Dipolar relaxation revisited : a complete derivation for the two spin case

Relaxation theory often forms the basis of various studies, like MRI contrast agents and ionic liquids, which require the understanding of the molecular dynamics or determination of molecular parameters (such as correlation times) or both. Such studies compare the explicit expressions of the relaxation times T1 and T2, derived many years ago by quantum physicists, with their experimental findings to understand various aspects of the molecular motion. However, because of the complicated terminology and widely scattered information, the task of understanding or re-deriving the relaxation theory for even the simplest cases becomes challenging for many students and researchers new to this field. In this article, the dipolar relaxation theory for the basic, yet practical case of a two spin system is presented with a detailed description of the concepts behind it. Aiming at a beginner level and making only modest assumptions on prior knowledge, we have discussed the basics of the quantum mechanics, spherical harmonics, stochastic processes and perturbation theories. It is hoped that this article will enable the reader to understand the general framework of the relaxation theory and will be beneficial in comprehending the relaxation in more complex systems as well, which is not discussed here.