# Modeling of the Resonant X-ray Response of a Chiral Cubic Phase

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Tensor Form Factor

#### 3.2. Resonant and Resonantly Enhanced Peaks

#### 3.3. Intensities of the Peaks

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Unit cells of (

**a**) a double gyroid cubic phase and (

**b**) chiral cubic phase used to model the resonant X-ray response. The double gyroid cubic phase consists of two entangled systems of channels (red and blue). Three channels meet in a planar junction, and the direction normal to the junction rotates in the opposite direction in the two systems of channels. The all-hexagon-continuous grid of a chiral cubic phase consists of eight nonplanar hexagons (blue lines) in which the sides are of the same length as in the double gyroid phase, and hexagons consisting of two sides of the blue hexagons connected to the centers of edges and centers of faces of the cube (black lines). There are micelles in the center and edges of the unit cell. The unit cell side in the $CC$ phase is always approximately $50\%$ longer than in the double gyroid phase.

**Figure 2.**(

**a**) A nonplanar hexagon in the coordinate system (${x}_{h},{y}_{h}$ and ${z}_{h}$ ) centered on the hexagon. (

**b**) A side view of the hexagon. Vector ${\stackrel{\rightharpoonup}{d}}_{1}$ connects the opposite vertices of the hexagon that lie on the ${x}_{h}$ axis, and vector ${\stackrel{\rightharpoonup}{d}}_{2}$, which is perpendicular to ${\stackrel{\rightharpoonup}{d}}_{1}$, connects the opposite midpoints of the hexagon edges and points along the ${y}_{h}$ axis. Anisotropic scatterers (blue and red lines) are positioned perpendicular to each side of a hexagon. Inside the hexagon, the polarization axis of the blue scatterers is tilted in the $-{z}_{h}$ direction by an angle $\varphi $ and along the $+{z}_{h}$ direction for the red scatterers. Scatterers, vertices and polarization axes are numbered from 1 to 6; ${\stackrel{\rightharpoonup}{s}}_{6}$ thus determines the polarization axis of the $6$-th scatterer.

**Figure 3.**Hexagons in the unit cell of the chiral cubic phase in the laboratory system. Red and blue lines represent different inclinations of anisotropic scatterers with respect to the hexagon sides. All hexagons are identical (see Figure 2); they differ only by location and orientation inside the unit cell.

**Figure 4.**Scattering geometry for the peak with Miller indices ($hkl$) in the laboratory system. (

**a**) ${\stackrel{\rightharpoonup}{Q}}_{hkl}$ is the scattering vector, ${\stackrel{\rightharpoonup}{k}}_{in}$ and ${\stackrel{\rightharpoonup}{k}}_{out}$ are the wave vectors of the incident and scattered wave, respectively; ${\stackrel{\rightharpoonup}{\pi}}_{in}$ and ${\stackrel{\rightharpoonup}{\pi}}_{out}$ are in-plane components of polarization of the incident and scattered wave, respectively, and ${\stackrel{\rightharpoonup}{\sigma}}_{in}$ and ${\stackrel{\rightharpoonup}{\sigma}}_{out}$ are perpendicular components. ${\theta}_{0}$ is the angle between the scattering vector and the $z$ axis and ${\theta}_{hkl}$ is the scattering angle. The angles between the wave vectors of the incident and scattered wave and the $xy$ plane are ${\alpha}_{in}$ and ${\alpha}_{out}$, respectively. (

**b**) Projection of the scattering vector on the $xy$ plane (${\stackrel{\rightharpoonup}{Q}}_{p}$ ) forms an angle ${\phi}_{0}$ with the $x$ axis.

**Figure 5.**Intensities (in arbitrary units) of the diffracted peaks, measured by the nonresonant X-ray scattering (black curve) [11] and the predicted resonant (red lines) and resonantly enhanced peaks (blue arrows) as a function of the normalized scattering vector magnitude ($Q/{Q}_{0}$), defined by the Miller indices $\left(hkl\right)$: $Q/{Q}_{0}=\sqrt{{h}^{2}+{k}^{2}+{l}^{2}}$. The measured and calculated intensities are not to scale. The intensities are calculated at the scatterers’ inclination $\varphi =0.1$.

