Role of Alloying-Atom Size Factor and System Shape Factor in Energetics of bcc Fe under Macroscopic Deformation

We present an \emph{ab initio} study of the effect of macroscopic deformation on energetics of twelve alloying elements in bcc Fe under three specially designed strain modes. We find that there exists a universal linear relation of describing the volume dependence of substitutional energy of alloying elements via introducing two factors --- the system shape factor ($f_{\scriptsize{ss}}$) and the size factor of alloying element $M$ ($\Omega^{M}_{\scriptsize{sf}}$): $E_{\scriptsize{sub}} \sim f_{\scriptsize{ss}}\Omega^{M}_{\scriptsize{sf}}V$. $\Omega^{M}_{\scriptsize{sf}}$ well describes the effect of intrinsic alloying-atom size and the influence of chemical interaction with matrix atom, and $f_{\scriptsize{ss}}$ characterizes the degree of system lattice distortion under deformation. This relation is further validated using the published data of stained-modulated doping in GaP