Bravais Lattices and Space Groups

Introduction

A photograph of a crystal may reveal repeating atoms, but repeating pattern and chemical identity are different pieces of information. Crystallographers separate the abstract lattice of equivalent translation points from the basis of atoms placed at each point. This separation makes it possible to classify very different substances with the same lattice and to understand why diffraction patterns obey systematic symmetry rules. Bravais lattices describe translation geometry; space groups describe the full symmetry of a periodic structure.

Core explanation

Three non-coplanar vectors a, b and c generate a lattice point at every position R = n₁a + n₂b + n₃c, with integer coefficients n₁, n₂ and n₃. Every lattice point is equivalent under translation. Attaching an identical basis to every point produces the ideal crystal structure. A basis may be one atom, several ions, a molecule or many atoms. The conventional unit cell is chosen to show symmetry clearly and may contain more than one lattice point; a primitive cell contains exactly one lattice point after fractional sharing is accounted for.

Cell geometry is grouped into seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. Possible centring patterns compatible with the different symmetries produce 14 Bravais lattice types , not seven times every conceivable centring. Examples are cubic P (primitive), I (body-centred) and F (face-centred). In a conventional cubic P cell, eight corner points each contribute one eighth, for one lattice point. Cubic I adds one at the body centre, giving two. Cubic F adds six face points each shared by two cells, giving 1 + 6/2 = 4. These are lattice-point counts, not necessarily atom counts: the latter multiply by the number of atoms in the basis.

Point operations such as proper rotations, reflections and inversion can map the entire structure onto itself. When translations are also included, one obtains a space group . Screw axes combine a rotation with a fractional translation along the axis. Glide planes combine reflection with a fractional translation within the plane. There are 230 crystallographic space-group types for three-dimensional periodic structures. The number is not the number of different crystals; countless chemical structures occupy each symmetry type. The International Tables for Crystallography provide the formal lattice classification, and the IUCr symmetry introduction explains how translations enlarge point symmetry to space-group symmetry.

Do not infer a space group from chemical formula alone. Sodium chloride and diamond have different bases and bonding despite both being associated with face-centred cubic translation arrangements in standard descriptions. Diffraction intensities and systematic absences constrain symmetry, but pseudosymmetry, disorder, twinning and measurement limitations can make an assignment difficult. A space-group symbol includes information about lattice centring and symmetry elements; its full meaning depends on the chosen conventional setting. Structural refinement tests whether atomic positions and observed reflections are compatible with that symmetry.

An atom may also occupy a special position fixed by part of the space group, giving fewer symmetry-equivalent copies than a general position. This is why simply multiplying an asymmetric-unit atom count by 230 or by the cell's lattice-point count is wrong. Crystallographic multiplicity is controlled by the particular Wyckoff position. The classification is geometric, but it has chemical consequences: symmetry constrains possible dipoles, optical responses and how many independent environments appear in spectroscopy.

Step-by-step reasoning

1. Identify the translation vectors and the conventional cell geometry. 2. Count lattice points after sharing corners, faces and body positions. 3. Attach the basis to each lattice point to count atoms or formula units. 4. Identify rotations, mirrors, inversion, screw axes and glide planes that map the complete basis onto itself. 5. Use diffraction evidence and refinement rather than formula alone to assign a space group.

Visual explanation

Draw three cubic boxes. The first has dots only at corners and totals one lattice point. Add a dot at the centre of the second box for the body-centred total of two. Add dots at the face centres of the third for the face-centred total of four. Then replace each dot with the same two-colour miniature motif, showing that changing the basis changes the crystal without changing the underlying translation-point pattern.

Real-world analogy

The lattice is like a regular grid of street addresses; the basis is the building placed at each address. A symmetry operation is a way of shifting, rotating or reflecting the whole city so it looks unchanged. Knowing that addresses are on a square grid does not tell whether the buildings are houses, shops or pairs of towers.

Real-world example

The conventional face-centred cubic cell of elemental copper contains four lattice points and, with one Cu atom as its basis, four copper atoms per conventional cell. The same four-point count would not equal four atoms if the basis contained several atoms. This distinction prevents mistakes in density and composition calculations.

Why?

Why are there only 14 Bravais lattices rather than arbitrarily many centring patterns? Many apparent centring choices are equivalent after a different choice of cell vectors, while others are incompatible with the rotational symmetry required for the crystal system. Classification retains only geometrically distinct translation lattices.

Common misconception

"Face-centred cubic means four molecules in every unit cell." It means four lattice points in its conventional cell. The number of molecules or atoms depends on the basis attached to each point and sometimes on how the crystallographic asymmetric unit is chosen.

Worked example

Question: A conventional cubic cell has identical lattice points at eight corners and at the centre of each of its six faces. Each lattice point carries a two-atom basis AB. How many A and B atoms are in the conventional cell?

Reasoning: Corners contribute 8 × 1/8 = 1 point in total. Faces contribute 6 × 1/2 = 3 more points. The cell therefore contains four lattice points. At each point the basis has one A and one B, so multiplying the lattice-point count by the basis composition gives four A atoms and four B atoms. The eight atoms do not make the lattice itself eight-pointed.

Answer: Four A and four B atoms, corresponding to four AB formula units.

Quick check

1. How many lattice points belong to a body-centred cubic conventional cell? Answer: Two, one from all corners combined and one from the body centre. An inversion centre is not compulsory; many space groups are non-centrosymmetric.

Exam focus

Show fractional sharing explicitly in cell-count problems. Define lattice, basis, primitive cell, Bravais type and space group in separate sentences. Mention screw and glide operations when explaining why space groups contain more information than the 14 Bravais lattices.

Advanced insight

The same periodic point set can be described by more than one choice of primitive vectors, and nonprimitive conventional cells are often preferred because they expose higher symmetry. Diffraction indexes and Miller planes are normally quoted in the conventional setting. Database structures may list a transformed setting, so comparing lattice constants without first checking the cell transformation can mislead.

Summary

Translation points plus a basis describe an ideal crystal. The seven crystal systems and allowed centring choices give 14 Bravais lattice types. Including rotations, mirrors, inversion, screw axes and glide planes yields 230 three-dimensional space-group types. Conventional-cell lattice-point counts and atom counts are separate quantities.

Practice questions

1. What is the difference between a lattice and a crystal structure? Answer: A lattice is a set of equivalent translation points; a structure attaches an atomic basis to each point. 2. How many lattice points are in a primitive cell by definition? Answer: One in total, including fractional sharing. 3. Why does an F-centred cubic conventional cell have four lattice points? Answer: Corners contribute one and six half-shared face points contribute three. 4. Name one symmetry operation present in space-group descriptions but absent from an ordinary finite point group. Answer: A pure lattice translation, screw rotation or glide reflection.