The longitudinal (or spin-lattice) relaxation time T is the decay constant for the recovery of the z component of the nuclear spin magnetization, M, towards its thermal equilibrium value, . In general,
In specific cases:
*If M has been tilted into the xy plane, then  and the recovery is simply
i.e. the magnetization recovers to 63% of its equilibrium value after one time constant T.
*In the inversion recovery experiment, commonly used to measure T values, the initial magnetization is inverted, , and so the recovery follows
T relaxation involves redistributing the populations of the nuclear spin states in order to reach the thermal equilibrium distribution. By definition, this is not energy conserving. Moreover, spontaneous emission is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of T relaxation. However, a variety of relaxation mechanisms allow nuclear spins to exchange energy with their surroundings, the lattice, allowing the spin populations to equilibrate. The fact that T relaxation involves an interaction with the surroundings is the origin of the alternative description, spin-lattice relaxation.
Note that the rates of T relaxation (i.e., 1/T) are generally strongly dependent on the NMR frequency and so vary considerably with magnetic field strength B. Small amounts of paramagnetic substances in a sample speed up relaxation very much. By degassing, and thereby removing dissolved oxygen, the T/T of liquid samples easily go up to an order of ten seconds.