International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 10-24   | 1 | 2 |
https://doi.org/10.1107/97809553602060000550

## Chapter 1.2. The structure factor

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

This chapter summarizes the mathematical development of the structure-factor formalism. It starts with the definition of the structure factor appropriate for X-ray and neutron scattering, and includes the derivation of the appropriate expressions. Beyond the isolated-atom case, the atom-centred 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.

### 1.2.1. Introduction

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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 X-ray 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 X-ray case, is the electron distribution, time-averaged over the vibrational modes of the solid.

In this chapter we will discuss structure-factor expressions for X-ray 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.

### 1.2.2. General scattering expression for X-rays

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The total scattering of X-rays contains both elastic and inelastic components. Within the first-order 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 X-ray beam by a free electron, which is equal to for an unpolarized beam of unit intensity, ψ is the n-electron space-wavefunction 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.

### 1.2.3. Scattering by a crystal: definition of a structure factor

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In a crystal of infinite size, is a three-dimensional 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 particle-size 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:

### 1.2.4. The isolated-atom approximation in X-ray diffraction

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To a reasonable approximation, the unit-cell 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 zero-order 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 second-order Born approximation, additional terms occur which are in particular of importance for X-ray 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 scattering-angle-dependent:

The treatment of resonance effects is beyond the scope of this chapter. We note however (a) that to a reasonable approximation the S-dependence 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 structure-factor expressions (1.2.4.2) can be simplified when the crystal class contains non-trivial 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 symmetry-related 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 symmetry-specific structure-factor expressions valid in the spherical-atom isotropic-temperature-factor approximation.

### 1.2.5. Scattering of thermal neutrons

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#### 1.2.5.1. Nuclear scattering

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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 X-ray 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 non-zero 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 structure-factor expression for elastic scattering can be written as by analogy to (1.2.4.2b).

#### 1.2.5.2. Magnetic scattering

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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 electron-magnetization 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 three-dimensional 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 spherical-atom approximation: the kappa formalism

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A first improvement beyond the isolated-atom 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 H2 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 valence-shell 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 structure-factor 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 anomalous-scattering contributions are incorporated in the core scattering.

### 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion

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#### 1.2.7.1. Direct-space description of aspherical atoms

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Even though the spherical-atom approximation is often adequate, atoms in a crystal are in a non-spherical environment; therefore, an accurate description of the atomic electron density requires non-spherical 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 charge-density 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 non-spherical 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 .

 Table 1.2.7.1| top | pdf | Real spherical harmonic functions (x, y, z are direction cosines)
lSymbol CAngular function, Normalization for wavefunctions, §Normalization for density functions,
ExpressionNumerical valueExpressionNumerical value
0 00 1 1 0.28209 0.07958
1 0.48860 0.31831
2 20 0.31539 0.20675
1.09255 0.75
3 30 0.37318 0.24485
0.45705 0.32033
1.44531 1 1
0.59004 0.42441
4 40 0.10579 ‡‡ 0.06942
0.66905 0.47400
0.47309 0.33059
1.77013 1.25
0.62584 0.46875
5 50 0.11695 0.07674
0.45295 0.32298
2.39677 1.68750
0.48924 0.34515
2.07566 1.50000
0.65638 0.50930
6 60 0.06357 0.04171
0.58262 0.41721
0.46060 0.32611
0.92121 0.65132
0.50457 0.36104
10395 2.36662 1.75000
10395 0.68318 0.54687
7 70 0.06828 0.04480
0.09033 0.06488
0.22127 0.15732
0.15646 0.11092
1.03783 0.74044
0.51892 0.37723
135135 2.6460 2.00000
135135 0.70716 0.58205
Common factor such that
, , .
§As defined by where are Cartesian functions.
Paturle & Coppens (1988), as defined by where are Cartesian functions.
††ar = arctan (2).
‡‡ where .

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 non-zero 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 .

Site-symmetry restrictions for the real spherical harmonics as given by Kara & Kurki-Suonio (1981) are summarized in Table 1.2.7.2.

 Table 1.2.7.2| top | pdf | Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981)
 λ, μ and j are integers.
SymmetryChoice of coordinate axesIndices of allowed ,
1 Any
Any
2
m
222 ,
mm2
mmm
4
,
422 ,
4mm
2m ,
,
3
32
,

3m

6
622
6mm

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 low-order terms are listed in Table 1.2.7.3. Both wavefunction and density-function normalization factors are specified in Table 1.2.7.3.

