= Bloch equations
{c}
{title2=$\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-\mathbf M_\perp/T_2-(M_z-M_0)\hat{\mathbf z}/T_1$}
{wiki}
The <Bloch equations> describe an ensemble <magnetization> $\mathbf M$ with <Larmor precession> plus phenomenological recovery and decay:
$$
\frac{d\mathbf M}{dt}=\gamma\mathbf M\times\mathbf B
-\frac{M_x\hat{\mathbf x}+M_y\hat{\mathbf y}}{T_2}
-\frac{M_z-M_0}{T_1}\hat{\mathbf z}.
$$
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.<Nuclear Induction by Felix Bloch>[§4, pp. 467–468, equations (32), (37) and (38): longitudinal recovery and assumed exponential transverse decay]{ref}
For positive $\gamma$ and no transverse drive, $M_x+iM_y$ rotates as $e^{-i\gamma B_0t}$: 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>.
Back to article page
