Crystal Lattices and Basis

Introduction

A crystal can be described by a periodic lattice plus a basis. The lattice is a mathematical pattern of equivalent points; the basis is the atom or group of atoms placed at each point. Together they generate the atomic arrangement. Separating these ideas prevents the mistake that every lattice point is one atom.

Core explanation

Imagine points repeated by translations along three independent vectors a, b and c. Any lattice point can be reached from another by an integer combination n₁a+n₂b+n₃c. The pattern is unchanged by those translations. A unit cell is a volume spanned by chosen translation vectors; repeating it fills space without gaps or overlaps. Different valid cells can describe the same lattice, so a drawn cell is a representation rather than a unique physical box cut out of the crystal.

The basis specifies what sits at each lattice point. A simple elemental crystal may have one atom in a primitive basis. An ionic crystal requires more than one chemical species in its structural motif. For a schematic one-dimensional chain alternating A and B, a lattice may repeat every A-to-A distance while the basis contains both A and B at defined positions. Calling both A and B separate equivalent lattice points would ignore their chemical difference.

The complete crystal structure is often summarized as lattice + basis. This phrase distinguishes periodic translation from local chemistry. A lattice can be mathematically identical for two compounds while their bases differ, giving different diffraction intensities and properties. Conversely, similar motifs can be arranged on different lattices.

A primitive cell contains one lattice point after fractional contributions from its boundaries are counted. A conventional cell may be larger and chosen to display symmetry clearly, such as a face-centered cubic cell with four lattice points in its conventional cube. The physical crystal does not have visible walls marking unit-cell boundaries; shared corners and faces are accounting devices.

Translation symmetry is not the only symmetry in crystals. Rotations, mirrors and inversions can also occur, but periodic translation is the defining feature of a lattice. Finite crystals have surfaces and defects, so exact infinite repetition is an ideal model. It is still useful because most interior atoms occupy environments well approximated by the repeating pattern.

Amorphous glass lacks long-range translational periodicity, even if local SiO₄ tetrahedra recur. A collection of similar motifs is not enough to define a crystal lattice unless their positions repeat periodically. This connects the lattice concept to the previous page's crystalline-amorphous distinction.

When analyzing a crystal, the unit cell gives geometry, the basis gives chemical occupancy and the combination predicts coordination and stoichiometry. The following pages use cubic cells to practice counting atoms shared among neighboring conventional cells.

Step-by-step reasoning

1. Identify a set of equivalent repeating points. 2. Choose translation vectors that map points onto points. 3. Specify the atom or group attached to each point as the basis. 4. Repeat lattice plus basis to reconstruct the crystal. 5. Count boundary contributions when using a conventional unit cell.

Visual explanation

Draw a row of equally spaced empty dots labeled lattice. Below attach a two-atom A–B motif to every dot. The result is an alternating chain. In three dimensions, sketch a cube translated side by side with one colored motif repeated at equivalent positions.

Real-world analogy

A city grid gives repeating street intersections, while a standardized building placed at each intersection supplies the motif. The street grid is the lattice; the building is the basis. Changing buildings changes the city appearance without changing the grid.

Real-world example

Rock-salt sodium chloride is not a set of identical atoms placed on every point. Its periodic geometry includes a basis that distinguishes Na and Cl positions, yielding the observed 1:1 composition and each ion's neighbor environment.

Why?

Why is a lattice point not necessarily one atom? A lattice point marks a position of equivalent local environment under translation. Its attached basis can contain several atoms at different relative positions.

Common misconception

“One visible conventional cell must contain one lattice point.” A nonprimitive conventional cell can contain multiple lattice points after boundary sharing is counted, while still describing one periodic lattice.

Worked example

In a two-dimensional sketch, lattice points occur at positions 0, a, 2a and so on. Attach A at each lattice point and B halfway to the next point. Repetition gives A–B–A–B along the line, with a repeat length a. The basis has two atoms, although each primitive repeat interval contains one lattice point. Choosing the shorter A-to-B distance as a translation would incorrectly swap chemically different A and B sites.

Quick check

1. What two ingredients define a crystal structure in this description? Answer: A periodic lattice and a basis attached to its points.

Exam focus

Define lattice, basis and cell separately. Explain equivalent-point translation and why chemical occupancy is not automatically one atom per lattice point.

Advanced insight

Diffraction peak positions primarily reflect periodic lattice spacings, while intensities depend strongly on the basis and scattering strengths of its atoms. Both are needed to infer a full crystal structure.

Summary

A lattice is a periodic set of equivalent points; a basis supplies the actual atom arrangement at each. Translating a unit cell reproduces the ideal crystal. Conventional cell choice and basis occupancy must be kept distinct.

Practice questions

1. Can two crystals share a lattice but have different structures? Answer: Yes. Different bases attached to the same lattice can produce different atomic structures. 2. Does a glass with repeated local tetrahedra necessarily have a lattice? Answer: No. Long-range translational repetition is required. 3. Why might a conventional cell be larger than a primitive cell? Answer: It may display the crystal's symmetry more clearly while containing multiple lattice points.