Lattice Vectors and the Unit Cell

Learning Objectives

By the end of this section, you will be able to:

1. Define lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ and explain how they generate an infinite lattice through integer linear combinations.
2. Translate between lattice parameters ($a, b, c, \alpha, \beta, \gamma$) and full Cartesian vector representations.
3. Distinguish between primitive cells, conventional cells, and the Wigner-Seitz cell, and explain when to use each one.
4. Compute the volume of a unit cell using the scalar triple product $V = \mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)$.
5. Write a correct VASP POSCAR file with either primitive or conventional lattice vectors and verify the cell volume.
6. Explain why the Wigner-Seitz cell in reciprocal space is the first Brillouin zone, and why that matters for DFT k-point sampling.

The Big Picture

The Problem: Describing Infinite Order with Finite Information

A macroscopic crystal of table salt contains roughly $10^{22}$ atoms. Writing down every atom's position is impossible. Yet the structure has a beautiful regularity: if you stand at any sodium atom and look around, you see the exact same arrangement of chlorine and sodium neighbors, no matter which sodium atom you chose. This translational symmetry means we can describe the entire infinite crystal using just a few numbers.

The key insight, formalized by Auguste Bravais in 1850, is to separate the repeating pattern (the lattice) from what repeats (the basis). In the previous section we introduced this decomposition. Now we make the lattice side precise: three vectors define the repeat directions and distances, and the region they enclose — the unit cell — is the minimal "tile" that can be copied to fill all of space.

From Wallpaper to Crystals

Consider a wallpaper pattern. Two vectors in the plane define how the design repeats horizontally and diagonally. Given these two vectors and one copy of the motif, you can reconstruct the entire infinite wallpaper. A 3D crystal is the same idea with three vectors instead of two. The vectors tell you the "step size and direction" of each repeat, and the motif (the basis) is the group of atoms that gets stamped at every lattice point.

The choice of repeat vectors is not unique. You could describe the same wallpaper with different pairs of vectors, producing different-shaped tiles that all cover the plane perfectly. Some tiles are as small as possible (primitive), others are larger but more symmetric (conventional). Understanding this freedom — and knowing which choice is best for a given purpose — is the central theme of this section.

Lattice Vectors

A Bravais lattice is fully specified by three lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$. These are three linearly independent vectors in $\mathbb{R}^3$ such that every lattice point $\mathbf{R}$ is located at:

$\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, \quad n_1, n_2, n_3 \in \mathbb{Z}$

Unpacking the Equation

Let us read this equation carefully, symbol by symbol:

• $\mathbf{R}$ is the position vector of a lattice point, measured from some chosen origin. It is a vector in $\mathbb{R}^3$ with units of length (typically angstroms, $\text{\AA}$).
• $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ are the three lattice vectors (also called basis vectors of the lattice — not to be confused with the atomic basis). Each is a vector in $\mathbb{R}^3$. They define the shape and size of the repeating unit.
• $n_1, n_2, n_3$ are integers (positive, negative, or zero). They act as "step counters": take $n_1$ steps along $\mathbf{a}_1$, $n_2$ steps along $\mathbf{a}_2$, and $n_3$ steps along $\mathbf{a}_3$. The resulting position is a lattice point.
• $\mathbb{Z}$ denotes the set of all integers: $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$. The requirement that $n_i \in \mathbb{Z}$ is what makes the lattice discrete rather than continuous.

Geometric Intuition

Think of the lattice vectors as three "step instructions" in 3D space. Starting at the origin, step $\mathbf{a}_1$ tells you how far and in what direction to walk to reach the nearest lattice point along the first direction. Similarly for $\mathbf{a}_2$ and $\mathbf{a}_3$. By taking integer combinations of these steps, you visit every point of an infinite 3D grid. The grid might be orthogonal (like a cubic lattice), skewed (like a triclinic lattice), or anything in between.

Cartesian Components

In practice, we express each lattice vector as three Cartesian components. For example, the FCC primitive vectors for a material with lattice constant $a$ are:

$\mathbf{a}_1 = \frac{a}{2}(0, 1, 1), \quad \mathbf{a}_2 = \frac{a}{2}(1, 0, 1), \quad \mathbf{a}_3 = \frac{a}{2}(1, 1, 0)$

For CdSe in the zinc blende structure with $a = 6.052$ $\text{\AA}$, each component evaluates to $6.052/2 = 3.026$ $\text{\AA}$. These are exactly the vectors you write in a VASP POSCAR file (after dividing by the scaling factor).

