# Analytic Binary Alloy Volume-Concentration Relations and the Deviation from Zen's Law
## 1 Introduction  
According to the Hume-Rothery rules [1], (a) extensive substitutional solid solution occurs only if the relative difference between the atomic diameters (radii) of two species is less than 15%; (b) for the appreciable solid solubility, the crystal structures of two elements must be identical; (c) a metal will dissolve a metal of a higher valency to a greater extent than one of a lower valency; (d) an electronegativity difference close to zero gives maximum solubility. The more electropositive one element and the more electronegative the other, the greater the likelihood is that they will form an intermetallic compound instead of a substitutional solid solution. According to [PERSON]'s law [2], the unit cell parameters should vary linearly with the composition for a continuous substitutional solid solution in which the atoms or ions that substitute for each other are randomly distributed. For ideal solutions, with excess energies and volumes equal to zero, the atomic volumes (\(\Omega\)) of disordered alloys vary linearly with the composition ([PERSON]'s law, [3]): \(\Omega(\mathrm{x})=\Omega_{\mathrm{A}}\mathrm{x}+\Omega_{\mathrm{B}}(1-\mathrm{ x})\), where x is the atomic composition of the element A. One should notice that even for an ideal solution (for which [PERSON]'s law is valid), a deviation from [PERSON]'s law can occur. Indeed, for a cubic structure, the lattice parameter, a, is related to the cell volume, V, by the relation \(\mathrm{a}=\Omega^{1/3}\), and the linear variation of the cell volume does not imply the linear variation of the lattice parameter. Models exist to predict the deviation from [PERSON]'s law; however, they are not sufficiently reliable. Even the accuracy for predicting the sign of the deviation from [PERSON]'s law does not exceed 60%. [PERSON] and [PERSON] [4] found a significant negative deviation from [PERSON]'s law for Cd-Mg solid solutions. [PERSON] and [PERSON] [5] found that the numerous intermetallic phases in the Cu-Zn system show a negative deviation from [PERSON]'s law. Indeed, negative deviations from [PERSON]'s law are observed for most of the ordered compounds. The excess volume of the alloy, \(\Delta\Omega=\Omega-\Omega_{\rm Zen^{\prime}s}\), usually called \"the superstructure contraction\", plays an important role in the stabilization of intermediate structures due to the free energy gain (\(\Omega_{\rm Zen^{\prime}s}\) is the atomic volume determined by [PERSON]'s law). [PERSON] et al. [6] introduced several parameters that define the geometry of binary alloys. In addition to the _dimensional size factor_, \(\delta=\rm R_{A}/R_{B}\), where \(\rm R_{A}\) and \(\rm R_{B}\) are the atomic radii of components A and B, respectively, _Lawes-Parthe space filling factor_, \(\psi\), and _superstructure contraction_, \(\Delta\Omega/\Omega\), have been introduced. The space filling factor of binary alloys is defined as \(\psi=[\Omega_{\rm A}+\Omega_{\rm B}(1-\chi)]/\Omega(\chi)\). By studying numerous binary alloys with AB and A\({}_{3}\)B stoichiometry, [PERSON] et al. [6] found that among AB stoichiometry structures, B2, B19, and L1\({}_{0}\) structures with Pm3m, Pmcm, and C4/mmm Space Groups, respectively, show a negative superstructure contraction, \(\Delta\Omega/\Omega\), equal to \(-0.100\), \(-\) 0.038, \(-0.045\), respectively, and only L1\({}_{1}\) structure, the Space Group R-3m, shows a very small but positive superstructure contraction equal to +0.003. Among A\({}_{3}\)B stoichiometry structures, L1\({}_{2}\) (Pm3m), D0\({}_{19}\) (P6\({}_{3}\)/mmc), D0\({}_{23}\) (4I/mmm), and D0\({}_{24}\) (P6\({}_{3}\)/mmc) structures show a negative superstructure contraction equal to \(-0.153\), \(-0.053\), \(-0.105\), and \(-0.144\), respectively. Two A\({}_{3}\)B structures, D0\({}_{22}\) (4I/mmm) and A\({}_{15}\) (I/mmm) show a positive superstructure contraction equal to +0.011 and +0.054, respectively. The existence of stable intermetallic compounds with a positive superstructure contraction is explained in terms of the space filling factor, \(\psi\).