Low-Temperature Specific Heat of Indium and Tin

Abstract

The heat capacities of indium and tin were measured between 0.4 and 4.2\ifmmode^\circ\else\textdegree\fi{}K. In the normal state, the specific heat could be represented by $A{T}^{\ensuremath{-}2}+\ensuremath{\gamma}T+\ensuremath{\alpha}{T}^{3}+\ensuremath{\beta}{T}^{5}+\ensuremath{\mu}{T}^{7}$. For Sn, in molar millijoule units, $A=0$; $\ensuremath{\gamma}$, the coefficient in the electronic term, is 1.80; $\ensuremath{\alpha}=0.242$, corresponding to a Debye temperature, ${\ensuremath{\theta}}_{0}$, of 200\ifmmode^\circ\else\textdegree\fi{}K; $\ensuremath{\beta}=0.004$; and $\ensuremath{\mu}=0.00014$. For In, $A$, the coefficient of a nuclear electric quadrupole term, is calculated to be 8.97\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ from resonance data; $\ensuremath{\gamma}=1.61$ for one ingot and 1.59 for another; ${\ensuremath{\theta}}_{0}=109\ifmmode^\circ\else\textdegree\fi{} \mathrm{and} 108\ifmmode^\circ\else\textdegree\fi{}$K; and $\ensuremath{\beta}=0.008$. In the superconducting state, the specific heat of Sn could be expressed as the normal lattice term plus an electronic term of the form $a\ensuremath{\gamma}{T}_{c}\mathrm{exp}(\ensuremath{-}\frac{b{T}_{c}}{T})$, with ${T}_{c}=3.70\ifmmode^\circ\else\textdegree\fi{}$K (0.02 deg lower than found in a magnetic measurement), $a=7.63$, and $b=1.41$ when $2l\frac{{T}_{c}}{T}l7$; the value of $b$ agrees with infrared measurements of the energy gap. This sort of analysis could not be applied to In, for below 0.8\ifmmode^\circ\else\textdegree\fi{}K the total superconducting specific heat was less than the normal lattice term. A possible interpretation is that ${\ensuremath{\theta}}_{0}$ is 9% higher in the superconducting state than in the normal metal at 0.4\ifmmode^\circ\else\textdegree\fi{}K; this is not supported, however, by the recent acoustic measurements of the elastic constants by Chandrasekhar and Rayne. The anomaly is not as yet understood, but a few plausible explanations are discussed.