# Chirality in the Solid State: Chiral Crystal Structures in Chiral and Achiral Space Groups

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## Abstract

**:**

## 1. Introduction

## 2. Chirality in Three Dimensions

## 3. Chirality in Two Dimensions

## 4. Chirality Measures

## 5. Chiral Systems: From Crystalline Elements to Compounds

#### 5.1. Elements with Chiral Structures

#### 5.2. Compounds with Chiral, Cubic Structures

#### 5.3. Compounds with Tetragonal Chiral Structures

#### 5.4. Compounds with Hexagonal or Trigonal, Chiral Structures

## 6. Chirality at Solid Surfaces

- achiral bulk with a chiral surface,
- chiral bulk with an achiral surface,
- chiral bulk with a chiral surface,
- achiral bulk with an achiral surface.

#### 6.1. Achiral Bulk with Chiral Surface

#### 6.2. Surfaces of Chiral Bulk Materials

## 7. Electronic Structure and Chirality

#### 7.1. Electronic Structure of Se

#### 7.2. Electronic Structure of FeSi and Other Compounds with B20 Structure

#### 7.3. Berry Curvature and Chirality

#### 7.4. Circular Dichroism, Chirality, and Electronic Structure

## 8. Discussion, Summary, and Conclusions

- (I)
- chirality, chirality measure, chirality sense, handedness, and helicity;
- (II)
- chiral structures and chiral crystallographic space groups;
- (III)
- chirality in two and three dimensions;
- (IV)
- material properties depending on chirality measures or chirality sense.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. First Principles Calculations

## Appendix B. Relation Between Chiral Structures, Symmorphic Space Groups, and Topological Spaces

**Table A1.**The 23 symmorphic Sohnke groups. Printed in bold are those groups belonging to ${S}^{3}$ (3 spheres) topological space.

Crystal System | Space Group |
---|---|

Triclinic | $P\phantom{\rule{0.222222em}{0ex}}1$ |

Monoclinic | $P\phantom{\rule{0.222222em}{0ex}}2$, $C\phantom{\rule{0.222222em}{0ex}}2$ |

Orthorhombic | $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{222}$, $\mathit{C}\phantom{\rule{0.222222em}{0ex}}\mathit{222}$, $I\phantom{\rule{0.222222em}{0ex}}222$, $\mathit{F}\phantom{\rule{0.222222em}{0ex}}\mathit{222}$ |

Tetragonal | $P\phantom{\rule{0.222222em}{0ex}}4$, $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{422}$, $I\phantom{\rule{0.222222em}{0ex}}4$, $\mathit{I}\phantom{\rule{0.222222em}{0ex}}\mathit{422}$ |

Trigonal | $P\phantom{\rule{0.222222em}{0ex}}3$, $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{312}$, $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{321}$, $R\phantom{\rule{0.222222em}{0ex}}3$, $\mathit{R}\phantom{\rule{0.222222em}{0ex}}\mathit{32}$ |

Hexagonal | $P\phantom{\rule{0.222222em}{0ex}}6$, $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{622}$ |

Cubic | $\mathit{P}\phantom{\rule{0.222222em}{0ex}}\mathit{23}$, $I\phantom{\rule{0.222222em}{0ex}}23$, $\mathit{I}\phantom{\rule{0.222222em}{0ex}}\mathit{432}$, $\mathit{F}\phantom{\rule{0.222222em}{0ex}}\mathit{23}$, $\mathit{F}\phantom{\rule{0.222222em}{0ex}}\mathit{432}$ |

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**Figure 1.**Chirality and handedness. (

**a**) A pair of shoes is chiral and handed. The right shoe is the mirror image of the left shoe, but none of the mirror images coincide with itself; (

**b**) potatoes are chiral because there exists no mirror operation that transform them into themselves; however, they are not handed, and there exist no left- or right-handed potatoes; (

**c**) a rotating ball can exhibit a left (clockwise) or a right (“handed”) (anti-clockwise) rotation and thus it may be called handed; however, one of its mirror images coincided with itself and therefore it is not chiral but helical.

