How to Read Crystallography Notation (Pearson symbol, Strukturbericht, Space Groups)

Crystallography is one of those sub-fields that developed independently in a dozen different places. Geologists, physicists, chemists, mathematicians, and metallurgists all developed their own set of rules for describing crystals. When the field of materials science emerged, there was some unification process for these rules, but standards for naming crystal structures still vary from journal to journal, and may be especially unconventional in older papers.

If you are reading an article which describes a crystal structure, that crystal may be noted by prototype, Strukturbericht, Pearson symbol, space group, Bravais lattice, crystal family, or common name.

In this article, I will explain how to read these crystallography notation systems, and also include a chart of the most common crystals. However, since there are hundreds of possible crystal structures, you may need to know how to read the notation on your own in case you encounter something rare.

Crystal Prototype Notation System

First, let’s discuss crystal prototypes, because in my opinion, prototypes are the most clear and have the least redundancy.

A crystal prototype is a real-world example of a material with a certain crystal structure. All known materials are indexed according to crystal structure, and one material is chosen to represent all others with the same structure.

For example, there are many metals that have the FCC crystal structure, such as aluminum, copper, nickel, γ-iron, silver, and gold. With the prototype crystal notation, I would just choose one example (such as Cu) to represent all FCC crystals. 

The AFLOW Encyclopedia of Crystallographic Prototypes has compiled this list of all crystal structures, and designated a prototype for each. The choice for FCC crystals is “Cu.”

Also note that many materials have polymorphism or allotropy, which means they can exist as multiple different crystal structures. For example, α-Po is the prototype for simple cubic, while β-Po is the prototype for one type of simple rhombohedral.

Common Crystals written with Prototypes Notation

Image	Common Name	Prototype Notation
	FCC	Cu
	BCC	W
	HCP	Mg
	L12	AuCu3
	Rock salt	NaCl

The advantage of the prototype system is that it includes every type of crystal and is easy to search on a database. The disadvantage of the prototype system is that if you see a prototype you are unfamiliar with, there is no way to “figure out” the crystal structure–you just have to look it up.

Additionally, the prototype system is fairly new, so most people don’t have common crystals memorized. For these reasons, the prototype system is usually used for rarer crystals, while the Strukturbericht system is used for more common crystal structures.

Strukturbericht Notation System

The Strukturbericht system was developed in the early 1900s and is probably the most common naming system in older scientific articles, although I see the prototype and Pearson naming systems (or a mix of all 3) in modern articles. In my own scientific articles, I typically use Strukturbericht to refer to intermetallic crystals made of more than 2 elements, such as L1₂ or B2.

Strukturbericht types are made of a letter and a number: the letter divides all crystals into some categories, and the letter acts as an index to arbitrarily distinguish crystals within these broad categories. Just like the prototype index, if you don’t have a particular Strukturbericht index memorized, you just have to look it up (although you can make some general guesses, as you’ll see later). However, the Strukturbericht is widely used, so crystallographers are likely to have common Strukturbericht indices memorized.

Meaning of Strukturbericht Letters

The first part of the Strukturbericht designation is a letter. These letters can be A-L, O, and S. Each letter means something about the crystal–for example, A means that the crystal is made of only a single type of element, B means that there are 2 elements, C means that there are 2 elements in a 2:1 ratio, and L means that the crystal is an intermetallic compound.

Strukturbericht Designation	Crystal Type
A	Elements
B	AB compounds
C	AB2 compounds
D	AmBn compounds
E – K	More complex compounds
L	Alloys
O	Organic compounds
S	Silicates

For an interesting history of the Strukturbericht system, and a more precise explanation of what the letters mean (for example, letters E-K depend on the “radicals”), check out this open access article.

Common Crystals written with Strukturbericht Notation

Image	Common Name	Strukturbericht Notation
	FCC	A1
	BCC	A2
	HCP	A3
	L12	L12
	Rock salt	B1

The advantage of the Strukturbericht system is mostly that it’s been the traditional crystallographic notation system for nearly 100 years, so the notation for common crystals are well-known. The first letter also tells you important information. The disadvantage of the Strukturbericht  system is that it doesn’t include every crystal (e.g. the ones discovered more recently) and notation hardly tells you anything about the crystal.

