# Motional Narrowing under Markovian and Non-Markovian Hopping Transitions in Inhomogeneous Broadened Absorption Line Shape

## TL;DR

This paper investigates motional narrowing effects in inhomogeneously broadened spectra, deriving exact solutions for spin correlation functions considering Markovian and non-Markovian transitions, and explaining experimental ESR/EPR line width minima.

## Contribution

It provides an exact analytical model for motional narrowing incorporating static inhomogeneous broadening and non-Markovian effects, linking theory with experimental observations.

## Key findings

- Exact solution for spin correlation functions considering both Markovian and non-Markovian processes.
- Demonstration that line width minima occur as a function of inverse temperature under thermal activation.
- Showing that Gaussian line shape narrowing does not necessarily produce Lorentzian line shapes, especially with heavy-tailed waiting time distributions.

## Abstract

Inspired by recent experiments showing a minimum of electron paramagnetic resonance (ESR/EPR) line width as a function of inverse temperature, we studied the motional narrowing effect by considering a combined model of carrier transitions and static dispersion of the angular frequency giving rise to an inhomogeneous broadening in the spectrum. The dispersion of the angular frequency results from the distribution of the local field. The transition between the sites under inhomogeneous static local field induces adiabatic relaxation of the spin. We also considered the on-site inherent (nonadiabatic) relaxation of the spin. We obtained the exact solution of the spin correlation function by explicitly considering transitions between two sites for both Markovian and non-Markovian transition processes. The absorption line shape is expressed in terms of the Voigt function, which is a convolution of a Gaussian function and a Lorentzian function. Using the known properties of the Voigt function, we discuss the correlation between the change in the full-width at half-maximum and the change in line shape, both of which are induced by motional narrowing. By assuming thermal activation processes for both the hopping transition and the on-site inherent relaxation, we show that the minimum of the width appears as a function of inverse temperature as observed experimentally in organic materials.Contrary to the general belief, we also show that the narrowing of the Gaussian line shape under a local random field did not necessarily lead to a Lorentzian line shape in particular under the presence of heavy tail property in the waiting time distribution of hopping transitions.