The electron-spin-echo envelope modulation (ESEEM) arising the electronic and geometric structure of paramagnetic sites from hyperfine coupling to a nucleus of arbitrary spin I is investithrough the measurement of nuclear transition frequencies. gated. The generic ESEEM pulse sequence, consisting of a succes-It is especially useful in the investigation of solid samples sion of nonselective microwave pulses and free-evolution periods, which often feature poorly resolved inhomogeneously broadis first considered. It is shown that, when the high-field approximaened spectra in the conventional electron paramagnetic resotion is valid, the ESEEM for a nucleus of arbitrary spin is a funcnance experiment (1). tion of the ESEEM for a I Å 1 2 nucleus subject to the same Hamilto-The envelope modulation of the primary (two-pulse) elecnian and that the functional relationship is provided by the Chebtron spin echo, first observed in 1961 by Mims et al. (2), yshev polynomials of the second kind. Such relationship also was theoretically described a few years later (3). A spin provides a very efficient method for the numerical simulation of system with arbitrary electron and nuclear spin quantum the ESEEM due to I ú 1 2 nuclei. Next, the experiments based on numbers (S and I, respectively) was considered and a genthe primary and on the stimulated electron-spin-echo pulse seeral recipe for the calculation of the ESEEM was reported quences are considered and explicit analytical formulas for arbitrary nuclear spin are given. The central result of the theory devel-which, however, neglected the nuclear quadrupole interacoped is that the modulation amplitudes are polynomials of degree tion. The explicit ESEEM formula was obtained for the spe-2I in the modulation depth parameter k. This nonlinearity introcial case S Å 1 2 , I Å 1 2 . Later, there was proposed another duces two differences with respect to the I Å 1 2 case. First, the general scheme (4) for the calculation of ESEEM based on amplitudes of the fundamental pure and combination modulathe matrix representation of operators. In this way, explicit tions, already present for spin-1 2 nuclei, are largely nonlinear funcformulas for the primary and for the stimulated (three-pulse) tions of k; second, harmonics of the fundamental modulations (up ESEEM were obtained for a spin system consisting of one to the 2Ith) occur in the ESEEM with amplitudes which can be electron and one nucleus with I Å 1 2 or 1. In the latter case, comparable with those of the fundamentals. As a general rule, the nuclear quadrupole interaction was treated as a small nonlinear effects are more important when I is large, provided perturbation affecting only the nuclear transition frequencies. that all the other factors are the same. Since terms of different order in k alternate in sign, the modulation amplitudes show an The problem of computing the ESEEM due to a nucleus of oscillating behavior and reach their maximum well before k Å 1. arbitrary spin was also considered, but an explicit formula While the amplitude of the pure modulations is always positive, was obtained only by neglecting the nuclear quadrupole inthat of the combination modulations (which occur only in primary teraction and by assuming that the modulation is very shaland 2-D stimulated ESEEM) can be either positive or negative, low (5). It is worthwhile to point out how the adjective depending on the value of k; thence a new type of suppression ''explicit'' is meant. Such formulas can be said to be explicit effect ensues which is independent of interpulse delays. Besides, because the echo amplitude is expressed as a linear combinathe well-known suppression effect in the 1-D stimulated ESEEM tion of cosine functions, their arguments containing the modof I Å 1 2 nuclei occurs also when I ú 1 2 in a similar way. Spectral ulation frequencies and the time intervals between the pulses simulations are presented to illustrate the characteristics of the and their amplitudes being proportional to a single parame-ESEEM arising from I ú 1 2 nuclei. The theory developed is comter, namely the modulation depth parameter k. On one hand, pared with an earlier analysis which neglected nonlinear terms, this parameter has a clear physical (and even geometrical) and its advantages are demonstrated. ᭧ 1997 Academic Press
interpretation and is readily related to the strength of the nuclear magnetic interactions; on the other hand, the echomodulation amplitude is completely determined by k and I.