Modulation Broadening of NMR and ESR Line Shapes

TLDR

Phase‑sensitive detection of magnetic resonances at multiple harmonics of the modulation frequency is analyzed. The study develops a mathematical treatment of modulation broadening. The authors derive expressions for the amplitude and line shape of Lorentzian and Gaussian resonances at the first three harmonics as functions of modulation amplitude, relate the Fourier coefficients of broadened lines to the true integrated shape, and provide a correction method compared with a convergent‑series approach. The derived expressions enable correction of modulation‑broadened line shapes in any phase‑sensitive detection experiment and are applicable in all practical cases, outperforming the Taylor‑expansion method.

Abstract

An analysis is made of the phase sensitive detection of magnetic resonances at various harmonics of the modulation frequency. The results are applicable to any experiment in which phase sensitive detection is employed. A mathematical treatment of modulation broadening is developed. Unlike the Taylor expansion method this is applicable in all practical cases. Signal amplitude and line shape of Lorentz and Gaussian resonances are calculated for the first three harmonics as functions of the modulation amplitude. Expressions which relate the Fourier coefficients of a modulation-broadened resonance line shape to those of the true integrated line shape are derived. The use of these expressions in correcting observed line shapes for modulation broadening is compared with that of another method involving the summation of a convergent series.