PART II - Papers - A Classical Model of Solid Solutions Based on Nearest-Neighbor Interactions Which Involve Both Central and Linked-Central Forces

A classical theory of solid solutions involving neavest-nergkbor intevactions with both central and linked-central forces between atoms has been developed. It has been found that the theory, where it can be checked quantitatively, is in ageement with experiment. The theory encompasses a description of many diverse pkenomena, such as antiphase shift structures, size effect, relative stabilities of various solutions, lattice para,neters, and order-disorder transitions. In particulav. a quantitative prediction not involving adjustable pavameters is made concevning the deviation of the Au-Cu interatonlic distance in long-range ordered (Ll,) Cu-Au I fronl the average distance based on the distances in pure gold and copper. This prediction, which is in agreement with expel-intent, has not been encompassed by any preuious theory. The theory of order-disorder is fragmentary. That is, no one theory exists that can explain the variety of qualitative phenomena observed. Further, many theories are not in good quantitative agreement with experiment. This subject has been reviewed by Muto and Takagi, Tuttman, and Oriani.3 There exists no doubt that the quasi-chemical approximation is not a complete description and that the inclusion of strain energy using macroscopic elasticity theory concepts leads to results in disagreement with experimenL4 The observation of antiphase domains and ordering systems such as Cu-Pt has led to Brillouin zone treat-ment of the order-disorder transition as opposed to the classical Ising model. The objective of this paper is to demonstrate that it is possible to develop a pairwise approximation model that can explain many of the observed order-disorder phenomena that have puzzled investigators in the past. This theory is based upon an empirical model due to ergmman' for the elastic constants of metals. This model is generalized for multicomponent systems. As will be shown, the theory yields a short-range ordering energy for the disordered solution which differs from the ordering energy calculated from the differences in energy of disordered and long-range ordered solutions. It will be demonstrated that there is no necessary correlation between heats of formation and the tendency to order or between size effect and the tendency to order. Also, the existence of antiphase domains and iso-short-range-order systems that form superlattices (Cu-Pt) is predicted on the basis of the theory. Further, the relative stability of competing superlattices is calculable from the theory. If single-crystal elastic-moduli data are available for the pure components and one superlattice then there exists but one adjustable parameter in the calculation of lattice parameters for both the disordered and ordered solid solutions. In one special case, no adjustable parameters are required and a quantitative prediction is made. For the calculation of energies and partial order, there exists but one additional adjustable parameter, the pair-exchange energy V used in the quasi-chemical approximation (or the Ising model.) However, in these calculations, much more precise values are required for the single-crystal elastic moduli than available if the quantitative uncertainties in the predicted values of the energies are to be sufficiently small. THEORY ~er~man' has developed a model with which he was able to obtain fair agreement with experiment for the relations between the elastic constants for metals. This model which we shall call Bergman's model is a linear combination of his models I and 11. In effect, Bergman, in this model, considers that each interatomic distortion is composed of two components: a classical central force distortion with an associated central force constant and what we shall call a linked-central force distortion with an associated linked central-force constant. The linked-central force distortion component obeys the constraint that the sum of such distortions over all the bonds equals zero. No constraint is imposed on the classical central force distortion component. Bergman' derives the constraint on the linked-central force distortion on the basis of application of Pauling's relation between bond distortion and bond number to metals.ga This assumption is not logically necessary, however, and the Bergman model may be taken as a mathematical model for elastic constants, e.g., a purely empirical model without a physical basis. In the present work, the method of Bergman has been applied to two-component systems (solid solutions). In place of an external strain—which would allow a calculation of the elastic constants for the two-component system—it is considered that internal interatomic distortions exist as a consequence of having three potentially unequal distortion-free interatomic distances and but one "average" interatomic distance. It is assumed that the distortion-free interatomic distances between atoms of the same element are those found in the pure element having the same undistorted crystal structure as the solid solution. The distortion-free interatomic distance between unlike atoms is in general not measurable except in the probably nonexistent