As discussed in Section 1.4 above,
diffraction studies yield information not only about mean atomic
positions but also about the probability density functions (pdf's) of atomic
displacements from these mean positions. If the atomic pdf is assumed to
be a trivariate Gaussian, the characteristic function
corresponding to this pdf - by definition, its Fourier transform - can
be described by the second moments of the pdf, which in the present
context are called anisotropic mean-square displacements. If
not, higher cumulants of a non-Gaussian pdf can, in principle, also be
determined; these are simple functions of moments (*e.g.*,
Kendall & Stuart, 1977), but there are difficulties. For
example, these higher terms are only likely to be important when the
second moments of the pdf's are relatively large. However, as can be seen
from the basic expression for the isotropic Debye-Waller factor,
,
the larger the second moment, the
more rapidly the scattering from the atomic center in question falls off with
increase in the scattering angle. Thus, just when the higher terms
become important, they become difficult to measure, for lack of
contribution by the scattering center to the Bragg intensities.

The mean-square displacements, which define the pdf in the various
Gaussian approximations, used to be known as *atomic vibration
parameters* or *thermal parameters* but have recently been designated as
*atomic displacement parameters*, isotropic or anisotropic, to allow
for the effects of static displacive disorder, as well as for those of the
always-present atomic motion.
There exists an extensive literature on the interpretation of these parameters
(*e.g.*, Dunitz, Schomaker & Trueblood, 1988, and references cited
therein).

The purpose of this section is to relate alternative forms of anisotropic
displacement parameters (ADPs) to
the expression for the Debye-Waller factor that is valid within the
framework of the assumptions underlying the harmonic approximation (*e.g.*,
Willis & Pryor, 1975). We also discuss anisotropic displacement
parameters in relation to different coordinate systems, outline the
transformation properties of the resulting quantities, present several forms of
equivalent isotropic displacement parameters, and describe briefly
graphical representations of the Gaussian
mean-square displacement matrix.

The usual expression for T(**h**) is [eq. (1.4.10) restated]:

These fundamental equations take on different forms according to the basis
vectors to which we refer the diffraction and displacement vectors. In
carrying out coordinate transformations in the formalism of tensor algebra,
quantities that transform like direct basis vectors are
called *covariant* and are indicated by subscripts, while quantities
transforming like reciprocal basis vectors are called
*contravariant* and are indicated by superscripts. The direct and
reciprocal bases are not *necessarily* those of the corresponding
lattices; they may be any pair of dual bases. Let us first assume that the
diffraction vector is referred to the basis of the reciprocal lattice and the
atomic displacement vector to the basis of the direct lattice, as follows

and

Note that the components of **h** and **u** are dimensionless. The first
scalar product appearing on the right-hand side of eq. (2.1.15) can
now be evaluated as

and similarly

We used here the definition of the dual (direct and reciprocal) bases:

If we insert eqs. (2.1.18) and (2.1.19) into eq. (2.1.15) we obtain for

with

The quantity defined by eqs. (2.1.21) and
(2.1.22) is one of
the frequently employed forms of the anisotropic displacement parameter;
note the use of *super*scripts for the indices, since the
components of are *contra*variant.
For an atom, each component is
2 times an average of a product of two
components of an atomic displacement vector, when the latter is referred to
the basis of the direct lattice.

We shall now retain **h** as defined by eq. (2.1.16), but redefine
**u** as follows

The components of **u** in this representation, , have
dimension
length, and the basis vectors
are dimensionless (see, *e.g.*, Hirshfeld & Rabinovich, 1966).
Only in orthorhombic, tetragonal and cubic crystal systems must these basis
vectors be mutually orthogonal unit vectors, *i.e.*, orthonormal,
since it is only in these systems that the equalities ,
,
and are necessarily true. The departures of these basis
vectors from orthonormality in other systems are associated with the
departures of the angles , , and from 90 . If we
now repeat the evaluation of the scalar products in eq. (2.1.15) with
**h** given by eq. (2.1.16) and **u** given by eq. (2.1.23),
we obtain for *T*

with

another well known form of the ADP. This
form is often preferred because the elements of the tensor
**U** have
dimension (length)^{2} and can be directly associated with the mean-square displacements of the atom considered in the corresponding directions. Note in
particular that the mean-square displacement in an arbitrary direction denoted
by the unit vector **n**, when **n** is referred to
unit vectors parallel to the *reciprocal*
basis vectors so that its components are covariant, is given by
(see section 2.3.2).
In any event, the dimensionless elements of
are also correctly associated with the general expression
for *T*(**h**), given by eq. (2.1.15).

Another form of the anisotropic displacement parameter, which is used in some conventional refinement calculations, especially in biomolecular crystallography, is

and the corresponding expression for *T* becomes

Since **B** and **U** are equivalent, apart from a
constant factor, and **U** has a more direct physical significance than
**B**, we recommend that the use of **B** be discouraged.

A brief discussion of the transformation properties of
and **U** may be helpful. The
corresponding representations of the atomic displacement vector are

and

respectively (Hirshfeld & Rabinovich, 1966). If the basis of the direct
lattice is changed in some manner, the new components of the displacement
vector **u** are related to the old ones by linear transformations, say

and

The elements of the transformation matrices depend on the old and new bases.
It follows from eqs. (2.1.22) and (2.1.25) that and **U** transform as products of the corresponding components of
the displacement vector. Hence, the transformation rules for and **U** become

and

and thus conform to those valid for tensors of the second order (*e.g.*,
Spain, 1956). The transformation matrices **R** and **Q** are obviously
different, since the basis vectors to which **u**, in its two representations,
is referred depend in a different manner on the basis of the direct lattice .
This transformation property will be illustrated in detail in
Subsection 2.1.2
by the orthogonalization of and **U**.

We comment finally on the form of the Debye-Waller factor when both the diffraction vector and the atomic displacement vector are referred to the same Cartesian basis, say , , . It is understood that the use of this representation is usually, in crystallographic practice, preceded by appropriate transformations (see below for a detailed example).

The **h** and **u** vectors, in the Cartesian representation, are given by

and

All the indices are given here as subscripts, since in the Cartesian
representation the position of the indices is irrelevant. Note that the
components of **h** in eq. (2.1.34) have dimension
(length)^{-1} .
The scalar products in eq. (2.1.15) are now readily evaluated and
we obtain for the Debye-Waller factor

with

an element of an atomic mean-square displacement tensor, with dimension
(length)^{2} , referred to a Cartesian basis. This representation avoids
the hazards associated with calculations in oblique coordinate systems and is
used almost always in lattice-dynamical studies and thermal motion
analysis, and very often in constrained refinement of atomic parameters.

To be precise, the symbols

- 2.1.1 Relationships between the anisotropic displacement parameters
- 2.1.2 Construction of Cartesian mean-square displacement tensors