 Table 1.2.7.3| top | pdf | Kubic Harmonic' functions
 (a) Coefficients in the expression with normalization (Kara & Kurki-Suonio, 1981).
Even l mp
l j0+2+4+6+8+10+
0 1 1
4 1
0.76376   0.64550
6 1
0.35355   −0.93541
6 2
0.82916   −0.55902
8 1
0.71807   0.38188   0.58184
10 1
0.41143   −0.58630   −0.69784
10 2
0.80202   0.15729   0.57622
l j   2− 4− 6− 8−
3 1   1
7 1
0.73598   0.41458
9 1
0.43301   −0.90139
9 2
0.84163   −0.54006
 (b) Coefficients and density normalization factors in the expression where (Su & Coppens, 1994).
Even l mp
l j   0+ 2+ 4+ 6+ 8+ 10+
0 1 1
4 1 0.43454 1
6 1 0.25220 1
6 2 0.020833   1
8 1 0.56292 1   1/5940
10 1 0.36490 1   1/5460
10 2 0.0095165 1
l j     2− 4− 6− 8−
3 1 0.066667   1
7 1 0.014612   1
9 1 0.0059569   1
9 2 0.00014800     1
 (c) Density-normalized Kubic harmonics as linear combinations of density-normalized spherical harmonic functions. Coefficients in the expression . Density-type normalization is defined as .
Even l mp
l j0+2+4+6+8+10+
0 1 1
4 1 0.78245   0.57939
6 1 0.37790   −0.91682
6 2   0.83848   −0.50000
l j 2− 4− 6− 8−
3 1 1
7 1 0.73145   0.63290
 (d) Index rules for cubic symmetries (Kurki-Suonio, 1977; Kara & Kurki-Suonio, 1981).
lj23 432
T O
0 1 × × × × ×
3 1 ×     ×
4 1 × × × × ×
6 1 × × × × ×
6 2 × ×
7 1 ×     ×
8 1 × × × × ×
9 1 ×     ×
9 2 ×   ×
10 1 × × × × ×
10 2 × ×

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 energy-optimized 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.

#### 1.2.7.2. Reciprocal-space description of aspherical atoms

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The aspherical-atom 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; Cohen-Tannoudji 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 lth-order 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 scattering-factor tables (IT IV, 1974). Expressions for the evaluation of using the radial function (1.2.7.5ac) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closed-form expressions are listed in Table 1.2.7.4.

 Table 1.2.7.4| top | pdf | Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990)
N
k 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7

Expressions (1.2.7.8) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant.

The scattering factors of the aspherical density functions in the multipole expansion (1.2.7.6) are thus given by

The reciprocal-space 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.

### 1.2.8. Fourier transform of orbital products

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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 multi-Slater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2) but with non-integer values for the coefficients .

The summation (1.2.8.1) consists of one- and two-centre 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.

#### 1.2.8.1. One-centre orbital products

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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 ClebschGordan 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 .

