Chemical Crystallography II: Macroscopic & Microscopic Symmetry

The external morphological faces of well-formed macroscopic crystals exhibit rigorous symmetry described by crystallographic point groups.

The Crystallographic Restriction Theorem

Unlike free molecules in the gas phase, which can exhibit arbitrary rotational symmetry (such as $C_5$ in ferrocene or $C_7$ in tropylium), periodic crystals in two and three dimensions can possess only 1-, 2-, 3-, 4-, and 6-fold rotational symmetry. Fivefold ($C_5$) and eightfold or higher ($C_{\ge 7}$) rotation axes are strictly forbidden by translational periodicity.

Mathematical Proof:

Consider a 1D row of identical lattice points along a translation vector $\mathbf{a}$ of length $a$. Let two neighboring lattice points be $A$ and $B$, separated by distance $a$. Assume a counterclockwise rotation by angle $\alpha$ about an axis perpendicular to the plane passing through $A$ maps the lattice onto itself. Then point $B$ rotates to $B'$. Similarly, rotating clockwise by $\alpha$ about an axis passing through $B$ maps point $A$ to $A'$. Because the crystal lattice is translationally periodic, the vector connecting $A'$ and $B'$ must itself be a lattice translation vector parallel to $AB$, having length equal to an integer multiple $m$ of the primitive translation $a$:

Projecting the segments onto the line $AB$:

Dividing throughout by $a$:

Because the cosine function is strictly bounded by $-1 \le \cos\alpha \le +1$:

Evaluating the permitted angles $\alpha$ and rotation orders $n = 360^\circ / \alpha$:

1. $N = -2 \implies \cos\alpha = -1 \implies \alpha = 180^\circ \implies n = 2$ (twofold rotation).
2. $N = -1 \implies \cos\alpha = -1/2 \implies \alpha = 120^\circ \implies n = 3$ (threefold rotation).
3. $N = 0 \implies \cos\alpha = 0 \implies \alpha = 90^\circ \implies n = 4$ (fourfold rotation).
4. $N = +1 \implies \cos\alpha = +1/2 \implies \alpha = 60^\circ \implies n = 6$ (sixfold rotation).
5. $N = +2 \implies \cos\alpha = +1 \implies \alpha = 360^\circ \implies n = 1$ (identity rotation).

Fivefold rotation requires $\cos(72^\circ) = (\sqrt{5}-1)/4 \approx 0.3090$, which is not a half-integer ($N/2$). Hence, pentagonal tiles cannot tile Euclidean 2D space without gaps or overlaps.

Crystallographic point symmetry comprises proper rotations and improper (roto-inversion) operations.

Proper Rotations ($n$)

A proper rotation of order $n$ ($n \in \{1, 2, 3, 4, 6\}$) rotates an object by angle $\alpha = 2\pi/n$ about an axis, preserving enantiomeric handedness (congruent transformation).

Center of Inversion ($\bar{1}$ or $i$)

An inversion center maps point $(x, y, z)$ through the origin to $(-x, -y, -z)$:

Crystals possessing an inversion center are centrosymmetric and cannot exhibit polar physical phenomena such as piezoelectricity, pyroelectricity, or optical second-harmonic generation (SHG).

Mirror Planes ($m$ or $\sigma$)

A reflection plane maps coordinates across a plane. A mirror plane normal to the $z$-axis maps $(x, y, z) \to (x, y, -z)$. In Hermann-Mauguin notation, a mirror plane is denoted by $m$, equivalent to a twofold rotary-inversion axis:

Rotary-Inversion Axes ($\bar{n}$)

Combines a proper rotation by $2\pi/n$ with an immediate inversion through the origin:

1. $\bar{1}$: Inversion center ($i$).
2. $\bar{2}$: Equivalent to a mirror plane $m$ perpendicular to the axis.
3. $\bar{3}$: A threefold rotation followed by inversion, equivalent to $3 + \bar{1}$ (a threefold axis with a center of inversion).
4. $\bar{4}$: An essential fourfold rotary-inversion axis (characteristic of tetrahedral and tetragonal groups such as $\text{CuFeS}_2$ chalcopyrite). It consists of two operations: $\bar{4}^1$ and $\bar{4}^3$, containing an embedded twofold axis $\bar{4}^2 = 2$.
5. $\bar{6}$: A sixfold rotation followed by inversion, equivalent to a threefold rotation with a perpendicular mirror plane ($3/m$).

Combining the 5 proper rotation axes ($1, 2, 3, 4, 6$) with inversion and reflection generates precisely 32 crystallographic point groups (crystal classes) distributed among the 7 crystal systems.