**Figure 2.**The 2-, 3-, 4-, and 6-fold screw axes. For each screw, the relative translation $\tau =\frac{n}{N}$ along the z axis after the rotation is assigned by an arrow. For the neutral axes a positive or negative rotation about the z-axis (e.g., rotation by $\pm \pi $ in case of the ${2}_{1}$ screw rotation) yields the same result.

**Figure 3.**Screw rotations and simple structure in space group 198. (

**a**) Arrangement of the atoms on position 4a. (

**b**) Position of the 2- (blue) and 3-fold (red) screw axes. The three ${2}_{1}$ screw axes are parallel to the principle axes. The three ${3}_{2}$ screw axes are parallel to $\u2329\overline{1}11\u232a$ type axes. The [111] axis shown in a) (green diagonal line) is a simple 3-fold rotational axis.

**Figure 4.**Hausdorff distance $h\left(u\right)$ and continuous chirality measure ${S}^{2}\left(u\right)$ in a chiral cubic compound based on space group 198 with a single 4a Wyckoff position occupied. Shown are the dependencies of $h\left(u\right)$ and ${S}^{2}\left(u\right)$ on the position parameter u of the Wyckoff position $4a$. The character (R and L) of structures in space group 198 is adopted from the handedness of the closest chiral space group 213 or 212.

**Figure 5.**Chirality of the $A8$ structure of $\gamma $-Se. Shown are the Hausdorff distance h and the continuous chirality measure ${S}^{2}$ as a function of the internal parameter u. Open symbols assign the values at optimized and experimental u values (Section 7.1) and closed symbols mark the parameters where the space group changes away from $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}21$. The range of u and ${S}^{2}$ under pressure in the experiment [47] is marked by the thick red line.

**Figure 6.**The $B20$ structure of FeSi. The enantiomorphous pair RL and LR is shown with $({u}_{\mathrm{Fe}},{u}_{\mathrm{Si}})=(0.3858,0.094)$ and $(0.6142,0.906)$, respectively. RL implies that Fe atoms are positioned in an R type and Si in an L type structure and the opposite for LR. Both structures are shown for views along the [111] axes and arbitrary axes. Fe (or Si) atoms in the triangles of the [111] view are in the same (111) plane; these planes appear at a distance of $(2-{u}_{i})/\sqrt{3}$ from the origin. Atoms in the center, on the [111] axis, are in a different plane ${u}_{i}\sqrt{3}$ away from the origin. See Figure 3 for the positions of the screw axes. Connections between atoms are drawn for better visibility and may not be confused with bonds.

**Figure 7.**Chirality and nearest neighbor distances of the compounds with FeSi structure. Shown are on top the nearest neighbor distances and on the bottom the continuous chirality measure ${S}^{2}({u}_{\mathrm{TM}},{u}_{\mathrm{MG}})$ for space group 198 with two 4a Wyckoff positions occupied ($B20$ compounds). The values for FeSi (LR), the space groups of the achiral structures where $S=0$, or the chiral space groups where $S=1$ are assigned in the graph. Important note: The borderline parameters $({u}_{\mathrm{TM}},{u}_{\mathrm{MG}})=(0,1)$, and $(1/2,1/2)$ are not possible as is evident from the graph of the nearest neighbor distances. Note that the largest possible nearest neighbor distance in $B20$ compounds is ${d}_{\mathrm{NN},\mathrm{max}}/a=\sqrt{3}/(1+\sqrt{5})$.

**Figure 8.**Chirality of the $B20$ compounds. Continuous chirality measures of $B20$ compounds where the complete structure determination is reported in Pearson’s database [20]. The full and dashed lines mark u values of 1/8 and 7/8, respectively.

**Figure 9.**Chirality of the $F{0}_{1}$ compounds. Continuous chirality measures of $F{0}_{1}$ compounds where the complete structure determination is reported in the Pearson database [20].

