So, you want to know what a crystal system is. If that’s the case, you might not be 100% clear on the difference between crystal system, crystal structure, crystal family, crystal lattices, or crystal bases. Do crystal systems include rhombohedral or trigonal crystals?

I don’t blame you for the confusion–most of the other articles I see on the internet mix these up as well! Even the textbook I’m referencing appears to be wrong (until I saw the tiny footnotes).

**Crystal systems are all the ways that rotational axes of symmetry can be combined and connected to a lattice. There are 7 crystal systems in 3D, which directly connect to 32 point groups when adding mirror planes and inversion.**

**The 7 crystal systems are:** **Cubic**, **Hexagonal**, **Tetragonal**, **Trigonal**, **Orthorhombic**, **Monoclinic**, **Triclinic**.

Crystal System | Visualization |

Cubic | |

Hexagonal | |

Tetragonal | |

Trigonal | |

Orthorhombic | |

Monoclinic | |

Triclinic |

The crystal systems are listed in order of decreasing symmetry. Note that “rhombohedral” is NOT a crystal system (but it is a crystal lattice).

This might create more questions for you–don’t worry, **this article will explain everything step-by-step.** I will walk through these topics a bit slowly, but be careful about skipping ahead! I see a lot of mistakes, even from experts like graduate students or other ”authorities” writing on the internet.

#### Outline

### Crystal Structures

Crystal structures are the most intuitive–and most complicated–category of crystallography.

**A crystal structure includes the position of every atom and the translational axes which allow the structure to repeat. There are an infinite number of crystal structures.**

Since crystal structures extend infinitely (that’s the definition of a crystal actually: a structure that can be repeated infinitely), we need to define some small section that can be repeated. We call this small section a “unit cell.”

Unit cells are made of a **lattice** and **basis**. Lattices tell you how the crystal is repeated. The basis tells you what is repeated.

For example, in the figure above, any of these purple boxes would be a valid unit cell. Since they are square/rectangular (with 90º corners), you can see that translating the box up, down, left, or right (90º angles) will result in the exact same arrangement of atoms.

The purple box is the lattice, and the yellow circles are the basis. You could describe this crystal structure as “yellow circle** translated the distance “ a” at 90º angles**.” The underlined portion refers to the basis, and the

**bold**portion refers to the lattice.

**You could easily get an infinite number of crystal structures because of an infinite number of bases.** For example, instead of 1 yellow circle, you could have 2. Or you could have 1 yellow circle and 1 blue circle. Or 1 yellow circle and 2 blue circles, etc.

Although there are only a finite number of lattice shapes, you can also have infinite permutations of lattice size (represented by “a” in the figure above),

When you’re talking about the crystal structure of real, 3D crystals you need to include the lattice and the basis. For example the Cu crystal structure is a cube with side length 0.3615 nm and 8 Cu atoms. There are 4 Cu atoms at positions (0,0,0), (½, ½, 0), (½, 0, ½,) and (0, ½, ½). These coordinates are in terms of the lattice parameter *a*, so the atom at (½, ½, 0) is ½*a* = 0.18075 nm to front and right of the atom at (0,0,0).

Another example, the NaCl crystal structure is a cube with side length 0.5639 nm and 8 atoms. There are 4 Cl atoms at positions (0,0,0), (½, ½, 0), (½, 0, ½,) and (0, ½, ½). There are 4 Na atoms at positions (½, 0, 0), (0, ½, 0), (0, 0, ½), and (½, ½, ½). Note that the ½ refers to ½ of the cube side length.

In practical terms, this information can be written in a “CIF” file, which stands for Crystallographic Information Framework. If you want to peruse thousands of known crystal structures, check out the collection of CIF files from the AFLOW library or International Union of Crystallography.

Even the relatively-simple NaCl crystal structure was pretty tedious to write this way, so crystallographers have different ways of omitting/simplifying information to convey the most important parts. For example, you could say that NaCl belongs to the cubic crystal system (I didn’t forget where we’re going), which tells you it has the important symmetries of a cube.

If you want to be even more specific, you could say the NaCl crystal structure has a “face-centered cubic” lattice–if you knew this, then you only need to say the position of 1 Na and 1 Cl atom, because the rest can be generated by the symmetry of the face-centered cubic lattice.

