Quasiparticle spin susceptibility in heavy-fermion superconductors : An NMR study compared with specific heat results

Quasi-particle spin susceptibility ($\chi^{qp}$) for various heavy-fermion (HF) superconductors are discussed on the basis of the experimental results of electronic specific heat ($\gamma_{el}$), NMR Knight shift ($K$) and NMR relaxation rate ($1/T_1$) within the framework of the Fermi liquid model for a Kramers doublet crystal electric field (CEF) ground state. $\chi^{qp}_{\gamma}$ is calculated from the enhanced Sommerfeld coefficient $\gamma_{el}$ and $\chi^{qp}_{T_1}$ from the quasi-particle Korringa relation $T_1T(K^{qp}_{T_1})^2=const.$ via the relation of $\chi^{qp}_{T_1}=(N_A\mu_B/A_{hf})K^{qp}_{T_1}$ where $A_{hf}$ is the hyperfine coupling constant, $N_A$ the Abogadoro's number and $\mu_B$ the Bohr magneton. For the even-parity (spin-singlet) superconductors CeCu$_2$Si$_2$, CeCoIn$_5$ and UPd$_2$Al$_3$, the fractional decrease in the Knight shift, $\delta K^{obs}$, below the superconducting transition temperature ($T_c$) is due to the decrease of the spin susceptibility of heavy quasi-particle estimated consistently from $\chi^{qp}_{\gamma}$ and $\chi^{qp}_{T_1}$. This result allows us to conclude that the heavy quasi-particles form the spin-singlet Cooper pairs in CeCu$_2$Si$_2$, CeCoIn$_5$ and UPd$_2$Al$_3$. On the other hand, no reduction in the Knight shift is observed in UPt$_3$ and UNi$_2$Al$_3$, nevertheless the estimated values of $\chi^{qp}_{\gamma}$ and $\chi^{qp}_{T_1}$ are large enough to be probed experimentally. The odd-parity superconductivity is therefore concluded in these compounds. The NMR result provides a convincing way to classify the HF superconductors into either even- or odd- parity paring together with the identification for the gap structure, as long as the system has Kramers degeneracy.