An exact analytical solution for the evolution of a dipole–dipole interacting system under spherical diffusion in magnetic resonance experiments

Probing with Nuclear Magnetic Resonance echo decay and relaxation: A study on nitrile butadiene rubber

“…Another similarity is that in both cases a broad dynamical distribution has been represented with a single average value. As it has been noted before, this is not exact, but it has also been shown in another paper [32] that this kind of analysis gives results equivalent to those obtained with the well known Anderson-Weiss model [36]. Therefore we think that the results hold at least a good value as a parametric description on which one can lay foundation for further analyses, as the spinlattice relaxation one.…”

“…(1), a more rigorous way to obtain information on the dynamics from the FID's has been attempted. This was carried out by searching a complete analytical solution for the time evolution of the observable magnetization in a spin system evolving under a dipolar interaction Hamiltonian, in the general case in which the axis linking the two spins diffuses rotationally in the solid angle, with a pseudo-diffusion constant D. The full derivation and testing of this new model is presented in a recently published paper [32]. The fittings of simulations and experimental data were performed with EDDIE, an open source software developed by one of the authors expressly for this purpose and now available to the public at the address https://sites.google.com/site/eddienmr/home.…”

“…(13) has already been published elsewhere [32]; here it is possible to find an abridged version for reference in Appendix A. The important thing to know is that the last step needed to find the final form of G(t) is to perform a Fourier antitransform of a kernel function defined as a continuous fraction (see Eq.…”

Small and large scale segmental motion in polymers: estimating cooperativity length by ordinary relaxation experiments

“…In order to treat the effect of motion on the FID in the case of polymers, reference is usually made to the chain‐like structure of these molecules, and pre‐averaging over both fast segmental and β motions is done (or assumed) on deriving suitable analytical expressions for the discussion of experimental data . In the present case, pre‐averaging from just β motion is assumed, and an expression recently derived for the FID will be used for data analysis . This expression is appropriate for a system consisting of an ensemble of spin pairs undergoing rotational diffusion, different pairs being considered as not mutually interacting.…”

“…Details on the derivation of the relaxation function can be found in Sturniolo and Pieruccini; for the present purposes it suffices to report the final result: …”