So let’s go into more depth about crystal lattices.

### Crystal Lattices and Bravais Lattices

Lattices systems are all the ways that translational symmetry can be combined, with “centering” operations removed.** There are 7 crystal lattices in 3D, which directly connect to the 14 Bravais lattices when adding face-, body-, and base-centering.**

You can read my full article about Bravais lattices (with lots of pictures) if you don’t already know what they are, so I’ll only repeat the parts that are important for distinguishing crystal systems.

The lattice is a set of vectors that tells you which direction the unit cell can be translated.

These vectors fall into certain finite patterns. For example, the 2D lattice above has vectors V_{1} and V_{2}, which are at 90º to each other but are not the same length. This is also the definition of a rectangle, so this is a rectangular lattice.

For the rest of the article, I’ll try to stick to 3 dimensions.

**In 3 dimensions, there are 14 Bravais lattices:**

- Simple cubic
- Face-centered cubic
- Body-centered cubic
- Hexagonal
- Rhombohedral
- Simple tetragonal
- Body-centered tetragonal
- Simple orthorhombic
- Base-centered orthorhombic
- Face-centered orthorhombic
- Body-centered orthorhombic
- Simple monoclinic
- Base-centered monoclinic
- Triclinic

You can see that these fall into 7 basic **lattice systems**, with centering operations, which include

- no centering (or “simple”)
- face-centering
- base-centering
- body-centering
- base-centering

In principle, you can add any kind of centering to any of the 7 lattice systems–so why do you only get 14 Bravais lattices, instead of 7 x 5 = 35? That is because some combinations of lattice system + centering result in the same Bravais lattice.

For example, base-centered cubic and simple tetragonal are identical Bravais lattices.

If you remove centering operations, you end up with **7 basic lattice systems** that are defined by their **translational symmetry**.

Basic LatticeSystems | Edge | Angle | Visualization |

Cubic | a=b=c | α=β=γ=90° | |

Hexagonal | a=b≠c | α=β=90°, γ=120° | |

Rhombohedral | a=b=c | α=β=γ<120° | |

Tetragonal | a=b≠c | α=β=γ=90° | |

Orthorhombic | a≠b≠c | α=β=γ=90° | |

Monoclinic | a≠b≠c | α=γ=90°<β | |

Triclinic | a≠b≠c | α≠β≠γ |

Notice that this is *almost* the same as the list of crystal systems I showed you at the beginning of the article–except here we have a rhombohedral lattice system, vs the trigonal crystal system. Nope, those don’t have a 1-1 correlation, so it’s finally time to dig into crystal systems!

### Crystal Systems and Point Groups

If you’re comfortable with the vector definition of lattice systems, you are probably poised to make a mistake that I see all the time.

**The 7 crystal systems are NOT defined by translational vectors.** This is where so much confusion comes from. While it is true that the cubic crystal system generates conventional unit cells with 3 perpendicular, equi-length axes, that is a *coincidence* (or at least, a correct fact which is derived from the true definition of a cubic crystal system).

Crystal systems are a classification system for point groups, and are defined by **rotational axes**, instead of translational axes.

Crystal System | Symmetry | Conventional cell |

Triclinic | No axes of symmetry | a≠b≠c; α≠β≠γ |

Monoclinic | 2-fold axis | a≠b≠c; α=γ=90°<β |

Orthorhombic | Three mutually perpendicular 2-fold axis | a≠b≠c; α=β=γ=90° |

Trigonal | 3-fold axis | a=b=c; α=β=γ<120°^{1}a=b≠c; α=β=90°, γ=120° ^{2} |

Tetragonal | 4-fold axis | a=b≠c; α=β=γ=90° |

Hexagonal | 6-fold axis | a=b≠c; α=β=90°, γ=120° |

Cubic | Four 3-fold axis | a=b=c; α=β=γ=90° |

2) This is also the conventional cell of the hexagonal system.

That table was adapted from Kelly and Knowles’s *Crystallography and Crystal Defects*.

Note that ≠ means “not necessarily equal to.” For each crystal system, there is a “conventional” cell which can be chosen. These conventional cells depend on the translational symmetry discussed regarding crystal lattices.

You see there’s also some confusion about the trigonal crystal system, because it can be mapped onto 2 different cells: rhombohedral and hexagonal.

