The **Pearson symbol**, or **Pearson notation**, is used in crystallography as a means of describing a crystal structure, and was originated by W. B. Pearson.^{ [1] } The symbol is made up of two letters followed by a number. For example:

- Diamond structure,
*cF*8 - Rutile structure,
*tP*6

The two (italicised) letters specify the Bravais lattice. The lower-case letter specifies the crystal family, and the upper-case letter the centering type. The number at the end of the Pearson symbol gives the number of the atoms in the conventional unit cell.^{ [2] }

a | triclinic = anorthic |

m | monoclinic |

o | orthorhombic |

t | tetragonal |

h | hexagonal |

c | cubic |

P | Primitive | 1 |

S, A, B, C | One side/face centred | 2 |

I | Body-centred (from German : )innenzentriert^{ [3] } | 2 |

R | Rhombohedral centring (see below) | 3 |

F | All faces centred | 4 |

The letters A, B and C were formerly used instead of S. When the centred face cuts the X axis, the Bravais lattice is called A-centred. In analogy, when the centred face cuts the Y or Z axis, we have B- or C-centring respectively.^{ [3] }

The fourteen possible Bravais lattices are identified by the first two letters:

Crystal family | Lattice symbol | Pearson-symbol letters |
---|---|---|

Triclinic | P | aP |

Monoclinic | P | mP |

S | mS | |

Orthorhombic | P | oP |

S | oS | |

F | oF | |

I | oI | |

Tetragonal | P | tP |

I | tI | |

Hexagonal | P | hP |

R | hR | |

Cubic | P | cP |

F | cF | |

I | cI |

The Pearson symbol does not uniquely identify the space group of a crystal structure. For example, both the NaCl structure (space group Fm3m) and diamond (space group Fd3m) have the same Pearson symbol *cF*8.

Confusion also arises in the rhombohedral lattice, which is alternatively described in a centred hexagonal (*a* = *b*, *c*, *α* = *β* = 90°, *γ* = 120°) or primitive rhombohedral (*a* = *b* = *c*, *α* = *β* = *γ*) setting. The more commonly used hexagonal setting has 3 translationally equivalent points per unit cell. The Pearson symbol refers to the hexagonal setting in its letter code (*hR*), but the following figure gives the number of translationally equivalent points in the primitive rhombohedral setting. Examples: *hR*1 and *hR*2 are used to designate the Hg and Bi structures respectively.

Because there are many possible structures that can correspond to one Pearson symbol, a prototypical compound may be useful to specify.^{ [2] } Examples of how to write this would be *hP*12-MgZn or *cF*8-C. Prototypical compounds for particular structures can be found on the Inorganic Crystal Structure Database (ICSD) or on the AFLOW Library of Crystallographic Prototypes.^{ [4] }^{ [5] }^{ [6] }

The Pearson symbol should only be used to designate simple structures (elements, some binary compound) where the number of atoms per unit cell equals, ideally, the number of translationally equivalent points.

In crystallography, **crystal structure** is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

In geometry, biology, mineralogy and solid state physics, a **unit cell** is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.

In crystallography, a **crystal system** is a set of point groups. A **lattice system** is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a **crystal family**.

In mathematics, physics and chemistry, a **space group** is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called **Bieberbach groups**.

In geometry and group theory, a **lattice** in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension which spans the vector space . For any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell.

In mathematics and solid state physics, the first **Brillouin zone** is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

In geometry and crystallography, a **Bravais lattice**, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by

In crystallography, a **crystallographic point group** is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected.

In crystallography, the **cubic****crystal system** is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

**Miller indices** form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.

In geometry, **Hermann–Mauguin notation** is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called **international notation**, because it was adopted as standard by the *International Tables For Crystallography* since their first edition in 1935.

A **crystallographic database** is a database specifically designed to store information about the structure of molecules and crystals. Crystals are solids having, in all three dimensions of space, a regularly repeating arrangement of atoms, ions, or molecules. They are characterized by symmetry, morphology, and directionally dependent physical properties. A crystal structure describes the arrangement of atoms, ions, or molecules in a crystal.

For elements that are solid at standard temperature and pressure the first table gives the crystalline structure of the most thermodynamically stable form(s) in those conditions. Each element is shaded by a color representing its respective Bravais lattice, except that all orthorhombic lattices are grouped together.

