Low-frequency spin dynamics as probed by ${}^{63}\mathrm{Cu}$ and ${}^{199}\mathrm{Hg}$ NMR in ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\mathrm{\delta }}$ superconductors with different oxygen content

${}^{63}\mathrm{Cu}$ and ${}^{199}\mathrm{Hg}$ nuclear magnetic resonance was performed on a series of powder ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\mathrm{\delta }}$ samples with different oxygen content δ in order to study the influence of oxygen doping on the spin dynamics in the normal and superconducting state. The spin-lattice relaxation time $T_1$ of ${}^{199}\mathrm{Hg}$ and ${}^{63}\mathrm{Cu}$ nuclei were measured over a wide range of temperature at different orientations of the crystallites with respect to the magnetic field. For the optimally doped sample ${(T}_c$=96 K) we compared our experimental results to numerical calculations of the spin-lattice relaxation rate as a function of temperature below $T_c$ for several types of the order parameter symmetry: isotropic s wave, anisotropic s wave without nodes, s wave with nodes, and d wave, using the different values of the gap parameter $2\Delta {/k}_B{T}_c$ within the AF fluctuation model. The comparison with our experimental relaxation data shows that for ${}^{199}\mathrm{Hg}$ nuclei in the orientation ${\mathbf{B}}_0\parallel \mathrm{}(a,b)\mathrm{}$ plane as well as for ${}^{63}\mathrm{Cu}$ nuclei in both orientations, ${\mathbf{B}}_0\parallel \mathrm{}(a,b)\mathrm{}$ plane and ${\mathbf{B}}_0\parallel c$ axis, the experimental results are in coincidence with the $d_{x^2-y^2}$-wave symmetry with the gap parameter $2\Delta {/k}_B{T}_c=7.02\mathrm{}$.