|
NMR
This site is running |
NMR /
PhenomenologicalRelaxation< Quantum rotating frame in matlab | Index | Bloch equations > Magnetic field produces a torque on a magnetic moment, yielding the cassical Larmor equation that corresponds to precession. {$$ \frac {d\mathbf m}{dt} = \mathbf m\times\mathbf B$$} Further interactions with the environment will lead to relaxation of the precessional motion. Spin relaxation is intrinsic to any nuclear system, since nuclear spins interact among themselves and with the lattice they are embedded into. It is also a feature that we exploit in order to produce any experimental result: as a matter of fact the sample is typically inserted in a superconducting magnet and we rely upon the fact that, after a short time, the nuclear magnetization reaches the relatively large equilibrium value appropriate to the magnetic field. This can only happen by virtue of energy exchanges with a thermal reservoir that allows the spin to reach their minimum statistical energy. Two types of relaxation are natually distinguished by the form of the Larmor equation, one that affects precessing components of {$\mathbf m$}, perpendicular to {$\mathbf B$} and is called {$T_2$} relaxation, plus another one that affects the parallel component, and is called {$T_1$} relaxation. The former is due to dephasing or loss of phase coherence in the precession, and leads to the vanishing of a coherent precession, whereas the latter is an independent process of recovery of thermodynamic equilibrium. As a matter of fact the minimum of {$E=-\mathbf m\cdot\mathbf B$} dictates that at equilibrim the moment lies along the magnetic field. This implies that {$T_1$} is the time scale governing the process of moment alignment to the external field, after it has been switched on, or after moment manipulations. We shall now give a similar phenomenological description of the macroscopic magnetization by means of the Bloch equations. < Quantum rotating frame in matlab | Index | Bloch equations > |