Question

Question: The critical temperature lead is 7.2 K. What is the binding energy of its Cooper pairs at zero temperature?

Short answer

Answer

The binding energy of the Cooper pairs is 217meV.

Step-by-step solution

Given data

The critical temperature of lead is, 7.2 K .

Temperature of cooper pairs is, 0K .

Definition of Binding Energy

The superconducting energy gap ${\mathit{E}}_{\mathbf{g}}$ , the energy needed to split a molecule into its individual atoms, must be equivalent to the energy needed to break a cooper pair.

As per BCS theory, energy at absolute zero shall be${\mathit{E}}_{\mathbf{g}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{5}{\mathit{k}}_{\mathbf{B}}{\mathit{T}}_{\mathbf{c}}$ .

Where K_B represents Boltzmann's constant andT_c represents critical temperature.

Determine the Binding Energy

The binding energy of the Cooper pairs is expressed as,

.$E_g=3.5k_B{T}_c$

Substitute $7.2 \text{K}\to T_c$, and $1.38\times {10}^{-23}\text{J/K}\to k_B$ in the above equation.

$$\begin{gathered} E_g=3.5k_B{T}_c \\ =3.5\left(1.38\times {10}^{-23} \text{J/K}\right)(7.2 \text{K}) \\ =3.48\times {10}^{-22} \text{J}\left(\frac{1 \text{cV}}{1.6\times {10}^{-19} \text{J}}\right) \\ =2.175\times {10}^{-3} \text{eV}\left(\frac{1 \text{mcV}}{{10}^{-3} \text{eV}}\right) \end{gathered}$$

Which gives, $E_g=2.17 \text{meV}$.

The binding energy of the Cooper pairs is $2.17 \text{meV}$.