Pearson symbol

Construction

The two letters in the Pearson symbol specify the Bravais lattice, and more specifically, the lower-case letter specifies the crystal family, while the upper-case letter the lattice type. The number at the end of the Pearson symbol gives the number of the atoms in the conventional unit cell (atoms which satisfy 1 x,y,z \geq 0 for the atom's position (x,y,z) in the unit cell). The following two tables give the six letters possible for the crystal family and the five letters possible for the lattice type:

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.

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

Pearson symbol and space group

The Pearson symbol does not uniquely identify the space group of a crystal structure. For example, both the NaCl structure (space group Fmm) and diamond (space group Fdm) have the same Pearson symbol cF8. Due to this constraint, 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.

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: hR1 and hR2 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. Examples of how to write this would be hP12-MgZn_2 or cF8-C. Prototypical compounds for particular structures can be found on the Inorganic Crystal Structure Database (ICSD) or on the AFLOW Library of Crystallographic Prototypes.

References

References

1. W. B. Pearson, "A Handbook of Lattice Spacings and Structures of Metals and Alloys", Vol. 2, Pergamon Press, Oxford, 1967.
2. Villars, Pierre. (1997). "Pearson's handbook: crystallographic data for intermetallic phases". ASM.
3. "Pearson symbol".
4. [http://old.iupac.org/publications/books/rbook/Red_Book_2005.pdf Nomenclature of Inorganic Chemistry IUPAC Recommendations 2005]; IR-3.4.4, pp. 49–51; IR-11.5, pp. 241–242. [[IUPAC nomenclature of inorganic chemistry 2005. IUPAC]].
5. 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.
6. (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". Computational Materials Science.
7. (2019). "The AFLOW Library of Crystallographic Prototypes: Part 2". Computational Materials Science.
8. (2021). "The AFLOW Library of Crystallographic Prototypes: Part 3". Computational Materials Science.