International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 1024
https://doi.org/10.1107/97809553602060000550 Chapter 1.2. The structure factor ^{a}Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 142603000, USA This chapter summarizes the mathematical development of the structurefactor formalism. It starts with the definition of the structure factor appropriate for Xray and neutron scattering, and includes the derivation of the appropriate expressions. Beyond the isolatedatom case, the atomcentred spherical harmonic (multipole) model is treated in detail. Expressions for thermal motion in the harmonic approximation and for treatments including anharmonicity are given and their relative merits are discussed. 
The structure factor is the central concept in structure analysis by diffraction methods. Its modulus is called the structure amplitude. The structure amplitude is a function of the indices of the set of scattering planes h, k and l, and is defined as the amplitude of scattering by the contents of the crystallographic unit cell, expressed in units of scattering. For Xray scattering, that unit is the scattering by a single electron , while for neutron scattering by atomic nuclei, the unit of scattering length of is commonly used. The complex form of the structure factor means that the phase of the scattered wave is not simply related to that of the incident wave. However, the observable, which is the scattered intensity, must be real. It is proportional to the square of the scattering amplitude (see, e.g., Lipson & Cochran, 1966).
The structure factor is directly related to the distribution of scattering matter in the unit cell which, in the Xray case, is the electron distribution, timeaveraged over the vibrational modes of the solid.
In this chapter we will discuss structurefactor expressions for Xray and neutron scattering, and, in particular, the modelling that is required to obtain an analytical description in terms of the features of the electron distribution and the vibrational displacement parameters of individual atoms. We concentrate on the most basic developments; for further details the reader is referred to the cited literature.
The total scattering of Xrays contains both elastic and inelastic components. Within the firstorder Born approximation (Born, 1926) it has been treated by several authors (e.g. Waller & Hartree, 1929; Feil, 1977) and is given by the expression where is the classical Thomson scattering of an Xray beam by a free electron, which is equal to for an unpolarized beam of unit intensity, ψ is the nelectron spacewavefunction expressed in the 3n coordinates of the electrons located at and the integration is over the coordinates of all electrons. S is the scattering vector of length .
The coherent elastic component of the scattering, in units of the scattering of a free electron, is given by
If integration is performed over all coordinates but those of the jth electron, one obtains after summation over all electrons where is the electron distribution. The scattering amplitude is then given by or where is the Fourier transform operator.
In a crystal of infinite size, is a threedimensional periodic function, as expressed by the convolution where n, m and p are integers, and δ is the Dirac delta function.
Thus, according to the Fourier convolution theorem, which gives
Expression (1.2.3.3) is valid for a crystal with a very large number of unit cells, in which particlesize broadening is negligible. Furthermore, it does not account for multiple scattering of the beam within the crystal. Because of the appearance of the delta function, (1.2.3.3) implies that with .
The first factor in (1.2.3.3), the scattering amplitude of one unit cell, is defined as the structure factor F:
To a reasonable approximation, the unitcell density can be described as a superposition of isolated, spherical atoms located at . Substitution in (1.2.3.4) gives or , the spherical atomic scattering factor, or form factor, is the Fourier transform of the spherically averaged atomic density , in which the polar coordinate r is relative to the nuclear position. can be written as (James, 1982) where is the zeroorder spherical Bessel function.
represents either the static or the dynamic density of atom j. In the former case, the effect of thermal motion, treated in Section 1.2.9 and following, is not included in the expression.
When scattering is treated in the secondorder Born approximation, additional terms occur which are in particular of importance for Xray wavelengths with energies close to absorption edges of atoms, where the participation of free and bound excited states in the scattering process becomes very important, leading to resonance scattering. Inclusion of such contributions leads to two extra terms, which are both wavelength and scatteringangledependent:
The treatment of resonance effects is beyond the scope of this chapter. We note however (a) that to a reasonable approximation the Sdependence of j′ and j″ can be neglected, (b) that j′ and j″ are not independent, but related through the Kramers–Kronig transformation, and (c) that in an anisotropic environment the atomic scattering factor becomes anisotropic, and accordingly is described as a tensor property. Detailed descriptions and appropriate references can be found in Materlick et al. (1994) and in Section 4.2.6 of IT C (2004).
The structurefactor expressions (1.2.4.2) can be simplified when the crystal class contains nontrivial symmetry elements. For example, when the origin of the unit cell coincides with a centre of symmetry the sine term in (1.2.4.2b) cancels when the contributions from the symmetryrelated atoms are added, leading to the expression where the summation is over the unique half of the unit cell only.
