\textit{Ab-initio} calculations on the energetics of vacancies in $\alpha $-Fe under strain field*

\textit{Ab-initio} calculations have been carried out to investigate the energetics of vacancies in $\alpha $-Fe under external deformation. We found that volumetric compression has a significant effect on the energetics of vacancies. The formation energy of mono-vacancy decreases with increasing compression monotonically. The $<$111$>$ di-vacancies experience a binding-repelling transition when volumetric compression exceeds 13.2{\%}. On the other hand, uniaxial deformation has weaker influence on the formation and binding energies. However, this type of deformation distinguishes $<$001$>$ di-vacancies: those which are parallel to the strain have larger (smaller) binding energies than those perpendicular ones under compression (expansion). Our results indicate that external strain field is a key factor to modulate the concentration of vacancy and/or vacancy clusters in $\alpha $-Fe.