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the ground state (for further discussion on the charge state of impurities in ionic
crystals, see Flynn 1972 , p. 579).
In view of the roles of the various types of crystal defects in mechanical properties,
their concentrations can be of particular interest. The dislocation content of a crystal
depends on its history and degree of annealing, that is, on the kinetics of rear-
rangement and annihilation of the dislocations and not on equilibrium thermody-
namic considerations; in fact, except possibly at temperatures in the neighborhood of
the melting point, a crystal in thermodynamic equilibrium is predicted to be free of
dislocations (Friedel 1964 ; Nabarro 1967 , p. 688). A similar situation probably
applies for many volume defects. However, point defects can exist in finite con-
centrations in crystals in thermodynamic equilibrium, and the equilibrium can often
be attained in laboratory times at temperatures well below the melting point. These
concentrations are governed by chemical equilibrium considerations (''defect
chemistry'') which can be lengthy and complicated but which involve basically the
following steps (Flynn 1972 , pp. 207-216; Kröger 1974 , Chaps. 9-14; Libowitz
1975 ; Madelung 1978 , pp. 397-406; Mrowec 1980 , Chap. 1 ; Smyth 2000 ; van Gool
1966 ):
1. Identify all relevant defect species, atomic and electronic, and the reactions
between them that are thought to be of interest, paying particular attention to the
states of charge and taking into account ionization of defects, formation of
associates, changes in occupancy of valence and conduction bands, and interac-
tions with the environment. Physical insights play an important part at this stage.
2. Apply the law of mass action to each reaction; that is, for the reaction
aA þ bB þ ... : ¼ cC þ dD þ ... :
Obtain
¼ K ¼ exp DG 0 = RT
a A a B ... := a C a D ... :
or
a ln a A þ b ln a B þ ... c ln a C d ln a D ... ¼ ln K ¼ DG 0 = RT
where a A ; a B ; ... are the activities of the defect species A ; B ; ... ; K is the ther-
modynamic equilibrium constant, DG 0 is the standard Gibbs function for the
reaction, R is the gas constant, and T is the absolute temperature (see, for example,
Atkins 1986 , pp. 213-218), noting that:
a. a A ¼ c A x A ... where c A ... are the activity coefficients and x A ... the mole
fractions of A...
b. if molar concentrations c A ... are used in place of mole fractions x A ... ; the
values of c A ... ; K and DG 0 must be adjusted accordingly, now expressing DG 0 ;
for example, in kJ m -3 instead of kJ mol -1 and taking care about definition of
standard states (Atkins 1986 , p. 186).
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