Quest for nuclear quadrupole relaxation in HgCa$_{2}$Ba$_{2}$Cu$_{3}$O$_δ$

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 nuclear spin $I$ = 3/2. 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$_{\delta}$, we estimated the nuclear electric quadrupole spin-lattice relaxation rate $^{63}W_Q$ and the upper limit.