NMR line-shape calculation for a linear dipolar chain

Numerical calculations have been performed to obtain exact results for the high-temperature space- and time-dependent transverse spin pair correlation functions $G_r^x\mathrm{}\left(t\right)\mathrm{}$, where $\mathcal{r}$ is the spatial separation of the spins, and the free-induction-decay function $F\left(t\right)\mathrm{}$, for a linear chain of $N$ dipolar coupled spins ($S=\frac{1}{2}$) for $N=7, 9, \mathrm{and} 11$ with periodic boundary conditions. The range $\Delta$ of the dipolar interaction was allowed to take separately the values 1, 2, and 3. The moments of the NMR line shape ($M_2$ through $M_{20}$) were obtained for the infinite chain with $\Delta =1\mathrm{}$ and $\Delta =2\mathrm{}$ and were found to be in excellent agreement with previous reliable moment calculations ($M_2$ through $M_8$). By an evaluation of the contribution of the four-particle terms to $M_6$, it is shown that the Bersohn-Das theorem fails in one dimension. We find that $G_r^x\mathrm{}\left(t\right)\mathrm{}$ for even (odd) values of $\mathcal{r}$ are predominantly positive (negative). This gives rise to a damped beat structure for $F\left(t\right)\mathrm{}$. A study of $F\left(t\right)\mathrm{}$ for $N=7, 9, \mathrm{and} 11$ shows that our 11-spin solution reproduces that of the infinite chain at least up through the fifth node, and is also in good agreement with recent ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ free-induction-decay measurement in fluoroapatite. The nodes of $F\left(t\right)\mathrm{}$ were found to shift towards shorter values of time as $\Delta$ was allowed to increase. Further, for $N=11\mathrm{}$, a comparison of our exact results for $F\left(t\right)\mathrm{}$ for $\Delta =3\mathrm{}$, with a Gaussian broadening of $F\left(t\right)\mathrm{}$ for $\Delta =2\mathrm{}$, shows that the effect of the weak interactions is not well approximated by a simple Gaussian broadening for this case.