Stereographic Projection Principles

A stereographic projection represents 3D angular relationships of crystal faces and symmetry elements on a 2D plane:

1. The crystal is placed at the center of a reference sphere.
2. Normals to crystal faces (poles) radiate outward, intersecting the sphere surface.
3. Points in the northern hemisphere ($z > 0$) are projected onto the equatorial plane by connecting them to the south pole; their intersections are marked with a solid dot ($\bullet$).
4. Points in the southern hemisphere ($z < 0$) are projected to the north pole and marked with an open circle ($\circ$).
5. Mirror planes perpendicular to the projection plane appear as straight diametral lines; mirror planes inclined to the plane appear as circular arcs.

Distribution of the 32 Point Groups Across Crystal Systems

• Triclinic ($2$): $1$ (non-centrosymmetric), $\bar{1}$ (centrosymmetric).
• Monoclinic ($3$): $2, m, 2/m$.
• Orthorhombic ($3$): $222, mm2, mmm$.
• Tetragonal ($7$): $4, \bar{4}, 4/m, 422, 4mm, \bar{4}2m, 4/mmm$.
• Trigonal ($5$): $3, \bar{3}, 32, 3m, \bar{3}m$.
• Hexagonal ($7$): $6, \bar{6}, 6/m, 622, 6mm, \bar{6}m2, 6/mmm$.
• Cubic ($5$): $23, m\bar{3}, 432, \bar{4}3m, m\bar{3}m$.

Centrosymmetric vs Non-Centrosymmetric Groups

• Centrosymmetric ($11$ groups): Contain $\bar{1}$. Include $\bar{1}, 2/m, mmm, 4/m, 4/mmm, \bar{3}, \bar{3}m, 6/m, 6/mmm, m\bar{3}, m\bar{3}m$.
• Non-Centrosymmetric ($21$ groups): Lack inversion. Of these, 20 groups are piezoelectric (polar or chiral), and 10 groups are polar / pyroelectric ($1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm$), exhibiting spontaneous electric polarization along a unique polar axis.

Two notation systems are used internationally to denote crystallographic symmetry:

Hermann-Mauguin vs Schoenflies Notation

1. Schoenflies Notation: Widely used in molecular spectroscopy and quantum chemistry:

• $C_n$: Cyclic group of order $n$.
• $C_{nv}$: Cyclic group with vertical mirror planes.
• $C_{nh}$: Cyclic group with horizontal mirror plane.
• $D_n$: Dihedral group ($C_n$ plus $n$ perpendicular $C_2$ axes).
• $D_{nh}, D_{nd}$: Dihedral groups with horizontal or diagonal mirror planes.
• $T, T_h, T_d$: Tetrahedral cubic groups.
• $O, O_h$: Octahedral cubic groups.

2. Hermann-Mauguin (International) Notation: Standard in solid-state chemistry and X-ray crystallography:

• Direct axes are denoted by numbers: $1, 2, 3, 4, 6$.
• Inversion axes are denoted with a bar: $\bar{1}, \bar{3}, \bar{4}, \bar{6}$.
• Mirror planes are denoted by $m$.
• A slash ($/$) indicates a mirror plane perpendicular to a rotation axis: $2/m, 4/m, 6/m$.
• Successive symbols indicate symmetry along conventional crystallographic directions:
• Orthorhombic ($mmm$): Mirror planes perpendicular to $\mathbf{a}, \mathbf{b}, \mathbf{c}$.
• Tetragonal ($4/mmm$): $4/m$ along $c$; $m$ along $a, b$; $m$ along $[110]$.
• Cubic ($m\bar{3}m$): $m$ along $\langle 100 \rangle$; $\bar{3}$ along body diagonals $\langle 111 \rangle$; $m$ along face diagonals $\langle 110 \rangle$.

Macroscopic Crystal Forms and Habits

A crystal form $\{hkl\}$ is a set of crystal faces related by the symmetry operations of the point group.

• Open Forms: Faces do not enclose space (e.g., pedions [1 face], pinacoids [2 parallel faces], prisms, pyramids). Must be combined with other forms to create a closed crystal.
• Closed Forms: Faces completely enclose space (e.g., octahedron $\{111\}$, cube $\{100\}$, rhombic dodecahedron $\{110\}$, tetrahedron).
• Crystal Habit: The external morphological shape developed during growth (e.g., acicular [needle-like], platy [tabular], dendritic, prismatic), dictated by relative face growth velocities under differing supersaturation and impurity conditions.

When point symmetry operations are combined with fractional translations of the crystal space lattice, new microscopic symmetry elements arise that operate only at the atomic scale: glide planes and screw axes.