 Table 1.2.8.1| top | pdf | Products of complex spherical harmonics as defined by equation (1.2.7.2a)
 Y 00 Y 00 = 0.28209479Y 00 Y 10 Y 00 = 0.28209479Y 10 Y 10 Y 10 = 0.25231325Y 20 + 0.28209479Y 00 Y 11 Y 00 = 0.28209479Y 11 Y 11 Y 10 = 0.21850969Y 21 Y 11 Y 11 = 0.30901936Y 22 Y 11 Y 11− = −0.12615663Y 20 + 0.28209479Y 00 Y 20 Y 00 = 0.28209479Y 20 Y 20 Y 10 = 0.24776669Y 30 + 0.25231325Y 10 Y 20 Y 11 = 0.20230066Y 31 − 0.12615663Y 11 Y 20 Y 20 = 0.24179554Y 40 + 0.18022375Y 20 + 0.28209479Y 00 Y 21 Y 00 = 0.28209479Y 21 Y 21 Y 10 = 0.23359668Y 31 + 0.21850969Y 11 Y 21 Y 11 = 0.26116903Y 32 Y 21 Y 11− = −0.14304817Y 30 + 0.21850969Y 10 Y 21 Y 20 = 0.22072812Y 41 + 0.09011188Y 21 Y 21 Y 21 = 0.25489487Y 42 + 0.22072812Y 22 Y 21 Y 21− = −0.16119702Y 40 + 0.09011188Y 20 + 0.28209479Y 00 Y 22 Y 00 = 0.28209479Y 22 Y 22 Y 10 = 0.18467439Y 32 Y 22 Y 11 = 0.31986543Y 33 Y 22 Y 11− = −0.08258890Y 31 + 0.30901936Y 11 Y 22 Y 20 = 0.15607835Y 42 − 0.18022375Y 22 Y 22 Y 21 = 0.23841361Y 43 Y 22 Y 21− = −0.09011188Y 41 + 0.22072812Y 21 Y 22 Y 22 = 0.33716777Y 44 Y 22 Y 22− = 0.04029926Y 40 − 0.18022375Y 20 + 0.28209479Y 00
 Table 1.2.8.2| top | pdf | Products of real spherical harmonics as defined by equations (1.2.7.2b) and (1.2.7.2c)
 y 00 y 00 = 0.28209479y 00 y 10 y 00 = 0.28209479y 10 y 10 y 10 = 0.25231325y 20 + 0.28209479y 00 y 11± y 00 = 0.28209479y 11± y 11± y 10 = 0.21850969y 21± y 11± y 11± = 0.21850969y 22+ − 0.12615663y 20 + 0.28209479y 00 y 11+ y 11− = 0.21850969y 22− y 20 y 00 = 0.28209479y 20 y 20 y 10 = 0.24776669y 30 + 0.25231325y 10 y 20 y 11± = 0.20230066y 31± − 0.12615663y 11± y 20 y 20 = 0.24179554y 40 + 0.18022375y 20 + 0.28209479y 00 y 21± y 00 = 0.28209479y 21± y 21± y 10 = 0.23359668y 31± + 0.21850969y 11± y 21± y 11± = ± 0.18467439y 32+ − 0.14304817y 30 + 0.21850969y 10 y 21± y 11∓ = 0.18467469y 32− y 21± y 20 = 0.22072812y 41± + 0.09011188y 21± y 21± y 21± = ± 0.18022375y 42+ ± 0.15607835y 22+ − 0.16119702y 40 + 0.09011188y 20 + 0.28209479y 00 y 21+ y 21− = −0.18022375y 42− + 0.15607835y 22− y 22± y 00 = 0.28209479y 22± y 22± y 10 = 0.18467439y 32± y 22± y 11± = ± 0.22617901y 33+ − 0.05839917y 31+ + 0.21850969y 11+ y 22± y 11∓ = 0.22617901y 33− ± 0.05839917y 31− ∓ 0.21850969y 11− y 22± y 20 = 0.15607835y 42± − 0.18022375y 22± y 22± y 21± = ± 0.16858388y 43+ − 0.06371872y 41+ + 0.15607835y 21+ y 22± y 21∓ = 0.16858388y 43− ± 0.06371872y 41− ∓ 0.15607835y 21− y 22± y 22± = ± 0.23841361y 44+ + 0.04029926y 40 − 0.18022375y 20 + 0.28209479y 00 y 22+ y 22− = 0.23841361y 44−

By using (1.2.8.5a) or (1.2.8.5c), the one-centre orbital products are expressed as a sum of spherical harmonic functions. It follows that the one-centre 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 charge-density functions, the right-hand side of (1.2.8.5c) has to be multiplied by the ratio of the normalization constants, as the wavefunctions and charge-density 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.

 Table 1.2.8.3| top | pdf | Products of two real spherical harmonic functions in terms of the density functions defined by equation (1.2.7.3b)
 y 00 y 00 = 1.0000d 00 y 10 y 00 = 0.43301d 10 y 10 y 10 = 0.38490d 20 + 1.0d 00 y 11± y 00 = 0.43302d 11± y 11± y 10 = 0.31831d 21± y 11± y 11± = 0.31831d 22+ − 0.19425d 20 + 1.0d 00 y 11+ y 11− = 0.31831d 22− y 20 y 00 = 0.43033d 20 y 20 y 10 = 0.37762d 30 + 0.38730d 10 y 20 y 11± = 0.28864d 31± − 0.19365d 11± y 20 y 20 = 0.36848d 40 + 0.27493d 20 + 1.0d 00 y 21± y 00 = 0.41094d 21± y 21± y 10 = 0.33329d 31± + 0.33541d 11± y 21± y 11± = ±0.26691d 32+ − 0.21802d 30 + 0.33541d 10 y 21± y 11∓ = −0.26691d 32− y 21± y 20 = 0.31155d 41± + 0.13127d 21± y 21± y 21± = ±0.25791d 42+ ± 0.22736d 22+ − 0.24565d 40 + 0.13747d 20 + 1.0d 00 y 21+ y 21− = 0.25790d 42− + 0.22736d 22− y 22± y 00 = 0.41094d 22± y 22± y 10 = 0.26691d 32± y 22± y 11± = ± 0.31445d 33+ − 0.083323d 31+ + 0.33541d 11+ y 22± y 11∓ = 0.31445d 33− ± 0.083323d 31− ∓ 0.33541d 11− y 22± y 20 = 0.22335d 42± − 0.26254d 22± y 22± y 21± = ± 0.23873d 43+ − 0.089938d 41+ + 0.22736d 21+ y 22± y 21∓ = 0.23873d 43− ± 0.089938d 41− ∓ 0.22736d 21− y 22± y 22± = ± 0.31831d 44+ + 0.061413d 40 − 0.27493d 20 + 1.0d 00 y 22+ y 22− = 0.31831d 44−

