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 ($f_{ss}$) and the size factor of alloying element $M$ (${\mathrm{\Omega }}_{sf}^M$): $E_{sub}\sim f_{ss}{\mathrm{\Omega }}_{sf}^{M}V$. ${\mathrm{\Omega }}_{sf}^M$ well describes the effect of intrinsic alloying-atom size and the influence of chemical interaction with matrix atom, and $f_{ss}$ characterizes the degree of system lattice distortion under deformation. This relation is further validated using the published data of stained-modulated doping in GaP