the distortion
of the crystal structure is within the linear elastic range and the classical theory of
elasticity can be applied, while within the core region the energy can only be
calculated using considerations at the atomic level.
The energy E
el
in the long-range stress field of a segment of length l of a
straight dislocation is approximately Gb
2
l if the dislocation density is low, where
b is the magnitude of the Burgers vector and G is the shear modulus in the
approximation of isotropic elasticity (the value of which is probably best obtained
by Reuss averaging of the actual anisotropic elastic properties: Hirth and Lothe
1982
, p. 424). More exactly, the energy depends somewhat on the character of the
dislocation line and on R
=
r
0
;
according to
1
E
el
l
¼
aGn
2
R
r
0
ln
ð
6
:
7
Þ
4p
where a
¼
cos
2
h
þ
sin
2
h
1
m
, h being the angle between the dislocation line and the
Burgers vector and v the Poisson ratio; the factor -1 arises when the surface at
radius R is taken to be stress free (for derivation, see Hirth and Lothe
1982
, Chap.
3). When full account is taken of the actual anisotropic elasticity of the crystal (for
example, Steeds
1973
) (Hirth and Lothe
1982
, Chap. 13), a more complicated
function K of the elastic constants replaces aG in (
6.7
). However, the effect on the
calculated energy is usually not very great. Thus, Heinisch et al. (
1975
) showed
that for olivine and orthopyroxene the isotropic approximation gives energies that
are mostly within 10 % of those calculated on anisotropic elasticity, while in the
more anisotropic cases of quartz and calcite the differences do not exceed about
30 % or so. The change in energy when curvature of the dislocation is taken into
account is also usually relatively small. Thus, the elastic energy per unit length for
a dislocation loop of radius R
l
is approximately
ð
Gb
2
=
2p
Þ
ln
ð
R
l
=
r
0
Þ
, giving a
similar value to (
6.7
) except for very small loops (Nabarro
1967
, p. 75).
In the long-range linear elastic field around any dislocation with an edge
component there are regions of volumetric expansion and contraction on opposite
of the dislocation. These effects closely compensate each other so that the overall
elastic dilatation is close to zero. This also applies for a screw dislocation, which,
to the first approximation, has no dilatational components in its long-range elastic
field, However, due to anharmonic effects in or near the core, there is actually a
small net volume increase, of the order of b
2
l for a dislocation segment of length l,
for both screw and edge dislocations (Hirth and Lothe
1982
, p. 231; Poirier
1985
,
p. 152; Seeger and Haasen
1958
).
The core energy, which depends on the structural arrangement and bonding
energies of the atoms in the core region, is more difficult to estimate (Hirth and
Lothe
1982
, Chap. 8). Earlier calculations based on assumed interatomic potentials
of conveniently simple form, such as the well-known Peierls-Nabarro model, can
be regarded as little more than illustrative. However, ab initio molecular orbital
