The term anisotropic displacement parameters (abbreviated
ADPs) should be used in referring to the individual atomic coefficients in
the exponent of the factor that describes the effects of
atomic motion and static displacement.
The elements of the tensors and
should
always be superscripted when the refinement is referred to a crystal system
rather than to a Cartesian system. This definition follows from the
definition of the elements of and
as contravariant tensor components (see Section 2.1,
). The frequent
use of subscripts for the ADPs, and specifically for those not referred to
Cartesian systems, is inconsistent with their tensorial properties.
With the common Gaussian approximation, use either
the quantities , which have dimension
(length)2 , defined in eq.
(2.1.25), or the
dimensionless , defined in eq. (2.1.22).
When the Gaussian approximation to the probability density function is
not deemed valid, the use of the Gram-Charlier expansion of eq. (3.1.62)
is recommended, although other formalisms may sometimes be advantageous for
special problems.
Standard uncertainties of ADPs obtained from a full-matrix
refinement are valid within the system in which the refinement is made.
If ADPs are transformed to any other axial system, Cartesian or not, then
the uncertainties may also be calculated by transforming the original
variance-covariance matrix to this new axial system and taking the square
roots of its diagonal elements, i.e., the variances. The required
variance-covariance matrix is usually not available for ADPs taken from
the literature. Hence, although ADPs can still be transformed, their
uncertainties cannot be. Calculations involving published ADPs and
their (published) uncertainties should therefore be referred to the same
system of coordinates as the original refinement in order to retain the
significance of the published uncertainties.
Avoid using the term ``temperature factor", both because the phenomenon
represented may not be due entirely to thermal motion and because that
phrase has in the past been used in several quite distinct senses (see
Section 1.5).
Avoid using the Gaussian
anisotropic parameters that are now usually
symbolized as and are defined in eq. (2.1.26).
These quantities are directly proportional to the
recommended , the ratio being 8 .
Avoid using ADPs that do not represent matrix elements. In some
early references and computer programs, cross-terms were sometimes
doubled in magnitude, being represented, for example, as instead of , for programming
convenience. This was possible because the matrix representing the ADP is
symmetric, with only six independent terms. This practice is not found
in modern crystallographic software.
Published values of
should always be accompanied by their standard
uncertainties. The ratio of the minimum to the maximum eigenvalues of the
corresponding anisotropic displacement tensors should also be published,
either in the primary publication itself or in the secondary (deposition)
publication.
Authors of crystallographic software and crystallographers who
maintain their own software are encouraged to introduce the minor
modifications that are required for the implementation of these
Recommendations.