1.2 Primitive cells and basis

Primitive cells and basis

Primitive cells and crystal basis are the two ingredients you need to fully describe any crystal structure. The lattice tells you where things repeat in space, and the basis tells you what sits at each repeat point. Together, they generate the entire crystal.

These concepts underpin nearly everything else in solid state physics: diffraction, band structure, phonon spectra, and symmetry analysis all depend on getting the lattice and basis right.

Primitive cells

A primitive cell is the smallest volume of space that, when translated repeatedly, fills all of space with no gaps and no overlaps. Each primitive cell contains exactly one lattice point.

Because it's the minimal repeating unit, the primitive cell captures all the structural information of the lattice. Any property that depends on periodicity can be studied within a single primitive cell.

The choice of primitive cell is not unique. You can draw it in many different ways, as long as it contains exactly one lattice point and tiles space perfectly under translation.

Wigner-Seitz primitive cell

The Wigner-Seitz cell is a specific, uniquely defined primitive cell that preserves the full point symmetry of the lattice. You construct it as follows:

1. Pick a lattice point as the origin.
2. Draw line segments from that point to all nearby lattice points.
3. Construct the perpendicular bisector plane of each segment.
4. The smallest enclosed region around the origin is the Wigner-Seitz cell.

This construction guarantees that every point inside the cell is closer to the chosen lattice point than to any other. Because it reflects the local symmetry of the lattice, the Wigner-Seitz cell is especially useful when studying electronic properties and phonon modes.

Primitive translation vectors

A set of three linearly independent vectors $\vec{a}_1$, $\vec{a}_2$, $\vec{a}_3$ are primitive translation vectors if every lattice point can be reached by an integer combination:

$\vec{R} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3$

where $n_1, n_2, n_3$ are integers. Like the primitive cell itself, the choice of primitive vectors is not unique.

• Simple cubic lattice: $\vec{a}_1 = a\hat{x}$, $\vec{a}_2 = a\hat{y}$, $\vec{a}_3 = a\hat{z}$, where $a$ is the lattice constant.
• FCC lattice: $\vec{a}_1 = \frac{a}{2}(\hat{y}+\hat{z})$, $\vec{a}_2 = \frac{a}{2}(\hat{x}+\hat{z})$, $\vec{a}_3 = \frac{a}{2}(\hat{x}+\hat{y})$

Notice that the FCC primitive vectors point along face diagonals, not along the cube edges. This is why the FCC primitive cell looks quite different from the familiar cubic conventional cell.

The volume of the primitive cell is given by the scalar triple product $V = |\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)|$.

Crystal basis

The basis is the set of atoms (or ions, or molecules) associated with each lattice point. While the lattice is a purely geometric object describing periodicity, the basis provides the actual physical content of the crystal.

Definition of crystal basis

Each atom in the basis is specified by its position relative to the lattice point, typically written as:

$\vec{r}_j = x_j \vec{a}_1 + y_j \vec{a}_2 + z_j \vec{a}_3$

where $x_j, y_j, z_j$ are fractional coordinates (numbers between 0 and 1) and $j$ labels the atoms in the basis.

A basis can be as simple as a single atom (monatomic basis) or contain many atoms of different species (polyatomic basis). The number of atoms in the basis, combined with the lattice, determines the total number of atoms per unit volume.

Relationship between basis and lattice

The key equation in crystallography is:

The lattice tells you the periodicity. The basis tells you what's being repeated. Different bases placed on the same lattice produce different crystal structures with different physical properties. For example, both diamond (C) and zinc blende (ZnS) are built on an FCC lattice, but they have different bases and therefore very different electronic properties.

Examples of crystal basis

• Diamond structure: The basis consists of two identical carbon atoms at positions $\vec{r}_1 = (0,0,0)$ and $\vec{r}_2 = \frac{1}{4}(1,1,1)$ relative to each FCC lattice point. This creates the tetrahedral bonding geometry characteristic of diamond.
• NaCl structure: The basis is one Na$^+$ ion and one Cl$^-$ ion, placed on an FCC lattice. The Cl$^-$ sits at the origin and the Na$^+$ is displaced by $\frac{a}{2}$ along one cube edge, giving each ion an octahedral coordination.
• Perovskite (ABX$_3$): A three-component basis on a simple cubic lattice, with the A atom at the corner, B at the body center, and three X atoms at the face centers.

Lattice with basis

Combining primitive cell and basis

To build the full crystal structure from a lattice and basis:

1. Define the primitive translation vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3$.
2. Specify the basis atoms and their positions relative to a lattice point.
3. Place a copy of the basis at every lattice point $\vec{R} = n_1\vec{a}_1 + n_2\vec{a}_2 + n_3\vec{a}_3$.
4. The resulting arrangement of atoms fills all of space and represents the complete crystal.