Meanwhile, the conventional cubic vectors are simply:

$\mathbf{a}_1 = a\hat{\mathbf{x}}, \quad \mathbf{a}_2 = a\hat{\mathbf{y}}, \quad \mathbf{a}_3 = a\hat{\mathbf{z}}$

Both sets of vectors describe the same infinite lattice, but the primitive set produces a smaller cell (fewer atoms, cheaper calculation).

Lattice Parameters

Instead of writing full Cartesian vectors, crystallographers often use six scalar lattice parameters to characterize the unit cell geometry:

• Three lengths: $a = |\mathbf{a}_1|$, $b = |\mathbf{a}_2|$, $c = |\mathbf{a}_3|$.
• Three angles:

The angles are computed from the dot products: for example, $\cos \gamma = \frac{\mathbf{a}_1 \cdot \mathbf{a}_2}{|\mathbf{a}_1| |\mathbf{a}_2|}$.

Crystal Systems and Their Constraints

The 7 crystal systems impose specific constraints on these 6 parameters. A few key examples:

For a cubic system, a single number $a$ suffices. For triclinic, all 6 parameters are independent. The fewer the constraints, the lower the symmetry.

Explore how these 6 parameters define the shape of the unit cell in 3D. Select any of the 7 crystal systems to see its characteristic parallelepiped, or adjust the sliders to watch the cell morph continuously between systems:

Now explore these lattice types in full 3D. Rotate the view to see how corner, body-center, and face-center atoms are arranged, and toggle between conventional and primitive cell representations:

The Unit Cell

A unit cell is a region of space that, when translated by all lattice vectors, tiles the entire crystal without gaps or overlaps. Formally, it is a fundamental domain of the translation group generated by $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$.

The Tiling Analogy

Think of a unit cell as a single tile on an infinite bathroom floor. The tile can be any shape that covers the floor with no gaps and no overlaps when you slide copies of it around (without rotating or reflecting). The simplest tile is the parallelepiped — the 3D analogue of a parallelogram — whose edges are the lattice vectors themselves. But there are infinitely many valid tile shapes for any given lattice.

Formal Properties

1. Space-filling: copies of the cell, translated by $\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3$ for all integers $n_1, n_2, n_3$, cover every point of $\mathbb{R}^3$ exactly once.
2. Integer lattice points: the cell contains an integer number of lattice points: 1 (primitive), 2 (body-centered), 4 (face-centered), or 2 (base-centered).
3. Non-uniqueness: for any lattice, there are infinitely many valid unit cells. The choice depends on your purpose — minimal cost (primitive), maximum symmetry display (conventional), or maximum symmetry (Wigner-Seitz).

Volume of the Unit Cell

The volume of the parallelepiped spanned by three lattice vectors is given by the scalar triple product:

$V = |\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)|$

Geometric Meaning

This formula has a beautiful geometric interpretation in two stages:

1. Cross product $\mathbf{a}_2 \times \mathbf{a}_3$: this gives a vector perpendicular to both $\mathbf{a}_2$ and $\mathbf{a}_3$, with magnitude equal to the area of the parallelogram they span. Think of it as the "base area" of the parallelepiped.
2. Dot product with $\mathbf{a}_1$: this projects $\mathbf{a}_1$ onto the perpendicular direction, giving the height of the parallelepiped. Area $\times$ height = volume.

Step through this decomposition in 3D to see the base parallelogram, cross product vector, and height projection that combine to give the parallelepiped volume:

Equivalently, the volume is the absolute value of the determinant of the lattice matrix:

$V = \left| \det \begin{pmatrix} a_{1x} & a_{1y} & a_{1z} \\ a_{2x} & a_{2y} & a_{2z} \\ a_{3x} & a_{3y} & a_{3z} \end{pmatrix} \right|$

Example: Simple Cubic

For a simple cubic lattice with lattice constant $a$: the vectors are $a\hat{\mathbf{x}}$, $a\hat{\mathbf{y}}$, $a\hat{\mathbf{z}}$, so $V = a \cdot a \cdot a = a^3$. For CdSe with $a = 6.052$ $\text{\AA}$, the conventional cube volume is $V = 6.052^3 \approx 221.7$ $\text{\AA}^3$.

Example: FCC Primitive Cell

For the FCC primitive vectors $\mathbf{a}_1 = \frac{a}{2}(0,1,1)$, $\mathbf{a}_2 = \frac{a}{2}(1,0,1)$, $\mathbf{a}_3 = \frac{a}{2}(1,1,0)$:

$V = \left| \det \frac{a}{2} \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \right| = \frac{a^3}{8} \cdot |0(0-1) - 1(0-1) + 1(1-0)| = \frac{a^3}{8} \cdot 2 = \frac{a^3}{4}$

This is exactly $1/4$ of the conventional cubic volume, consistent with $Z = 4$ lattice points in the FCC conventional cell. For CdSe: $V_{\text{prim}} = 221.7/4 \approx 55.4$ $\text{\AA}^3$.

