BCS coherence factors ($u_k^2+v_k^2=1$)

For real gap and the particle-hole block $\left(\begin{array}{cc}{\xi }_k& -{\Delta }_k\\ -{\Delta }_k& -{\xi }_k\end{array}\right)$, the BCS coherence factors obey $u_k^2=(1+{\xi }_k/E_k)/2$, $v_k^2=(1-{\xi }_k/E_k)/2$ and $u_k{v}_k={\Delta }_k/(2E_k)$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }_k^2}$. They diagonalize the Bogoliubov--de Gennes Hamiltonian and give the empty/pair amplitudes of the BCS ground state. Their relative sign depends on the pairing convention.

For an original check, write $x={\xi }_k/\Delta$ with constant real $\Delta >0$. At $x=0$, the BCS coherence factors give $u_k^2=v_k^2=1/2$. At $x=1$, they give $(u_k^2,v_k^2)=(0.853553,0.146447)$; at $x=-1$ these values exchange places. Thus the zero-temperature occupation $v_k^2$ rounds the normal Fermi sea's step over an energy interval of order $\Delta$. This mixing is strongest at the Fermi surface, where the BCS quasiparticle spectrum has its minimum.

For a uniform isotropic weak-gap BCS state, the zero-temperature electron occupation is $n(\xi )=(1-\xi /\sqrt{{\xi }^2+{\Delta }^2})/2$. The number above the normal-state Fermi surface is approximately $D(\mu ){\int }_0^{\infty }n(\xi )d\xi$. The integral equals $\Delta /2$, while a three-dimensional parabolic band including both spins has $D(\mu )=3N/(2\mu )$. This gives the displayed leading fraction.

The result measures redistribution relative to a normal Fermi sea, not bosonic single-mode condensate depletion or quasiparticle occupation. A different band dimension or anisotropic gap requires its actual density-of-states or momentum integral. Material estimates need specified gap and chemical-potential scales.

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]