Correlation functions for simple hopping in a simple cubic lattice

We calculate the correlation functions for a single specific particle and for a pair of specific particles in an ensemble of particles making nearest-neighbor simple jumps in a simple cubic lattice. These correlation functions can be used to obtain tracer correlation factors and NMR relaxation times due to motionally altered $I-S$ (unlike spin) or $I-I$ (like spin) dipolar interactions. In particular we calculate $T_1$ due to the $I-I$ dipolar interaction for various concentrations of particles as may be appropriate for some metal hydrides at different concentrations. Our results show that the $T_1$ calculated using the random-walk model is in error by more than 15% at low concentrations and by almost 100% at high concentrations.