Quest for nuclear quadrupole relaxation in HgCa${}_2$Ba${}_2$Cu${}_3$O${}_{8+\delta }$

We derived a simple reduced form from the exact solution of the mixed magnetic and quadrupolar nuclear spin–lattice relaxation function at the central transition line of a quadrupole-split NMR spectrum for the nuclear spin $I$ = 3/2. The reduced form of the exact solution could be extended to include inhomogeneous relaxation effects. From the application of the reduced form to the ${}^{63}$Cu nuclear spin–lattice relaxation curves in the high-$T_{\mathrm{c}}$ superconductors of ${}^{63}$Cu-enriched triple-layer HgBa${}_2$Ca${}_2$Cu${}_3$O${}_{8+\delta }$, we estimated the predominant magnetic relaxation rate ${}^{63}{W}_M$ and the upper limit of a weak nuclear electric quadrupole spin–lattice relaxation rate ${}^{63}{W}_Q$.