Low-Temperature Specific Heats of Titanium, Zirconium, and Hafnium

The specific heats of titanium, zirconium, and hafnium were found to obey the relation $c=\ensuremath{\gamma}T+\ensuremath{\beta}{T}^{3}$ from 1.1 to 4.5\ifmmode^\circ\else\textdegree\fi{}K within the experimental error. As in other transition metals, the electronic term is large and for the Group IV-A metals decreases with increasing atomic number indicating a progressively larger degree of electronic interaction in the sequence titanium, zirconium, and hafnium. The Debye temperatures decrease with increasing atomic mass as would be expected from the central-force model; however, the ratio of the Debye temperatures indicate that the average atomic force constant for hafnium is some 50% larger than for titanium and zirconium in agreement with the unusually small atomic volume of hafnium.