Optical Phonons in Electron Spin Relaxation. II. Field Dependence

The general Van Vleck-Hebel-Slichter formalism is used to calculate the magnetic field dependence of the spin-lattice relaxation time attributed to the two-optical-phonon processes for salts doped with the rare-earth ions. For the case in which only the applied field Hₒ, the hyperfine field Hhyp, and the dipolar field Hdip are important, we find results similar to those obtained by Orbach. The relaxation rate in this case is proportional to ${[{{H}ₒ}²+{{μ}}₁{{H}hyp}²+{{{μ}}₁}^{{'}}1/2{{H}dip}²]}{[{{H}ₒ}²+{{H}hyp}²+1/2{{H}dip}²]}$, where ${{μ}}₁=1$, ${{{μ}}₁}^{{'}}=2$ for ${g}_{{⊥}}=0$, and ${{μ}}₁<0$ for ${g}_{{⊥}}{≠}0$. For an isotropic crystal, we find $1/3{}{{μ}}₁{}1/2$, and $7/15{}{{{μ}}₁}^{{'}}{}73/120$. If only ${H}ₒ$, ${H}hyp$, and the exchange field ${H}ₑₓ$ are important, the relaxation rate is proportional to ${[{{H}ₒ}²+{{μ}}₂{{H}hyp}²+{{{μ}}₂}^{{'}}1/2{{H}ₑₓ}²]}{[{{H}ₒ}²+{{H}hyp}²+1/2{{H}ₑₓ}²]}$, where ${{μ}}₂{}1$, ${{{μ}}₂}^{{'}}{}0$ for ${g}_{{⊥}}=0$, and ${{μ}}₂<1$ bor ${g}_{{⊥}}{≠}0$. If ${g}_{{∥}}={g}_{{⊥}}$, and the exchange interaction is isotropic, then $1/3{}{{μ}}₂{}1/2$, $0{}{{{μ}}₂}^{{'}}{}1/60$.