The Glide Plane Operation

A glide plane combines a mirror reflection across a plane with a simultaneous translation $\mathbf{t}$ parallel to that plane by a fraction of a unit cell vector:

Applying the operation twice corresponds to two reflections (identity) plus two translations $2\mathbf{t}$:

Types of Glide Planes in Hermann-Mauguin Notation

1. Axial Glide Planes ($a, b, c$):

Translation is exactly half of a primitive unit cell vector parallel to the glide plane:

• $a$-glide: Reflection across plane, followed by translation $\mathbf{t} = \mathbf{a}/2$.
• $b$-glide: Translation $\mathbf{t} = \mathbf{b}/2$.
• $c$-glide: Translation $\mathbf{t} = \mathbf{c}/2$.

2. Diagonal Glide Plane ($n$):

Translation is half the face diagonal of the cell:

• For an $n$-glide perpendicular to $c$: $\mathbf{t} = \frac{\mathbf{a} + \mathbf{b}}{2}$.

3. Diamond Glide Plane ($d$):

Translation is one-quarter of a face or body diagonal, occurring in centered lattices (such as the diamond cubic lattice $Fd\bar{3}m$):

• $\mathbf{t} = \frac{\mathbf{a} \pm \mathbf{b}}{4}$ or $\frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{4}$.

Glide planes leave no invariant macroscopic face angles and are therefore invisible in morphological mineralogy. However, their translation components introduce destructive phase interference in diffracted X-ray beams, generating systematic absences in $(hkl)$ reflections that allow crystallographers to determine space groups unambiguously.

A screw axis combines a proper rotation by angle $2\pi/n$ with a fractional translation parallel to the rotation axis.

Mathematical Formulation

A screw axis is denoted by the symbol:

The operation consists of:

1. Rotation by $2\pi/n$ about the axis.
2. Translation parallel to the axis by fractional vector:

Applying the $n_m$ operation $n$ times yields an overall rotation of $n \times (2\pi/n) = 2\pi$ (identity) and a total translation of $n \times (m/n)\mathbf{T} = m\mathbf{T}$, which is an integer lattice translation.

The Eleven Screw Axes

• Twofold:
• $2_1$: Rotation $180^\circ$, translation $\mathbf{t} = \mathbf{c}/2$.
• Threefold:
• $3_1$: Rotation $120^\circ$, translation $\mathbf{t} = \mathbf{c}/3$ (right-handed helix).
• $3_2$: Rotation $120^\circ$, translation $\mathbf{t} = 2\mathbf{c}/3$ (left-handed enantiomorph of $3_1$).
• Fourfold:
• $4_1$: Rotation $90^\circ$, translation $\mathbf{t} = \mathbf{c}/4$ (right-handed).
• $4_2$: Rotation $90^\circ$, translation $\mathbf{t} = 2\mathbf{c}/4 = \mathbf{c}/2$ (neutral).
• $4_3$: Rotation $90^\circ$, translation $\mathbf{t} = 3\mathbf{c}/4$ (left-handed enantiomorph of $4_1$).
• Sixfold:
• $6_1$ and $6_5$: Enantiomorphic pair with translations $\mathbf{c}/6$ and $5\mathbf{c}/6$.
• $6_2$ and $6_4$: Enantiomorphic pair with translations $2\mathbf{c}/6$ and $4\mathbf{c}/6$.
• $6_3$: Translation $3\mathbf{c}/6 = \mathbf{c}/2$.

Enantiomorphism and Chiral Crystals

Chiral molecules (such as L-amino acids and D-sugars) cannot crystallize in space groups possessing inversion centers, mirror planes, or glide planes. They crystallize exclusively in the 65 Sohncke space groups, which contain only proper rotations and screw axes ($n_m$). Quartz ($\alpha$-quartz) crystallizes as enantiomorphic pairs in space groups $P3_1 21$ (right-handed) and $P3_2 21$ (left-handed), exhibiting macroscopic optical rotation.

The complete symmetry of any crystalline solid is rigorously described by one of the 230 crystallographic space groups, derived independently in 1891 by Arthur Schoenflies and Evgraf Fedorov.

Construction of Space Groups

A space group $\mathcal{G}$ is the infinite discrete group of all symmetry operations that map a 3D periodic crystal onto itself. It is mathematically an extension of the translational group $\mathcal{T}$ (14 Bravais lattices) by a point group $\mathcal{P}$ (32 point groups):

• Symmorphic Space Groups (73): Contain only point group operations and pure lattice translations (no glide planes or screw axes).
• Non-Symmorphic Space Groups (157): Contain glide planes and/or screw axes with fractional translations.

