On the theory of lattice vibrations in mixed crystals. The connection between the REI model, ATA, and CPA

For an isotopically disordered one‐dimensional model of a mixed crystal a hierarchy of equations for conditionally averaged Green functions is derived. A definite single‐site approximation leads to the random element isodisplacement model (REI‐model). This approximation is shown to be equivalent to a somewhat modified average t ‐matrix approximation (ATA) and, for a self‐consistent formulation, to a somewhat modified coherent potential approximation (CPA).