Nuclear relaxation rate enhancement by a ¹⁴N quadrupole nucleus in a fluctuating electric-field gradient

We consider the longitudinal quadrupole relaxation rate enhancement (QRE) of a ¹H nucleus due to the time fluctuations of the local dipolar magnetic field created by a close quadrupole ¹⁴N nucleus, the electric-field gradient (EFG) Hamiltonian of which changes with time because of vibrations/distortions of its chemical environment. The QRE is analytically expressed as a linear combination of the cosine Fourier transforms of the three quantum time auto-correlation functions G_AA(t) of the ¹⁴N spin components along the principal axes A = X, Y, and Z of the mean (time-averaged) EFG Hamiltonian. Denoting the three transition frequencies between the energy levels of this mean Hamiltonian by ν_A, the functions G_AA(t) oscillate at frequencies ν_A + s_A/(2π) with mono-exponential decays of relaxation times τ_A, where the frequency dynamic shifts s_A and the relaxation times τ_A are closed expressions of the magnitude of the fluctuations of the instantaneous EFG Hamiltonian about its mean and of the characteristic fluctuation time. Thus, the theoretical QRE is the sum of three Lorentzian peaks centered at ν_A + s_A/(2π) with full widths at half maxima 1/(πτ_A). The predicted peak widths are nearly equal. The predicted dynamic shifts of the peaks are much smaller than their widths and amazingly keep proportional to the transition frequencies ν_A for reasonably fast EFG fluctuations. The theory is further improved by correcting the transition frequencies by the ¹⁴N Zeeman effects of second order. It is successfully applied to reinterpret the QRE pattern measured by Broche, Ashcroft, and Lurie [Magn. Reson. Med. 68, 358 (2012)] in normal cartilage.