Many compound materials exhibit polymorphism, that is they can exist in different structures called polymorphs. Silicon carbide (SiC) is unique in this regard as more than 250 **polymorphs of silicon carbide** had been identified by 2006, with some of them having a lattice constant as long as 301.5 nm, about one thousand times the usual SiC lattice spacings.

In crystallography, the **hexagonal crystal family** is one of the 7 crystal families, which includes two crystal systems and two lattice systems. While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent. In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice.

Few compounds of californium have been made and studied. The only californium ion that is stable in aqueous solutions is the californium(III) cation. The other two oxidation states are IV (strong oxidizing agents) and II (strong reducing agents). The element forms a water-soluble chloride, nitrate, perchlorate, and sulfate and is precipitated as a fluoride, oxalate or hydroxide. If problems of availability of the element could be overcome, then CfBr_{2} and CfI_{2} would likely be stable.

Berkelium forms a number of chemical compounds, where it normally exists in an oxidation state of +3 or +4, and behaves similarly to its lanthanide analogue, terbium. Like all actinides, berkelium easily dissolves in various aqueous inorganic acids, liberating gaseous hydrogen and converting into the trivalent oxidation state. This trivalent state is the most stable, especially in aqueous solutions, but tetravalent berkelium compounds are also known. The existence of divalent berkelium salts is uncertain and has only been reported in mixed lanthanum chloride-strontium chloride melts. Aqueous solutions of Bk^{3+} ions are green in most acids. The color of the Bk^{4+} ions is yellow in hydrochloric acid and orange-yellow in sulfuric acid. Berkelium does not react rapidly with oxygen at room temperature, possibly due to the formation of a protective oxide surface layer; however, it reacts with molten metals, hydrogen, halogens, chalcogens and pnictogens to form various binary compounds. Berkelium can also form several organometallic compounds.

In crystallography, a **Strukturbericht designation** or **Strukturbericht type** is a system of detailed crystal structure classification by analogy to another known structure. The designations were intended to be comprehensive but are mainly used as supplement to space group crystal structures designations, especially historically. Each Strukturbericht designation is described by a single space group, but the designation includes additional information about the positions of the individual atoms, rather than just the symmetry of the crystal structure. While Strukturbericht symbols exist for many of the earliest observed and most common crystal structures, the system is not comprehensive, and is no longer being updated. Modern databases such as Inorganic Crystal Structure Database index thousands of **structure types** directly by the prototype compound. These are essentially equivalent to the old Stukturbericht designations.

In solid state physics, the **magnetic space groups**, or **Shubnikov groups**, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone.

- ↑ W. B. Pearson, "A Handbook of Lattice Spacings and Structures of Metals and Alloys", Vol. 2, Pergamon Press, Oxford, 1967.
- 1 2 Nomenclature of Inorganic Chemistry IUPAC Recommendations 2005; IR-3.4.4, pp. 49–51; IR-11.5, pp. 241–242. IUPAC.
- 1 2 Page 124 in chapter 3. "Crystallography: Internal order and symmetry" in Cornelius Klein & Cornelius S. Hurlbut, Jr.: "Manual of Mineralogy", 21st edition, 1993, John Wiley & Sons, Inc., ISBN 0-471-59955-7.
- ↑ Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1".
*Computational Materials Science*.**136**: S1-S828. arXiv: 1806.07864 . doi:10.1016/j.commatsci.2017.01.017 . Retrieved 13 December 2022. - ↑ Hicks, David; Mehl, Michael J.; Gossett, Eric; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2019). "The AFLOW Library of Crystallographic Prototypes: Part 2".
*Computational Materials Science*.**161**: S1-S1011. arXiv: 1806.07864 . doi:10.1016/j.commatsci.2018.10.043 . Retrieved 13 December 2022. - ↑ Hicks, David; Mehl, Michael J.; Esters, Marco; Oses, Corey; Levy, Ohad; Hart, Gus L. W.; Toher, Cormac; Curtarolo, Stefano (2021). "The AFLOW Library of Crystallographic Prototypes: Part 3".
*Computational Materials Science*.**199**: 110450. arXiv: 2012.05961 . doi: 10.1016/j.commatsci.2021.110450 . Retrieved 13 December 2022.

- "Inorganic crystal structure database (ICSD)" . Retrieved 13 December 2022.
- "AFLOW Library of Crystallographic Prototypes" . Retrieved 13 December 2022.

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