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BlochEquations

< Phenomenology of relaxation and radio frequency pulses | Index | One pulse and the Free Induction Decay >

These equations describe both the precession of the average nuclear magnetization {$\mathbf M$} and the distinct relaxation dynamics of its equilibrium and out-of-equilibrium components. The description is entirely phenomenological as for a purely classical quantity. For a field {$\mathbf{B} = B \hat z$} the equations read

{$ \begin{eqnarray} \frac {dM_x}{dt} & = & \gamma B M_y - \frac {M_x}{T_2} \\ \frac {dM_y}{dt} &=& -\gamma B M_x - \frac {M_y}{T_2}\\ \frac {dM_z}{dt} &=& - \frac {M_z-\chi H}{T_1}\end{eqnarray}$}

These equations become particularly simple if one recognizes, in the first two up to the first term in the rhs, the precession at the Larmor frequency {$\omega=\gamma B$} of the magnetization component perpendicular to {$\hat z$}. The precession is removed by switching to the frame {$x',y',z$} rotating around {$\hat z$} at frequency {$\omega$}, where le Bloch equations read

{$ \begin{eqnarray} \frac {dM_x'}{dt} & = & - \frac {M_x'}{T_2} \\ \frac {dM_y'}{dt} &=& - \frac {M_y'}{T_2}\\ \frac {dM_z}{dt} &=& - \frac {M_z-\chi H}{T_1}\end{eqnarray}$}

< Phenomenology of relaxation and radio frequency pulses | Index | One pulse and the Free Induction Decay >