Pearson Notation System

The Pearson crystal notation system tells you the fundamental symmetry of the crystal: Bravais lattice + centering conditions. For this reason, I think it is the most useful designation in my subfield of materials science (metallurgy) where I usually work with simple crystal structures.

The Pearson system consists of 3 parts:

1. The crystal family (such as cubic, hexagonal, triclinic, etc) in lowercase.
2. The symmetry centering (such as body-centered or face-centered) in uppercase
3. The number of atoms per unit cell.

We use “c” for cubic, “h” for hexagonal, “t” for tetragonal, “o” for orthorhombic, “m” for monoclinic, and “a” for triclinic (think “a” for asymmetrical).

For centering operations, we use “F” for face-centered, “I” for body-centered, “S” for base-centered, and “P” for primitive (no centering). Additionally, there is some overlap between the rhombohedral and hexagonal crystal families–the Pearson system solves this by naming the primitive rhombohedral cell “hR” for rhombohedral centering in a hexagonal family.

As you might realize, the combination of crystal family and centering (1 & 2 above) defines the 14 Bravais lattices.

Crystal Family	Full Bravais Lattice	Lattice Symbol	Pearson Symbol
Cubic	Simple Cubic	P	cP
	Face-Centered Cubic	F	cF
	Body-Centered Cubic	I	cI
Hexagonal	Simple Hexagonal	P	hP
	Rhombohedral	R	hR
Tetragonal	Simple Tetragonal	P	tP
	Body-Centered Tetragonal	I	tI
Orthorhombic	Simple Orthorhombic	P	oP
	Base-Centered Orthorhombic	S	oS (or oC)
	Face-Centered Orthorhombic	F	oF
	Body-Centered Orthorhombic	I	oI
Monoclinic	Simple Monoclinic	P	mP
	Base-Centered Monoclinic	S	mS (or mC)
Triclinic	Simple Triclinic	P	aP

If you know what the crystal structure looks like, it is extremely easy to write the notation in the Pearson system. Just determine the Bravais lattice, and count how many atoms are in the unit cell.

For example, FCC is cubic, face-centered, and there are 4 atoms per unit cell. Thus: cF4. The diamond crystal structure is also face-centered and cubic, but with 8 atoms per unit cell: cF8.

One downside of the Pearson system is that there are many different crystals with the same name. For example, NaCl is also face-centered, cubic, and has 8 atoms per unit cell, so it is also written as “cF8.” There is no way to distinguish which atoms appear, nor where those atoms appear (as long as symmetry is followed).

Common Crystals written with Pearson Notation

Image	Common Name	Pearson Notation
	FCC	cF4
	BCC	cI2
	HCP	hP2
	L12	cP4
	Rock salt	cF8

The advantage of the Pearson system is that it’s easy to determine the system if you know the structure, and it communicates the essential symmetry. The disadvantage of the Pearson system is that there are many crystals that fall into the same name, so if you know the Pearson symbol but nothing else, you may not have enough information to know which crystal structure is being discussed. For this reason, the Pearson system is not ideal for describing complex crystals.

Using Bravais Lattices to Identify Crystals

The best “casual” way of identifying crystals is by Bravais lattice, since the 14 Bravais lattices convey the most important symmetry operation (translation).

I’ve written an in-depth article just about Bravias lattices, so I won’t go into too much detail here, except to remind you that Bravais lattices describe all the translation symmetry possible. Crystals are made of a “lattice” and “basis,” although many of the most common crystal structures have a 1-atoms basis, so the crystal structure is the same as the Bravais lattice.

For example, the FCC crystal structure is just the FCC Bravais lattice, with a single-atom basis. On the other hand, the diamond cubic crystal structure also has the FCC Bravais lattice, but with a two-atom basis.

Whenever you see people list the crystal structure of each element, they typically get this wrong, by mixing-and-matching Bravais lattices with crystal structures. For example, HCP has a hexagonal Bravais lattice, but with a two-atom basis. While “Face-centered cubic” could be a crystal structure or a Bravais lattice, “Hexagonal close-packed” is not a Bravais lattice.

