Keith A. Earle

The spin echo experiment is an important tool in magnetic resonance for exploring the coupling of spin systems to their local environment. The strong couplings in a typical Electron Spin Resonance (ESR) experiment lead to rapid relaxation effects that puts significant technical constraints on the kinds of time domain experiments that one can perform in ESR. Recent developments in high frequency ESR hardware have opened up new possibilities for utilizing phase-modulated or composite phase slice (CPS) pulses at 95 GHz and higher. In particular, we report preliminary results at 95 GHz on experiments performed with CPS pulses in studies of rapidly relaxing fluid state systems. In contemporary ESR, this has important consequences for the design of pulse sequences where, due to finite excitation bandwidths, contributions from the Hamiltonian dynamics and relaxation processes must be considered together in order to achieve a quantitative treatment of the effects of selective, finite bandwidth pulses on the spin system under study. The approach reported on here is generic and may be expected to be of use for solid state and fluid systems. In particular we indicate how our approach may be extended to higher frequencies, e.g., 240 GHz.

In this paper, we use Boltzmann statistics and the maximum likelihood distribution derived from Bayes’ Theorem to infer parameter values for a Pake Doublet Spectrum, a lineshape of historical significance and contemporary relevance for determining distances between interacting magnetic dipoles. A Metropolis Hastings Markov Chain Monte Carlo algorithm is implemented and designed to find the optimum parameter set and to estimate parameter uncertainties. The posterior distribution allows us to define a metric on parameter space that induces a geometry with negative curvature that affects the parameter uncertainty estimates, particularly for spectra with low signal to noise.

We have shown in previous work that magnetic resonance spectra may be profitably analyzed by using concepts derived from information geometry. The systems that have been studied so far have been restricted to the linear response regime where constraints, in the form of sum rules for example, induced a geometry on the line shape function treated as a probability density function (PDF). Although careful analysis of the linear response line shape is a useful tool for quantifying important details of molecular structure and dynamics, complementary information is available from the non-linear response line shape which is often sensitive to processes on longer time scales than the transient coherences probed in the linear regime. The non-linear response of a system satisfies constraints that do not allow for a direct interpretation in terms of a PDF, as the non-linear response may have absorptive and emissive character and thus cannot be constrained to be non-negative in all circumstances. Nevertheless, entropic methods and concepts from information geometry may be used to compare simulated and measured non-linear spectra in order to infer appropriate model parameter values. We demonstrate how this can be achieved for a model system described by the Zeeman interaction of a magnetic moiety with an applied magnetic field modulated by stochastic processes such as rotational diffusion. For the case of Electron Paramagnetic Resonance (EPR), we will show how the methods presented here may be extended to include hyperfine interactions which arise due to couplings of the electron magnetic moment with nearby nuclei.

In this work, we explore the connections between parameter fitting and statistical thermodynamics using the maxent principle of Jaynes as a starting point. In particular, we show how signal averaging may be described by a suitable one particle partition function, modified for the case of a variable number of particles. These modifications lead to an entropy that is extensive in the number of measurements in the average. Systematic error may be interpreted as a departure from ideal gas behavior. In addition, we show how to combine measurements from different experiments in an unbiased way in order to maximize the entropy of simultaneous parameter fitting. We suggest that fit parameters may be interpreted as generalized coordinates and the forces conjugate to them may be derived from the system partition function. From this perspective, the parameter fitting problem may be interpreted as a process where the system (spectrum) does work against internal stresses (non‐optimum model parameters) to achieve a state of minimum free energy/maximum entropy. Finally, we show how the distribution function allows us to define a geometry on parameter space, building on previous work[1, 2]. This geometry has implications for error estimation and we outline a program for incorporating these geometrical insights into an automated parameter fitting algorithm.

Methods of statistical geometry are introduced which allow one to estimate, on the basis of computable criteria, the conditions under which maximally informative data may be collected. We note the important role of constraints which introduce curvature into parameter space and discuss the appropriate mathematical tools for treating curvature effects. Channel capacity, a term from communication theory, is suggested as a useful figure of merit for estimating the information content of spectra in the presence of noise. The tools introduced here are applied to the case of a model nitroxide system as a concrete example, but we stress that the methods described here are of general utility.