crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...

, a Wyckoff position is a point belonging to a set of points for which site

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s are conjugate

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s of the

space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...

. Crystallography tables give the Wyckoff positions for different space groups. For any point in a

unit cell 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 (unlike a unit vector, for example) does not necessaril ...

, given by fractional coordinates, you can apply a

symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...

to this point. In some cases it will move to new coordinates, while in other cases the point will remain unaffected. For example, reflecting across a mirror plane will switch all the points left and right of the mirror plane, but points exactly on the mirror plane itself will not move. We can test every symmetry operation in the crystal's

point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...

and keep track of whether the specified point is invariant under the operation or not. The finite list of all symmetry operations which leave the given point invariant taken together make up another group, which is known as the ''site symmetry group'' of that point. By definition, all points with the same site symmetry group (or a site symmetry group in the same

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...

) are assigned the same Wyckoff position. Wyckoff positions are used in calculations of

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...

properties. There are two types of positions: general and special. *General positions are left invariant only for the

identity operation Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

(''E''). Each space group has only one general position. *Special positions are left invariant by the identity operation and at least one other operation of the space group. General positions have a site symmetry of the

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...

and all correspond to the same Wyckoff position. Special positions have a non-trivial site symmetry group. For a particular space group, one can check the Wyckoff positions using International Tables of Crystallography. The table presents the multiplicity, Wyckoff letter and site symmetry for Wyckoff positions. The Wyckoff positions are named after

Ralph Walter Graystone Wyckoff Ralph Walter Graystone Wyckoff, Sr. (August 9, 1897 in Geneva, New York – November 3, 1994 in Tucson, Arizona) was an American scientist and pioneer of X-ray crystallography. He was elected member of the National Academy of Sciences in 1949 and ...

, an American X-ray crystallographer who authored several books in the field. His 1922 book, The Analytical Expression of the Results of the Theory of Space Groups, contained tables with the positional coordinates, both general and special, permitted by the symmetry elements. This book was the forerunner of International Tables for X-ray Crystallography, which first appeared in 1935.

External links

Wyckoff Positions at Bilbao Crystallographic ServerMaterials Project DatabaseAmerican Mineralogical Society Crystal Structure Database

References

Crystallography Group theory {{CMP-stub