Further simplifications occur when other symmetry elements are present. They are treated in Chapter 1.4 , which also contains a complete list of symmetryspecific structurefactor expressions valid in the sphericalatom isotropictemperaturefactor approximation.
The scattering of neutrons by atomic nuclei is described by the atomic scattering length b, related to the total cross section by the expression . At present, there is no theory of nuclear forces which allows calculation of the scattering length, so that experimental values are to be used. Two types of nuclei can be distinguished (Squires, 1978). In the first type, the scattering is a resonance phenomenon and is associated with the formation of a compound nucleus (consisting of the original nucleus plus a neutron) with an energy close to that of an excited state. In the second type, the compound nucleus is not near an excited state and the scattering length is essentially real and independent of the energy of the incoming neutron. In either case, b is independent of the Bragg angle θ, unlike the Xray form factor, since the nuclear dimensions are very small relative to the wavelength of thermal neutrons.
The scattering length is not the same for different isotopes of an element. A random distribution of isotopes over the sites occupied by that element leads to an incoherent contribution, such that effectively . Similarly for nuclei with nonzero spin, a spin incoherent scattering occurs as the spin states are, in general, randomly distributed over the sites of the nuclei.
For free or loosely bound nuclei, the scattering length is modified by , where M is the mass of the nucleus and m is the mass of the neutron. This effect is of consequence only for the lightest elements. It can, in particular, be of significance for hydrogen atoms. With this in mind, the structurefactor expression for elastic scattering can be written as by analogy to (1.2.4.2b).
The interaction between the magnetic moments of the neutron and the unpaired electrons in solids leads to magnetic scattering. The total elastic scattering including both the nuclear and magnetic contributions is given by where the unit vector describes the polarization vector for the neutron spin, is given by (1.2.4.2b) and Q is defined by is the vector field describing the electronmagnetization distribution and is a unit vector parallel to H.
Q is thus proportional to the projection of M onto a direction orthogonal to H in the plane containing M and H. The magnitude of this projection depends on , where α is the angle between Q and H, which prevents magnetic scattering from being a truly threedimensional probe. If all moments are collinear, as may be achieved in paramagnetic materials by applying an external field, and for the maximum signal (H orthogonal to M), (1.2.5.2a) becomes and (1.2.5.1a) gives and for neutrons parallel and antiparallel to , respectively.
1.2.6. Effect of bonding on the atomic electron density within the sphericalatom approximation: the kappa formalism
A first improvement beyond the isolatedatom formalism is to allow for changes in the radial dependence of the atomic electron distribution.
Such changes may be due to electronegativity differences which lead to the transfer of electrons between the valence shells of different atoms. The electron transfer introduces a change in the screening of the nuclear charge by the electrons and therefore affects the radial dependence of the atomic electron distribution (Coulson, 1961). A change in radial dependence of the density may also occur in a purely covalent bond, as, for example, in the H_{2} molecule (Ruedenberg, 1962). It can be expressed as (Coppens et al., 1979), where ρ′ is the modified density and κ is an expansion/contraction parameter, which is > 1 for valenceshell contraction and < 1 for expansion. The factor results from the normalization requirement.
The valence density is usually defined as the outer electron shell from which charge transfer occurs. The inner or core electrons are much less affected by the change in occupancy of the outer shell and, in a reasonable approximation, retain their radial dependence.
The corresponding structurefactor expression is where and are the number of electrons (not necessarily integral) in the core and valence shell, respectively, and the atomic scattering factors and are normalized to one electron. Here and in the following sections, the anomalousscattering contributions are incorporated in the core scattering.
Even though the sphericalatom approximation is often adequate, atoms in a crystal are in a nonspherical environment; therefore, an accurate description of the atomic electron density requires nonspherical density functions. In general, such density functions can be written in terms of the three polar coordinates r, θ and ϕ. Under the assumption that the radial and angular parts can be separated, one obtains for the density function:
The angular functions Θ are based on the spherical harmonic functions defined by with , where are the associated Legendre polynomials (see Arfken, 1970).
The real spherical harmonic functions , , are obtained as a linear combination of : and The normalization constants are defined by the conditions which are appropriate for normalization of wavefunctions. An alternative definition is used for chargedensity basis functions: The functions and differ only in the normalization constants. For the spherically symmetric function , a population parameter equal to one corresponds to the function being populated by one electron. For the nonspherical functions with , a population parameter equal to one implies that one electron has shifted from the negative to the positive lobes of the function.
