Relaxation of anisotropically oriented I=3/2 nuclei in the multipole basis: Evolution of the second rank tensor in the double quantum filtered nuclear magnetic resonance experiment

Abstract

The relaxation of an I=3/2 spin system in an anisotropic environment characterized by a finite residual quadrupolar splitting ωq is modeled by analytically solving for the density operator from Redfield’s relaxation theory. The resulting equations are cast into the multipole basis in order to describe the tensorial components of the spin density matrix. Included in the relaxation matrix are off-diagonal elements J1 and J2, which account for anisotropic systems with ωq values less than the width of the resonant line. With the Wigner rotation matrices simulating hard pulses, the response to an arbitrary pulse sequence can be determined. An analytical expression for the response to the double quantum filtered (DQF) pulse sequence (π/2)−(τ/2)−π−(τ/2)−θ−δ−θ−AQ for θ=π/2 is presented, showing explicitly the formation of a second rank tensor owing only to the presence of a finite ωq. This second rank tensor displays asymptotic behavior when the (reduced) quadrupole splitting is equal to either of the off-diagonal spectral densities J2 and J1. Line shape simulations for ωq values of less than a linewidth reproduce the general features of some recently reported 23Na DQF line shapes from biological systems. Distinct relaxation dynamics govern each of the tensorial components of the resonant signal revealing the influence of the experimental variables on the line shape.

Thermal relaxation and coherence dynamics of spin 3/2. I. Static and fluctuating quadrupolar interactions in the multipole basis

“…The density operator is expanded in 16 basis operators: T 00 (the identity), T 10 ϭ 1/ ͌ 5I z (proportional to longitudinal magnetization), T 11 (a) ϭ 1/ ͌ 5I x and T 11 (s) ϭ Ϫi/ ͌ 5I y (proportional to the xand y-magnetization, respectively), T 20 (quadrupolar spin polarization), T 21 (s) and T 21 (a) (rank-two singlequantum coherences), T 22 (s) and T 22 (a) (rank-two double-quantum coherences), T 30 (octopolar spin polarization), T 31 (s) and T 31 (a) (rank-three single-quantum coherences), T 32 (s) and T 32 (a) (rank-three double-quantum coherences), and T 33 (s) and T 33 (a) (rank-three triple-quantum coherences).…”

“…The relaxation functions are given by Eqs. [30] and [33] with rates [37] where the Ϯ sign in the superscript has been dropped to express the equality of the dynamics of the positive and negative coherence orders.…”

“…These differential equations can be integrated in analytical form (30) and the resulting time dependencies read with the evolution functions…”

Dynamics of $I = {3 \over 2}$ nuclei in isotropic slow motion, anisotropic and partially ordered phases

“…λ, g(t), h(t) are given in the previous section. Table 2 and setting D k = 0; k = 1, 2, we arrive at the following results (23). where…”

“…In the past, a number of different experiments were used to study separately the effects of the relaxation and the quadrupolar interaction, but the analysis of the effects together is relatively complicated . Moreover, the techniques developed for the study of 23 Na NMR relaxation may port over to the study of other quadrupolar nuclei in general (I > 1 2 ), and to I = 3 2 in particular (e.g., 7 Li NMR and MRI). In this work, we review the theoretical basis for describing these effects, as well as, provide convenient tools for the analysis of experimental results.…”