# Analytic Binary Alloy Volume-Concentration Relations and the Deviation from Zen's Law
## 2 Theoretical Background  
[PERSON] [11] derived an expression for the effective lattice parameter of binary solid solutions by using an elasticity inclusion model, in conjunction with an _apparent_ size of the solute atom when resolved in the _solvent_ matrix. Assuming the R\({}_{1}\) is the Wigner-Seitz radius of the solvent material, \(\Omega_{1}=\frac{4\pi\rm R_{1}^{3}}{3}\), the volume increase produced by insertion of the solute atom to the solvent is expressed as  
\[\Delta\Omega=4\pi\rm R_{1}^{3}\gamma_{1}C, \tag{1}\]where the _misfit coefficient_, C, is given by the expression  
\[\mathrm{C}=\frac{(\mathrm{R}_{2}^{*}-\mathrm{R}_{1})}{\mathrm{R}_{1}\gamma_{2}} \tag{2}\]  
and \(\mathrm{R}_{2}^{*}\) is the _apparent_ Wigner-Seitz radius of the _solute_ material, which is introduced to approximately account for the electronic interactions between the outermost quantum shells of the solute and solvent atoms. The parameters \(\gamma_{1}\) and \(\gamma_{2}\) are defined by  
\[\gamma_{1}=1+\frac{4\mu_{1}}{3\mathrm{K}_{1}},\ \gamma_{2}=1+\frac{4\mu_{1}}{3 \mathrm{K}_{2}}, \tag{3}\]  
where \(\mu_{1(2)}\) and \(\mathrm{K}_{1(2)}\) are the shear and bulk moduli of the solvent (solute). If x is the atomic concentration of the solute, the total volume increase produced by insertion of the \(\mathrm{N}_{2}=\mathrm{x}\mathrm{N}_{1}\) solute atoms (\(\mathrm{N}_{1}\) is the total number of the solvent atoms) is  
\[\Delta\mathrm{V}=4\pi\mathrm{R}_{1}^{3}\mathrm{x}\mathrm{N}_{1}\gamma_{1} \mathrm{C} \tag{4}\]  
and the total volume is  
\[\mathrm{V}=\mathrm{V}_{1}+4\pi\mathrm{R}_{1}^{3}\mathrm{x}\mathrm{N}_{1} \gamma_{1}\mathrm{C} \tag{5}\]  
where \(\mathrm{V}_{1}\) is the volume of the solvent. The atomic volume of the system with \(\mathrm{N}_{2}=\mathrm{x}\mathrm{N}_{1}\) solute atoms will be  
\[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\mathrm{x}\gamma_{1}\mathrm{C}. \tag{6}\]  
The _apparent_ radius of the solute atom, \(\mathrm{R}_{2}^{*}\), is estimated using one piece of experimental information about the solid solution, i.e., the initial slope of the lattice spacing vs. composition curve, \(\left(\frac{\mathrm{d}\mathrm{d}}{\mathrm{d}x}\right)_{\mathrm{x}=0}\). [PERSON] [12] introduced the volume size factor:  
\[\omega_{2}=\frac{1}{\Omega_{1}}\left(\frac{\mathrm{d}\Omega}{\mathrm{d}x} \right)_{\mathrm{x}=0}=\frac{3}{\mathrm{a}_{1}}\left(\frac{\mathrm{d}\mathrm{ a}}{\mathrm{d}x}\right)_{\mathrm{x}=0}. \tag{7}\]  
By differentiating Equation (6), \(\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\mathrm{x}\gamma_{1}\mathrm{C}\),  
\[\left(\frac{\mathrm{d}\Omega}{\mathrm{d}x}\right)_{\mathrm{x}=0}=4\pi\mathrm{ R}_{1}^{3}\gamma_{1}\mathrm{C}. \tag{8}\]  
Thus, combining Expressions (7) and (8), one gets \(\omega_{2}\Omega_{1}=\left(\frac{\mathrm{d}\Omega}{\mathrm{d}x}\right)_{ \mathrm{x}=0}=4\pi\mathrm{R}_{1}^{3}\gamma_{1}\mathrm{C}=4\pi\mathrm{R}_{1}^{3 }\gamma_{1}\frac{(\mathrm{R}_{2}^{*}-\mathrm{R}_{1})}{\mathrm{R}_{1}\gamma_{2}}\) or \(\omega_{2}\Omega_{1}=3\frac{4\pi\mathrm{R}_{1}^{3}}{3}\gamma_{1}\frac{(\mathrm{ R}_{2}^{*}-\mathrm{R}_{1})}{\mathrm{R}_{1}\gamma_{2}}=3\Omega_{1}\gamma_{1}\frac{( \mathrm{R}_{2}^{*}-\mathrm{R}_{1})}{\mathrm{R}_{1}\gamma_{2}}\) and \(\omega_{2}=3\gamma_{1}\frac{(\mathrm{R}_{2}^{*}-\mathrm{R}_{1})}{\mathrm{R}_{ 1}\gamma_{2}}\). Thus, the _apparent_ Wigner-Seitz radius of the solute atom can be calculated from  
\[\mathrm{R}_{2}^{*}= \mathrm{R}_{1}\left(1+\frac{\gamma_{2}}{3\gamma_{1}}\omega_{2}\right) \tag{9}\]  
and an analogous expression is derived for \(\mathrm{R}_{1}^{*}=\mathrm{R}_{2}\left(1+\frac{\gamma_{1}}{3\gamma_{2}} \omega_{1}\right)\). [PERSON] [12], using the high-precision lattice parameter data available in the literature, presented numerical values of \(\omega_{2}\) for 469 metallic solid solutions.