On the theory of spin-spin relaxation II

Abstract

The long time behaviour of an isolated spin system, analyzed in a previous paper for the case of weak coupling (exchange interaction large compared to dipole-dipole interaction), is now investigated for the case of strong coupling (exchange and dipole-dipole interaction of the same order or no exchange) and strong external field. It is shown that, due to the fact that what is to be considered as unperturbed part of the Hamiltonian in a perturbation treatment now has a discrete eigenvalue spectrum, a simple reduction of the relaxation function to exponential form, as was carried out in the weak coupling case, cannot be made, contrary to assertions of Tjon. However, for sufficiently strong field the Fourier spectrum of the relaxation function can be decomposed into five lines, centered around 0, ±ω H and ±2ω H resp., ω H being the Larmor frequency. Explicit expressions for these lines are derived and with the approximation of an effective internal field it is shown that the broadened Larmor lines have widths much larger than the line at the origin, this line being, in good approximation, of Lorentzian shape, corresponding to an exponential decay of the relaxation function for long times. A formal expression for the relaxation time, containing an undetermined effective field parameter, is given and the results are discussed in connection with the theories of Caspers and Tjon.