**Figure 10.**The CrSi${}_{2}$ structure of quartz (SiO${}_{2}$) and transition metal $C40$ compounds. The enantiomorphous pair of structures in the chiral space groups $P\phantom{\rule{0.222222em}{0ex}}{6}_{2}22$ (180) and $P\phantom{\rule{0.222222em}{0ex}}{6}_{4}22$ (181) are shown. Transition metal atoms (Cr) are green (dark), and the main group elements (Si) are shown in grey (light). (For quartz, green atoms correspond to Si and grey ones to O).

**Figure 11.**The chiral $fcc\left(643\right)$ surface. The two modifications $fcc{\left(643\right)}^{\mathrm{S}}$ and $fcc{\left(643\right)}^{\mathrm{R}}$ with opposite handedness from different perspectives are shown. Different colors are used to better distinguish edge and kink atoms from those on terraces.

**Figure 12.**The FeSi(001) surface. The evolution of the surface structure and symmetry is shown by increasing the number of layers in (

**a**–

**c**). The upper row shows the side view and the lower row the top view. Fe atoms are drawn in red (

**dark**) and Si atoms in grey (

**light**).

**Figure 13.**An Fe terminated FeSi(111) surface. The evolution of the surface structure and symmetry are shown for increasing number of layers in (

**a**–

**d**), assuming that the surface is terminated by Fe with layer type A (see text).

**Figure 14.**Different CrSi${}_{2}$(0001) surfaces and their ($000\overline{1}$) counterparts. The upper row shows the right-handed and the lower row the left-handed crystal, with the indicated space group. The stacking order is sketched in the middle using [$11\overline{2}0$] as viewing directions.

**Figure 15.**Electronic structure of chiral and achiral ($u=1/3$) Se. The band structures are calculated for variation of the chirality by changing the position parameter u. From left to right: (

**a**) $u=1/6$ (${S}^{2}=100$%), (

**b**) $u=0.22$ (optimized) (${S}^{2}=45.2$%), (

**c**) $u=0.23$ (experiment) (${S}^{2}=38.4$%), (

**d**) $u=1/3$ (${S}^{2}=0$), (

**e**) $u=1/2$ (${S}^{2}=100$%). The corresponding chirality measures are given in brackets (compare also Figure 5). Note that the parameters $u=1/3$ ($R\phantom{\rule{0.222222em}{0ex}}\overline{3}m$) and $u=1/2$ ($P\phantom{\rule{0.222222em}{0ex}}{6}_{4}22$) result in different space groups and symmetry. Calculations are for variation of the position parameter at optimized lattice parameters.

**Figure 16.**Pressure dependence of the chirality of FeSi. The position parameters (symbols connected by lines) for Fe and Si and the continuous chirality measure ${S}^{2}\left(p\right)$ are shown. The position parameters resulting in ${S}^{2}=1$ are marked in the upper part by full (1/8) and dashed (3/8) lines (blue).

**Figure 17.**Band structure of FeSi, calculated for variation of the chirality by changing the position parameters ${u}_{\mathrm{Fe}}$ and ${u}_{\mathrm{Si}}$. From left to right: (

**a**) $F\phantom{\rule{0.222222em}{0ex}}\overline{4}3m$ achiral, (

**b**) $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ at 60 GPa pressure, (

**c**) $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ optimized, (

**d**) $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ with ${S}^{2}=1$, and (

**e**) $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}21$ also with ${S}^{2}=1$. Calculations are for variation of the position parameter at optimized lattice parameter a for each structure. Note that the $\Delta $ direction ($\overline{\Gamma X}$) has in space group 198 only a 2-fold rotational symmetry, therefore, the perpendicular directions $\overline{MX}$ and $\overline{X{M}^{\prime}}$ are not equivalent. (Please note the different Brillouin zone of the face-centered space group in (

**a**)).

**Figure 18.**Berry curvature of FeSi, calculated for the two enantiomers of FeSi. The x and y components ${\Omega}_{x,y}({k}_{x},{k}_{y})$ in the (001) plane through the $\Gamma $-point (${k}_{z}=0$) and the corresponding in-plane vector field $\overrightarrow{\Omega}({k}_{x},{k}_{y})$ are shown.

