The structure factor of reflection h is given in a fairly general form by the Fourier transform of the average density of scattering matter
with the integration extending over the repeating structural motif, confined to a single unit cell. The brackets denote a double averaging over the possible displacements of the atoms from their mean positions - a time average over the atomic vibrations in each cell, followed by a space average that consists of projecting all the time-averaged cells onto one and dividing by the number of cells, h is a diffraction vector obeying the Laue equations, and is the static density of the motif, consistent with the instantaneous local configuration of the nuclei in a unit cell.
To reduce the above general picture to what is used in conventional crystal structure analysis, we first assume that the average density of matter in (1.4.1) can be regarded as a superposition of averaged atomic densities. This so-called isolated-atom approximation is essentially equivalent to assuming independently displaced atoms, a fair initial approximation, although not generally valid. The average density of scattering matter at the point r in a unit cell can then be approximated as
Here N is the number of atoms in the unit cell, is the occupancy factor of the kth atom, is the density (electron density for X-rays, or a delta function weighted with the scattering length for neutrons) due to atom k at a point r when the nucleus of atom k is at , and is the probability density function (pdf) corresponding to the probability for having atom k displaced by the vector ( - ) from its reference position in an average unit cell, which will be the mean position if is sufficiently symmetrical. It is important to remember that the approximations in (1.4.2) include the assumption that atoms are not deformable, by bonding or otherwise, even though at this stage the static atomic electron density, , has not been assumed to be spherically symmetrical.
If eq. (1.4.2) is now substituted into eq. (1.4.1), and the order of the summation and integration is interchanged, the structure factor becomes
If the substitutions and are made, the integral in (1.4.4) becomes
The inner integral in (1.4.5) has the form of a conventional convolution of the density of atom k with the pdf for a displacement of this atom from its mean position; the outer integral is a Fourier transform of this convolution. This transform is multiplied by an exponential that depends on the mean position, , of atom k.
By the convolution theorem, the Fourier transform of a convolution equals the product of Fourier transforms of the functions involved. When this theorem is applied to the outer integral in (1.4.5), we obtain the conventional approximation for the structure factor of a Bragg reflection
If we let and (as before) , then in (1.4.6)
is the scattering factor or form factor of atom k (for neutrons this is replaced by the scattering length ), and
is the Fourier transform of the pdf, , for the displacement of the kth atom from its reference position, r . This term contains the dependence of the structure factor on atomic displacements, and has been known by the names ``atomic Debye-Waller factor" and ``atomic temperature factor" (see section 1.5). There are no restrictions on the functional form of the pdf in the integrand of (1.4.8).
Let us now recall that the structure factor equation used in routine refinement of atomic parameters is further simplified in two ways:
First, for X-rays, the static atomic electron density is assumed to have spherical symmetry. This reduces the atomic scattering factor to the form
which has been computed and extensively tabulated for all the neutral elements and many ions (Maslen, Fox & O'Keefe, 1992). The spherical-atom approximation necessarily removes fine details of the (calculated) electron density, but may be used routinely, and serve as a starting point for more refined determinations of atomic positions and studies of charge density (e.g., Coppens & Becker, 1992; Coppens, 1993).
Second, the pdf for atomic displacement is most frequently approximated by a univariate or trivariate Gaussian, depending on whether the atomic displacements are assumed to be isotropic or anisotropic respectively. If a trivariate Gaussian is assumed, and the atomic subscript k is omitted, the resulting expression for from (1.4.8) is
Equation (1.4.10) can be derived from the theory of lattice dynamics in the harmonic approximation, which considers only the (always present) contribution of motion to the atomic displacement (e.g., Willis & Pryor, 1975). However, this equation may also be applied to static displacive disorder. The form of the atomic Debye-Waller factor, , represented in (1.4.10) is the most common one in standard structure refinements and will be discussed in Section 2. Various other approximations have been proposed for situations in which the Gaussian formalism is not adequate, e.g., when the anharmonic contribution to the crystal dynamics is significant; the most common are discussed in Section 3.
We present now a short discussion of common variants of eq. (1.4.10), which can be rewritten as
This shows that the exponent is proportional to minus the mean-square projection of the atomic displacement u on the direction of the diffraction vector h times the squared magnitude of h. If we denote the projection of u on the direction of h by , and make use of the relation: , (1.4.11) becomes
As long as the atomic displacements are anisotropic, the value of the average in (1.4.12) depends on the direction of h. This is then the anisotropic Gaussian Debye-Waller factor, , which is discussed in detail in Section 2. If, however, the atomic displacements are isotropic, the average in (1.4.12) is a constant determined by the structure alone, but possibly different for non-equivalent atoms, and the left-hand side of this equation no longer depends on the direction of h, but only on its magnitude. This is then the atomic isotropic Gaussian Debye-Waller factor,
The lowest-order approximation to is the overall isotropic Debye-Waller factor. It has the same form as (1.4.13), and presumes that all the atoms have the same isotropic mean-square displacement, . The whole crystal structure is assigned, in this approximation, a single displacement parameter. This approximation is used in initial stages of crystal structure determination by direct methods.
We conclude this section with some remarks on the structure factor for electron diffraction by a crystal. The density of scattering matter, , is here interpreted as the distribution of electrostatic potential within the unit cell. This potential is then approximated by a superposition of electrostatic potentials contributed by individual atoms, and the effects of motion are taken into account, as for X-rays and neutrons, by the convolution of the potential of an atom at rest with the probability density function describing the atomic motion (e.g., Vainshtein and Zvyagin, 1993). The atomic (spherical) scattering factor for electron diffraction, , for an atom at rest and diffraction vector , is related to that for X-rays by the Mott formula (e.g., Vainshtein, 1964) which has the form
with the atomic number and is the X-ray form factor of atom k [see (1.4.9)]. This formula, with the correct proportionality constants, has been used along with other techniques in extensive tabulations of spherical form factors for electron-diffraction (see, e.g., Cowley, 1992). The Debye-Waller factor, here expressing the `smearing out' of the electrostatic potential, is given by the same expression as that quoted above for X-rays and neutrons (e.g., Vainshtein, 1964; Vainshtein & Zvyagin, 1993). The structure factor for electron diffraction is therefore analogous to that appearing in (1.4.6) but is often given in a different notation.
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