  Understanding the effect of vibrations on the relaxation process of
individual spins is crucial for implementing nano systems for quantum
information and quantum metrology applications. In this work, we present a
theoretical microscopic model to describe the spin-lattice relaxation of
individual electronic spins associated to negatively charged nitrogen-vacancy
centers in diamond, although our results can be extended to other spin-boson
systems. Starting from a general spin-lattice interaction Hamiltonian, we
provide a detailed description and solution of the quantum master equation of
an electronic spin-one system coupled to a phononic bath in thermal
equilibrium. Special attention is given to the dynamics of one-phonon processes
below 1 K where our results agree with recent experimental findings and
analytically describe the temperature and magnetic-field scaling. At higher
temperatures, linear and second-order terms in the interaction Hamiltonian are
considered and the temperature scaling is discussed for acoustic and
quasi-localized phonons when appropriate. Our results, in addition to
confirming a $T^5$ temperature dependence of the longitudinal relaxation rate
at higher temperatures, in agreement with experimental observations, provide a
theoretical background for modeling the spin-lattice relaxation at a wide range
of temperatures where different temperature scalings might be expected.