**Figure 19.**Polarization dependent photoelectron spectra and CDAD from FeSi(001). The polarization-dependent spectra are shown in comparison to the band structure along $\Delta $ for one of the enantiomers. The calculations are for normal incident photons of 21.2 eV energy and opposite helicity. For the enantiomorphous pair, the resulting circular dichroism is compared to the total intensity.

**Figure 20.**CDAD from VSi${}_{2}$(0001). The intensity distributions for excitation by circularly polarized photons of opposite helicity with normal incidence (that is along the (0001) surface normal) are shown. The intensities and circular dichroism (CDAD) from crystals belonging to the two space groups with opposite handedness are compared.

**Table 1.**Structure–symmetry–property relations in 7 non-centrosymmetric classes composed from the 21 non-centrosymmetric Laue classes.

Property | |||||
---|---|---|---|---|---|

No. | Laue Class | E | PY | O | PI |

1 | 1, 2, 3, 4, 6 | ⊠ | ⊠ | ⊠ | ⊠ |

2 | 222, 32, 422, 622, 23 | ⊠ | ⊠ | ⊠ | |

3 | 432 | ⊠ | ⊠ | ||

4 | $m,mm2$ | ⊠ | ⊠ | ⊠ | |

5 | $3m,4mm,6mm$ | ⊠ | ⊠ | ||

6 | $\overline{4},\overline{4}2m$ | ⊠ | ⊠ | ||

7 | $\overline{6},\overline{6}2m,\overline{4}3m$ | ⊠ |

**E**= enantiomorphism (chirality),

**PY**= polar (pyroelectric, ferroelectric),

**O**= optical active, and

**PI**= piezoelectric and nonlinear optics.

Crystal System | Laue Class | Point Group | Hermann-Mauguin Symbol | Space Group Number |
---|---|---|---|---|

Triclinic | 1 | ${C}_{1}$ | $P\phantom{\rule{0.222222em}{0ex}}1$ | 1 |

Monoclinic | 2 | ${C}_{2}$ | $P\phantom{\rule{0.222222em}{0ex}}121$, $P\phantom{\rule{0.222222em}{0ex}}{12}_{1}1$, $C\phantom{\rule{0.222222em}{0ex}}121$ | 3–5 |

Orthorhombic | 222 | ${D}_{2}$ | $P\phantom{\rule{0.222222em}{0ex}}222$, $P\phantom{\rule{0.222222em}{0ex}}{222}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}{2}_{1}2$, $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}{2}_{1}{2}_{1}$, | 16–… |

$C\phantom{\rule{0.222222em}{0ex}}{222}_{1}$, $C\phantom{\rule{0.222222em}{0ex}}222$, $F\phantom{\rule{0.222222em}{0ex}}222$, $I\phantom{\rule{0.222222em}{0ex}}222$, $I\phantom{\rule{0.222222em}{0ex}}{2}_{1}{2}_{1}{2}_{1}$ | …–24 | |||

Tetragonal | 4 | ${C}_{4}$ | $P\phantom{\rule{0.222222em}{0ex}}4$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{2}$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}$, $I\phantom{\rule{0.222222em}{0ex}}4$, $I\phantom{\rule{0.222222em}{0ex}}{4}_{1}$ | 75–80 |

422 | ${D}_{4}$ | $P\phantom{\rule{0.222222em}{0ex}}422$, $P\phantom{\rule{0.222222em}{0ex}}{42}_{1}2$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}22$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}{2}_{1}2$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{2}22$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{2}{2}_{1}2$, | 89–… | |

