Geology Reference
In-Depth Information
in which the dislocation content at equilibrium is zero to a state in which it is very
large (Nabarro
1967
, p. 688).
Since the energy of a dislocation line is increased or decreased by increasing or
decreasing its length, respectively, the line can be regarded as being under tension,
the derivative of the energy with respect to the length being known as the line
tension. The line tension is identical with the energy per unit length, as given by
the sum of (
6.7
) and the core energy per unit length, and hence is of the order of
Gb
2
;
with the dimensions of force. However, analogy with a string under tension is
not an exact one because there is interaction between the parts of a dislocation
(
Sect. 6.3.4
) and so the line tension depends on the dislocation configuration (Hirth
and Lothe
1982
, p. 174).
6.2.3 The Peierls Potential
In considering the mobility of a dislocation line in the absence of the influence of
thermal agitation or extrinsic factors such as interaction with other dislocations or
the presence of impurities, it is the fluctuation in the dislocation energy with
position in the crystal that is of primary importance rather than the absolute value
of the energy. The fluctuating component of the energy, which is associated with
the crystalline periodicity and has maxima at spacings equal to the Burgers vector
or submultiples of it, arises entirely within the core energy and is known as the
Peierls energy or Peierls potential since Peierls (1940), at the instigation of
Orowan, was the first to attempt to calculate it (Hirth and Lothe
1982
, p. 217).
In view of the difficulty of modeling the dislocation core, it is conceptually
useful to begin by assuming an empirical expression for the dislocation energy E
d
of the form
E
d
¼
E
0
þ
E
p
sin
2
pnx
ð
6
:
8
Þ
where E
0
is the non fluctuating part of E
d
and includes the energy of the long-
range elastic stress field, E
p
is the amplitude of the Peierls potential, xb is the
displacement of the dislocation core from a position of minimum energy, and n is
an integer (usually 1 or 2) that allows for different forms of the Peierls potential
(Hirth and Lothe
1982
, Chap. 8). The force needed to move a straight segment of
dislocation line over the Peierls barrier E
p
;
the Peierls force, is then given by the
maximum slope of the potential E
d
;
that is, by
ð
1
=
b
Þð
dE
d
=
dx
Þ
Max
¼
pnE
p
:
From
(
6.5
), the corresponding stress, the Peierls stress s
p
;
is
ð
pn
=
b
2
Þð
E
p
=
l
Þ:
If we write
pn
ð
E
p
=
l
Þ¼
bGb
2
;
we obtain
s
p
¼
bG
ð
6
:
9
Þ
where b is an empirical numerical constant. If we also write
ð
E
0
=
l
Þ¼
aGb
2
where
is of the order of the ratio of E
p
to
a
2
to 1 (
Sect. 6.2.2
), then b
¼
apnE
p
=
E
0
the total dislocation energy.
