# The Difference Between Crystal Systems and Crystal Families

This is another of those articles that clarifies subtle terminology differences, plus a bonus chart that discusses the relationship between Bravais lattices, point groups, and space groups categorization.

What’s the difference between the terms “crystal system” and “crystal family?”

Crystal systems are determined by the underlying symmetry of point groups (rotation, reflection, inversion), while crystal families expand one family (hexagonal) to incorporate the underlying translational symmetry of lattice systems. In 3-dimensions, there are 7 crystal systems and 6 crystal families.

All rhombohedral crystals (lattice system) belong to the trigonal crystal system, but trigonal crystals may belong to the rhombohedral or to the hexagonal lattice system. Crystal families are defined such that any rhombohedral or trigonal crystal is part of the hexagonal crystal family.

Technically, the hexagonal crystal family encompasses any point group which has at least one associated space group that has a hexagonal lattice.

For example, there are four space groups which are generated from point group 3: space group P3, P31, P32, and R3. Of these four space groups, R3 has a rhombohedral lattice, but P3, P31, and P32 have hexagonal lattices. Since at least one of point group 3’s space groups has a hexagonal lattice, point group 3 belongs to the hexagonal crystal family.

The 6 3-dimensional crystal families are:

1. Cubic
2. Hexagonal
3. Tetragonal
4. Orthorhombic
5. Monoclinic
6. Triclinic

This may seem very similar to the Bravais lattice systems

1. Cubic,
2. Hexagonal
3. Rhombohedral
4. Tetragonal
5. Orthorhombic
6. Monoclinic
7. Triclinic

or the point group crystal systems

1. Cubic,
2. Hexagonal
3. Trigonal
4. Tetragonal
5. Orthorhombic
6.  Monoclinic
7.  Triclinic

except both of these systems have an additional 7th category. (Note that the trigonal point group system and rhombohedral lattice system are not exactly related).

The crystal families are an attempt to combine the 7 lattice systems and 7 point group systems, by integrating the rhombohedral lattice system and trigonal point group system into the existing hexagonal crystal family.

This handy chart, inspired by the Practical Electron Microscopy and Database, illustrates this relationship between crystal family, crystal system, and lattice system.

In some sense, it is a coincidence that crystal systems and lattice systems line up so well, because they each have different symmetry definitions.

 Crystal System Essential Symmetry of Crystal System Crystal Family Essential Symmetry of Lattice System Lattice System Triclinic No axes of symmetry Triclinic a≠b≠c; α≠β≠γ Triclinic Monoclinic 2-fold axis Monoclinic a≠b≠c; α=γ=90°≠β Monoclinic Orthorhombic Three mutually perpendicular 2-fold axis Orthorhombic a≠b≠c; α=β=γ=90° Orthorhombic Tetragonal 4-fold axis Tetragonal a=b≠c; α=β=γ=90° Tetragonal Trigonal 3-fold axis Hexagonal a=b=c; α=β=γ<120° Rhombohedral a=b≠c; α=β=90°, γ=120° Hexagonal Hexagonal 6-fold axis Cubic Four 3-fold axis Cubic a=b=c; α=β=γ=90° Cubic

Final Thoughts

At the end of the day, crystal family, crystal system, and lattice system are mostly interchangeable. They have only a few categories different–the hexagonal crystal family splits into hexagonal + trigonal crystal systems, or hexagonal + rhombohedral lattice systems, which comes from the symmetry approach used to create these categories.

You can think of “crystal families” as a simplification/unification of crystal systems and lattice systems.