Nuclear Spin-Lattice Relaxation in Dilute Paramagnetic Sapphire

Abstract

The exchange of energy between nuclear spin system and lattice has been theoretically and experimentally studied for circumstances in which the nuclear Zeeman energy levels are not necessarily equally spaced. Starting from the master rate equations for the nuclear energy level populations, expressions are found for the population difference of an adjacent pair of energy levels as a function of time. For nuclear spin $I$, this population difference in general returns to thermal equilibrium with the lattice as a sum of ($2I$) exponential terms. Under certain conditions, exact solutions of the rate equations may be obtained. As an example, detailed exact solutions are found for an artificial physical situation, in which the nuclear spins ($I=\frac{5}{2}$) are presumed to interact, independently of each other, with a rapidly fluctuating paramagnetic ion (the lattice). From the solutions to this model system, some conclusions are drawn which are consistent with more sophisticated statistical arguments. First, in the limiting case of equally spaced energy levels, these solutions reduce to a single exponential term; a unique spin-lattice relaxation time $T_1$ may then be defined. Second, it is found that even for unequally spaced levels, any pair of level populations recovers to thermal equilibrium asymptotically as an exponential with this same time constant $T_1$.