# Analytic Binary Alloy Volume-Concentration Relations and the Deviation from Zen's Law
## 2 Theoretical Background
In cases of limited mutual solubility of the alloy components, it is reasonable to consider the atomic volume of the solvent to be constant and equal to its _real_ value, \(\Omega_{1(2)}\), \[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\chi\gamma\mathrm{C}=\Omega_{1}+4\pi \mathrm{R}_{1}^{3}\ \gamma_{2}\mathrm{C}\mathrm{x}\left(\frac{\gamma}{\gamma_{2}}\right)=\Omega_{ 1}+(\Omega_{2}-\Omega_{1})\frac{\gamma}{\gamma_{2}}\mathrm{x}, \tag{26}\]0 (35) The atomic volume of the solute, \(\Omega_{1(2)}\), undergoes a linear change with composition, x,  
\[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\chi\gamma\mathrm{C}=\Omega_{1}+4\pi \mathrm{R}_{1}^{3}\ \gamma_{2}\mathrm{C}\mathrm{x}\left(\frac{\gamma}{\gamma_{2}}\right)=\Omega_{ 1}+(\Omega_{2}-\Omega_{1})\frac{\gamma}{\gamma_{2}}\mathrm{x}, \tag{26}\]1  
The experimental (_real_) atomic volumes of selected elements at the room temperature, together with the bulk and shear moduli, are listed in \[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\chi\gamma\mathrm{C}=\Omega_{1}+4\pi \mathrm{R}_{1}^{3}\ \gamma_{2}\mathrm{C}\mathrm{x}\left(\frac{\gamma}{\gamma_{2}}\right)=\Omega_{ 1}+(\Omega_{2}-\Omega_{1})\frac{\gamma}{\gamma_{2}}\mathrm{x}, \tag{26}\]2 The atomic volumes correspond to the Wigner-Seitz radii reported in table one of Ref. [(11)], and the elastic constants are the same as in table three of Ref. [(11)]. In addition to the binary alloys studied in Ref. [(11)], we present data for Mg-Cd and Fe-Cr solid solutions. The volume size factors, \(\omega_{1}\) and \(\omega_{2}\), for the alloy systems under consideration are reproduced in \[\Omega=\Omega_{1}+4\pi\mathrm{R}_{1}^{3}\chi\gamma\mathrm{C}=\Omega_{1}+4\pi \mathrm{R}_{1}^{3}\ \gamma_{2}\mathrm{C}\mathrm{x}\left(\frac{\gamma}{\gamma_{2}}\right)=\Omega_{ 1}+(\Omega_{2}-\Omega_{1})\frac{\gamma}{\gamma_{2}}\mathrm{x}, \tag{26}\]3 The values of the real and _apparent_ atomic volumes are listed in Table 3 (table four of Ref. [(11)]). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & \(\Omega\) (\(\lambda^{3}\)) & **K (GPa)** & \(\mu\) (GPa)** \\ \hline Mg & 23.2396 & 35.6 & 17.3 \\ Cd & 22.0210 & 46.8 & 19.1 \\ Al & 16.6036 & 72.6 & 26.0 \\ Si & 20.0182 & 97.6 & 66.2 \\ Ti & 17.6542 & 108.2 & 45.6 \\ V & 13.8256 & 157.9 & 46.7 \\ Cr & 12.0064 & 160.0 & 115.1 \\ Mn & 12.2199 & 98.0 & 39.0 \\ Fe & 11.7771 & 169.6 & 81.4 \\ Co & 11.0732 & 82.3 & 88.8 \\ Ni & 10.9415 & 183.0 & 80.0 \\ Cu & 11.8072 & 136.4 & 46.8 \\ Zn & 15.2123 & 69.6 & 41.9 \\ Ge & 22.6345 & 75.0 & 54.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The experimental atomic volumes and elastic constants for polycrystalline metals [(11)]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & \(\Omega\) (Å\({}^{3}\)) & **K (GPa)** & \(\mu\) (GPa) \\ \hline Zr & 23.2790 & 94.0 & 30.0 \\ Nb & 17.8715 & 170.3 & 37.5 \\ Mo & 15.5834 & 261.3 & 125.5 \\ Ag & 17.0578 & 103.4 & 30.3 \\ Sn & 27.3255 & 58.2 & 18.4 \\ Ta & 18.0173 & 196.5 & 69.0 \\ W & 15.8566 & 311.0 & 160.6 \\ Au & 16.9618 & 170.7 & 27.5 \\ Pb & 30.3246 & 45.9 & 5.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Alloy** & \(\omega_{1}\) & \(\omega_{2}\) \\ \hline Al-Ag & \(-0.0918\) & \(+0.0012\) \\ Al-Cu & \(+0.2000\) & \(-0.3780\) \\ Al-Mg & \(-0.3580\) & \(+0.4082\) \\ Al-Mn & \(+0.1620\) & \(-0.4681\) \\ Al-Ti & \(-0.2009\) & \(-0.1506\) \\ Al-Zn & \(-0.0625\) & \(-0.0574\) \\ Cu-Ag & \(-0.2775\) & \(+0.4352\) \\ Cu-Au & \(-0.2781\) & \(+0.4759\) \\ Cu-Fe & \(+0.1753\) & \(+0.0457\) \\ Cu-Ni & \(+0.0718\) & \(-0.0845\) \\ Cu-Zn & \(-0.5457\) & \(+0.1710\) \\ Fe-Co & \(+0.0524\) & \(+0.0154\) \\ Fe-Cr & \(-0.0207\) & \(+0.0436\) \\ Fe-V & \(-0.1886\) & \(+0.1051\) \\ Ag-Au & \(-0.0064\) & \(-0.0178\) \\ Ag-Mg & \(-0.6342\) & \(+0.0713\) \\ Mg-Cd & \(-0.0160\) & \(-0.2108\) \\ Si-Ge & \(-0.2065\) & \(+0.0468\) \\ Nb-Ta & \(-0.0023\) & \(-0.0026\) \\ Pb-Sn & \(+0.2905\) & \(-0.0825\) \\ Ti-Zr & \(-0.2233\) & \(+0.3008\) \\ Cr-W & \(-0.2173\) & \(+0.3735\) \\ \hline \hline \end{tabular}
\end{table}