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

ImageCommon NamePrototype Notation
FCCCu
BCCW
HCPMg
L12AuCu3
Rock saltNaCl

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
AElements
BAB compounds
CAB2 compounds
DAmBn compounds
E – KMore complex compounds
LAlloys
OOrganic compounds
SSilicates

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

ImageCommon NameStrukturbericht Notation
FCCA1
BCCA2
HCPA3
L12 L12
Rock saltB1

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 FamilyFull Bravais LatticeLattice SymbolPearson Symbol
CubicSimple CubicPcP
Face-Centered CubicFcF
Body-Centered CubicIcI
HexagonalSimple HexagonalPhP
RhombohedralRhR
TetragonalSimple TetragonalPtP
Body-Centered TetragonalItI
OrthorhombicSimple Orthorhombic PoP
Base-Centered Orthorhombic SoS (or oC)
Face-Centered Orthorhombic FoF
Body-Centered Orthorhombic IoI
MonoclinicSimple MonoclinicPmP
Base-Centered MonoclinicSmS (or mC)
TriclinicSimple TriclinicPaP

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

ImageCommon NamePearson 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 Fm\bar{3}m space group.” (Although it’s important for me to know that FCC belongs to the Fm\bar{3}m 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 Fm\bar{3}m ).

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!

PrototypeStrukturberichtPearson symbolBravais LatticeSpace Group
CuA1cF4Face-Centered Cubic (FCC)Fm\bar{3}m \ (225)
WA2CI2Body-Centered Cubic (BCC)Im\bar{3}m \ (229)
MgA3hP2HexagonalP6_{3}/mmc \ (194)
CA4cF8Face-Centered Cubic (FCC)Fd\bar{3}m \ (227)
SnA5tI4Body-Centered Tetragonal (BCT)I4_{1}/amd \ (141)
InA6tI2Body-Centered Tetragonal (BCT)I4/mmm \ (139)
AsA7hR2Rhombohedral R\bar{3}m \ (166)
SeA8hP3HexagonalP3_{1}21 \ (152)
CA9hP4HexagonalP6_{3}/mmc \ (194)
HgA10hR1RhombohedralR\bar{3}m \ (166)
GaA11oC8Base-Centered OrthorhombicCmca \ (64)
α-MnA12cI58Body-Centered Cubic (BCC)I\bar{4}3m \ (217)
β-MnA13cP20Simple CubicP4_{1}32 \ (213)
β-WA15cP8Simple CubicPm\bar{3}n \ (223)
α-UA20oC4Base-Centered Orthorhombic Cmcm \ (63)
NaClB1cF8Face-Centered Cubic (FCC)Fm\bar{3}m \ (225)
CsClB2cP20Simple CubicPm\bar{3}m \  (221)
ZnSB3cF8Face-Centered Cubic (FCC)F\bar{4}3m \ (216)
ZnSB4hP4HexagonalP6\bar{3}mc \ (186)
AsNiB81hP4HexagonalP6_{3}/mmc \ (194)
InNi2B82hP6HexagonalP6_{3}/mmc \ (194)
HgSB9hP6HexagonalP3\bar{2}21 \ (154)
PBOB10tP4Simple TetragonalP4/nmm \ (129)
γ-CuTiB11tP4Simple TetragonalP4/nmm \ (129)
NiSB13hR6Rhombohedral R3m \ (160)
GeSB16oP8Simple OrthorhombicPnma \ (62)
PtSB17tP4Simple TetragonalP4\bar{2}/mmc \ (131)
CuSB18hP12HexagonalP6_{3}/mmc \ (194)
AuCdB19oP4Simple OrthorhombicPmma \ (51)
FeSiB20cP8Simple CubicP2\bar{1}3 \ (198)
BFeB27oP8Simple OrthorhombicPnma \ (62)
MnPB31oP8Simple OrthorhombicPnma \ (62)
NaTlB32cF16Face-Centered Cubic (FCC)Fd\bar{3}m \ (227)
PdSB34tP15Simple TetragonalP4\bar{2}/m \ (84)
CoSnB35hP6HexagonalP6/mmm \ (191)