Anatomy of Hermann-Mauguin Space Group Symbols

The standard full international symbol specifies:

1. First Character: Bravais lattice centering type:

• $P$ (Primitive), $I$ (Body-centered), $F$ (Face-centered), $C, A, B$ (Base-centered), $R$ (Rhombohedral).

2. Subsequent Characters: Symmetry elements along principal crystallographic directions:

• Monoclinic ($P2_1/c$):
• $P$: Primitive lattice.
• $2_1/c$: Along the unique $b$-axis, a $2_1$ screw axis is perpendicular to a $c$-glide plane.
• Orthorhombic ($Pnma$):
• $P$: Primitive lattice.
• $n$: $n$-glide plane perpendicular to $a$.
• $m$: Mirror plane perpendicular to $b$.
• $a$: $a$-glide plane perpendicular to $c$.
• Cubic ($Fm\bar{3}m$, Rock Salt):
• $F$: Face-centered cubic lattice.
• $m$: Mirror plane perpendicular to $\langle 100 \rangle$.
• $\bar{3}$: Threefold rotary-inversion axis along body diagonals $\langle 111 \rangle$.
• $m$: Mirror plane perpendicular to face diagonals $\langle 110 \rangle$.

The Asymmetric Unit and Wyckoff Positions

• Asymmetric Unit: The minimal fraction of unit cell volume from which the entire crystal can be generated by applying all space group symmetry operations.
• Wyckoff Positions: Sites within the unit cell classified by their site symmetry:
• General Position: A point $(x, y, z)$ having no site symmetry ($1$). Its multiplicity equals the order of the space group.
• Special Positions: Points lying on symmetry elements (such as mirror planes, rotation axes, or inversion centers). Their site symmetry is higher, and their multiplicity is a fraction of the general position.

The structural response of solid materials to temperature, pressure, and chemical substitution is categorized into four fundamental crystallographic relationships:

1. Polymorphism

The ability of a chemical compound of fixed stoichiometry to exist in two or more distinct crystal structures (e.g., $\text{CaCO}_3$ as calcite [trigonal] vs aragonite [orthorhombic]; $\text{TiO}_2$ as rutile, anatase, and brookite).

• Enantiotropic Polymorphism: Phase transition is thermodynamically reversible at a specific transition temperature ($T_{\text{tr}}$) and pressure ($P_{\text{tr}}$) where $\Delta G = 0$.
• Monotropic Polymorphism: One polymorph is thermodynamically stable under all conditions; other polymorphs are metastable and convert irreversibly to the stable phase.

2. Polytypism

A special one-dimensional sub-case of polymorphism in which distinct structures arise solely from different close-packed stacking sequences of identical two-dimensional modular layers:

• Silicon Carbide (SiC): More than $250$ distinct polytypes exist:
• $3C$: Cubic Zinc Blende stacking ($ABCABC\dots$, space group $F\bar{4}3m$).
• $4H$: Hexagonal stacking ($ABCB\dots$, space group $P6_3 mc$).
• $6H$: Hexagonal stacking ($ABCACB\dots$, space group $P6_3 mc$, repeat distance $c = 15.1\text{ \AA}$).
• $15R$: Rhombohedral stacking with 15-layer repeat ($c = 37.8\text{ \AA}$).

3. Isomorphism

Distinct chemical compounds possessing identical crystal structures, space groups, and closely similar unit cell dimensions (discovered by Eilhard Mitscherlich in 1819).

• Examples:
• $\text{K}_2\text{SO}_4$, $\text{K}_2\text{SeO}_4$, and $\text{K}_2\text{CrO}_4$ (all orthorhombic $Pnma$).
• Alums: $M^I M^{III}(\text{SO}_4)_2 \cdot 12\text{H}_2\text{O}$ (where $M^I = \text{K}^+, \text{NH}_4^+, \text{Rb}^+$ and $M^{III} = \text{Al}^{3+}, \text{Cr}^{3+}, \text{Fe}^{3+}$).

Isomorphous salts readily form continuous substitutional solid solutions and undergo epitaxy.

4. Allotropy

Polymorphism occurring in elemental substances:

• Carbon: Diamond ($sp^3$ covalent network), graphite ($sp^2$ layered hexagonal sheets), fullerenes ($\text{C}_{60}$ molecular crystals), carbon nanotubes, and graphene.
• Tin ($\text{Sn}$): Grey tin ($\alpha\text{-Sn}$, diamond cubic semiconductor) converts below $13.2^\circ\text{C}$ to white tin ($\beta\text{-Sn}$, body-centered tetragonal ductile metal), a volume expansion of $27\%$ responsible for "tin pest".