The functions and can be expressed in Cartesian coordinates, such that and where the are Cartesian functions. The relations between the various definitions of the real spherical harmonic functions are summarized by in which the direction of the arrows and the corresponding conversion factors define expressions of the type (1.2.7.4) . The expressions for with are listed in Table 1.2.7.1, together with the normalization factors and .

The spherical harmonic functions are mutually orthogonal and form a complete set, which, if taken to sufficiently high order, can be used to describe any arbitrary angular function.
The spherical harmonic functions are often referred to as multipoles since each represents the components of the charge distribution , which gives nonzero contribution to the integral , where is an electrostatic multipole moment. Terms with increasing l are referred to as monopolar , dipolar , quadrupolar , octapolar , hexadecapolar , triacontadipolar and hexacontatetrapolar .
Sitesymmetry restrictions for the real spherical harmonics as given by Kara & KurkiSuonio (1981) are summarized in Table 1.2.7.2.

In cubic space groups, the spherical harmonic functions as defined by equations (1.2.7.2) are no longer linearly independent. The appropriate basis set for this symmetry consists of the `Kubic Harmonics' of Von der Lage & Bethe (1947). Some loworder terms are listed in Table 1.2.7.3. Both wavefunction and densityfunction normalization factors are specified in Table 1.2.7.3.

A related basis set of angular functions has been proposed by Hirshfeld (1977). They are of the form , where is the angle with a specified set of polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics with , , for , as shown elsewhere (Hirshfeld, 1977).
The radial functions can be selected in different manners. Several choices may be made, such as where the coefficient may be selected by examination of products of hydrogenic orbitals which give rise to a particular multipole (Hansen & Coppens, 1978). Values for the exponential coefficient may be taken from energyoptimized coefficients for isolated atoms available in the literature (Clementi & Raimondi, 1963). A standard set has been proposed by Hehre et al. (1969). In the bonded atom, such values are affected by changes in nuclear screening due to migrations of charge, as described in part by equation (1.2.6.1).
Other alternatives are: or where L is a Laguerre polynomial of order n and degree .
In summary, in the multipole formalism the atomic density is described by in which the leading terms are those of the kappa formalism [expressions (1.2.6.1), (1.2.6.2)]; the subscript p is either + or −.
The expansion in (1.2.7.6) is frequently truncated at the hexadecapolar level. For atoms at positions of high site symmetry the first allowed functions may occur at higher l values. For trigonally bonded atoms in organic molecules the terms are often found to be the most significantly populated deformation functions.
The asphericalatom form factor is obtained by substitution of (1.2.7.6) in expression (1.2.4.3a): In order to evaluate the integral, the scattering operator must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959; CohenTannoudji et al., 1977)
The Fourier transform of the product of a complex spherical harmonic function with normalization and an arbitrary radial function follows from the orthonormality properties of the spherical harmonic functions, and is given by where is the lthorder spherical Bessel function (Arfken, 1970), and θ and ϕ, β and γ are the angular coordinates of r and S, respectively.
For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics: which leads to Since occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions
In (1.2.7.8b) and (1.2.7.8c), , the Fourier–Bessel transform, is the radial integral defined as of which in expression (1.2.4.3) is a special case. The functions for Hartree–Fock valence shells of the atoms are tabulated in scatteringfactor tables (IT IV, 1974). Expressions for the evaluation of using the radial function (1.2.7.5a–c) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closedform expressions are listed in Table 1.2.7.4.

Expressions (1.2.7.8) show that the Fourier transform of a directspace spherical harmonic function is a reciprocalspace spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fouriertransform invariant.
The scattering factors of the aspherical density functions in the multipole expansion (1.2.7.6) are thus given by
The reciprocalspace spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1, except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S.
If the wavefunction is written as a sum over normalized Slater determinants, each representing an antisymmetrized combination of occupied molecular orbitals expressed as linear combinations of atomic orbitals , i.e. , the electron density is given by (Stewart, 1969a) with . The coefficients are the populations of the orbital product density functions and are given by
For a multiSlater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2) but with noninteger values for the coefficients .
The summation (1.2.8.1) consists of one and twocentre terms for which and are centred on the same or on different nuclei, respectively. The latter represent the overlap density, which is only significant if and have an appreciable value in the same region of space.