$P\phantom{\rule{0.222222em}{0ex}}{4}_{3}22$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}{2}_{1}2$, $I\phantom{\rule{0.222222em}{0ex}}422$, $I\phantom{\rule{0.222222em}{0ex}}{4}_{1}22$ | …–98 | |||

Trigonal | 3 | ${C}_{3}$ | $P\phantom{\rule{0.222222em}{0ex}}3$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}$, $R\phantom{\rule{0.222222em}{0ex}}3$ | 143–146 |

32 | ${D}_{3}$ | $P\phantom{\rule{0.222222em}{0ex}}312$, $P\phantom{\rule{0.222222em}{0ex}}321$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}12$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}21$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}12$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}21$, $R\phantom{\rule{0.222222em}{0ex}}32$ | 149–155 | |

Hexagonal | 6 | ${C}_{6}$ | $P\phantom{\rule{0.222222em}{0ex}}6$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{5}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{2}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{4}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{3}$ | 168–173 |

622 | ${D}_{6}$ | $P\phantom{\rule{0.222222em}{0ex}}622$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{1}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{5}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{2}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{4}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{3}22$ | 177–182 | |

Cubic | 23 | T | $P\phantom{\rule{0.222222em}{0ex}}23$, $F\phantom{\rule{0.222222em}{0ex}}23$, $I\phantom{\rule{0.222222em}{0ex}}23$, $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$, $I\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ | 195–199 |

432 | O | $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}2$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{2}32$, $F\phantom{\rule{0.222222em}{0ex}}432$, $F\phantom{\rule{0.222222em}{0ex}}{4}_{1}32$, $I\phantom{\rule{0.222222em}{0ex}}432$, | 207–… | |

$P\phantom{\rule{0.222222em}{0ex}}{4}_{3}32$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}32$, $I\phantom{\rule{0.222222em}{0ex}}{4}_{1}32$ | … –214 |

**Table 3.**The 22 chiral space groups of class II in 11 enantiomorphous pairs (columns as in Table 2).

Tetragonal | 4 | ${C}_{4}$ | $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}$ | (76,78) |

422 | ${D}_{4}$ | $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}22$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}22$ | (91,95) | |

$P\phantom{\rule{0.222222em}{0ex}}{4}_{1}{2}_{1}2$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}{2}_{1}2$ | (92,96) | |||

Trigonal | 3 | ${C}_{3}$ | $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}$ | (144,145) |

32 | ${D}_{3}$ | $P\phantom{\rule{0.222222em}{0ex}}{3}_{1}12$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}12$ | (151,153) | |

$P\phantom{\rule{0.222222em}{0ex}}{3}_{1}21$, $P\phantom{\rule{0.222222em}{0ex}}{3}_{2}21$ | (152,154) | |||

Hexagonal | 6 | ${C}_{6}$ | $P\phantom{\rule{0.222222em}{0ex}}{6}_{1}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{5}$ | (169,170) |

$P\phantom{\rule{0.222222em}{0ex}}{6}_{2}$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{4}$ | (171,172) | |||

622 | ${D}_{6}$ | $P\phantom{\rule{0.222222em}{0ex}}{6}_{1}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{5}22$ | (178,179) | |

$P\phantom{\rule{0.222222em}{0ex}}{6}_{2}22$, $P\phantom{\rule{0.222222em}{0ex}}{6}_{4}22$ | (180,181) | |||

Cubic | 432 | O | $P\phantom{\rule{0.222222em}{0ex}}{4}_{3}32$, $P\phantom{\rule{0.222222em}{0ex}}{4}_{1}32$ | (212,213) |

**Table 4.**Chiral and polar axes of the Sohncke groups. c is assumed to be a unique axis for the monoclinic system. The rhombohedral axes are assumed for trigonal systems (e.g., 32), for hexagonal setting Laue class 321 must be distinguished from 312. Single axes are denoted by [], groups of axes are denoted by <>.