It’s perfectly okay to say that your crystal structure is “hexagonal” (name of the Bravais lattice) without committing to a specific crystal structure, such as HCP or DHCP.

Using Bravais lattices to identify crystals conveys strictly less information than using Pearson notation, because Pearson notation includes the Bravais lattice + number of atoms per unit cell.

Using Space Groups to Identify Crystals

Just like there are 14 ways to arrange 3D stuff by translation (Bravais lattices), there are 32 ways to arrange 3D stuff by reflection, rotation, or rotoinversion (point groups). Point groups aren’t important for today’s purpose, but you can learn all about point groups in this article.

When you combine point groups with Bravais lattices, you get 230 3-dimensional space groups, which are mathematical representations of all the ways stuff can repeat.

Not every space group actually exists in real crystals, but all crystals can be defined by one of these 230 combinations of symmetry operations.

In many cases, crystal symmetry is an important theme, so crystals are referred to by their space group. This is typically done when you’re not quite sure what kind of crystal you have–it would be silly for me to call a known FCC crystal “a structure with space group.” (Although it’s important for me to know that FCC belongs to the space group if I want to calculate, for example, diffraction patterns).

Space groups can be referred to by a few different methods, but the two most common are by number (i.e. 1 to 230) or short name (such as ).

Naming space groups is complex and you’re better off using a list instead of trying to memorize or figure out the rationale behind different short names. If you do want to see the explanation of space groups’ Hermann–Mauguin notation, however, you can check out my article all about space groups.

Summary Table

Putting everything together, here is a summary table for each common crystal structure, showing the Bravais lattice, prototype, Strukturbericht designation, Pearson symbol, and space group. In order to put this table together, we used the AFLOW library which is an amazing resource for crystallographic prototypes!

