Part II – February 1968 - Papers - The Influence of the Density of States on the Thermodynamic Activity of Zinc in the Epsilon Phase of Ag-Zn System

A dew-point technique was used to determine the thermodynamic activity of zinc at 430°C in a series of e phase Ag-Zn alloys. The composition of the alloys ranged from 72 to 88 at. pct Zn. This range included the composition at which Massalski and King3 found a reversal in the composition variation of the crystallo-graphic c/a ratio, which they attributed to an overlap of the Fermi surface across the (002) faces of the Brillouin zone. The data is presented in a graph in which RT In yz,, where yz, is the activity coefficient of zinc, is shown as a function of atomic percent zinc. This curve has an unmistakable change in slope at the same composition that Massalski and King observed the beginning of the reversal in the c/a ratio. This change in slope of the thermodynamic data is also attributed to a Brillouin zone overlap. Equations are presented to demonstrate that the thermodynamic activity can be related to the density of states of the conduction electrons, and that the observed phenomena are consistent with this model. It is also demonstrated that the contribution of the density of states can be related to the excess stability, a phenomenological parameter recently shown by Darkeen" to be significant in the interpretation of thermodynamic data of metallic phases. The data seem to indicate that zone overlap has caused a spinodal point, and the resulting misci-bility gap, in the phase diagram. THE problem of developing an adequate thermodynamic model of solid solutions has proven to be difficult, and is still only partially solved. The main approach has been to develop a statistical model, such as that of an ideal solution, regular solution, and so forth, to which can be attached corrections for electronic, vibrational, magnetic, ordering, or other contributions. Such corrective terms are usually derived on an ad hoc basis, and it is difficult to predict in advance what their relative importance will be. This problem has been discussed in general terms by Oriani and Alcock,' who have reviewed several thermodynamic models and a few empirical correlations. The measurements described in the present paper were made to demonstrate in a special case the importance of one such corrective term, the contribu- tion of the energy of the conduction electrons of an alloy. It was our premise that the contribution of the energy of the conduction electrons to the thermodynamic activity of the alloy components could be detected; further, that such an effect would be observed at alloy compositions where other phenomena, also ascribed to the energy and density of states of the conduction electrons, are observed. The idea of the importance of the conduction electrons is hardly new. Hume-Rothery and his adherents have developed the well-known theory of alloy phases in which the sequence of phase fields in binary equilibrium diagrams, especially those involving the noble metals with the IIB, IIIB, and IVB subgroups, can be correlated by replacing the composition variable with the ratio of conduction electrons to atoms, e/a. Jones and others have developed a physical explanation for this correlation, in which they consider the solubility limits of phase fields to be restricted by an intersection between the Fermi surface and a Brillouin zone. The general features of the model are also quite well-known—presumably zone intersection causes the density of states to decrease at critical alloy compositions. The attendant increase in energy of conduction electrons in the original crystal structure allows an alternate structure to become more stable as the concentration of polyvalent solute is increased. In spite of the wide acceptance of these ideas on phase stability, there is only indirect* evidence, such as the variations in lattice parameter recorded extensively by Massalskizy3 and others, that Brillouin zone interactions occur. There are few experimental measurements, other than the correlations of the phase sequence, that substantiate the premise that the energy of conduction electrons affects the solubility limits of alloy phases. Much thermodynamic data of alloys has been found to be consistent with the theory; yet there is a lack of detailed data at compositions where zone intersection and overlap are thought to occur. One would expect that the energy of the conduction electrons would make a measurable contribution to the thermodynamic properties of alloys at compositions near zone intersection and overlap if the theory of Hume-Rothery and Jones is correct. This conclusion cannot be avoided, because the phase boundaries are determined by the requirement that the chemical potential of the components be equal in both phases at equilibrium. An electronic effect large enough to alter the stability of a phase should also affect the thermodynamic activity by a measurable amount.