Small deviation from stoichiometry. I. Metal vacancies

3 Small deviation from stoichiometry. I. Metal vacancies 

Let us consider the compounds which show a small deviation from the stoichiometric composition and whose non-stoichiometry is derived from metal vacancies. The free energy of these compounds, which take the composition MX in the ideal or non-defect state, can be calculated by the method proposed by Libowitz. To readers who are well acquainted with the Fowler-Guggenheim style of statistical thermodynamics, the method here adopted may not be quite satisfactory however, the Libowitz method is understandable even to beginners who know only elementary thermodynamics and statistical mechanics. It goes without saying that the result calculated by the Libowitz method is essentially coincident with that calculated by the Fowler-Guggenheim method. 

Consider a crystal in which M atoms randomly occupy lattice points in N lattice points and anions occupy lattice points in N lattice points. The free energy of the crystal may be written 

This example, which may be the most simple model of non-stoichiometric compounds, is important for understanding the following examples. 

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