Homology of interatomic forces and Debye temperatures in transition metals

Abstract

Using experimental data on the entropy S at intermediate temperatures, corrected for an electronic term to get the vibrational entropy, we obtain well-defined and accurate ‘‘entropy Debye temperatures’’ FTHETA=${\mathrm{FTHETA}}^S$. From ${\mathrm{FTHETA}}^S$ we define a quantity with the dimension of a force constant, $k^S$==M($k_B$FTHETA/ħ${)}^2$, where M is the atomic mass. Similarly, data on the low-temperature vibrational heat capacity and the elastic coefficients yield Debye temperatures ${\mathrm{FTHETA}}^C$ and ${\mathrm{FTHETA}}^{\mathrm{elas}}$ and corresponding force constants $k^C$ and $k^{\mathrm{elas}}$. The ratios of $k^S$ for the 4d-5d pairs Zr,Hf, Nb,Ta, Mo,W, Tc,Re, Ru,Os, Rh,Ir, and Pd,Pt are remarkably constant, ($k^S$ ${)}^{4d}$/($k^S$ ${)}^{5d}$ =0.76±0.01, in spite of varying crystal structures. Further, ($k^C$ ${)}^{4d}$/($k^C$ ${)}^{5d}$ =0.70±0.07 and ($k^{\mathrm{elas}{)}^{4d}/(k^{\mathrm{elas}{)}^{5d}}}$ =0.69±0.08. The corresponding force-constant ratios $k^{3d}$/$k^{4d}$ vary more. This correlates with an approximately constant atomic-volume ratio ${\Omega }^{4d}$/${\Omega }^{5d}$=0.99±0.02, while ${\Omega }^{3d}$/${\Omega }^{4d}$=0.81±0.05 shows irregularities, mainly of magnetic origin. As a basis for our analysis, we review low-temperature Debye temperatures ${\mathrm{FTHETA}}^{\mathrm{elas}}$ and ${\mathrm{FTHETA}}^C$ and the bulk modulus B of the transition metals.