Sixth Moment of the Magnetic-Resonance Line Shape

The sixth moment of the magnetic-resonance line shape is presented as the sum of nine lattice sums instead of ten as reported by Das and Bersohn. The sixth moment is evaluated for simple-cubic, bcc, and fcc symmetry and the applied magnetic field along the [100], [110], and [111] directions. The three-index lattice sums are evaluated two different ways: the usual way by summing each index over a sphere of lattice points and secondly by rewriting the three-index sums as a combination of one- and two-index sums. The two-index sums were then evaluated by summing the second index over a sphere centered on the first index. Lattice sums containing odd powers of $B_{\mathrm{ij}}$ are not always much smaller than lattice sums containing only even powers, as has been assumed in previous free-induction-decay (FID) calculations. A moment analysis of the Lowe-Norberg and Evans-Powles FID expansions shows that they contain relatively small parts of $M_6$ and the higher moments, so that they cannot be improved by the addition of complete higher-moment terms.