where 𝜏 SL is the spin-lattice relaxation time, 𝜏 LB is the lattice-bath relaxation time, 𝑐 𝑠 is the specific heat of the spin system, 𝑐 𝐿 is the lattice specific heat and the first term is presumed to have power-law temperature dependence (BT -m ). We fit the temperature-dependence of relaxation rates for 4-Dy to Equation S9 (derivation using Equation (1) based on Rousset et al. ) which considers Raman and QTM processes to be subject to a phonon bottleneck, while any effect of the phonon bottleneck on the Orbach process is absorbed into the value of 𝜏 0 . 65