An intuitive approach to the unified theory of spin-relaxation

Spin-relaxation is conventionally discussed using two different approaches for materials with and without inversion symmetry. The former is known as the Elliott-Yafet (EY) theory and for the latter the D'yakonov-Perel' (DP) theory applies, respectively. We discuss herein a simple and intuitive approach to demonstrate that the two seemingly disparate mechanisms are closely related. A compelling analogy between the respective Hamiltonian is presented and that the usual derivation of spin-relaxation times, in the respective frameworks of the two theories, can be performed. The result also allows to obtain the less canonical spin-relaxation regimes; the generalization of the EY when the material has a large quasiparticle broadening and the DP mechanism in ultrapure semiconductors. The method also allows a practical and intuitive numerical implementation of the spin-relaxation calculation, which is demonstrated for MgB$_2$ that has anomalous spin-relaxation properties.