Anisotropy of hardness from first principles: The cases of ReB2 and OsB2

The expression for hardness related to crystal orientation is proposed. Because all quantities in the equation are inherently coupled to the atomistic structure of matter, the anisotropy of hardness can be determined by first-principles methods. The calculations show the highest hardness is in ReB₂ and OsB₂ single crystals along the c axis, 50.3 and 45.5 GPa, respectively, and are in agreement with measurements. The presented model implies that hardness predominantly reflects the strength of bonds transversely oriented to the direction of indentation.