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In 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 ...
, thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoid
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s used to indicate the magnitudes and directions of the thermal vibration of atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s in crystal structure
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
s. Since the vibrations are usually anisotropic
Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
(different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and therefore the symmetry and time averaged position of an atom in a crystal. Their theoretical framework was introduced by D. W. J. Cruickshank in 1956 and the concept was popularized through the program ORTEP (Oak Ridge Thermal-Ellipsoid Plot Program), first released in 1965.
Thermal ellipsoids can be defined by a tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, a mathematical object which allows the definition of magnitude and orientation of vibration with respect to three mutually perpendicular axes. The three principal axes of the thermal vibration of an atom are denoted , , and , and the corresponding thermal ellipsoid is based on these axes. The size of the ellipsoid is scaled so that it occupies the space in which there is a particular probability of finding the electron density of the atom. The particular probability is usually 50%.
See also
* Debye–Waller factorReferences
{{Crystallography Crystallography Ellipsoids