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Q. 10.7

Estimate the energy gap $E_{ g }$ for niobium at T = 0, and find the minimum photon wavelength needed to break the Cooper pair.

Strategy From Table 10.5 we see that $T_{ c }=9.25$ K for Nb. Therefore Equation (10.46) can be used to compute the energy gap. The photon wavelength can then be found using the usual formula relating energy and wavelength, $E=h c / \lambda$.

$E_{ g }(0) \approx 3.54 k T_{ c }$ (10.46)

Verified Solution

From Equation (10.46),

\begin{aligned}E_{ g } \approx 3.54\left(1.38 \times 10^{-23} J / K \right)(9.25 K ) &=4.52 \times 10^{-22} J \\&=2.82 meV\end{aligned}

This small energy corresponds to a photon wavelength of

$\lambda=h c / E_{ g }=4.39 \times 10^{-4} m$

This is in the far-infrared region of the electromagnetic spectrum. Photons with this wavelength or lower have sufficient energy to break the Cooper pair in niobium. Note that this estimate of $E_{g}$ is close to the experimental value of 3.03 meV.