Theoretical Study on the Anisotropy of the Spin-Lattice Relaxation for Paramagnetic Ions in MgO Crystal

The anisotropy of the spin-lattice relaxation time with respect to the orientation of the Zeeman field about the crystal axes has been studied theoretically for ${\mathrm{Co}}^{2+}$, ${\mathrm{Ni}}^{2+}$, and ${\mathrm{Cr}}^{3+}$ ions in an MgO crystal for both single- and two-phonon processes, using a Debye model of the phonon distribution. For the single-phonon process, the anisotropy in the relaxation time is large for ${\mathrm{Co}}^{2+}$ but small for ${\mathrm{Ni}}^{2+}$ and ${\mathrm{Cr}}^{3+}$. For the double-phonon process, where the calculations are done without making the long-wavelength approximation, the relaxation time is totally isotropic for ${\mathrm{Co}}^{2+}$, but for ${\mathrm{Ni}}^{2+}$ and ${\mathrm{Cr}}^{3+}$ it is slightly anisotropic. The deviation from the expected $T^{-7\mathrm{}}$ law is found to be significant for the ${\mathrm{Cr}}^{3+}$ ion. The effect of the dipolar interaction on the anisotropic factor for both single- and two-phonon processes is discussed.