Electron-Spin-Echo Envelope Modulation Arising from Hyperfine Coupling to a Nucleus of Arbitrary Spin

Abstract

The electron-spin-echo envelope modulation (ESEEM) arising from hyperfine coupling to a nucleus of arbitrary spinIis investigated. The generic ESEEM pulse sequence, consisting of a succession of nonselective microwave pulses and free-evolution periods, is first considered. It is shown that, when the high-field approximation is valid, the ESEEM for a nucleus of arbitrary spin is a function of the ESEEM for aI= 12 nucleus subject to the same Hamiltonian and that the functional relationship is provided by the Chebyshev polynomials of the second kind. Such relationship also provides a very efficient method for the numerical simulation of the ESEEM due toI>12 nuclei. Next, the experiments based on the primary and on the stimulated electron-spin-echo pulse sequences are considered and explicit analytical formulas for arbitrary nuclear spin are given. The central result of the theory developed is that the modulation amplitudes are polynomials of degree 2Iin the modulation depth parameterk. This nonlinearity introduces two differences with respect to theI= 12 case. First, the amplitudes of the fundamental pure and combination modulations, already present for spin-12 nuclei, are largely nonlinear functions ofk; second, harmonics of the fundamental modulations (up to the 2Ith) occur in the ESEEM with amplitudes which can be comparable with those of the fundamentals. As a general rule, nonlinear effects are more important whenIis large, provided that all the other factors are the same. Since terms of different order inkalternate in sign, the modulation amplitudes show an oscillating behavior and reach their maximum well beforek= 1. While the amplitude of the pure modulations is always positive, that of the combination modulations (which occur only in primary and 2-D stimulated ESEEM) can be either positive or negative, depending on the value ofk; thence a new type of suppression effect ensues which is independent of interpulse delays. Besides, the well-known suppression effect in the 1-D stimulated ESEEM ofI= 12 nuclei occurs also whenI>12 in a similar way. Spectral simulations are presented to illustrate the characteristics of the ESEEM arising fromI>12 nuclei. The theory developed is compared with an earlier analysis which neglected nonlinear terms, and its advantages are demonstrated.