Thermodynamic calculations of the phase equilibrium boundaries for two- and three-component body-centred cubic solid solutions of uranium

The authors put forward models and give basic approximations for a thermodynamic description of two-component and three-component isomorphous solid solutions. They consider the various contributions made to the free energy of solid solutions formed by isomorphous components, namely the chemical contribution (using Becker's model), the contribution made by the energy of elastic deformation of the crystal lattice, the vibrational contribution, and that of thermally excited electrons. Further, the authors demonstrate the predominant part played in shaping the decomposition curves by the elastic component of the free energy, and analyse the effect of the concentration and temperature dependence of the modulus of elasticity on the shape of the decomposition curves for the solid solutions. Results are given for the calculation of: (a) decomposition curves (solubility curves) for eight systems of γ-uranium (with β-Zr, β-Hf, V, Nb, Ta, Cr, Mo and W) and also for the system β-Zr-Nb; and (b) phase boundaries for the separation regions of body-centred cubic solution in the ternary systems U-Nb-Mo and U-Nb-Zr. A comparison of the results of calculations with experimental phase diagrams indicates, as a whole, that there is satisfactory quantitative concordance. The authors establish the interrelationship between the shape of the phase equilibrium boundaries for isomorphous solid solutions in two-component and three-component systems, on the one hand, and the concentration relationships for some of the simplest physical characteristics, such as the coefficient of electron thermal capacity, Debye temperature, moduluo of elasticity, and anharmonic characteristics, on the other. The difference in principle between the present calculations and the numerous computer calculations of phase diagrams is the maximum generality of the problem as formulated and its solution in the mathematical sense. Indeed, the phase equilibrium curves were calculated by direct computer plotting of the general tangent to the isothermal sections of the free energy surface, and computation of the points of tangency. The main factors responsible for the shape of the asymmetric curves computed for decomposition and solubility are, for the systems U-Zr, U-Hf, U-Nb and U-Ta, as follows: the elastic component of the free energy, which decreases as the temperature rises, the mixing energy at OK, and the configuration-entropy term. A further contribution is made by the vibration component of the free energy for the systems U-V, U-Cr, U-Mo, U-W and Zr-Nb; the system U-Mo exhibits an additional factor of electron origin