Abstract:
Second-rank tensor interactions, such as quadrupolar interactions between the spin-
1 deuterium nuclei and the electric field gradients created by chemical bonds, are
affected by rapid random molecular motions that modulate the orientation of the
molecule with respect to the external magnetic field. In biological and model membrane
systems, where a distribution of dynamically averaged anisotropies (quadrupolar
splittings, chemical shift anisotropies, etc.) is present and where, in addition,
various parts of the sample may undergo a partial magnetic alignment, the numerical
analysis of the resulting Nuclear Magnetic Resonance (NMR) spectra is a mathematically
ill-posed problem. However, numerical methods (de-Pakeing, Tikhonov regularization)
exist that allow for a simultaneous determination of both the anisotropy
and orientational distributions. An additional complication arises when relaxation
is taken into account. This work presents a method of obtaining the orientation
dependence of the relaxation rates that can be used for the analysis of the molecular
motions on a broad range of time scales. An arbitrary set of exponential decay
rates is described by a three-term truncated Legendre polynomial expansion in the
orientation dependence, as appropriate for a second-rank tensor interaction, and a
linear approximation to the individual decay rates is made. Thus a severe numerical
instability caused by the presence of noise in the experimental data is avoided. At
the same time, enough flexibility in the inversion algorithm is retained to achieve a
meaningful mapping from raw experimental data to a set of intermediate, model-free