Crystal System | Laue Class | Point Group | Chiral Axes | Polar Axes |
---|---|---|---|---|

Monoclinic | 2 | ${C}_{2}$ | [001] | [001] |

Orthorhombic | 222 | ${D}_{2}$ | [001], [100], [010] | None |

Tetragonal | 4 | ${C}_{4}$ | [001] | [001] |

422 | ${D}_{4}$ | [001], [100], [010], | None | |

[110], [$\overline{1}10$] | ||||

Trigonal | 3 | ${C}_{3}$ | [001] | [001] |

32 | ${D}_{3}$ | [111], [$1\overline{1}0$], | [$1\overline{1}0$], | |

[$01\overline{1}$], [$\overline{1}01$] | [$01\overline{1}$], [$\overline{1}01$] | |||

Hexagonal | 6 | ${C}_{6}$ | [001] | [001] |

622 | ${D}_{6}$ | [001], [100], [010], | None | |

[$\overline{1}10$], [$\overline{1}\overline{1}0$], | ||||

[210], [120] | ||||

Cubic | 23 | T | $\u2329111\u232a$, $\u2329100\u232a$ | $\u2329111\u232a$ |

432 | O | $\u2329111\u232a$, $\u2329100\u232a$, $\u2329110\u232a$ | None |

**Table 5.**Plane lattices and space groups. The 5 groups without mirror operations, that are hosts for chiral objects, are marked in bold. The numbering of the groups is according to Ref. [22].

Bravais Lattice | Point Group | Plane Group | Number |
---|---|---|---|

Oblique | ${C}_{1}$ | $\mathit{p}\mathit{1}$ | 1 |

${C}_{2}$ | $\mathit{p}\mathit{2}$ | 2 | |

Rectangular | ${C}_{s}$ | $pm,pg$ | 3, 4 |

${C}_{2v}$ | $p2mm,p2mg,p2gg$ | 6, 7, 8 | |

Rhombic | ${C}_{s}$ | $cm$ | 5 |

${C}_{2v}$ | $c2mm$ | 9 | |

Square | ${C}_{4}$ | $\mathit{p}\mathit{4}$ | 10 |

${C}_{4v}$ | $p4mm,p4gm$ | 11, 12 | |

Hexagonal | ${C}_{3}$ | $\mathit{p}\mathit{3}$ | 13 |

${C}_{6}$ | $\mathit{p}\mathit{6}$ | 16 | |

${C}_{3v}$ | $p3m1,p31m$ | 14, 15 | |

${C}_{6v}$ | $p6mm$ | 17 |

**Table 6.**Atom positions in cubic, achiral and chiral structures related to space group $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$. Tabulated are the positions for $u\le 1/8$. Others may be found using a shift vector $(\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)$ ($n=1,2,3$) and the equivalence of positions with $-u$ and $1-u$. The minimum distance of the positions between $P\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ and $F\phantom{\rule{0.222222em}{0ex}}m\overline{3}m$ is in all four cases $u\sqrt{3}$.

$\mathit{F}\phantom{\rule{0.222222em}{0ex}}\mathit{m}\overline{3}\mathit{m}$ | $\mathit{P}\phantom{\rule{0.222222em}{0ex}}{2}_{1}3$ | $\mathit{P}\phantom{\rule{0.222222em}{0ex}}{4}_{3}32$ |
---|---|---|

$(0,0,0)$ | $(u,u,u)$ | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.)$ |

$(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,0)$ | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+u,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-u,-u)$ | $(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.)$ |

$(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-u,-u,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+u)$ | $(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$8$}\right.)$ |

$(0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ | $(-u,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+u,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-u)$ | $(\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.)$ |

**Table 7.**Chirality of WO${Z}_{4}$ compounds ($Z=$ Br, Cl). Given are the unnormalized Hausdorff distances, h, and continuous chirality measures, s, with respect to different possible achiral structures.

