Dynamical Theory of Nuclear Induction. II

Abstract

This paper represents the generalization of an earlier theory by Wangsness and the author in which the phenomena of relaxation were treated by considering the interaction of individual nuclear moments with the molecular system and assuming that the latter remains always in thermal equilibrium. Instead of a single nuclear moment, the representative spin system is here allowed to consist of several moments, interacting with each other, and the corresponding general Boltzmann equation for the distribution matrix is developed. A first application is given by investigating the effect of a weak alternating field in the vicinity of resonance conditions. It is seen that the phenomenon of saturation is closely related to the change of populations in states, other than the two between which the resonance transitions occur. This general type of Overhauser effect is shown to be equivalent to that of a dc circuit and it is illustrated by a special example. The general formalism is adapted to the treatment of a nuclear spin system in a strong constant field with particular attention to the structure of resonance spectra in liquids, due to chemical shift and spin coupling. A special case is that where the spin coupling causes a splitting of the lines, large compared to their natural width and their broadening due to the alternating field. An expression for the signal, obtained in this case, is developed and the effect of the spin coupling upon the effective longitudinal and transverse relaxation time is illustrated by the particular example of the two coupled nuclei of spin ½ and with independent dipole relaxation. New phenomena appear if the rate of relaxation-transitions is comparable or large compared to the frequency separation of resonance lines, due to spin coupling. The effect of such transitions by some nuclei upon the line width and structure of the resonance of others is investigated, assuming the spin coupling to be small compared to the chemical shift. Similar effects occur to the resonances of a nucleus in a weak alternating field, if other nuclei are at the same time irradiated by an alternating field of different frequency and sufficiently strong so that its effect is comparable or large compared to that of the spin coupling. It is shown that, even for the case of a single kind of nucleus, the presence of the strong field causes a doubling of the resonance with the weak field which can be used to calibrate the strength of the former by a frequency measurement. Another illustration is given in the case of two nuclei and explicit expressions for line width and intensity are given for the example of two nuclei with spin ½ and independent dipole relaxation.