Assessment of the spherical-averaging approximation for the interpretation of electron spin-echo modulation for cupric ion complexes

Abstract

A comparison is made between three methods for the simulation of deuterated ligand modulation from cupric ion complexes as powders using electron spin-echo modulation (ESEM) spectroscopy: (a) the spherical-averaging approximation which assumes both isotropic g and a random distribution of deuterium nuclei about the paramagnetic center, (b) an average which is performed over specific orientations of the field H0 with respect to g∥ using analytical expressions for the spin-echo intensity for an axially symmetric g tensor, and (c) a similar approach to (b) but with full geometrical correlation between the deuterium nuclei. Approach (c) is found to be unnecessary because of the degree of freedom of the ligand molecules to rotate about the ligand axis before freezing locks them into a particular configuration. This, in effect, removes any strict correlation between the deuterium nuclei. Approaches (a) and (b) are found to be equivalent when (i) the electron–nuclear distance r is greater than 0.35 nm, (ii) r is less than 0.35 nm but the number of interacting nuclei, N, is 6 or greater, and (iii) r is less than 0.35 nm, N is less than 6, and the angle θI between g∥ and r is greater than 45°. In all cases microwave irradiation must be near g⊥. When the situation is none of the above, then parameters defined by the spherical-averaging approximation should be interpreted with caution.