If the atomic basis consists of hydrogenic type s, p, d, f, … orbitals, the basis functions may be written as or which gives for corresponding values of the orbital products and respectively, where it has been assumed that the radial function depends only on l.
Because the spherical harmonic functions form a complete set, their products can be expressed as a linear combination of spherical harmonics. The coefficients in this expansion are the Clebsch–Gordan coefficients (Condon & Shortley, 1957), defined by or the equivalent definition The vanish, unless is even, and .
The corresponding expression for is with and for , and and for and .
Values of C and for are given in Tables 1.2.8.1 and 1.2.8.2. They are valid for the functions and with normalization and .


By using (1.2.8.5a) or (1.2.8.5c), the onecentre orbital products are expressed as a sum of spherical harmonic functions. It follows that the onecentre orbital product density basis set is formally equivalent to the multipole description, both in real and in reciprocal space. To obtain the relation between orbital products and the chargedensity functions, the righthand side of (1.2.8.5c) has to be multiplied by the ratio of the normalization constants, as the wavefunctions and chargedensity functions are normalized in a different way as described by (1.2.7.3a) and (1.2.7.3b). Thus where . The normalization constants and are given in Table 1.2.7.1, while the coefficients in the expressions (1.2.8.6) are listed in Table 1.2.8.3.

Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the twocentre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slatertype (Bentley & Stewart, 1973; Avery & Ørmen, 1979) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian (Stewart, 1969b, 1970; Stewart & Hehre, 1970; Hehre et al., 1970) and Slatertype (Clementi & Roetti, 1974) functions are available for many atoms.
Since the crystal is subject to vibrational oscillations, the observed elastic scattering intensity is an average over all normal modes of the crystal. Within the Born–Oppenheimer approximation, the theoretical electron density should be calculated for each set of nuclear coordinates. An average can be obtained by taking into account the statistical weight of each nuclear configuration, which may be expressed by the probability distribution function for a set of displacement coordinates .
In general, if is the electron density corresponding to the geometry defined by , the timeaveraged electron density is given by
When the crystal can be considered as consisting of perfectly following rigid entities, which may be molecules or atoms, expression (1.2.9.1) simplifies:
In the approximation that the atomic electrons perfectly follow the nuclear motion, one obtains The Fourier transform of this convolution is the product of the Fourier transforms of the individual functions: Thus , the atomic temperature factor, is the Fourier transform of the probability distribution .
1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation
For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian, centred at the equilibrium position. For the threedimensional isotropic harmonic oscillator, the distribution is where is the meansquare displacement in any direction.
The corresponding trivariate normal distribution to be used for anisotropic harmonic motion is, in tensor notation, Here σ is the variance–covariance matrix, with covariant components, and is the determinant of the inverse of σ. Summation over repeated indices has been assumed. The corresponding equation in matrix notation is where the superscript T indicates the transpose.
The characteristic function, or Fourier transform, of is or With the change of variable , (1.2.10.3a) becomes
The treatment of rigidbody motion of molecules or molecular fragments was developed by Cruickshank (1956) and expanded into a general theory by Schomaker & Trueblood (1968). The theory has been described by Johnson (1970b) and by Dunitz (1979). The latter reference forms the basis for the following treatment.
The most general motions of a rigid body consist of rotations about three axes, coupled with translations parallel to each of the axes. Such motions correspond to screw rotations. A libration around a vector , with length corresponding to the magnitude of the rotation, results in a displacement , such that with or in tensor notation, assuming summation over repeated indices, where the permutation operator equals +1 for i, j, k a cyclic permutation of the indices 1, 2, 3, or −1 for a noncyclic permutation, and zero if two or more indices are equal. For , for example, only the and terms occur. Addition of a translational displacement gives
When a rigid body undergoes vibrations the displacements vary with time, so suitable averages must be taken to derive the meansquare displacements. If the librational and translational motions are independent, the cross products between the two terms in (1.2.11.4) average to zero and the elements of the meansquare displacement tensor of atom n, , are given by where the coefficients and are the elements of the libration tensor and the translation tensor , respectively. Since pairs of terms such as and correspond to averages over the same two scalar quantities, the and tensors are symmetrical.
If a rotation axis is correctly oriented, but incorrectly positioned, an additional translation component perpendicular to the rotation axes is introduced. The rotation angle and the parallel component of the translation are invariant to the position of the axis, but the perpendicular component is not. This implies that the tensor is unaffected by any assumptions about the position of the libration axes, whereas the tensor depends on the assumptions made concerning the location of the axes.
