3.2.1 General

The effective interactions (3.2,3.3) between the conduction electrons of a metal and a nuclear or muonium electron spin may be treated as first order scattering problems[94] According to Fermi's golden rule, the rate of transitions between spin states is determined by

where ${\bf k}$ and $\sigma$ label the conduction electron momentum and spin states; $E_{{\bf k}}$ is the corresponding kinetic energy and $f_{{\bf k},\sigma}$ the occupation probability; a and b label the nuclear or muonium spin states; and $\alpha_m(\sigma_i)$ is the change in magnetic energy. One may convert the sum in Eq. (3.6) to an integral over energy in the usual way, using the (normal state) electronic density of states (DOS), g_N(E), where E is measured relative to E_F. Using $g_N(E)\approx g_N(0)$ within kT of the E_F to simplify the integral, one obtains the Korringa law:

In the superconducting state, the expression for the nuclear transition probability (3.6) is formally the same, but the scattering is accomplished by the bogolons (quasi-particle excitations of the superconducting state), which differ in two important respects from conduction electrons of the normal state: i) there are phase correlations between states of opposite momentum and spin which necessitate combination of pairs of matrix elements before squaring (see, e.g. §3.9 of Tinkham[103]) and give rise to the ``coherence factors''; and ii) the excitation spectrum near E_F is strongly modified: the DOS is gapped and the gap is flanked on either side by singular peaks, predicted by BCS[42], the DOS (see Fig. 3.12a) is:

where $\Re$ is the real part, E is the energy measured from E_F, and $\Delta(T)$is the order parameter, which for the moment we consider to be real, isotropic and homogeneous. Applying these modifications, we get the following integral for the T₁ relaxation rate in the superconducting state (normalized to the rate in the normal state)[104]

where $\beta = (kT)^{-1}$, f is the Fermi-Dirac distribution function, and $E'-E = \alpha_m$.Neglecting any spin-polarization of the quasiparticles, the exothermic and endothermic scattering events will be equally probable, and we take the ratio R_S/R_N to be the simple average of the integrals (3.9) with $\alpha_m$ both positive and negative. If the inelasticity of the collisions is neglected ($\alpha_m = 0$), the two singularities in the integrand coalesce, and the integral becomes logarithmically divergent; however, the singularity is not a practical problem because $\alpha_m$ is finite, and, more importantly, the peak in the DOS is broadened from the BCS result (3.8), as will be discussed in detail in section C. As a function of decreasing temperature, (3.9) exhibits a peak just below T_c, due to the peaked DOS factors, and at lower temperatures, falls off exponentially. For $\alpha_m \approx 0.002 \Delta(0)$(appropriate to the case of Rb₃C₆₀, if it is a BCS superconductor), and assuming the BCS temperature dependence $\Delta(T)$, the maximum of R_S/R_N is about 4 (see Fig. 3.12b).