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
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
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
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 superscripts for the indices, since the components of are contravariant. 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
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
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
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 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
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
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.
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