The quadratic correlation between librational and translational motions can be allowed for by including in (1.2.11.5) cross terms of the type , or, with (1.2.11.3), which leads to the explicit expressions such as
The products of the type are the components of an additional tensor, , which unlike the tensors and is unsymmetrical, since is different from . The terms involving elements of may be grouped as or As the diagonal elements occur as differences in this expression, a constant may be added to each of the diagonal terms without changing the observational equations. In other words, the trace of is indeterminate.
In terms of the and tensors, the observational equations are The arrays and involve the atomic coordinates , and are listed in Table 1.2.11.1. Equations (1.2.11.9) for each of the atoms in the rigid body form the observational equations, from which the elements of and can be derived by a linear leastsquares procedure. One of the diagonal elements of must be fixed in advance or some other suitable constraint applied because of the indeterminacy of . It is common practice to set equal to zero. There are thus eight elements of to be determined, as well as the six each of and , for a total of 20 variables. A shift of origin leaves invariant, but it intermixes and .

If the origin is located at a centre of symmetry, for each atom at r with vibration tensor there will be an equivalent atom at −r with the same vibration tensor. When the observational equations for these two atoms are added, the terms involving elements of disappear since they are linear in the components of r. The other terms, involving elements of the and tensors, are simply doubled, like the components.
The physical meaning of the and tensor elements is as follows. is the meansquare amplitude of translational vibration in the direction of the unit vector l with components along the Cartesian axes and is the meansquare amplitude of libration about an axis in this direction. The quantity represents the mean correlation between libration about the axis l and translation parallel to this axis. This quantity, like , depends on the choice of origin, although the sum of the two quantities is independent of the origin.
The nonsymmetrical tensor can be written as the sum of a symmetric tensor with elements and a skewsymmetric tensor with elements . Expressed in terms of principal axes, consists of three principal screw correlations . Positive and negative screw correlations correspond to opposite senses of helicity. Since an arbitrary constant may be added to all three correlation terms, only the differences between them can be determined from the data.
The skewsymmetric part is equivalent to a vector with components , involving correlations between a libration and a perpendicular translation. The components of can be reduced to zero, and made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes also minimizes the trace of . In terms of the coordinate system based on the principal axes of , the required origin shifts are in which the carets indicate quantities referred to the principal axis system.
The description of the averaged motion can be simplified further by shifting to three generally nonintersecting libration axes, one each for each principal axis of . Shifts of the axis in the and directions by respectively, annihilate the and terms of the symmetrized tensor and simultaneously effect a further reduction in (the presuperscript denotes the axis that is shifted, the subscript the direction of the shift component). Analogous equations for displacements of the and axes are obtained by permutation of the indices. If all three axes are appropriately displaced, only the diagonal terms of remain. Referred to the principal axes of , they represent screw correlations along these axes and are independent of origin shifts.
The elements of the reduced are
The resulting description of the average rigidbody motion is in terms of six independently distributed instantaneous motions – three screw librations about nonintersecting axes (with screw pitches given by etc.) and three translations. The parameter set consists of three libration and three translation amplitudes, six angles of orientation for the principal axes of and , six coordinates of axis displacement, and three screw pitches, one of which has to be chosen arbitrarily, again for a total of 20 variables.
Since diagonal elements of enter into the expression for , the indeterminacy of introduces a corresponding indeterminacy in . The constraint is unaffected by the various rotations and translations of the coordinate systems used in the course of the analysis.
The probability distribution (1.2.10.2) is valid in the case of rectilinear harmonic motion. If the deviations from Gaussian shape are not too large, distributions may be used which are expansions with the Gaussian distribution as the leading term. Three such distributions are discussed in the following sections.
The threedimensional Gram–Charlier expansion, introduced into thermalmotion treatment by Johnson & Levy (1974), is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958). If is the operator , where is the harmonic distribution, or 3, and the operator is the rth partial derivative . Summation is again implied over repeated indices.
The differential operators D may be eliminated by the use of threedimensional Hermite polynomials defined, by analogy with the onedimensional Hermite polynomials, by the expression which gives where the first and second terms have been omitted since they are equivalent to a shift of the mean and a modification of the harmonic term only. The permutations of here, and in the following sections, include all combinations which produce different terms.
