Effects of Finite Lattice Heat Capacity on Spin-Lattice Relaxation

The spin-lattice approach to equilibrium is studied for situations in which the littice heat capacity is not infinite. It is assumed that the spins and the lattice each have a well-defined temperature, and that a bath either has been removed or is only very weakly coupled to the lattice. It is shown that asymptotically the magnetization relaxes exponentially with a relaxation time which can be very much different from $T_1$ (the spin-lattice relaxation time for an infinite lattice heat capacity), not only in numerical value but in its functional dependence upon temperature, magnetic field, and spin concentration. A general formula yielding the final common temperature in terms of initial conditions is given. Some examples are presented in which the effect of the finite lattice is to "lock" the magnetization in at its initial value for times 1 to 2 orders of magnitude longer than the ultimate relaxation time.