**Table 1.**Position (${\stackrel{\rightharpoonup}{r}}_{o}^{\left(i\right)}$) of the $i$-th vertex of the hexagon and direction ${\widehat{s}}^{\left(i\right)}$ of the polarization axis for the $i$-th scatterer in the case of an achiral hexagon ($\varphi =0$ in Figure 2) in the coordinate system centered on the hexagon (see Figure 2). Coordinates of vertices are given in units of the unit cell length ($a$ ).

$\mathit{i}$ | $\raisebox{1ex}{${\stackrel{\rightharpoonup}{\mathit{r}}}_{\mathit{o}}^{\left(\mathit{i}\right)}$}\!\left/ \!\raisebox{-1ex}{$\mathit{a}$}\right.$ | ${\widehat{\mathit{s}}}^{\left(\mathit{i}\right)}$ |
---|---|---|

1 | $\left\{-0.2244,\text{}0,\text{}0\right\}$ | $\left\{-0.8643,\text{}0.5,-0.0539\right\}$ |

2 | $\left\{-0.1109,\text{}0.1940,-0.0209\right\}$ | $\left\{0,\text{}1,\text{}0\right\}$ |

3 | $\left\{0.1109,\text{}0.1940,\text{}0.0209\right\}$ | $\left\{0.8643,\text{}0.5,\text{}0.0539\right\}$ |

4 | $\left\{0.2244,\text{}0,\text{}0\right\}$ | $\left\{0.8643,-0.5,\text{}0.0539\right\}$ |

5 | $\left\{0.1109,-0.1940,\text{}0.0209\right\}$ | $\left\{0,-1,\text{}0\right\}$ |

6 | $\left\{-0.1109,-0.1940,-0.0209\right\}$ | $\left\{-0.8643,-0.5,-0.0539\right\}$ |

**Table 2.**Coordinates of the centers ${r}_{s}^{\left(i\right)}$ of hexagons within the unit cell with a side length $a$, unit vectors ${\widehat{d}}_{1}^{lab}$ and ${\widehat{d}}_{2}^{lab}$ along the direction of vectors ${\stackrel{\rightharpoonup}{d}}_{1}$ and ${\stackrel{\rightharpoonup}{d}}_{2}$ (see Figure 2) given in the laboratory coordinate system ($x$,$y$,$z$) (Figure 3).

$\mathit{i}$ (Color on Figure 3) | $\raisebox{1ex}{$4{\stackrel{\rightharpoonup}{\mathit{r}}}_{\mathit{s}}^{\left(\mathit{i}\right)}$}\!\left/ \!\raisebox{-1ex}{$\mathit{a}$}\right.$ | ${\widehat{\mathit{d}}}_{1}^{\mathit{l}\mathit{a}\mathit{b}}$ | $\sqrt{2}{\widehat{\mathit{d}}}_{2}^{\mathit{l}\mathit{a}\mathit{b}}$ |
---|---|---|---|

1 (magenta) | $\left\{1,-1,1\right\}$ | $\left\{0.4434,0.7790,0.4434\right\}$ | $\left\{1,0,-1\right\}$ |

2 (gray) | $\left\{1,1,1\right\}$ | $\left\{0.4434,-0.7790,0.4434\right\}$ | $\left\{1,0,-1\right\}$ |

3 (black) | $\left\{-1,1,1\right\}$ | $\left\{0.4434,0.7790,-0.4434\right\}$ | $\left\{-1,0,-1\right\}$ |

4 (brown) | $\left\{-1,-1,1\right\}$ | $\left\{0.4434,-0.7790,-0.4434\right\}$ | $\left\{1,0,1\right\}$ |

