Nuclear-spin relaxation due to translational diffusion in a hexatic-Band crystalline-Bphase

Abstract

The general theory of nuclear-spin relaxation induced by translational self-diffusion in liquid crystals, developed in previous papers, is applied to the crystalline-$B$ and hexatic-$B$ phases of smectics by taking into account their characteristic features concerning molecular positional ordering, the inter-and intralayer molecular jumps, and the correlation in the layer stacking. The evaluated anisotropy and dispersion of $T_1$ and $T_{1\rho }$ are presented graphically for a variety of parameters. It is shown that in most cases $T_{1\rho }$, due to translational diffusion, has a characteristic anisotropy and dispersion while $T_1$ is expected to be nearly isotropic and with ${\omega }^2$-type dispersion at usual NMR frequencies. The relaxation study cannot discriminate between various types of the interlayer and intralayer correlations occurring in the crystalline-$B$ and hexatic-$B$ phases of the smectics except in the case when relative layer motions are fast compared to the diffusion. The comparison of the theoretical relation and experimental data enables the determination of diffusion constant $D_{\perp }$ in the smectic-$B$ phases while the ratio $\frac{D_{\perp }}{D_{\parallel }}$ could be determined only via precise study of the $T_{1\rho }$ angular dependence. With the application of the theory to available $T_1$ data of the smectic-$B$ phase of N-[4-($n$-dodecanoyl) benzylidene]-4′-aminoazobenzene (${\mathrm{C}}_{12}$BAA) and of the smectic-$B_c$ phase of terephthal-bis(4-$n$-butylaniline) (TBBA), the corresponding $D_{\perp }$ are estimated.