Unit cells vs primitive cells

A unit cell is any region that reproduces the crystal when translated. A primitive cell is the special case where the unit cell contains exactly one lattice point.

Non-primitive (conventional) unit cells are often larger and contain multiple lattice points. They're chosen because they make the symmetry of the structure more obvious and are easier to visualize.

Conventional unit cells

Conventional unit cells sacrifice minimality for clarity. Two important examples:

• FCC conventional cell: A cube with atoms at corners and face centers. It contains 4 lattice points (8 corners × $\frac{1}{8}$ + 6 faces × $\frac{1}{2}$ = 4). The primitive cell has only 1 lattice point and is a rhombohedron, which is harder to visualize.
• BCC conventional cell: A cube with atoms at corners and the body center. It contains 2 lattice points (8 × $\frac{1}{8}$ + 1 = 2).

When you see a crystal structure drawn as a cube, you're almost always looking at the conventional cell, not the primitive cell.

Primitive reciprocal lattice vectors

The reciprocal lattice is the set of wave vectors $\vec{K}$ whose plane waves $e^{i\vec{K}\cdot\vec{r}}$ have the same periodicity as the real-space lattice. It plays a central role in diffraction theory, electronic band structure, and phonon dispersion.

Definition of reciprocal lattice

A wave vector $\vec{K}$ belongs to the reciprocal lattice if it satisfies:

$e^{i\vec{K} \cdot \vec{R}} = 1 \quad \text{for all lattice vectors } \vec{R}$

This means $\vec{K} \cdot \vec{R} = 2\pi \times \text{integer}$ for every $\vec{R}$. Each reciprocal lattice point corresponds to a family of parallel lattice planes in real space, which is why the reciprocal lattice is so useful for understanding X-ray diffraction.

Reciprocal lattice translation vectors

The primitive reciprocal lattice vectors $\vec{b}_1, \vec{b}_2, \vec{b}_3$ are defined by the orthogonality condition:

$\vec{a}_i \cdot \vec{b}_j = 2\pi\delta_{ij}$

They can be computed explicitly:

$\vec{b}_1 = \frac{2\pi(\vec{a}_2 \times \vec{a}_3)}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}, \quad \vec{b}_2 = \frac{2\pi(\vec{a}_3 \times \vec{a}_1)}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}, \quad \vec{b}_3 = \frac{2\pi(\vec{a}_1 \times \vec{a}_2)}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$

• Simple cubic: $\vec{b}_1 = \frac{2\pi}{a}\hat{x}$, $\vec{b}_2 = \frac{2\pi}{a}\hat{y}$, $\vec{b}_3 = \frac{2\pi}{a}\hat{z}$. The reciprocal lattice is also simple cubic.
• FCC: The reciprocal lattice is BCC, with primitive vectors $\vec{b}_1 = \frac{2\pi}{a}(-\hat{x}+\hat{y}+\hat{z})$, $\vec{b}_2 = \frac{2\pi}{a}(\hat{x}-\hat{y}+\hat{z})$, $\vec{b}_3 = \frac{2\pi}{a}(\hat{x}+\hat{y}-\hat{z})$.

Note the important duality: the reciprocal of FCC is BCC, and vice versa.

Reciprocal lattice primitive cell

The primitive cell of the reciprocal lattice, constructed via the Wigner-Seitz method, is called the first Brillouin zone. It contains all the unique wave vectors needed to describe the electronic and vibrational properties of the crystal.

Brillouin zones

Brillouin zones partition reciprocal space into regions based on proximity to reciprocal lattice points. They're essential for analyzing band structures, Fermi surfaces, and phonon dispersion.

First Brillouin zone

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all wave vectors $\vec{k}$ that are closer to the origin than to any other reciprocal lattice point.

Its boundaries are the Bragg planes: the perpendicular bisectors of the vectors connecting the origin to neighboring reciprocal lattice points. At these boundaries, diffraction conditions are satisfied, which is why energy gaps open up in the electronic band structure at zone boundaries.

Higher order Brillouin zones

Higher-order zones are constructed by extending the bisector-plane construction to more distant reciprocal lattice points. The $n$th Brillouin zone is the set of points reached from the origin by crossing exactly $n-1$ Bragg planes.

Each higher zone has the same volume as the first zone, and when "folded back" into the first zone (by translation through reciprocal lattice vectors), it fills it exactly. In practice, most analysis is done within the first zone.

