Seven shapes a unit cell can take

Every crystal in nature is built from one of 7 unit-cell shapes: cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic. They differ by how the cell's three side lengths (a, b, c) and three angles (α, β, γ) relate to each other.

CrystalSim focuses on the four families that actually matter for next-gen semiconductors:

Key Concept

Cubic

All three sides equal (a = b = c), all angles 90°. The simplest, most symmetric system. Silicon (Si) and InGaOₓ live here.

Key Concept

Hexagonal

Two equal short sides (a = b), one longer side c. Two angles 90°, one 120°. Six-fold symmetry around the c-axis. Quartz, MoS₂, and LiNbO₃ live here.

Key Concept

Wurtzite

A specific hexagonal stacking with two interpenetrating hexagonal sublattices. Famous for being non-centrosymmetric — which is the secret to its built-in piezoelectricity. ZnO and GaN are wurtzite.

Key Concept

Perovskite

Cubic-like ABO₃ structure: a central A cation (e.g. Pb), corner B cations (e.g. Zr/Ti), and face-centered O atoms. PZT — the strongest piezoelectric in the set — has this shape.

Diagram · Unit cells in 3D

interactive

System:drag to rotate

Lattice parameters

a = b = c, α=β=γ=90°

CrystalSim materials

Silicon (Si), InGaOₓ

Drag to rotate. Switch systems with the dropdown — notice how the lattice parameters change.

Key Concept

Unit-Cell Parameters

The six numbers (a, b, c, α, β, γ) that define a unit cell. Crystallographers use them to classify any crystal into one of the 7 systems.

Checkpoint · +5 XP

Which system is GaN?

Why does the system matter? Because symmetry determines properties. A centrosymmetric lattice (like cubic silicon) cannot be piezoelectric — physics forbids it. A non-centrosymmetric lattice (like wurtzite or perovskite) can be. That single distinction is why crystals like PZT and GaN are at the heart of the Crystal-EM thesis, and silicon isn't.

Try It Yourself

Group the 8 crystals by system

Open Crystal Lab. Click each crystal and check its 'Crystal System' field. Make a mental map: which are cubic, which are hexagonal, which are wurtzite, which are perovskite?

Open Crystal Lab

Lesson Summary

There are 7 possible crystal systems, defined by unit-cell side lengths and angles.
CrystalSim uses 4 of them: cubic, hexagonal, wurtzite, and perovskite.
Cubic = silicon and InGaOₓ. Hexagonal = quartz, MoS₂, LiNbO₃. Wurtzite = ZnO, GaN. Perovskite = PZT.
Unit-cell parameters (a, b, c, α, β, γ) tell you which system a crystal belongs to.

Test Your Knowledge · +50 XP

How many crystal systems exist in total?

Silicon's crystal system is:

Which is the perovskite material in CrystalSim?

Which two materials share the wurtzite system?

What do unit-cell parameters describe?