Nuclear Magnetic Resonance Line-Shape Calculations for a Spin System in a Fixed Lattice

A theoretical formula is derived for the free-induction decay of a system of identical particles of spin I. The exponential terms containing noncommuting operators are expanded by the method used in the paper by Lowe and Norberg. Only enough terms are kept to make the expansion rigorous through $t^4$. The resulting formula is evaluated for a system of identical particles for which: (1) the spins form a simple cubic lattice, a face-centered cubic lattice, and a body-centered cubic lattice; (2) there is pure magnetic dipole-dipole interaction between the spins; (3) the applied magnetic field is along the [100], [110], and [111] axes of the lattice; (4) $I=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} \infty$. The Fourier transforms of the free-induction decays are also plotted. The computations show that the free-induction decay shape is remarkably insensitive to the value of $I$.