The coefficients c, defined by (1.2.12.1) and (1.2.12.2), are known as the quasimoments of the frequency function (Kutznetsov et al., 1960). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958) and are invariant to permutation of indices. There are 10, 15, 21 and 28 components of c for orders 3, 4, 5 and 6, respectively. The multivariate Hermite polynomials are functions of the elements of and of , and are given in Table 1.2.12.1 for orders (IT IV, 1974; Zucker & Schulz, 1982).

The Fourier transform of (1.2.12.3) is given by where is the harmonic temperature factor. is a powerseries expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.
A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958; Johnson, 1969). It expresses the function as
Like the moments μ of a distribution, the cumulants κ are descriptive constants. They are related to each other (in the onedimensional case) by the identity When it is substituted for t, (1.2.12.5b) is the characteristic function, or Fourier transform of (Kendall & Stuart, 1958).
The first two terms in the exponent of (1.2.12.5a) can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by .
The Fourier transform of (1.2.12.5a) is, by analogy with the lefthand part of (1.2.12.5b) (with t replaced by ), where the first two terms have been omitted. Expression (1.2.12.6) is similar to (1.2.12.4) except that the entire series is in the exponent. Following Schwarzenbach (1986), (1.2.12.6) can be developed in a Taylor series, which gives
This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4), and corresponds to the probability distribution [analogous to (1.2.12.3)]
The relation between the cumulants and the quasimoments are apparent from comparison of (1.2.12.8) and (1.2.12.4):
The sixth and higherorder cumulants and quasimoments differ. Thus the thirdorder cumulant contributes not only to the coefficient of , but also to higherorder terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a finite truncation of (1.2.12.6), the probability distribution cannot be represented by a finite number of terms. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type.
When an atom is considered as an independent oscillator vibrating in a potential well , its distribution may be described by Boltzmann statistics. with N, the normalization constant, defined by . The classical expression (1.2.12.10) is valid in the hightemperature limit for which .
Following Dawson (1967) and Willis (1969), the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates: The equilibrium condition gives . Substitution into (1.2.12.10) leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by : in which etc. and the normalization factor N depends on the level of truncation.
The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters , for example, are linear combinations of the seven octapoles and three dipoles (Coppens, 1980). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially confined to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model.
The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970a; Scheringer, 1985a) where is the harmonic temperature factor and G represents the Hermite polynomials in reciprocal space.
If the OPP temperature factor is expanded in the coordinate system which diagonalizes , simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates (Dawson et al., 1967; Coppens, 1980; Tanaka & Marumo, 1983).
The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982), Kuhs (1983), and by Scheringer (1985b). In general, the Gram–Charlier expression is found to be preferable because it gives a better fit in the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the oneparticle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real and reciprocalspace expressions. The terms of the OPP model have a specific physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969; Coppens, 1980), provided the potential function itself can be assumed to be temperature independent.
It has recently been shown that the Edgeworth expansion (1.2.12.5a) always has negative regions (Scheringer, 1985b). This implies that it is not a realistic description of a vibrating atom.
In the generalized structurefactor formalism developed by Dawson (1975), the complex nature of both the atomic scattering factor and the generalized temperature factor are taken into account. We write for the atomic scattering factor: and where the subscripts c and a refer to the centrosymmetric and acentric components, respectively. Substitution in (1.2.4.2) gives for the real and imaginary components A and B of and (McIntyre et al., 1980; Dawson, 1967).
Expressions (1.2.13.3) illustrate the relation between valencedensity anisotropy and anisotropy of thermal motion.
This chapter summarizes mathematical developments of the structurefactor formalism. The introduction of atomic asphericity into the formalism and the treatment of thermal motion are interlinked. It is important that the complexities of the thermal probability distribution function can often be reduced by very low temperature experimentation. Results obtained with the multipole formalism for atomic asphericity can be used to derive physical properties and dorbital populations of transitionmetal atoms (IT C, 2004). In such applications, the deconvolution of the charge density and the thermal vibrations is essential. This deconvolution is dependent on the adequacy of the models summarized here.
Acknowledgements
The author would like to thank several of his colleagues who gave invaluable criticism of earlier versions of this manuscript. Corrections and additions were made following comments by P. J. Becker, D. Feil, N. K. Hansen, G. McIntyre, E. N. Maslen, S. Ohba, C. Scheringer and D. Schwarzenbach. Z. Su contributed to the revised version of the manuscript. Support of this work by the US National Science Foundation (CHE8711736 and CHE9317770) is gratefully acknowledged.
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