SeTlB37tI16Body-Centered Tetragonal (BCT)I4/mcm \ (140)
CdSbBeoP16Simple OrthorhombicPbca \ (61)
ξ-BCrBf (B33)oC8Base-Centered Orthorhombic Cmcm \ (63)
BMoBgtI16Body-Centered Tetragonal (BCT)I4\bar{1}/amd \ (141)
CWBhhP2HexagonalP\bar{6}m2 \ (187)
AsTiBihP8HexagonalP6_{3}/mmc \ (194) 
CuF2C1cF12Face-Centered Cubic (FCC)Fm\bar{3}m \ (225)
AgAsMgC1b cF12Face-Centered Cubic (FCC)F\bar{4}3m \ (216)
FeS2C2cF12Face-Centered Cubic (FCC)Pa\bar{3} \ (205)
Cu2OC3cP6Simple CubicPn\bar{3}m \ (224)
TiO2C4tP6Simple TetragonalP4\bar{2}/mnm \ (136)
CdI2C6hP3HexagonalP\bar{3}m1 \ (164)
MoS2C7hP6HexagonalP6_{3}/mmc \ (194)
CaC2C11atI6Body-Centered Tetragonal (BCT)I4/mmm \ (139)
MoSi2C11btI6Body-Centered Tetragonal (BCT)I4/mmm \ (139)
CaSi2C12hR6Rhombohedral R\bar{3}m \ (166)
MgZn2C14hP12HexagonalP6_{3}/mmc \ (194)
Cu2MgC15cF24Face-Centered Cubic (FCC)Fd\bar{3}m \ (227)
AuBe3C15bcF24Face-Centered Cubic (FCC)F\bar{4}3m \ (216)
Al2CuC16tI12Body-Centered Tetragonal (BCT)I4/mcm \ (140)
FeS2C18oP6Simple OrthorhombicPnnm \ (58)
CdCl2C19hR3Rhombohedral R\bar{3}m \ (166)
Fe2PC22hP9HexagonalP321 \ (150)
Cl2PbC23oP12Simple OrthorhombicPnma \ (62)
AlB2C32hP3HexagonalP6/mmm \ (191)
Bi2STe2C33hR5Rhombohedral R\bar{3}m \ (166)
AuTe2C34mC6Base-Centered MonoclinicC2/m \ (12)
MgNi2C36hP24HexagonalP6_{3}/mmc \ (194)
Cu2SbC38tP6Simple TetragonalP4/nmm \ (129)
CrSi2C40hP9HexagonalP6_{2}22 \ (180)
GeS2C44oF72Face-Centered OrthorhombicFdd2 \ (43)
AuTe2C46mC6Base-Centered MonoclinicPma2 \ (28)
SiZrC49oC12Base-Centered Orthorhombic Cmcm \ (63)
Si2TiC54oF24Face-Centered Orthorhombic Fddd \ (70)
SiThCctI12Body-Centered Tetragonal (BCT)I4\bar{1}/amd \ (141)
CoGe2CeoC23Base-Centered Orthorhombic Aba2 \ (41)
As3CoD02cI32Body-Centered Cubic (BCC)Pnma \ (62)
BiF3D03cF16Face-Centered Cubic (FCC)Fm\bar{3}m \ (225)
ReO3D09cP4Simple CubicPm\bar{3}m \ (221)
Fe3CD011oP16Simple OrthorhombicPnma \ (62)
AsNa3D018hP8HexagonalP6_{3}/mmc \ (194)
Ni3SnD019hP8HexagonalP6_{3}/mmc \ (194)
Al3NiD020oP16Simple OrthorhombicPnma \ (62)
Cu3PD021hP24HexagonalP\bar{3}c1 \ (165)
Al3TiD022tI8Body-Centered Tetragonal (BCT)P6_{3}/mmc \ (194)
Al3ZrD023tI16Body-Centered Tetragonal (BCT)I4/mmm  \ (139)
Ni3TiD024hP16HexagonalP6_{3}/mmc \ (194)
SiU3D0ctI16Body-Centered Tetragonal (BCT)I4/mcm \ (140)
Ni3PD0etI32Body-Centered Tetragonal (BCT)I\bar{4} \ (82)
Al4BaD13tI10Body-Centered Tetragonal (BCT)I4/mmm \ (139)
MoNi4D1atI10Body-Centered Tetragonal (BCT)I4/m \ (87)
Al4UD1boI20Body-Centered OrthorhombicImma \ (74)
PtSn4D1coC20Base-Centered OrthorhombicAba2 \ (41)
B4ThD1etP20Simple TetragonalP4/mbm \ (127)
BMn4D1foF40Face-Centered OrthorhombicFddd \ (70)
B6CaD21cP7Simple CubicPm\bar{3}m \ (221)
NaZn13D23cF112Face-Centered Cubic (FCC)Fm\bar{3}c \ (226)
CaCu5D2dhP6HexagonalP6/mmm \ (191)

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

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

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