LAUSR

We present a first-principles approach for computing the phonon-limited ${T}_{1}$ spin relaxation time due to the Elliott-Yafet mechanism. Our scheme combines fully relativistic spin-flip electron-phonon interactions with an approach to compute the effective spin of band electrons in materials with inversion symmetry. We apply our method to silicon and diamond, for which we compute the temperature dependence of the spin relaxation times and analyze the contributions to spin relaxation from different phonons and valley processes. The computed spin relaxation times in silicon are in excellent agreement with experiment in the 50--300 K temperature range. In diamond, we predict intrinsic spin relaxation times of 540 $\ensuremath{\mu}\mathrm{s}$ at 77 K and 2.3 $\ensuremath{\mu}\mathrm{s}$ at 300 K. We show that the spin-flip and momentum relaxation mechanisms are governed by distinct microscopic processes. Our work enables precise predictions of spin-phonon relaxation times in a wide range of materials, providing microscopic insight into spin relaxation and guiding the development of spin-based quantum technologies.