Apparent Two Energy Gaps in Pure Niobium

It is found experimentally that two energy gaps are obtained for some pure superconductors when ultrasonic attenuation data are analyzed in terms of a simple BCS expression $\frac{{\alpha }_s}{{\alpha }_n}=\frac{2}{(e^{\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{T}}+1)}$ for the ratio of the attenuation coefficient in the normal and in the superconducting state. The energy gap $\Delta \mathrm{}\left(T\right)\mathrm{}$ determined in the vicinity of the transition temperature $T_c$ is usually larger than that expected from the low-temperature value $\Delta \mathrm{}\left(0\right)\mathrm{}$. The above BCS expression is valid either in the limit $\mathrm{ql}\gg 1$ or if the electronic mean free path $l$ is mainly determined by impurity scattering, where $q$ is the sound wave vector. However, when $\mathrm{ql}\ll 1$ and when the sample becomes pure enough that electron-phonon scattering dominates the electron lifetime, we expect an important deviation from the above relation. A theory is proposed which takes into account the effect of the electron-phonon scattering explicitly. This theory predicts that the ratio of the energy gap deduced from high-temperature data (i.e., $T\cong T_c$ the transition temperature) to the one deduced from low-temperature data is a universal function of $x=l_{0}B{T}^3$, where $l_0$ is the electron mean free path due to impurity scattering while ${\left(B{T}^3\right)}^{-1\mathrm{}}$ is that due to electron-phonon scattering; the electron mean free path in the normal state is given by $l_n^{-1\mathrm{}}=l_0^{-1\mathrm{}}+B{T}^3$. In order to have a $T^3$ dependence, it is assumed that niobium has two distinct electron bands with different effective masses. Making use of the ultrasonic attenuation data in the normal state, we can determine $l_n$ and therefore the parameter $x$. The ratio of the two energy gaps is then calculated by substituting this value of $x$ into a universal function $f\left(x\right)\mathrm{}$ which is theoretically derived. It is found that the predicted ratio accounts for roughly one-half of the observed deviation from unity of this ratio. Alternatively, if we use the ratio of the energy gaps to determine $x$, we find that $x$ (i.e. the electron-phonon scattering contribution) in the superconducting state is three times as large as that determined in the normal state. We are unable to account for these discrepancies at present, although we feel that the observed ratio of the two energy gaps is mostly due to the electron-phonon interaction. However, we cannot completely disregard other possible explanations like the anisotropy of the energy gap or the strong-coupling effect. A numerical estimate of $B$ making use of the known parameters like the electron-phonon coupling constant ${\lambda }_0$ and the Debye frequency ${\omega }_D$ of niobium, on the other hand, yields $B\cong \frac{1.2}{K_0^3}$ cm in semiquantative agreement with the value determined experimentally.