MW Pulses and Spin Dynamics

In this section we first introduce the concepts of macroscopic magnetization and the rotating frame to describe the effects of MW pulses on a given spin system.

The macroscopic magnetization M of a sample is obtained by adding up the contribution of all magnetic moments in a given sample volume. The precession of the magnetic moments about the external magnetic field B₀ (taken along the z-axis) being out-of-phase in the absence of an external perturbation, the magnetic xy components are randomly distributed and a net magnetization (M_z) is observed only along B₀ in the Boltzmann equilibrium.
In the rotating frame the coordinate system rotates with the frequency ν_MW of the MW excitation about the z-axis. In this frame the excitation becomes time-independent and the resonances frequency for the electron spins is given by the offset Ω = ν-ν_MW of the true resonance frequency ν, obtained from the resonance condition in Eq. (3), from the excitation frequency ν_MW. The large static field B₀ thus appears to be considerably reduced or even zero for the on-resonant case (ν = ν_MW).

Fig. 9: Illustration of the simplest pulse experiment with a single MW π/2 pulse (top) and the behavior of the magnetization vector M in the rotating frame (bottom).

It is the interaction of M with the applied electromagnetic fields and the detector which gives observable pulse EPR spectra. In Fig. 9 the most simple pulse experiment, a single π/2 pulse, and the behavior of the magnetization under this pulse in the rotating frame are illustrated. The application of a MW pulse of frequency ν_MW perpendicular to the static field B₀ leads to a nutation of M about the resultant of the static and the perturbing magnetic field during the pulse. Under the influence of a strong MW π/2 pulse applied along the y axis, the magnetization M is rotated to the x-axis⁵. This is shown for the on-resonant case in Fig. 9, where the z-component of the resulting field is zero. After the MW pulse is switched off, the magnetization vector relaxes back to its equilibrium position along z, thereby precessing with the offset frequency Ω. The precession leads to oscillating components M_x and M_y of the magnetization vector in the xy-plane which can be recorded as free induction decay (FID) after the MW pulse.

⁵ The flip angle of a MW pulse (e.g. 90° for a π/2 pulse) depends on its length t_p and its amplitude. The axis about which the magnetization vector is rotated is given by the phase of the pulse.