Brillouin zone vs Wigner-Seitz cell

Both are Wigner-Seitz constructions, but they live in different spaces:

• The Wigner-Seitz cell is built in real space around a direct lattice point. It describes the spatial extent of the primitive cell.
• The Brillouin zone is built in reciprocal space around the origin. It describes the range of independent wave vectors.

The Wigner-Seitz cell tells you about real-space geometry. The Brillouin zone tells you about allowed $\vec{k}$-states for electrons and phonons.

Crystal structures

A crystal structure is the periodic arrangement of atoms in a solid, fully specified by its lattice and basis. Different structures lead to different mechanical, electronic, and optical properties.

Simple crystal structures

These have small bases (often a single atom) and high symmetry:

• Simple cubic (SC): One atom per lattice point on a cubic lattice. Rare in nature (polonium is the only element with this structure at ambient conditions). Coordination number: 6.
• Body-centered cubic (BCC): Can be described as a simple cubic lattice with a two-atom basis, or equivalently as a BCC Bravais lattice with a one-atom basis. Coordination number: 8. Examples: Fe, W, Cr.
• Face-centered cubic (FCC): One atom per FCC lattice point. Coordination number: 12 (the highest for cubic structures). Examples: Al, Cu, Au.
• Hexagonal close-packed (HCP): A hexagonal lattice with a two-atom basis. Also has coordination number 12. Examples: Mg, Ti, Zn.

FCC and HCP are both close-packed structures with the same packing fraction ($\approx 74\%$), but they differ in stacking sequence: FCC is ABCABC, while HCP is ABABAB.

Complex crystal structures

These involve multi-atom bases, often with more than one atomic species:

• Diamond structure: FCC lattice with a two-atom basis of identical atoms at $(0,0,0)$ and $\frac{1}{4}(1,1,1)$. Each atom is tetrahedrally coordinated. Examples: C (diamond), Si, Ge.
• Zinc blende structure: Same geometry as diamond, but the two basis atoms are different species. Examples: GaAs, ZnS. The loss of inversion symmetry compared to diamond has important consequences for piezoelectricity.
• Perovskite (ABX$_3$): A simple cubic lattice with a five-atom basis. The A cation sits at corners, B at the body center, and X anions at face centers. Examples: BaTiO$_3$, SrTiO$_3$. Perovskites are important for ferroelectricity and, more recently, solar cells.

Crystal structure notation

Several notation systems are used to classify structures:

• Pearson symbols encode the crystal system, centering type, and number of atoms per conventional cell. For example, cF8 means cubic (c), face-centered (F), 8 atoms per cell (diamond structure).
• Strukturbericht notation assigns letter-number labels to common structure types: A1 = FCC, A2 = BCC, A3 = HCP, B1 = NaCl, B3 = zinc blende.
• Space group notation (Hermann-Mauguin symbols) describes the full symmetry. For example, Fm$\bar{3}$m for FCC, Fd$\bar{3}$m for diamond.

Symmetry in crystal lattices

Symmetry constrains which physical properties a crystal can have and simplifies calculations enormously. Three levels of symmetry matter: translational, point group, and space group.

Translational symmetry

Every crystal lattice is invariant under translation by any lattice vector $\vec{R}$. This is the most fundamental symmetry of a crystal and has deep consequences:

• It leads to Bloch's theorem, which states that electronic wavefunctions in a periodic potential can be written as a plane wave modulated by a function with the lattice periodicity.
• It justifies the use of periodic boundary conditions and working in reciprocal space.
• It's the reason we can describe the entire crystal by studying just one unit cell.

Point group symmetry

Point group operations leave at least one point fixed. They include rotations, reflections, inversions, and improper rotations (rotation followed by reflection).

There are 32 crystallographic point groups in 3D, constrained by the requirement of compatibility with translational symmetry (only 1-, 2-, 3-, 4-, and 6-fold rotations are allowed).

Point group symmetry determines the anisotropy of physical properties like electrical conductivity, thermal expansion, and the dielectric tensor. If a crystal has high point-group symmetry, many of these tensor components are related or zero.

Space group symmetry

A space group combines point group operations with translations (including screw axes and glide planes, which are combinations of rotation/reflection with fractional translations).

There are exactly 230 space groups in three dimensions. Every crystal belongs to one of them. The space group dictates:

• Which electronic and vibrational states are allowed
• Whether certain phenomena like piezoelectricity or ferroelectricity can occur (these require the absence of inversion symmetry)
• Systematic absences in diffraction patterns, which help identify the structure experimentally

Examples of space groups:

• FCC metals (Cu, Al): Fm$\bar{3}$m
• Diamond/Si: Fd$\bar{3}$m
• Graphite: P6$_3$/mmc