Inverse Laplace Transform Approaches to βNMR Relaxation

Spin lattice relaxation is the simplest type of beta NMR measurement. The usual approach is to implant a pulse of hyperpolarized nuclei and monitor the time-resolved beta-decay asymmetry, yielding the ensemble average spin-lattice relaxation. In the simplest case, the asymmetry decays exponentially with a characteristic time constant T-1, but this ideal is rarely obtained in practice. In most data, the relaxation is more complicated. This can be the result of multiple crystallographic sites for the implanted probe each having a distinct T-1. The sample may also be inhomogeneous due to: impurities or defects (including interfaces that are particularly important for thin films), intrinsic phase separation, or, if it is a glass. There may also be a background signal from probe ions that stop outside the sample. The general approach to this problem has been the ad hoc development of an appropriate relaxation model that avoids overparametrization. Given the prevalence of more complicated relaxation, it is crucial to develop a systematic approach to relaxation modelling. The decomposition of a relaxing signal into exponentials is, however, a mathematically ill-posed problem[1]. This feature is intrinsic and unavoidable, but there are a number of methods to accommodate it for noisy real-world data, including nuclear spin relaxation[2, 3, 4]. Here we demonstrate one of the best and most commonly used methods, Tikhonov regularization for the inverse Laplace transform, implemented for the particular features of fiNMR relaxation data, most importantly the strong time dependence of the statistical uncertainty stemming from the radioactive lifetime of the probe.