General treatment of paramagnetic relaxation enhancement associated with translational diffusion.
A theory of nuclear spin relaxation in isotropic liquids for nuclear spins interacting with electron spins, residing in other molecules (the outer-sphere relaxation), is presented. The approach, valid outside of the Redfield limit for electron spin relaxation, is an extension of the Swedish slow motion theory [Benetis et al., Mol. Phys. 48, 329 (1983); Nilsson and Kowalewski, J. Magn. Reson. 146, 345 (2000)] for inner-sphere relaxation. It is demonstrated that the outer-sphere relaxation rate can be expressed as an integral of a product of a translational diffusion correlation function and a function analogous to the inner-sphere spectral density. A numerical implementation of the theory is described and applied to a large number of realistic parameter sets for S = 7/2 and S = 1, which may correspond to Gd(III) and Ni(II) systems. It is shown that the outer-sphere contribution is relevant and should be included into the analysis of nuclear magnetic relaxation dispersion relaxation profiles, especially for slow relative translational diffusion and fast molecular tumbling.
Outer-sphere nuclear spin relaxation in paramagnetic systems: a low-field theory
A low-field theory for paramagnetic relaxation enhancement (PRE), appropriate for the outer-sphere relaxation, is presented for the electron spin quantum number S = 1, 3/2, 2, 5/2, 3 and 7/2. The theory is used to calculate the PRE at low magnetic field, as a function of the translational diffusion coefficient, for various values of the electron spin quantum number, for small and fairly large values of the static zero-field splitting (ZFS), and for a given set of parameters determining the electron spin relaxation. We have found earlier that the static ZFS has a profound influence on the electron spin relaxation; such effects are also evident in the present study. Comparisons are made with other existing models for the outer-sphere PRE, and significant differences are found for slowly diffusing systems with large ZFS. The theory is also used to obtain a novel interpretation of experimental data for an acetone solution of a Mn(III) complex.
The quantum particle in a box: what we can learn from classical electrodynamics
The problem of a charged particle enclosed in an infinite square potential well is analysed from the point of view of classical theory with the addition of the electromagnetic zero-point radiation field, with the aim to explore the extent to which such an analysis can contribute to enhance our understanding of the quantum behavior. First a proper treatment is made of the freely moving particle subject to the action of the radiation field, involving a frequency cutoff ωc. The jittering motion and the effective structure of the particle are sustained by the permanent action of the zero-point field. As a result, the particle interacts resonantly with the traveling field modes of frequency ωc in its proper frame of reference, which superpose to give rise to a modulated wave accompanying the particle. This is identified with the de Broglie wave, validating the choice of Compton’s frequency for ωc. For the stationary states of particles confined in the potential well, the Lorentz force produced by the accompanying field is shown to lead to discrete values for the mean speed and to an uneven probability distribution that echoes the corresponding quantum distribution. The relevance of the results obtained and the limitations of the classical approach used, are discussed in the context of present-day stochastic electrodynamics.