# Analytic Binary Alloy Volume-Concentration Relations and the Deviation from Zen's Law
## 2 Theoretical Background
\begin{array}{c}\mu_{\mathrm{v}}(\mathrm{x})=\left(1-\mathrm{x}\right)\mu _{1}+\mathrm{x}\mu_{2}\\ \mu_{\mathrm{R}}(\mathrm{x})=\left[\frac{(1-\mathrm{x})}{\mu_{1}}+\frac{ \mathrm{x}}{\mu_{2}}\right]^{-1}\\ \mu(\mathrm{x})=\mu_{\mathrm{vRH}}(\mathrm{x})=\frac{1}{2}(\mu_{\mathrm{v}}( \mathrm{x})+\mu_{\mathrm{R}}(\mathrm{x}))\end{array}  \tag{24}\]  
Equations (22) and (23) are solved self-consistently. For a special case of the small atomic volume misfit, [PERSON] [11] assumed:  
\[\Omega_{2}-\Omega_{1}=4\pi\mathrm{R}_{1}^{3}\gamma_{2}\mathrm{C}, \tag{25}\]  
which allows rewriting Equation (23):  
\[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\chi\gamma\mathrm{C}=\Omega_{1}+4\pi \mathrm{R}_{1}^{3}\ \gamma_{2}\mathrm{C}\mathrm{x}\left(\frac{\gamma}{\gamma_{2}}\right)=\Omega_{ 1}+(\Omega_{2}-\Omega_{1})\frac{\gamma}{\gamma_{2}}\mathrm{x}, \tag{26}\]  
[PERSON] wrote this expression in a slightly different form:  
\[\Omega=\left(1-\frac{\gamma}{\gamma_{2}}\mathrm{x}\right)\Omega_{1}+\frac{ \gamma}{\gamma_{2}}\mathrm{x}\Omega_{2}, \tag{27}\]  
emphasizing that in the case of \(\gamma=\left(1+\frac{4\mu_{1}}{3\mathrm{K}}\right)=\gamma_{2}\)\(=\left(1+\frac{4\mu_{1}}{3\mathrm{K}_{2}}\right)\), which, according to Equations (3) and (11), means if K = K\({}_{2}\), [PERSON]'s mixture rule of additive atomic volumes of the solute and solvent will be fulfilled. In this paper, several additional assumptions have been made to achieve a continuous solution for the alloy atomic volume within the whole composition range. As pointed out above, see Equations (10) and (15), [PERSON] assumed that calculations should be performed for two opposite _terminal_ solid solutions located at the ends of the binary phase diagram. In that case the calculated lattice constants typically have a discontinuity in the middle, at the equiatomic composition. To avoid this problem, we redefine (symmetrize) the coefficient \(\gamma_{1}\), Equation (3), as well as assuming that the _effective_ coefficient, \(\gamma\), defined by Equation (11), should be recalculated at each alloy composition, x, and be expressed through the _effective_ bulk, K(x), and shear, \(\mu(\mathrm{x})\), moduli calculated by Equations (22) and (24):  
\[\gamma_{1}=1+\frac{4\mu_{2}}{3\mathrm{K}_{1}},\ \gamma_{2}=1+\frac{4\mu_{1}}{3 \mathrm{K}_{2}}, \tag{28}\]  
\[\gamma(\mathrm{x})=1+\frac{4\mu(\mathrm{x})}{3\mathrm{K}(\mathrm{x})}. \tag{29}\]  
In addition, we rewrote Equation (26) for the two _terminal_ solid solutions:  
\[\Omega^{(1)}(\mathrm{x})=\Omega_{1}+(\Omega_{2}-\Omega_{1})\frac{\gamma( \mathrm{x})}{\gamma_{2}}\mathrm{x}, \tag{30}\]\[\Omega^{(2)}(x)=\Omega_{2}+(\Omega_{1}-\Omega_{2})\frac{\gamma(x)}{\gamma_{1}}(1-x). \tag{31}\]  
Additionally, the atomic volume of the alloy is defined as a function of composition, x,  
\[\Omega(x)=\Omega^{(1)}(x)(1-x)+\ x\Omega^{(2)}(x)=\Omega^{(1)}(x)+x\Big{(}\Omega ^{(2)}(x)-\Omega^{(1)}(x)\Big{)}. \tag{32}\]  
[PERSON] (1977) used the _apparent_ Wigner-Seitz radius, Equation (9), in calculations of the lattice parameters of the alloy. In our calculations we introduce two approximations for the atomic volumes of the alloy components which, in turn, are used as input parameters in Equation (32). 1. _Continuum_ approximation. In cases where the field of the disordered solid solution spans throughout the whole composition range, we assume that the atomic volume of the solvent (\(\Omega_{1}(x)\)) changes linearly with composition from the _real_ value, \(\Omega_{1}\), x = 0, to its _apparent_ value, \(\Omega_{1}^{*}\), in the pure solute, x = 1. \[\Omega_{1}(x)= \Omega_{1}(1-x)+\Omega_{1}^{*}x\] (33) Similarly, the atomic volume of the solute, \(\Omega_{2}(x)\), changes linearly with composition, from its apparent value in the pure solvent, \(\Omega_{2}^{*}\), x = 0, to the real value, \(\Omega_{2}\), x = 1. \[\Omega_{2}(x)=\Omega_{2}x+\Omega_{2}^{*}(1-x)\] (34)
2. _Terminal_ approximation.