Direct reconstruction of the two-dimensional pair distribution function in partially ordered systems with angular correlations

An x-ray scattering approach to determine the two-dimensional (2D) pair distribution function (PDF) in partially ordered 2D systems is proposed. We derive relations between the structure factor and PDF that enable quantitative studies of positional and bond-orientational (BO) order in real space. We apply this approach in the x-ray study of a liquid crystal (LC) film undergoing the smectic-$A$-hexatic-$B$ phase transition, to analyze the interplay between the positional and BO order during the temperature evolution of the LC film. We analyze the positional correlation length in different directions in real space.

Figure 1

(a)–(d) Diffraction patterns measured at different temperatures from the LC film undergoing the smectic-$A$-hexatic-$B$ phase transition. (e)–(h) The PDFs $g\left(\mathbf{r}\right)$ determined from the diffraction patterns (a)–(d).

Figure 2

Temperature dependence of the correlation length determined in different ways: $\xi$ from the radial section of scattered intensity (red circles) [23] and ${\xi }_{\text{pr}}$ from the projection of the PDF $g\left(\mathbf{r}\right)$ onto the direction of the diffraction peak (blue triangles).

Figure 3

Projection of the PDF $g_{\text{pr}}\left(x\right)$ on the direction of the diffraction peak $A$ [see Fig. 1] at different temperatures. The projection $g_{\text{pr}}\left(x\right)$ is shown with the blue line and the envelope function in the form of an exponent $Aexp(-\gamma x)$ is shown with the red line.

Figure 4

The radial section of the PDF $g\left(\mathbf{r}\right)$ in the direction of the diffraction peak $A$ [see Fig. 1] at different temperatures. In the insets the radial section of $g\left(\mathbf{r}\right)$ in the direction $B$ between the diffraction peaks [see Fig. 1] is shown at the same temperature. The envelope function of the form $1+B{r}^{-0.5}exp(-\gamma r)$, which was used for calculation of the correlation lengths ${\xi }_{\text{1,2}}=1/{\gamma }_{1,2}$, is shown with the red line.