X-ray diffraction from smectic multilayers: crossover from kinematical to dynamical regime

TL;DR

This work shows that freely suspended smectic liquid-crystal multilayers often exhibit kinematical X-ray reflectivity even for relatively thick stacks, but a crossover to dynamical diffraction occurs when the number of layers exceeds a critical value $N_c$ that depends on the optical contrast, layer depth ratio, and surface roughness. The authors derive an analytical criterion for the kinematical-dynamical crossover using both kinematical and dynamical theories and validate it with numerical simulations and experimental data on an $\sim$80-layer film, revealing $N_c$ on the order of $10^3$ layers and a Darwin plateau beyond that thickness. They further discuss signatures such as Yoneda peaks as potential indicators of dynamical scattering and outline implications for LC-based Bragg mirrors, including tunable adaptive optics, with practical guidelines for thickness and material parameters. Overall, the study provides a quantitative framework to decide when to apply kinematical vs dynamical XRR in smectic multilayers and informs design of LC-based Bragg mirrors for X-ray applications.

Abstract

We study X-ray diffraction in smectic liquid crystal multilayers. Such systems are fabricated as freely suspended films and have a unique layered structure. As such, they can be described as organic Bragg mirrors with sub-nanometer roughness. However, an interesting peculiarity arises in the diffraction on these structures: the characteristic shape of diffraction peaks associated with dynamical scattering effects is not observed. Instead, the diffraction can be well described kinematically, which is atypical for Bragg mirrors. In this article we investigate the transition between the kinematical and dynamical regimes of diffraction. For this purpose, we analyze the reflection of synchrotron radiation on a real liquid crystal sample with both kinematical and dynamical theories. Furthermore, based on these theories, we derive a quantitative criterion for the transition from the kinematical to the dynamical regime. This, in turn, allows us to explain the peculiar diffraction behavior in smectic films with thicknesses exceeding thousands of molecular layers.

Figures (6)

• Figure 1: A free-standing smectic film is drawn across the opening in a holder. The inset shows the cross-section of the film, with the molecules of the liquid crystal arranged in a stack of smectic layers. Here, $\bm{k}_i$ and $\bm{k}_f$ are the incoming and outgoing wave vectors, respectively (the specular reflectivity geometry is shown), $\bm{q}$ is the scattering vector, and $\theta$ is the angle of scattering.
• Figure 2: The two-step box model is used to represent the electron density of a single layer. The smectic periodicity is formed by two 4O.8 molecules with opposite up-down orientations. $\delta_{\text{core}}$, $\delta_{\text{tail}}$, $L_{\text{core}}$, $L_{\text{tail}}$ define the electron density and the length of the molecular fragments within the two-step box model, respectively. The red line illustrates the result of convoluting the box model with a Gaussian kernel, which is performed to account for layer displacement fluctuations.
• Figure 3: The best-fit of the XRR data using different models of electron density distrbution across the smectic layer stack. Model (a) corresponds to fit made with three independent parameters: $\delta_{\text{core}}/\delta_{\text{tail}}$, $L_{\text{core}} / L_{\text{tail}}$ and $\sigma$ (violet curve), while model (b) contains an additional delta-like function that describes a small area with enhanced electron density between the layers in the model, as discussed in the text (red curve). Both models provide a good fit along the whole reflectivity curve, as well as around the 1-st Bragg peak -- see inset in the middle-left of the graph, showing the electron density distribution within the single smectic layer, where the vertical dashed lines indicate the boundaries of one period. However, model (a) fails to reproduce the 2-nd order Bragg peak, while model (b) makes this perfectly -- see inset in the up-right of the graph. The lower part of the graph shows the discrepancy $u$ between model b and the experimental data.
• Figure 4: Electron density $\delta$-profile across the smectic film (solid blue line), obtained by convoluting a set of box-like functions with a Gaussian kernel of varying roughness $\sigma$ (dashed red line). The layer displacement fluctuations are partly "frozen" at the FSSF borders due to interaction with the air-film interfaces in accordance with dejeu2003membranesfera1999filmsfera2001scattering.
• Figure 5: Dependence of the ratio of the FWHM of the first Bragg peak for kinematic and dynamic XRR curves on the number of layers $N$ in free standing smectic films. Each panel shows the effect of varying a single parameter while keeping the other two fixed: (top) thermal fluctuation profile $\sigma$, (middle) $\delta_{\rm core}$, and (bottom) two-step depth ratio: $\Gamma$. Diamond markers indicate a critical number of layers $N_{\rm c}$.