Prototype	Strukturbericht	Pearson symbol	Bravais Lattice	Space Group
Cu	A1	cF4	Face-Centered Cubic (FCC)
W	A2	CI2	Body-Centered Cubic (BCC)
Mg	A3	hP2	Hexagonal
C	A4	cF8	Face-Centered Cubic (FCC)
Sn	A5	tI4	Body-Centered Tetragonal (BCT)
In	A6	tI2	Body-Centered Tetragonal (BCT)
As	A7	hR2	Rhombohedral
Se	A8	hP3	Hexagonal
C	A9	hP4	Hexagonal
Hg	A10	hR1	Rhombohedral
Ga	A11	oC8	Base-Centered Orthorhombic
α-Mn	A12	cI58	Body-Centered Cubic (BCC)
β-Mn	A13	cP20	Simple Cubic
β-W	A15	cP8	Simple Cubic
α-U	A20	oC4	Base-Centered Orthorhombic
NaCl	B1	cF8	Face-Centered Cubic (FCC)
CsCl	B2	cP20	Simple Cubic
ZnS	B3	cF8	Face-Centered Cubic (FCC)
ZnS	B4	hP4	Hexagonal
AsNi	B81	hP4	Hexagonal
InNi2	B82	hP6	Hexagonal
HgS	B9	hP6	Hexagonal
PBO	B10	tP4	Simple Tetragonal
γ-CuTi	B11	tP4	Simple Tetragonal
NiS	B13	hR6	Rhombohedral
GeS	B16	oP8	Simple Orthorhombic
PtS	B17	tP4	Simple Tetragonal
CuS	B18	hP12	Hexagonal
AuCd	B19	oP4	Simple Orthorhombic
FeSi	B20	cP8	Simple Cubic
BFe	B27	oP8	Simple Orthorhombic
MnP	B31	oP8	Simple Orthorhombic
NaTl	B32	cF16	Face-Centered Cubic (FCC)
PdS	B34	tP15	Simple Tetragonal
CoSn	B35	hP6	Hexagonal
SeTl	B37	tI16	Body-Centered Tetragonal (BCT)
CdSb	Be	oP16	Simple Orthorhombic
ξ-BCr	Bf (B33)	oC8	Base-Centered Orthorhombic
BMo	Bg	tI16	Body-Centered Tetragonal (BCT)
CW	Bh	hP2	Hexagonal
AsTi	Bi	hP8	Hexagonal
CuF2	C1	cF12	Face-Centered Cubic (FCC)
AgAsMg	C1b	cF12	Face-Centered Cubic (FCC)
FeS2	C2	cF12	Face-Centered Cubic (FCC)
Cu2O	C3	cP6	Simple Cubic
TiO2	C4	tP6	Simple Tetragonal
CdI2	C6	hP3	Hexagonal
MoS2	C7	hP6	Hexagonal
CaC2	C11a	tI6	Body-Centered Tetragonal (BCT)
MoSi2	C11b	tI6	Body-Centered Tetragonal (BCT)
CaSi2	C12	hR6	Rhombohedral
MgZn2	C14	hP12	Hexagonal
Cu2Mg	C15	cF24	Face-Centered Cubic (FCC)
AuBe3	C15b	cF24	Face-Centered Cubic (FCC)
Al2Cu	C16	tI12	Body-Centered Tetragonal (BCT)
FeS2	C18	oP6	Simple Orthorhombic
CdCl2	C19	hR3	Rhombohedral
Fe2P	C22	hP9	Hexagonal
Cl2Pb	C23	oP12	Simple Orthorhombic
AlB2	C32	hP3	Hexagonal
Bi2STe2	C33	hR5	Rhombohedral
AuTe2	C34	mC6	Base-Centered Monoclinic
MgNi2	C36	hP24	Hexagonal
Cu2Sb	C38	tP6	Simple Tetragonal
CrSi2	C40	hP9	Hexagonal
GeS2	C44	oF72	Face-Centered Orthorhombic
AuTe2	C46	mC6	Base-Centered Monoclinic
SiZr	C49	oC12	Base-Centered Orthorhombic
Si2Ti	C54	oF24	Face-Centered Orthorhombic
SiTh	Cc	tI12	Body-Centered Tetragonal (BCT)
CoGe2	Ce	oC23	Base-Centered Orthorhombic
As3Co	D02	cI32	Body-Centered Cubic (BCC)
BiF3	D03	cF16	Face-Centered Cubic (FCC)
ReO3	D09	cP4	Simple Cubic
Fe3C	D011	oP16	Simple Orthorhombic
AsNa3	D018	hP8	Hexagonal
Ni3Sn	D019	hP8	Hexagonal
Al3Ni	D020	oP16	Simple Orthorhombic
Cu3P	D021	hP24	Hexagonal
Al3Ti	D022	tI8	Body-Centered Tetragonal (BCT)
Al3Zr	D023	tI16	Body-Centered Tetragonal (BCT)
Ni3Ti	D024	hP16	Hexagonal
SiU3	D0c	tI16	Body-Centered Tetragonal (BCT)
Ni3P	D0e	tI32	Body-Centered Tetragonal (BCT)
Al4Ba	D13	tI10	Body-Centered Tetragonal (BCT)
MoNi4	D1a	tI10	Body-Centered Tetragonal (BCT)
Al4U	D1b	oI20	Body-Centered Orthorhombic
PtSn4	D1c	oC20	Base-Centered Orthorhombic
B4Th	D1e	tP20	Simple Tetragonal
BMn4	D1f	oF40	Face-Centered Orthorhombic
B6Ca	D21	cP7	Simple Cubic
NaZn13	D23	cF112	Face-Centered Cubic (FCC)
CaCu5	D2d	hP6	Hexagonal

Final Thoughts

Unfortunately, there is no “best” way to name crystal structures. Each naming system is an attempt to abbreviate by removing information, so different naming systems preserve different types of information. In many scientific papers, it’s common to include several names for a crystal structure, if the structure plays an important role.

If you don’t want to lose any information, nowadays you can make a digital file which contains each atom and their exact position–these files are called “CIF” files and they can be read by any crystal visualizing program.

References and Further Reading

AFLOW library is a great resource for crystals.

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
What are Space Groups
List of Space Groups
The 7 Crystal Systems

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