Bloch equations ($\dot{\mathbf{M}}=\gamma \mathbf{M}\times \mathbf{B}-{\mathbf{M}}_{\perp }/T_2-(M_z-M_0)\hat{\mathbf{z}}/T_1$)

The Bloch equations describe an ensemble magnetization $\mathbf{M}$ with Larmor precession plus phenomenological recovery and decay:Here $T_1,T_2>0$ are model parameters, not a microscopic explanation of relaxation. The precession term follows from the torque on a magnetic moment; the two dissipative terms prescribe exponential decay. Bloch explicitly introduced this macroscopic relaxation description as a simplifying approximation.^[ref]

For positive $\gamma$ and no transverse drive, $M_x+i{M}_y$ rotates as $e^{-i\gamma B_{0}t}$: its signed rotation is negative about $z$, although the resonance angular frequency ${\omega }_0=\gamma B_0$ is positive. This distinction prevents a hidden sign change when choosing a rotating frame.

With no transverse drive, the Bloch equations giveLongitudinal nuclear spin relaxation restores the population imbalance and thus the magnetization parallel to the static magnetic field. It corresponds to population relaxation for the two-level model. Starting from $M_z(0)=0$, waiting $T_1$ recovers $1-e^{-1}$ of $M_0$, whereas waiting $5T_1$ recovers about $99.3\%$. This is why repeated measurements require a recovery-time decision.

Transverse relaxation describes loss of ensemble transverse magnetization. It must not automatically be identified with pure dephasing. In a simple completely positive two-level relaxation model,where $T_{\varphi }$ describes additional pure dephasing. Thus population recovery itself contributes to transverse decay. A spread of static magnetic fields can further cancel ensemble phases without the same microscopic relaxation; an observed free signal may have an effective $T_2^{\ast }$ rather than homogeneous $T_2$. The simple exponential examples below assume a uniform field and a single homogeneous $T_2$.