Nuclear pair electron spin echo envelope modulation

The interaction of electron spins with homonuclear spin pairs in their vicinity is one of the dominating mechanisms of electron spin echo decay at low temperature and low concentration. Here, we study this mechanism using established concepts of electron spin echo envelope modulation (ESEEM). We obtain analytical expressions for the Hahn echo, the refocused echo, the stimulated echo, and Carr–Purcell pulse trains with small numbers of $\pi$ pulses. Hahn echo decay is well approximated by the product of nuclear pair ESEEM functions. The same approximation can explain dependence of stimulated echo decay on the first interpulse delay and provides reasonable time scale estimates for decay of Carr–Purcell echos after an odd number of $\pi$ pulses. Carr–Purcell echoes after an even number of $\pi$ pulses are rather sensitive to correlations within larger nuclear spin clusters. Approximations improve for both odd and even numbers of $\pi$ pulses by factorising the nuclear spin bath into disjoint clusters, provided that modulation due to pairs of spins belonging to different clusters is considered in addition to cluster modulation. The analytical ESEEM expressions for the Hahn echo and the Carr–Purcell echo after two $\pi$ pulses have the same mathematical form as the filter functions of these sequences of spin noise spectroscopy. This coincidence provides a computationally very efficient way of predicting Hahn echo decay induced by homonuclear spin pairs. The analytical pair product approximation predicts the previously observed (Bahrenberg et al., 2021) increase of the refocused echo amplitude when one refocusing time is incremented and other one is fixed but longer. In contrast, the spin-noise concept fails to predict this effect.