1.3 Crystal systems

Crystal systems provide the classification scheme for how atoms arrange themselves in crystalline solids. By grouping crystals according to their symmetry, you can predict a wide range of physical properties and make sense of diffraction data, electronic structure, and mechanical behavior.

Crystal Systems

Every crystal belongs to one of seven crystal systems, defined by the symmetry relationships among the unit cell's edges and angles. The crystal system tells you what constraints exist on the lattice parameters and which symmetry operations the structure must possess.

Bravais Lattices

A Bravais lattice is one of the 14 distinct ways to arrange points periodically in three dimensions such that every point has an identical environment. Each Bravais lattice is defined by a unit cell, the smallest box that, when repeated by translation in all three directions, reproduces the entire lattice.

The 14 Bravais lattices arise because each of the seven crystal systems can support one or more centering types (primitive, body-centered, face-centered, or base-centered), but not every combination produces a unique lattice.

Primitive vs. Non-Primitive Cells

• A primitive cell (P) contains exactly one lattice point. It's the smallest possible unit cell for a given lattice.
• A non-primitive cell (also called a conventional cell) contains more than one lattice point. These are often chosen because they display the full symmetry of the lattice more clearly than the primitive cell does.

Common non-primitive types:

• Body-centered (I): one lattice point at each corner plus one at the center of the cell, giving 2 lattice points per cell.
• Face-centered (F): one lattice point at each corner plus one at the center of each face, giving 4 lattice points per cell.
• Base-centered (C): one lattice point at each corner plus one at the center of two opposite faces, giving 2 lattice points per cell.

Crystal Symmetry

Symmetry describes every transformation you can apply to a crystal that leaves it looking identical. The symmetry a crystal possesses directly constrains its optical, electrical, thermal, and mechanical properties. For example, a crystal that lacks inversion symmetry can exhibit piezoelectricity, while one that has it cannot.

Translation vs. Point Symmetry

• Translation symmetry is the repetition of the unit cell throughout space. Every crystal has this by definition.
• Point symmetry refers to operations that leave at least one point fixed. These operations describe the shape symmetry of the unit cell itself.

Symmetry Elements and Operations

A symmetry element is the geometric entity (an axis, a plane, or a point) about which a symmetry operation acts. A symmetry operation is the actual transformation performed.

Seven Crystal Systems

Each system is defined by minimum symmetry requirements, which in turn constrain the lattice parameters ($a, b, c$ and $\alpha, \beta, \gamma$).

Triclinic System

The lowest symmetry system. There are no constraints on edges or angles:

$a \neq b \neq c, \quad \alpha \neq \beta \neq \gamma \neq 90°$

The only required symmetry is either the identity or an inversion center. Example: copper(II) sulfate pentahydrate ($\text{CuSO}_4 \cdot 5\text{H}_2\text{O}$).

Monoclinic System

One twofold rotation axis or one mirror plane:

$a \neq b \neq c, \quad \alpha = \gamma = 90°, \quad \beta \neq 90°$

Example: gypsum ($\text{CaSO}_4 \cdot 2\text{H}_2\text{O}$).

Orthorhombic System

Three mutually perpendicular twofold rotation axes or mirror planes:

$a \neq b \neq c, \quad \alpha = \beta = \gamma = 90°$

Example: olivine ($(\text{Mg,Fe})_2\text{SiO}_4$).

Tetragonal System

One fourfold rotation axis:

$a = b \neq c, \quad \alpha = \beta = \gamma = 90°$

Example: rutile ($\text{TiO}_2$).

Trigonal (Rhombohedral) System

One threefold rotation axis. When described with a rhombohedral cell:

$a = b = c, \quad \alpha = \beta = \gamma \neq 90°$

Note: the trigonal system can also be described using a hexagonal cell (with $a = b \neq c$, $\gamma = 120°$), which is why some references list only six crystal systems. The distinguishing feature is always the threefold axis. Example: quartz ($\text{SiO}_2$).

Hexagonal System

One sixfold rotation axis:

$a = b \neq c, \quad \alpha = \beta = 90°, \quad \gamma = 120°$

Example: graphite (C).

Cubic System

The highest symmetry system, defined by four threefold rotation axes along the body diagonals:

$a = b = c, \quad \alpha = \beta = \gamma = 90°$

Examples: sodium chloride (NaCl), diamond (C).

Lattice Parameters

Lattice parameters specify the size and shape of the unit cell. There are six in total.

Axial Lengths

$a$, $b$, and $c$ are the lengths of the three unit cell edges, typically measured in angstroms (Å, where 1 Å = $10^{-10}$ m) or nanometers.

Interaxial Angles

• $\alpha$: angle between $b$ and $c$
• $\beta$: angle between $a$ and $c$
• $\gamma$: angle between $a$ and $b$

Getting these angle definitions right matters. A common mistake is mixing up which angle goes with which pair of axes.

Crystal Planes

Crystal planes are sets of parallel, equally spaced planes that pass through lattice points. They're central to understanding X-ray diffraction, cleavage behavior, and surface properties.

Miller Indices

Miller indices ($hkl$) label the orientation of a crystal plane. To determine them:

1. Find where the plane intercepts the $a$, $b$, and $c$ axes (in units of the lattice parameters).
2. Take the reciprocal of each intercept.
3. Clear fractions by multiplying through by the smallest common factor.
4. The resulting integers $(hkl)$ are the Miller indices.

If a plane is parallel to an axis (intercept at infinity), the corresponding Miller index is 0. Negative intercepts are written with a bar over the index, e.g., $\bar{1}$.

