Modulation-Effect Corrections for Moments of Magnetic Resonance Line Shapes

Abstract

Corrections are derived for the calculation of the second and fourth moments of magnetic resonance lines from experimental data, obtained by using the low-frequency modulation method. The results for 0\ifmmode^\circ\else\textdegree\fi{} phase shift between field modulation and the lock-in reference are ${〈\ensuremath{\Delta}{{\ensuremath{\omega}}_{\mathrm{exp}}}^{2}〉}_{\mathrm{Av}}={〈\ensuremath{\Delta}{\ensuremath{\omega}}^{2}〉}_{\mathrm{Av}}+\frac{1}{3}{{\ensuremath{\omega}}_{M}}^{2}+\frac{1}{4}{(\ensuremath{\gamma}{H}_{M})}^{2},$ ${〈\ensuremath{\Delta}{{\ensuremath{\omega}}_{\mathrm{exp}}}^{4}〉}_{\mathrm{Av}}={〈\ensuremath{\Delta}{\ensuremath{\omega}}^{4}〉}_{\mathrm{Av}}+{〈\ensuremath{\Delta}{\ensuremath{\omega}}^{2}〉}_{\mathrm{Av}}[2{{\ensuremath{\omega}}_{M}}^{2}+\frac{3}{2}{(\ensuremath{\gamma}{H}_{M})}^{2}]+\frac{1}{5}{{\ensuremath{\omega}}_{M}}^{4}+\frac{3}{4}{{\ensuremath{\omega}}_{M}}^{2}{(\ensuremath{\gamma}{H}_{M})}^{2}+\frac{1}{8}{(\ensuremath{\gamma}{H}_{M})}^{4}.$Furthermore, it is found that no corrections are necessary for the calculation of intensities despite distortion of the line shape resulting from modulation effects.The equivalence of field and frequency modulation is proved for signals describable by Bloch's equations and the discussion of the general case strongly supports the general validity of this equivalence.