Volume in Terms of Lattice Parameters

If you only know the six lattice parameters rather than the full vectors, the volume is:

$V = abc\sqrt{1 - \cos^2\alpha - \cos^2\beta - \cos^2\gamma + 2\cos\alpha\cos\beta\cos\gamma}$

For the cubic case ($a=b=c$, all angles $90^\circ$), this simplifies to $V = a^3$ since all cosines are zero. For hexagonal ($\gamma = 120^\circ$), it gives $V = a^2 c \sin(120^\circ) = \frac{\sqrt{3}}{2} a^2 c$.

Primitive Cells

A primitive cell is the smallest unit cell that contains exactly one lattice point. It has the minimum possible volume for the given lattice:

$V_{\text{prim}} = \frac{V_{\text{conv}}}{Z}$

where $Z$ is the number of lattice points in the conventional cell.

Properties of Primitive Cells

• Minimal volume: contains exactly 1 lattice point per cell.
• Computationally efficient: fewest atoms means fastest DFT calculation.
• May hide symmetry: the parallelepiped shape often looks "tilted" and does not obviously display the crystal system's symmetry.
• Not unique: there are infinitely many valid primitive cells for any lattice. The Niggli cell is a standardized choice.

FCC Primitive Vectors and Cell

The FCC lattice has $Z = 4$ in the conventional cubic cell. Its standard primitive vectors are:

$\mathbf{a}_1 = \frac{a}{2}(0, 1, 1), \quad \mathbf{a}_2 = \frac{a}{2}(1, 0, 1), \quad \mathbf{a}_3 = \frac{a}{2}(1, 1, 0)$

Each vector connects a corner of the conventional cube to the center of one of its faces. The resulting primitive cell is a rhombohedron (a parallelepiped with all edges equal and all face angles equal).

BCC Primitive Vectors and Cell

The BCC lattice has $Z = 2$. Its primitive vectors are:

$\mathbf{a}_1 = \frac{a}{2}(-1, 1, 1), \quad \mathbf{a}_2 = \frac{a}{2}(1, -1, 1), \quad \mathbf{a}_3 = \frac{a}{2}(1, 1, -1)$

Each vector connects a corner of the cube to the body center. The primitive cell volume is $a^3/2$.

Conventional Cells

A conventional cell is a larger unit cell chosen for convenience. It displays the full symmetry of the crystal system at the cost of containing multiple lattice points.

Why Use Conventional Cells?

1. Symmetry is visually obvious: a cube for cubic, a hexagonal prism for hexagonal. Primitive cells often hide this symmetry in a skewed parallelepiped.
2. Database compatibility: the ICSD, Materials Project, and AFLOW all report conventional cells by default. Comparing your structure to databases requires matching conventions.
3. Supercell construction: building a $2 \times 2 \times 2$ supercell is trivial with orthogonal conventional vectors. With primitive FCC vectors, the same supercell requires careful matrix manipulation.
4. Defect modeling: to study a vacancy, substitution, or interface, you typically need a supercell. Starting from the conventional cell keeps the geometry intuitive.

The Wigner-Seitz Cell

The Wigner-Seitz (WS) cell is a special primitive cell with a remarkable property: it has the full point-group symmetry of the lattice. It is constructed by a purely geometric recipe:

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

Mathematically, the WS cell is the Voronoi cell of the origin: the set of all points in space that are closer to the origin than to any other lattice point:

$\text{WS} = \{ \mathbf{r} \in \mathbb{R}^3 : |\mathbf{r}| \leq |\mathbf{r} - \mathbf{R}| \text{ for all lattice vectors } \mathbf{R} \neq 0 \}$

Properties of the Wigner-Seitz Cell

• It is a primitive cell: contains exactly 1 lattice point, has volume $V_{\text{prim}}$.
• It has the full point-group symmetry of the lattice (unlike generic primitive cells that may look asymmetric).
• It is always convex.
• It is unique for a given lattice (unlike parallelepiped primitive cells).

