Elastic constants of osmium between 5 and 300 K

Abstract

Using two measurement methods, pulse-echo ultrasound and resonance ultrasound spectroscopy, we measured the elastic constants of both monocrystal and polycrystal osmium between 5 and 300 K. Our measurements help to resolve the current measurement-and-theory controversy concerning whether osmium's bulk modulus exceeds diamond's. It does not at any temperature (for osmium, we find a zero-temperature bulk modulus of 410 GPa and a 300 K value of 405 GPa, while diamond's value being 442 GPa). From the zero-temperature elastic constants, we extract a Debye temperature of 477 K. From Gr\"uneisen's first rule, we extract a Gr\"uneisen parameter of 2.1, agreeing well with handbook values. Osmium shows near elastic anisotropy and small elastic constant changes with temperature (for example, the bulk modulus increases only about 1.2% upon cooling through the studied temperature interval). In all cases, the ${C}_{\text{ij}}(T)$ measurements agree well with an Einstein-oscillator model. We consider especially the Poisson ratio, which is low and anisotropic (${\ensuremath{\nu}}_{12}=0.242$, ${\ensuremath{\nu}}_{13}=0.196$) and suggests some covalent interatomic bonding, which may account for osmium's extreme high hardness and the departure of the $5d$ elements from Friedel's parabolic bulk-modulus/atomic-number model.