  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.