5 (green) | $\left\{1,1,-1\right\}$ | $\left\{0.4434,-0.7790,-0.4434\right\}$ | $\left\{1,0,1\right\}$ |

6 (orange) | $\left\{1,-1,-1\right\}$ | $\left\{0.4434,0.7790,-0.4434\right\}$ | $\left\{-1,0,-1\right\}$ |

7 (pink) | $\left\{-1,-1,-1\right\}$ | $\left\{0.4434,-0.7790,0.4434\right\}$ | $\left\{-1,0,1\right\}$ |

8 (cyan) | $\left\{-1,1,-1\right\}$ | $\left\{0.4434,0.7790,0.4434\right\}$ | $\left\{-1,0,1\right\}$ |

**Table 3.**Miller indices ($hkl$) of the resonant (green) and resonantly enhanced (orange) peaks predicted by the model and the corresponding tensor form factor (${\underset{\_}{F}}^{\left(CC\right)}$ ) in units of the tensor element ($f$ ), for two inclinations of the polarization axis of scatterers ($\varphi $ ). Only those resonantly enhanced peaks are included that are not observed in the nonresonant X-ray scattering.

$\raisebox{1ex}{${\underset{\_}{\mathit{F}}}^{\left(\mathit{C}\mathit{C}\right)}$}\!\left/ \!\raisebox{-1ex}{$\mathit{f}$}\right.$ | ||
---|---|---|

$\left(\mathit{h}\mathit{k}\mathit{l}\right)$ | $\mathit{\varphi}=0.01$ | $\mathit{\varphi}=0.1$ |

(011) | $\left(\begin{array}{ccc}-9.2& 0& 0\\ 0& 4.6& -10\\ 0& -10& 4.6\end{array}\right)$ | $\left(\begin{array}{ccc}-9.2& 0& 0\\ 0& 4.6& -9.8\\ 0& -9.8& 4.6\end{array}\right)$ |

$\left(002\right)$ | $\left(\begin{array}{ccc}9.2& 0& 0\\ 0& 9.2& 0\\ 0& 0& -18.4\end{array}\right)$ | $\left(\begin{array}{ccc}9.2& 0& 0\\ 0& 9.2& 0\\ 0& 0& -18.4\end{array}\right)$ |

$\left(012\right)$ | $\left(\begin{array}{ccc}0& -0.12i& -0.12i\\ -0.12i& 0& 0\\ -0.12i& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& -1.2i& -1.2i\\ -1.2i& 0& 0\\ -1.2i& 0& 0\end{array}\right)$ |

$\left(113\right)$ | $\left(\begin{array}{ccc}0.41i& 0& 0\\ 0& -0.41i& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}4.1i& 0& 0\\ 0& -4.1i& 0\\ 0& 0& 0\end{array}\right)$ |

$\left(023\right)$ | $\left(\begin{array}{ccc}0& 0.16i& -0.084i\\ 0.16i& 0& 0\\ -0.084i& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 1.6i& -0.83i\\ 1.6i& 0& 0\\ -0.83i& 0& 0\end{array}\right)$ |

$\left(014\right)$ | $\left(\begin{array}{ccc}0& -0.037i& 0.20i\\ -0.037i& 0& 0\\ 0.20i& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& -0.36i& 2.0i\\ -0.36i& 0& 0\\ 2.0i& 0& 0\end{array}\right)$ |

$\left(223\right)$ | $\left(\begin{array}{ccc}-0.41i& 0& 0.084i\\ 0& 0.41i& -0.084i\\ 0.084i& -0.084i& 0\end{array}\right)$ | $\left(\begin{array}{ccc}-4.1i& 0& 0.83i\\ 0& 4.1i& -0.83i\\ 0.83i& -0.83i& 0\end{array}\right)$ |

