crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

, a Wyckoff position is any point in a set of points whose site symmetry groups (see below) are all conjugate subgroups one of another. Crystallography tables give the Wyckoff positions for different

space groups 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 (its symmetry operations) are the rigid transformations of the pattern that ...

History

The Wyckoff positions are named after

Ralph Wyckoff Ralph Walter Graystone Wyckoff, Sr. (August 9, 1897 – November 3, 1994), or simply Ralph Wyckoff, was an American chemist and pioneer of X-ray crystallography. He also made contributions to vaccine developments against epidemic typhus and othe ...

, 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.

Definition

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 In mathematics, a unit vector i ...

, given by

fractional coordinates In crystallography, a fractional coordinate system (crystal coordinate system) is a coordinate system in which basis vectors used to describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis d ...

, one can apply a

symmetry operation In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a turn rotation of a regular triangle about its center (geometry), center, a refle ...

to the 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 group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...

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 conjugate site symmetry group, are assigned the same Wyckoff position. The Wyckoff positions are designated by a letter, often preceded by the number of positions that are equivalent to a given position with that letter, in other words the number of positions in the unit cell to which the given position is moved by applying all the elements of the space group. For instance, 2a designates the positions left where they are by a certain subgroup, and indicates that other symmetry elements move the point to a second position in the unit cell. The letters are assigned in alphabetical order with earlier letters indicating positions with fewer equivalent positions, or in other words with larger site symmetry groups. Some designations may apply to a finite number of points per unit cell (such as

inversion point In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...

improper rotation In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...

points, and intersections of rotation axes with mirror planes or other rotation axes), but other designations apply to infinite sets of points (such as generic points on rotation axes,

screw axes A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...

, mirror planes, and

glide plane In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Bec ...

s, as well as general points not lying on any symmetry axis or plane). 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, macros ...

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 that consists 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 usu ...

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.

External links

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

References

Crystallography Group theory {{CMP-stub