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

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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.