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Derivation of the equilibrium concentration of the point defects
The change in the Gibbs energy of a crystal due to the formation of vacancies is assumed to be and can be split into enthalpy and entropy terms:
We are now looking for the number or concentration of vacancies in a given crystal for which adopts a minimum. To do this you have to and as a function of the number of point defects be known.
The formation of point defects in the ideal crystal is always endothermic, so is positive. That becomes negative at all, is therefore entirely on the entropy term and its dependence on the point defect concentration.
contains two posts. On the one hand, the entropy of the crystal changes because the bond strengths and thus the oscillation frequencies of the atoms in the vicinity of a point defect change. This contribution corresponds to a change in the vibrational entropy of the entire crystal by . However, it is small and can also be negative as well as positive.
The second post is crucial too . It results from the large number of possibilities to distribute the point defects to the various lattice sites. If one looks at a crystal with Atoms of the variety in which is still There are spaces Total grid places. All spaces and all - Atoms are individually indistinguishable. In this case, the number of arrangements is equal to the number of permutations. One denotes with the number of spaces available To distribute grid positions, the following applies to the resulting contribution to entropy:
is always positive and is referred to as configuration entropy or as a contribution to entropy in terms of position statistics. The mole fraction of the vacancies in the crystal at the given values of pressure and temperature can be calculated using the Gibbs-Helmholtz equation be calculated:
The size denotes the change of per vacancy, which results from the change in the enthalpy of binding and the entropy of oscillation. She is so long from regardless of how the defects do not interact. This applies in general. for small defect concentrations. The term contains the contribution of the configuration entropy. The addition of the contributions leads to a minimum of the Gibbs energy at the equilibrium number the defects.