Achiral Group | Polar | WOBr${}_{4}$ | WOCl${}_{4}$ | ||
---|---|---|---|---|---|

$\mathit{h}$ | $\mathit{s}$ | $\mathit{h}$ | $\mathit{s}$ | ||

$I\phantom{\rule{0.222222em}{0ex}}4/m$ | N | 0.049 | 0.0650 | 0.045 | 0.0642 |

$I\phantom{\rule{0.222222em}{0ex}}4/mm$ | Y | 0.049 | 0.0654 | 0.045 | 0.0645 |

$I\phantom{\rule{0.222222em}{0ex}}4/mmm$ | N | 0.049 | 0.0818 | 0.045 | 0.0800 |

**Table 8.**Chirality of compounds with $C40$ structure. Given are the lattice ($a,c$) and position (u) parameters, as well as the continuous chirality measures (${S}^{2}$). Lattice parameters are from the Pearson database [20].

a [Å] | c [Å] | u | ${\mathit{S}}^{2}$ | |
---|---|---|---|---|

CrSi${}_{2}$ | 4.4283 | 6.3680 | 0.1658 | 0.2706 |

MoSi${}_{2}$ | 4.6220 | 6.6460 | 0.1642 | 0.2702 |

NbSi${}_{2}$ | 4.7974 | 6.5923 | 0.1593 | 0.2690 |

TaSi${}_{2}$ | 4.7839 | 6.5700 | 0.1590 | 0.2689 |

VSi${}_{2}$ | 4.5726 | 6.3744 | 0.1626 | 0.2698 |

WSi${}_{2}$ | 4.6180 | 6.6740 | 0.1640 | 0.2702 |

NbGe${}_{2}$ | 4.9670 | 6.7830 | 0.1631 | 0.2699 |

TaGe${}_{2}$ | 4.9380 | 6.7300 | 0.1640 | 0.2702 |

WAl${}_{2}$ | 4.7422 | 6.6057 | 0.1618 | 0.2696 |

**Table 9.**Symmetry of special projections of the chiral space groups. Chiral projections are marked by printing their plane groups ($\mathit{p}i$, $\mathit{i}=1,3,4,6$) in bold. The cubic, achiral Sohncke group 198 is given for comparison.

cubic | [001] | [110] | [111] |

212,213 | $p4gm$ | $p2gm$ | $p3m$ |

198 | $p2gg$ | $pg$ | $\mathit{p}3$ |

hexagonal | [0001] | [11$\overline{2}$0] | |

180,181 | $p6mm$ | $p2mm$ | |

178,179 | $p6mm$ | $p2gm$ | |

171,172 | $\mathit{p}6$ | $pm$ | |

169,170 | $\mathit{p}6$ | $pg$ | |

trigonal | [001] | [100] | [210] |

152,154 | $p3m$ | $\mathit{p}2$ | $pm$ |

151,153 | $p3m$ | $pm$ | $\mathit{p}2$ |

144,145 | $\mathit{p}3$ | $\mathit{p}1$ | $\mathit{p}1$ |

tetragonal | [001] | [100] | [110] |

92,96 | $p4gm$ | $p2gg$ | $p2gm$ |

91,95 | $p4mm$ | $p2gm$ | $p2gm$ |

76,78 | $\mathit{p}4$ | $pg$ | $pg$ |

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**MDPI and ACS Style**

Fecher, G.H.; Kübler, J.; Felser, C. Chirality in the Solid State: Chiral Crystal Structures in Chiral and Achiral Space Groups. *Materials* **2022**, *15*, 5812.
https://doi.org/10.3390/ma15175812

**AMA Style**

Fecher GH, Kübler J, Felser C. Chirality in the Solid State: Chiral Crystal Structures in Chiral and Achiral Space Groups. *Materials*. 2022; 15(17):5812.
https://doi.org/10.3390/ma15175812

**Chicago/Turabian Style**

Fecher, Gerhard H., Jürgen Kübler, and Claudia Felser. 2022. "Chirality in the Solid State: Chiral Crystal Structures in Chiral and Achiral Space Groups" *Materials* 15, no. 17: 5812.
https://doi.org/10.3390/ma15175812