# Interpolation formula for BCS superconducting gap

In BCS theory, the superconducting gap is given by solving at different temperatures the integral $$\frac{1}{N(0)V}=\int_0^{\hbar\omega_c}\frac{\tanh\frac{1}{2}\beta(\xi^2+\Delta^2)^{1/2}}{(\xi^2+\Delta^2)^{1/2}}$$ In textbooks like Tinkham (2nd edition, page 63) and Phillips (Advanced Solid State Physics, page 246) you can find approximate formulas for certain temperature ranges (typically $T\approx T_C$).

In some other references, such as here and here (for the latter, I couldn't find the arxiv version, sorry), it is mentioned an interpolation formula valid in the whole temperature range, that is $$\Delta(T)=\Delta_0\tanh (k\sqrt{\frac{T_{C}-T}{T}})$$ with $k=1.74$ or $k=2.2$.

Is there someone who can link a reference to this formula, and how it is obtained?

• Possible duplicates: physics.stackexchange.com/q/54200/2451 , physics.stackexchange.com/q/122781/2451 and links therein. – Qmechanic Jul 3 '15 at 12:40
• In the post you link there's a numerical method to solve the integral I wrote in the first line; in the comments the interpolation formula I'm looking for is quoted, but without references. References (and the derivation method for this interpolation formula) are actually what I'm asking in this post. – casx Jul 3 '15 at 13:00
• I'm not sure you can really prove an interpolation formula, but you can numerically show it's correct, up to some minor numerical difference. An interpolation formula is something which pass through all the known points of a curve. For the superconducting gap, one only knows $\Delta\left(T=0\right)=\Delta_{0}$ and $\Delta\left(T=T_{c}\right)=0$. The interpolation formula obviously pass through these points. You might fix the factor $k$ depending on the region you want to focus on. – FraSchelle Jul 9 '15 at 13:49