WS Cells for Common Lattices

The 2D construction above generalizes to 3D. Explore the actual Wigner-Seitz cell shapes for cubic lattices — the cube (SC), truncated octahedron (BCC), and rhombic dodecahedron (FCC) — and step through the 3D construction:

Real-World Examples

Example 1: X-Ray Diffraction and the Unit Cell

When X-rays hit a crystal, the diffraction pattern encodes the unit cell geometry. The positions of diffraction peaks give the lattice parameters, while their intensities encode the atomic basis. A powder X-ray diffraction (PXRD) experiment on CdSe zinc blende yields peaks at specific $2\theta$ angles. From the peak positions, we extract the d-spacings using Bragg's law $n\lambda = 2d\sin\theta$, and from the d-spacings we determine the lattice parameter $a$. For cubic systems, the relationship is $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$.

Example 2: Choosing Cells for Defect Calculations

To model a single Mn dopant substituting for Cd in CdSe, you need a supercell large enough that the periodic images of the defect do not interact. A typical choice is a $3 \times 3 \times 3$ supercell of the conventional cell (216 atoms) or a $4 \times 4 \times 4$ supercell of the primitive cell (128 atoms). The primitive-cell supercell has fewer atoms and is cheaper, but the conventional-cell supercell has a cleaner geometry that makes it easier to analyze the defect's local environment.

Example 3: Strain Analysis from Lattice Parameters

When a thin film of GaN is grown on a sapphire substrate, the mismatch in lattice parameters causes strain. The in-plane parameter $a$ is compressed or stretched to match the substrate, while the out-of-plane parameter $c$ changes in response (Poisson effect). By measuring both $a$ and $c$ via X-ray diffraction and comparing to the bulk values, you can quantify the biaxial strain: $\varepsilon_{\parallel} = (a_{\text{film}} - a_{\text{bulk}})/a_{\text{bulk}}$.

Example 4: Nanoparticle Size from Peak Broadening

For nanocrystalline CdSe quantum dots, the X-ray diffraction peaks are broadened. The Scherrer equation relates peak width to crystallite size: $D = \frac{K\lambda}{\beta \cos\theta}$, where $\beta$ is the full width at half maximum (FWHM) in radians and $K \approx 0.9$. This is a direct application of lattice periodicity: fewer repeat units means broader peaks.

Modern Computing and Materials Informatics

Why Crystallography Is the Language of Periodic Matter

Every atom-scale simulation of a solid must deal with periodicity. The unit cell is the computational box: atoms inside it are explicitly simulated, and periodic boundary conditions replicate it infinitely. This is not an approximation — a perfect crystal really is infinite repetition of a finite motif. The unit cell is the natural "quantum of crystal structure."

DFT and k-Point Sampling

In density functional theory (DFT), Bloch's theorem says that electronic wave functions in a periodic potential can be written as $\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_{n\mathbf{k}}(\mathbf{r})$, where $u_{n\mathbf{k}}$ has the periodicity of the lattice. The $\mathbf{k}$-points live in the first Brillouin zone — the Wigner-Seitz cell of the reciprocal lattice. A smaller primitive cell means a larger Brillouin zone, but you solve the Schrödinger equation with fewer atoms per k-point. This is the fundamental trade-off:

The primitive cell wins: although you need 4x more k-points, each is 64x cheaper, giving a net 16x advantage.

Machine Learning for Crystal Structure Prediction

Modern ML models for materials discovery must encode crystal structure as input features. The lattice parameters and atomic positions within the unit cell are the raw ingredients. Several approaches exist:

• Graph Neural Networks (GNNs): atoms are nodes, bonds are edges. The lattice vectors define periodic boundary conditions that determine which atoms are neighbors across cell boundaries.
• Symmetry-aware representations: models like CGCNN, MEGNet, and M3GNet use lattice vectors to compute interatomic distances correctly under periodic boundaries. Getting the lattice vectors wrong means wrong distances and wrong predictions.
• Generative models: diffusion models for crystal structure generation (like CDVAE) must learn to generate valid lattice vectors alongside atomic positions. The 6 lattice parameters are part of the model's output.

High-Throughput Screening

Databases like the Materials Project contain over 150,000 computed crystal structures. Each entry specifies the lattice vectors and atomic positions — exactly the POSCAR format we discuss below. Automated workflows use pymatgen to convert between primitive and conventional cells, compute volumes, and identify space groups. Understanding lattice vectors and unit cells is essential for navigating and using these databases.

Molecular Dynamics and Structure Factors

In molecular dynamics (MD) simulations, the simulation box is a unit cell with periodic boundary conditions. The structure factor $S(\mathbf{q})$ — the Fourier transform of the pair correlation function — directly reflects the lattice geometry. Sharp Bragg peaks appear at reciprocal lattice points $\mathbf{G} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$, where $\mathbf{b}_i$ are the reciprocal lattice vectors (Chapter 3). Computing $S(\mathbf{q})$ from an MD trajectory is a standard way to verify that the simulated crystal has the expected structure.