#### 1.2.8.2. Two-centre orbital products

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Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the two-centre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slater-type (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 Slater-type (Clementi & Roetti, 1974) functions are available for many atoms.

### 1.2.9. The atomic temperature factor

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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 time-averaged 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

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For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian, centred at the equilibrium position. For the three-dimensional isotropic harmonic oscillator, the distribution is where is the mean-square 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

### 1.2.11. Rigid-body analysis

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The treatment of rigid-body 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 non-cyclic 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 mean-square 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 mean-square 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 least-squares 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 .

 Table 1.2.11.1| top | pdf | The arrays and to be used in the observational equations [equation (1.2.11.9)]
ij kl
112233233112
11 0 0 0
22 0 0 0
33 0 0 0
23 yz 0 0 xy xz
31 0 xz 0 xy yz
12 0 0 xy xz yz
ij kl
112233233112321321
11 0 0 0 0 0 0 0 2z
22 0 0 0 0 0 2x 0 0
33 0 0 0 0 0 0 2y 0
23 0 x x 0 0 y 0 z 0
31 y 0 y z 0 0 0 0 x
12 z z 0 0 x 0 y 0 0

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 mean-square amplitude of translational vibration in the direction of the unit vector l with components along the Cartesian axes and is the mean-square 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 non-symmetrical tensor can be written as the sum of a symmetric tensor with elements and a skew-symmetric 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 skew-symmetric 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 non-intersecting 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 rigid-body motion is in terms of six independently distributed instantaneous motions – three screw librations about non-intersecting 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.

### 1.2.12. Treatment of anharmonicity

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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.

#### 1.2.12.1. The Gram–Charlier expansion

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The three-dimensional Gram–Charlier expansion, introduced into thermal-motion 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 three-dimensional Hermite polynomials defined, by analogy with the one-dimensional 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).

 Table 1.2.12.1| top | pdf | Some Hermite polynomials (Johnson & Levy, 1974; Zucker & Schulz, 1982)
 H(u) = 1 Hj(u) = wj Hjk(u) = wjwk − pjk Hjkl(u) = wjwkwl − (wjpkl + wkplj + wlpjk) = wjwkwl − 3w( jpkl) Hjklm(u) = wjwkwlwm − 6w( jwkplm) + 3pj( kplm) Hjklmn(u) = wjwkwlwmwn − 10w( lwmwnpjk) + 15w( npjkplm) Hjklmnp(u) = wjwkwlwmwnwp − 15w( jwkwlwmpjk) + 45w( jwkplmpnp) − 15pj( kplmpnp) where are the elements of , defined in expression (1.2.10.2) . Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that as illustrated for .

The Fourier transform of (1.2.12.3) is given by where is the harmonic temperature factor. is a power-series expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.

#### 1.2.12.2. The cumulant expansion

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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 one-dimensional 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 left-hand 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 higher-order cumulants and quasimoments differ. Thus the third-order cumulant contributes not only to the coefficient of , but also to higher-order 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.

#### 1.2.12.3. The one-particle potential (OPP) model

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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 high-temperature 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).

#### 1.2.12.4. Relative merits of the three expansions

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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 one-particle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real- and reciprocal-space 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.

### 1.2.13. The generalized structure factor

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In the generalized structure-factor 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 valence-density anisotropy and anisotropy of thermal motion.

### 1.2.14. Conclusion

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This chapter summarizes mathematical developments of the structure-factor 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 d-orbital populations of transition-metal 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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