Thermal ellipsoid
Encyclopedia
Thermal ellipsoids, more formally termed atomic displacement parameters, are ellipsoids used in crystallography
to indicate the magnitudes and directions of the thermal vibration of atom
s in crystal structure
s. Since the vibrations are usually anisotropic (different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and therefore the time-averaged position of an atom in a crystal.
Thermal ellipsoids can be defined by a tensor
, 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%.
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
to indicate the magnitudes and directions of the thermal vibration of atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s in crystal structure
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
s. Since the vibrations are usually anisotropic (different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and therefore the time-averaged position of an atom in a crystal.
Thermal ellipsoids can be defined by a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
, a mathematical object which allows the definition of magnitude and orientation of vibration with respect to three mutually perpendicular axes
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
. 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%.