Crystal systems are used to categorize point groups. Just like Bravais lattices are all the ways you can combine translational symmetry with centering, point groups are all the ways you can combine rotational axes, reflection, and inferversion.

If you want a full explanation of point groups, check out this article. Otherwise, let’s move on to our final category, crystal families.

### Crystal Families and Space Groups

Yes, for most of the materials scientists searching for “crystal system” on the internet, I think you *should *be searching for crystal family, or possibly lattice system. (Chemists, you are fine to search for crystal system, because you often deal with molecules that lack translational symmetry.)

Crystal families are a unification between lattice systems and crystal systems. It combines the rhombohedral lattice system and trigonal point group system into the existing hexagonal crystal family. Thus, you only end up with 6 crystal families.

- Cubic
- Hexagonal
- Tetragonal
- Orthorhombic
- Monoclinic
- Triclinic

Technically, the hexagonal crystal family is a classification for space groups, just like crystal systems are a classification for point groups, and lattice systems are a classification for Bravais lattices. (This should make sense because space groups are a combination of point groups + Bravais lattices.)

If you’re not quite sure what point groups are, you can check out my full explanation of point groups. Or if you already know what they are but you want to follow along with a reference list of all 230 space groups, click here.

The crystal families encompass 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, P3_{1}, P3_{2}, and R3. Of these four space groups, R3 has a rhombohedral lattice, but P3, P3_{1}, and P3_{2} 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.

Point Group | Associated Space Groups(H-M Notation) | Associated Space Groups (Number) |

If you’ve gotten through this article, you should know all about the differences between crystal family and crystal systems, most of which is repeated here. However, I did write that condensed article with a couple neat comparison tables/charts, so I’d be remiss not to let you know about it!

### Final Thoughts

Well now you know everything about crystal systems, and associated lattice systems and lattice families. For one final overview:

- The
**7****crystal systems**are a classification for 32 point groups, based on rotational symmetry. - The
**7****crystal lattices**are a classification for 14 Bravais lattices, based on translational symmetry. - The
**6 crystal families**are a classification for 230 point groups, based on rotational and translational symmetry (since space groups are a combination of Bravais lattices and point groups). - There are infinitely many
**crystal structures**, which can be classified according to the crystal system, crystal lattice, and crystal family.

Classification | Number | Describes | Attributes |

Crystal Structure | Infinite | number and position of atoms | Conveys most information; no shortcuts |

Crystal Lattice | 7 | Bravais Lattices (14) | Translational Symmetry |

Crystal System | 7 | Point Groups (32) | Rotational Symmetry |

Crystal Family | 6 | Space Groups (230) | Avoids trigonal / rhombohedral confusion |

### References and Further Reading

For a rigorous, mathematical approach to crystallography (and the textbook against which I checked this article), check out Kelly and Knowles’s *Crystallography and Crystal Defects.*

If you want a library of crystal structures, check out the AFLOW library.

If you want to see the other crystallography-related articles I’ve written, here is this list, in recommended reading order:

Introduction to Bravais Lattices

What is the Difference Between “Crystal Structure” and “Bravais Lattice”

Atomic Packing Factor

How to Read Miller Indices

How to Read Hexagonal Miller-Bravais Indices

Close-Packed Crystals and Stacking Order

Interstitial Sites

Primitive Cells

How to Read Crystallography Notation

What are Point Groups

List of Point Groups

If you are interested in more details about any specific crystal structure, I have written individual articles about simple crystal structures which correspond to each of the 14 Bravais lattices:

1. Simple Cubic

2. Face-Centered Cubic

2a. Diamond Cubic

3. Body-Centered Cubic

4. Simple Hexagonal

4a. Hexagonal Close-Packed

4b. Double Hexagonal Close-Packed (La-type)

5. Rhombohedral

5a. Rhombohedral Close-Packed (Sm-type)

6. Simple Tetragonal

7. Body-Centered Tetragonal

7a. Diamond Tetragonal (White Tin)

8. Simple Orthorhombic

9. Base-Centered Orthorhombic

10. Face-Centered Orthorhombic

11. Body-Centered Orthorhombic

12. Simple Monoclinic

13. Base-Centered Monoclinic

14. Triclinic