$\left(124\right)$ | $\left(\begin{array}{ccc}0& -0.037i& -0.12i\\ -0.037i& 0.13i& -0.084i\\ -0.12i& -0.084i& -0.13i\end{array}\right)$ | $\left(\begin{array}{ccc}0& -0.36i& -1.2i\\ -0.36i& 1.3i& -0.83i\\ -1.2i& -0.83i& -1.3i\end{array}\right)$ |

$\left(233\right)$ | $\left(\begin{array}{ccc}4.4& -0.00074& -0.00074\\ -0.00074& -2.2& 17\\ -0.00074& 17& -2.2\end{array}\right)$ | $\left(\begin{array}{ccc}4.4& -0.073& -0.073\\ -0.073& -2.2& 17\\ -0.073& 17& -2.2\end{array}\right)$ |

$\left(034\right)$ | $\left(\begin{array}{ccc}0& 0.025i& 0.27i\\ 0.025i& 0& 0\\ 0.27i& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0.25i& 2.6i\\ 0.25i& 0& 0\\ 2.6i& 0& 0\end{array}\right)$ |

**Table 4.**Relative intensities of peaks with Miller indices ($hkl$) and multiplicity $M$, calculated at ${\lambda}_{C}=4.4\mathrm{nm}$ (RSoXS) and ${\lambda}_{S}=0.5\mathrm{nm}$ (TReXS) for the scatterer’s inclinations ${\varphi}_{1}=0.01$ (${I}^{\left({\varphi}_{1}\right)}$) and ${\varphi}_{2}=0.1$ (${I}^{({\varphi}_{2})}$). Intensities are normalized with the intensity of the peak ($113$) at the scatterer’s inclination ${\varphi}_{1}=0.01$ (${I}_{113}^{({\varphi}_{1})}$).

$\mathit{M}$ | $\left(\mathit{h}\mathit{k}\mathit{l}\right)$ | ${\mathit{I}}_{\mathit{R}\mathit{S}\mathit{o}\mathit{X}\mathit{S}}^{({\mathit{\varphi}}_{1})}/{\mathit{I}}_{113}^{({\mathit{\varphi}}_{1})}$ | ${\mathit{I}}_{\mathit{R}\mathit{S}\mathit{o}\mathit{X}\mathit{S}}^{({\mathit{\varphi}}_{2})}/{\mathit{I}}_{113}^{({\mathit{\varphi}}_{1})}$ | |
---|---|---|---|---|

12 | (011) | $947$ | $908$ | RSoXS |

6 | (002) | $1.51\xb7{10}^{3}$ | $1.49\xb7{10}^{3}$ | |

24 | (012) | $0.57$ | $57$ | |

24 | (012) | $0.59$ | $58$ | TReXS |

24 | (113) | $1$ | $99$ | |

24 | (023) | $0.96$ | $95$ | |

24 | (014) | $0.012$ | $1.1$ | |

24 | (223) | $2.8$ | $2.8\xb7{10}^{2}$ | |

48 | (124) | $1.6$ | $1.6\xb7{10}^{2}$ | |

24 | (233) | $1.5\xb7{10}^{3}$ | $1.5\xb7{10}^{3}$ | |

24 | (034) | $0.83$ | $82$ |

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**MDPI and ACS Style**

Grabovac, T.; Gorecka, E.; Pociecha, D.; Vaupotič, N.
Modeling of the Resonant X-ray Response of a Chiral Cubic Phase. *Crystals* **2021**, *11*, 214.
https://doi.org/10.3390/cryst11020214

**AMA Style**

Grabovac T, Gorecka E, Pociecha D, Vaupotič N.
Modeling of the Resonant X-ray Response of a Chiral Cubic Phase. *Crystals*. 2021; 11(2):214.
https://doi.org/10.3390/cryst11020214

**Chicago/Turabian Style**

Grabovac, Timon, Ewa Gorecka, Damian Pociecha, and Nataša Vaupotič.
2021. "Modeling of the Resonant X-ray Response of a Chiral Cubic Phase" *Crystals* 11, no. 2: 214.
https://doi.org/10.3390/cryst11020214