Structure of the two-dimensional relaxation spectra seen within the eigenmode perturbation theory and the two-site exchange model

The form of the two-dimensional (2D) NMR-relaxation spectra - which allow to study interstitial fluid dynamics in diffusive systems by correlating spin-lattice (T1) and spin-spin (T2) relaxation times - has given rise to numerous conjectures. Herein we find analytically a number of fundamental structural properties of the spectra: within the eigen-modes formalism, we establish relationships between the signs and intensities of the diagonal and cross-peaks in spectra obtained by various 1 and 2D NMR-relaxation techniques, reveal symmetries of the spectra and uncover interdependence between them. We investigate more specifically a practically important case of porous system that has sets of T1- and T2-eigenmodes and eigentimes similar to each other by applying the perturbation theory. Furthermore we provide a comparative analysis of the application of the, mathematically more rigorous, eigen-modes formalism and the, rather more phenomenological, first-order two-site exchange model to diffusive systems. Finally we put the results that we could formulate analytically to the test by comparing them with computer-simulations for 2D porous model systems. The structural properties, in general, are to provide useful clues for assignment and analysis of relaxation spectra. The most striking of them - the presence of negative peaks - underlines an urgent need for improvement of the current 2D Inverse Laplace Transform (ILT) algorithm used for calculation of relaxation spectra from NMR raw data.