Nuclear Magnetic Resonance and Relaxation of Four Spin Molecules in a Liquid

Abstract

The semiclassical form of the density operator theory of relaxation is employed to calculate the longitudinal and transverse relaxation and the resonance line shape of a liquid sample whose molecules contain four identical spin \textonehalf{} nuclei at the corners of an equilateral tetrahedron. The relaxation is assumed to be due to the time dependence of the intramolecular dipole-dipole interactions that results from the classical rotational diffusion of the molecules. The calculation is not restricted to correlation times shorter than the Larmor period, but it is restricted to correlation times much shorter than the reciprocal coupling frequency. The second-order correction to the Zeeman energy due to the dipole-dipole interactions is included. The longitudinal relaxation is found to be, in general, the sum of three exponentials decaying with different time constants, but in the limit of either short or long correlation time there are only two exponentials. The expression for short correlation time agrees with the author's previous calculation. The transverse magnetization ${M}_{x}+i{M}_{y}$, in the absence of a radio-frequency field, is, in general, found to be the sum of three terms each precessing with slightly different frequencies and decaying exponentially with different time constants. In the case of short correlation time the precession frequencies are the same, and the amplitude decays in the same manner as the longitudinal magnetization for short correlation time. The resonance line shape, correct to first order in the magnitude of the radio frequency field, can be expressed in terms of a complex susceptibility $\ensuremath{\chi}={\ensuremath{\chi}}^{\ensuremath{'}}+i{\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$, which is shown to be proportional to $\ensuremath{\int}{0}^{\ensuremath{\infty}}[{M}_{x}(t)+i{M}_{y}(t)]\mathrm{exp}(i\ensuremath{\omega}t)dt$, where ${M}_{x}(t)+i{M}_{y}(t)$ is the transverse magnetization after a 90\ifmmode^\circ\else\textdegree\fi{} pulse. The saturation behavior of the resonance line shape is calculated for the case of short correlation time. The numerical values occurring in the expressions for the longitudinal and transverse relaxation and the resonance line shape are such that there is little difference between the results calculated here and the results obtained by neglecting in the calculation the effects of the correlations of different dipole-dipole interactions with one another.