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calculations have made some progress and recent calculations based on more
realistic potentials and structural relaxation procedures appear to yield results that
are useful approximations, especially in simple ionic crystals (Puls 1981 ). For
example, Bucher ( 1982 ) has calculated the core energy to be approximately
0.7 9 10 -9 Jm -1 (1.2 eV per length b) for edge dislocations in sodium chloride,
while Heggie and Jones ( 1986 ) have obtained approximately 1 9 10 -9 Jm -1
(3 eV per length b) for a basal 60 dislocation in quartz, to be compared with
values of Gb 2 of about 2 9 10 -9 and 11 9 10 -9 Jm -1 , respectively. The general
indication is that, at least for simple structures, the core energy does not exceed a
few tenths of Gb 2 per unit length, that is, that rather less energy is associated with
the core than with the long-range stress field. Also the core energy may be reduced
by structural changes in the core such as the incorporation of impurity atoms.
In summary, the total energy of a dislocation can be taken as being of the order
1
2 Gb 2 to Gb 2 per unit length, the greater part of which is normally associated with
the long-range stress field. For a typical mineral with b * 0.5 nm and
G * 50 GPa, the dislocation energy is therefore of the order of 10 -8 Jm -1
(or *30 eV per length b). It follows that, at a dislocation density of 10 12 m -2 , the
contribution of the dislocations to the internal energy of the crystal will tend to be
rather less than 1 J mol -1 , and even at extremely high dislocation densities of
10 15 -10 16 m -2 the contribution will not exceed the order of 1 kJ mol -1
(cf. measurements of stored energy in heavily cold-worked calcite by Gross 1965 ).
The energy that we have been considering so far is the increment in the Gibbs
energy of the crystal due to introducing an individual dislocation segment (it is
Gibbs energy because temperature and pressure, or stress, have been taken as the
independent variables, Sect. 2.2 ) . This energy will include a TDS term involving
entropy DS that can be associated mainly with the thermal vibrations of the atoms,
and hence is expressed in the temperature dependence of the elastic constants.
When we now consider an assemblage of dislocations, there is an additional
configurational entropy to be taken into account, but it can be shown that the
corresponding TDS term is relatively small and so the increment in the Gibbs
energy of the crystal due to the presence of an assemblage of dislocations will be
slightly smaller than the sum of the contributions of the individual dislocation
segments as calculated above (Cottrell 1953 , p. 39; Friedel 1964 , p. 73; Nabarro
1967 , p. 683). If the assemblage is viewed as consisting of a number of dislocation
loops (or independent segments) and the equilibrium concentration of loops is
calculated by minimizing the Gibbs energy of the assemblage, the concentration is
found to be negligibly small at any temperature below the melting point unless the
loops are exceedingly small (Friedel 1964 , p. 74, 1982 ). That is, the introduction of
dislocations into a crystal normally increases its Gibbs energy and so the dislo-
cation content of a crystal at equilibrium in respect of its defect structure at any
given temperature and pressure will be negligibly small. This argument has also
been extended to form the basis of a theory of melting, according to which the
melting point is the temperature at which there is an abrupt transition from a state
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