Families of Planes

Planes that are equivalent by symmetry form a family, denoted with curly brackets $\{hkl\}$. For instance, in a cubic crystal, $\{100\}$ includes $(100)$, $(010)$, $(001)$, $(\bar{1}00)$, $(0\bar{1}0)$, and $(00\bar{1})$. All six planes are physically identical due to cubic symmetry.

Reciprocal Lattice

The reciprocal lattice is a mathematical construction in which each point corresponds to a set of planes in the real-space lattice. It's indispensable for interpreting diffraction experiments and for describing electronic band structures.

Relationship to Real-Space Lattice

The reciprocal lattice vectors are defined as:

$\vec{a}^* = \frac{2\pi(\vec{b} \times \vec{c})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$

$\vec{b}^* = \frac{2\pi(\vec{c} \times \vec{a})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$

$\vec{c}^* = \frac{2\pi(\vec{a} \times \vec{b})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$

The denominator $\vec{a} \cdot (\vec{b} \times \vec{c})$ is the volume of the real-space unit cell. Each reciprocal lattice vector is perpendicular to two of the real-space vectors: $\vec{a}^*$ is perpendicular to both $\vec{b}$ and $\vec{c}$, and so on.

A key property: $\vec{a}_i \cdot \vec{a}_j^* = 2\pi \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta.

Wigner-Seitz Cell

The Wigner-Seitz cell is a particular choice of primitive cell. You construct it by:

1. Choosing a lattice point as the origin.
2. Drawing lines to all nearest-neighbor lattice points.
3. Constructing the perpendicular bisector plane of each line.
4. The smallest enclosed volume around the origin is the Wigner-Seitz cell.

Every point inside this cell is closer to the chosen lattice point than to any other. The Wigner-Seitz cell always has the full point symmetry of the lattice.

First Brillouin Zone

The first Brillouin zone is simply the Wigner-Seitz cell constructed in the reciprocal lattice rather than the real-space lattice. It plays a central role in solid-state physics because electronic band structures and phonon dispersion relations are conventionally plotted within it.

Space Groups

While point groups describe symmetry operations that leave a point fixed, space groups combine point symmetry with translational symmetry. There are exactly 230 space groups in three dimensions, and every crystal structure belongs to one of them.

Screw Axes

A screw axis combines rotation with translation along the rotation axis. The notation $n_m$ means:

• Rotate by $360°/n$
• Translate by $m/n$ of the lattice parameter along the axis

For example, a $2_1$ screw axis rotates by 180° and translates by half the lattice parameter. Screw axes produce helical arrangements of atoms.

Glide Planes

A glide plane combines reflection with translation parallel to the mirror plane. The type of glide is labeled by the direction of translation:

• $a$, $b$, $c$: translation by half the corresponding lattice parameter
• $n$: translation by half the sum of two lattice parameters (diagonal glide)
• $d$: translation by a quarter of a face or body diagonal (diamond glide)

Screw axes and glide planes are the symmetry elements that distinguish space groups from point groups. They only exist in periodic structures.

Examples of Crystal Structures

A crystal structure = a Bravais lattice + a basis (the set of atoms placed at each lattice point). The structures below are among the most commonly encountered.

Simple Cubic (SC)

One atom per unit cell, located at each corner. This is the simplest possible structure but quite rare because it packs atoms inefficiently (packing fraction $\approx 52\%$). The only element with this structure at ambient conditions is polonium (Po).

Body-Centered Cubic (BCC)

Two atoms per unit cell: corners plus one at the body center $(1/2, 1/2, 1/2)$. Packing fraction $\approx 68\%$. Each atom has 8 nearest neighbors. Examples: iron (Fe) at room temperature, tungsten (W), chromium (Cr).

Face-Centered Cubic (FCC)

Four atoms per unit cell: corners plus one at the center of each face. Packing fraction $\approx 74\%$, which is the maximum for identical spheres. Each atom has 12 nearest neighbors. Examples: copper (Cu), aluminum (Al), gold (Au).

Hexagonal Close-Packed (HCP)

Also achieves $74\%$ packing, like FCC, but with a different stacking sequence (ABAB vs. ABCABC for FCC). The unit cell contains 2 atoms. Each atom has 12 nearest neighbors. Examples: magnesium (Mg), zinc (Zn), titanium (Ti).

Diamond Cubic

An FCC lattice with a two-atom basis: one atom at $(0,0,0)$ and another at $(1/4, 1/4, 1/4)$. This gives 8 atoms per conventional unit cell. Each atom is tetrahedrally coordinated with 4 nearest neighbors, resulting in a relatively open structure (packing fraction $\approx 34\%$). Examples: diamond (C), silicon (Si), germanium (Ge).

Zinc Blende

Identical to the diamond cubic structure, but with two different atom types: one species at $(0,0,0)$ and the other at $(1/4, 1/4, 1/4)$. This breaks inversion symmetry, which is why zinc blende compounds can be piezoelectric. Examples: gallium arsenide (GaAs), zinc sulfide (ZnS).

Sodium Chloride (Rock Salt)

Two interpenetrating FCC sublattices, offset by $(1/2, 0, 0)$ (equivalently, one species at $(0,0,0)$ and the other at $(1/2, 1/2, 1/2)$). Each ion is octahedrally coordinated by 6 ions of the opposite type. Examples: NaCl, MgO, FeO.

Cesium Chloride

A simple cubic lattice with a two-atom basis: one atom at $(0,0,0)$ and the other at $(1/2, 1/2, 1/2)$. This is not BCC because the two sites are occupied by different atom types. Each ion has 8 nearest neighbors of the opposite type. Examples: CsCl, CsBr.