Excitation and relaxation

In thermal equilibrium, the net magnetization $\vec M_0$ is oriented along $\vec B_0$, i.e., along the z direction. In a typical MR experiment, the equilibrium is disturbed through a resonant radio frequent (RF) pulse. Resonant energy absorption by the spin system occurs with an angular frequency $\omega_0 = \gamma B_0$, often referred to as the Larmor frequency.
After this excitation, the net magnetization $\vec M$ relaxes back to its original state, during which energy is transferred from the spin system to the molecular environment. The spin-lattice energy transfer only contributes to the relaxation of M_z and occurs exponentially with a time constant T₁. Therefore, T₁ is called the spin-lattice or the longitudinal relaxation time. The initial phase coherence of the spins, immediately after excitation, is disturbed through spin-spin interaction such that M_x,y decreases to zero exponentially with a time constant T₂. T₂ is called the spin-spin or the transversal relaxation time. Excitation and relaxation are phenomenologically described by the Bloch equations [6] from a frame rotating with angular frequency $\omega$:

$${dM_x \over dt} \gamma M_y (B_0 - \omega/\gamma) - {M_x\over T_2}$$ = (1.8)

$${dM_y \over dt} \gamma M_z B_1 - \gamma M_x (B_0 - \omega/\gamma) - {M_y\over T_2}$$ = (1.9)

$${dM_z \over dt} \gamma M_y B_1 - {M_z - M_0\over T_1} \quad ,$$ = (1.10)

where B₁ denotes the magnitude of the RF pulse magnetic induction, which is conventionally directed along the x-axis. In case the rotating frame precesses with the Larmor frequency, after RF perturbation, relaxation occurs according to:

$$M_{x,y}(0)\exp\left(-{t\over T_2}\right)$$ M_x,y(t) = (1.11)

$$M_z(0)\exp\left(-{t\over T_1}\right) + M_0\left(1 - \exp\left(-{t\over T_1}\right)\right) \quad .$$ M_z (t) = (1.12)

Placing a couple of conductive coils around the system, the relaxation and rotation of $\vec M_{xy}$ will induce a complex electric signal in the coils: the Free Induction Decay (FID) signal.