Nuclear magnetic resonance spin-spin relaxation

Abstract

A general method is described for the calculation of the relaxation function G(t) for a macroscopic nuclear magnetization perpendicular to a strong steady magnetic field, under the influence of the nuclear dipole-dipole interaction. This function is the Fourier transform of the nuclear magnetic resonance absorption line shape. Approximations are introduced, the form of which depends upon whether the short-term or the long-term development of G(t) is required. A third type of approximation leads to the equation of Lowe and Norberg. The theory is applied to the case of the fluorine resonance in calcium fluoride with the steady field in the [100] direction. The first approximation gives a curve that differs only slightly from that of Lowe and Norberg over the range of t for which the latter is in good agreement with experiment. For larger t the disagreement with experiment is less severe, the characteristic oscillation of G(t) being present, though with too large an amplitude. In contrast the second approximation gives a curve which though overdamped provides a better description of the approach of the spin system to equilibrium. The theory may be extended to problems in which the dipolar Hamiltonian possesses an explicit time dependence. In this form it is applied to the case of motional narrowing due to rapid rotation of the sample about an axis inclined at 54.7° to the magnetic field, for which the absorption spectrum consists of a narrow central line flanked by an infinite series of sidebands spaced at intervals corresponding to the rotation frequency. It leads to a separation of the two components of the second moment of the spectrum due respectively to the generation of sidebands and to the finite width of the lines. Though the sum of these contributions is invariant to rotation they themselves are not. An assumed form for the shape of the lines leads to a linewidth which for rapid rotation depends inversely on the cube of the rotation frequency and is in substantial agreement with an earlier statistical theory.