VASP Connection: The POSCAR File

The POSCAR file is where you translate crystallographic knowledge into VASP input. It encodes the lattice vectors, atomic species, and positions — everything we have discussed in this section. Let us walk through a complete example using the primitive FCC cell of CdSe zinc blende.

Si diamond - primitive cell
GaN wurtzite - conventional hexagonal

If line 2 is 1.0, the vectors on lines 3–5 must be in absolute angstroms.
If line 2 is 6.052, the vectors are in units of a.

Cartesian: (0.000, 3.026, 3.026) Å

Cartesian: (3.026, 0.000, 3.026) Å

Cartesian: (3.026, 3.026, 0.000) Å

Wrong order: Se Cd  (if POTCAR has Cd first)
Correct order: Cd Se (matching POTCAR)

Cartesian: (1.513, 1.513, 1.513) Å
Bond length: a√3/4 ≈ 2.621 Å

1CdSe zinc blende - primitive cell
26.052
3  0.000  0.500  0.500
4  0.500  0.000  0.500
5  0.500  0.500  0.000
6Cd Se
71  1
8Direct
9  0.000  0.000  0.000
10  0.250  0.250  0.250

Verifying the Cell Volume

Always verify the cell volume from your POSCAR. The primitive FCC cell with $a = 6.052$ $\text{\AA}$ should have:

$V = \frac{a^3}{4} = \frac{6.052^3}{4} \approx 55.4$ $\text{\AA}^3$

VASP prints the cell volume in the OUTCAR file (search for "volume of cell"). If your OUTCAR volume does not match this expectation, your lattice vectors are wrong.

Building Structures Programmatically

For production work, writing POSCAR files by hand is error-prone. Use pymatgen to build, verify, and convert structures programmatically:

lattice.matrix =
[[ 0.    3.026 3.026]
 [ 3.026 0.    3.026]
 [ 3.026 3.026 0.   ]]

1from pymatgen.core import Structure, Lattice
2
3# Build CdSe zinc blende from primitive FCC cell
4a = 6.052  # conventional lattice constant in angstroms
5
6lattice = Lattice(
7    [[0.0, a/2, a/2],   # a1 = (a/2)(0,1,1)
8     [a/2, 0.0, a/2],   # a2 = (a/2)(1,0,1)
9     [a/2, a/2, 0.0]]   # a3 = (a/2)(1,1,0)
10)
11
12cdse = Structure(
13    lattice,
14    ["Cd", "Se"],
15    [[0.0, 0.0, 0.0],       # Cd at origin
16     [0.25, 0.25, 0.25]]    # Se at (1/4,1/4,1/4)
17)
18
19# Key quantities for verification
20print(f"Volume: {cdse.volume:.2f} A^3")
21# Expected: 6.052^3 / 4 = 55.43 A^3
22
23print(f"Density: {cdse.density:.3f} g/cm^3")
24
25# Convert to conventional cell
26from pymatgen.symmetry.analyzer import SpacegroupAnalyzer
27sga = SpacegroupAnalyzer(cdse)
28conv = sga.get_conventional_standard_structure()
29print(f"Conventional cell atoms: {len(conv)}")
30# Expected: 8 atoms (4 Cd + 4 Se)
31
32print(f"Space group: {sga.get_space_group_symbol()}")
33# Expected: F-43m (space group 216)

Common POSCAR Mistakes

Summary

This section made the abstract lattice concrete through vectors, cells, and volumes:

• Lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ generate the entire lattice through integer linear combinations $\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3$. They are equivalently described by six lattice parameters $(a, b, c, \alpha, \beta, \gamma)$.
• The unit cell is the parallelepiped spanned by the lattice vectors. Its volume is $V = |\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)|$, computable as a determinant.
• A primitive cell contains exactly 1 lattice point and has the minimum volume. It is the most computationally efficient choice for DFT.
• A conventional cell is larger (Z = 2 for I, 4 for F) but displays full symmetry and matches database conventions.
• The Wigner-Seitz cell is the unique primitive cell with full point-group symmetry. Its reciprocal-space counterpart is the first Brillouin zone — the domain where electronic band structures and phonon dispersions live.
• In VASP, the POSCAR file encodes the lattice vectors and atomic positions. Choosing primitive vs conventional cells dramatically affects computational cost ($\sim Z^3$ scaling).
• In modern computing, lattice vectors and unit cells appear everywhere: DFT k-point sampling, ML crystal representations, MD simulation boxes, and high-throughput materials databases.

In the next section, we will enumerate all possible lattice types: the 7 crystal systems and the 14 Bravais lattices, establishing the complete classification of translational symmetry in three dimensions.