Relaxation Times and Linewidths

Here the effects which bring the magnetization vector back to its equilibrium position are explained. In an S = 1/2 system the spins can be oriented parallel or anti-parallel to the external magnetic field and a perturbation (e.g. pulse) induces transitions between the two energy levels. The M_z magnetization in equilibrium is the result of a small surplus of spins parallel to their quantization axis. Changing the M_z magnetization involves reorientation of the microscopic magnetic moments. Transversal magnetization M_x and M_y on the other hand is given by an in-phase precession of the magnetic moments about the external field, induced e.g. by an MW pulse.

Upon relaxation the Boltzmann M_z magnetization is restored and the transversal magnetic components M_x and M_y vanish. These changes in magnetization are associated with the spin-lattice or longitudinal relaxation time T₁ and the spin-spin relaxation time T₂, respectively. The spin-lattice relaxation relates to the characteristic lifetime of the spin state and is determined by the dissipation of energy via the thermal vibration of the lattice. T₁ is related through the Heisenberg uncertainty principle to the linewidth of an individual spin packet. A small T₁ leads to a smearing out of the energy levels and thus a broad resonance line. Large T₁ values are usually found for systems with isolated electronic ground states, well separated from the lowest excited states. The spin-spin relaxation is concerned with the mutual spin flips caused by dipolar and exchange interactions between the assembly of spin in the sample. T₂ is usually much shorter than T₁ and thus the dominant contribution to the linewidth. The two contributions are often summarized by relating the resultant linewidth of a single spin packet to a relaxation time T'₂ given by