International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 8.7, pp. 713734
https://doi.org/10.1107/97809553602060000615 Chapter 8.7. Analysis of charge and spin densities^{a}732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 142603000, USA,^{b}Digital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 017542122, USA, and ^{c}Ecole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F92295 Châtenay Malabry CEDEX, France This chapter describes the interaction of Xrays, unpolarized and polarized neutrons with the charge and spin densities of a periodic system. The methods developed for analysis of the electron densities from the diffraction intensities are described in detail and the concepts used in the interpretation of the results are defined. 
Knowledge of the electron distribution is crucial for our understanding of chemical and physical phenomena. It has been assumed in many calculations (e.g. Thomas, 1926; Fermi, 1928; and others), and formally proven for nondegenerate systems (Hohenberg & Kohn, 1964) that the electronic energy is a functional of the electron density. Thus, the experimental measurement of electron densities is important for our understanding of the properties of atoms, molecules and solids. One of the main methods to achieve this goal is the use of scattering techniques, including elastic Xray scattering, Compton scattering of Xrays, magnetic scattering of neutrons and Xrays, and electron diffraction.
Meaningful information can only be obtained with data of the utmost accuracy, which excludes most routinely collected crystallographic data sets. The present chapter will review the basic concepts and expressions used in the interpretation of accurate data in terms of the charge and spin distributions of the electrons. The structurefactor formalism has been treated in Chapter 1.2 of Volume B (IT B, 2001).
A wavefunction for a system of n electrons is a function of the 3n space and n spin coordinates of the electrons. The wavefunction must be antisymmetric with respect to the interchange of any two electrons. The most general density function is the nparticle density matrix, of dimension , defined as In this expression, each index represents both the continuous space coordinates and the discontinuous spin coordinates of each of the n particles. Thus, the nparticle density matrix is a representation of the state in a 6ndimensional coordinate space, and the ndimensional (discontinuous) spin state.
The pth reduced density matrix can be derived from (8.7.2.1) by integration over the space and spin coordinates of n − p particles, According to a basic postulate of quantum mechanics, physical properties are represented by their expectation values obtained from the corresponding operator equation, where is a (Hermitian) operator. As almost all operators of interest are one or twoparticle operators, the one and twoparticle matrices and are of prime interest. The charge density, or oneelectron density, can be obtained from by setting 1′ = 1 and integrating over the spin coordinates, i.e.
An electron density ρ(r) that can be represented by an antisymmetric Nelectron wave function and is derivable from that wave function through (8.7.2.4) is called Nrepresentable. As in any real system, the particles will undergo vibrations, so (8.7.2.4) must be modified to allow for the continuous change in the configuration of the nuclei. The commonly used Born–Oppenheimer approximation assumes that the electrons rearrange instantaneously in the field of the oscillating nuclei, which leads to the separation where r and R represent the electronic and nuclear coordinates, respectively, and is the electronic wavefunction, which is a function of both the electronic and nuclear coordinates. The timeaveraged, oneelectron density <ρ(r)>, which is accessible experimentally through the elastic Xray scattering experiment, is obtained from the static density by integration over all nuclear configurations: where P(R) is the normalized probability distribution function of the nuclear configuration R. The total Xray scattering can be derived from the twoparticle matrix Γ(1, 2, 1′, 2′) through use of the twoparticle scattering operator. The total Xray scattering includes the inelastic incoherent Compton scattering, which is related to the momentum density π(p).
The wavefunction in momentum space, defined by the coordinates of the jth particle (here the caret indicates momentumspace coordinates), is given by the Dirac–Fourier transform of ψ, leading, in analogy to (8.7.2.2), to the oneelectron density matrix and the momentum density The spin density distribution of the electrons, s(r), can be obtained from Γ^{1}(1, 1′) by use of the operator for the z component of the spin angular momentum. If where M, the total magnetization, is the eigenvalue of the operator
The charge density is related to the elastic Xray scattering amplitude F(S) by the expression or, for scattering by a periodic lattice, As F(h) is in general complex, the Fourier transform (8.7.3.1) requires calculation of the phases from a model for the charge distribution. In the centrosymmetric case, the freeatom model is in general adequate for calculation of the signs of F(h). However, for noncentrosymmetric structures in which phases are continuously variable, it is necessary to incorporate deviations from the freeatom density in the model to obtain estimates of the experimental phases.
Since the total density is dominated by the core distribution, differences between the total density ρ(r) and reference densities are important. The reference densities represent hypothetical states without chemical bonding or with only partial chemical bonding. Deviations from spherical, freeatom symmetry are obtained when the reference state is the promolecule, the superposition of freespace spherical atoms centred at the nuclear positions. This difference function is referred to as the deformation density (or standard deformation density) where is the promolecule density. In analogy to (8.7.3.1), the deformation density may be obtained from where and are in general complex.
Several other different density functions analogous to (8.7.3.3), summarized in Table 8.7.3.1, may be defined. Particularly useful for the analysis of effects of chemical bonding is the fragment deformation density, in which a chemical fragment is subtracted from the total density of a molecule. The fragment density is calculated theoretically and thermally smeared before subtraction from an experimental density. `Prepared' atoms rather than spherical atoms may be used as a reference state to emphasize the electrondensity shift due to covalent bond formation.

The electron density ρ(r) in the structurefactor expression can be approximated by a sum of nonnormalized density functions with scattering factor centred at Substitution in (8.7.3.4a) gives
When is the spherically averaged, freeatom density, (8.7.3.4b) represents the freeatom model. A distinction is often made between atomcentred models, in which all functions g(r) are centred at the nuclear positions, and models in which additional functions are centred at other locations, such as in bonds or lonepair regions.
A simple, atomcentred model with spherical functions g(r) is defined by This `kappa model' allows for charge transfer between atomic valence shells through the population parameter , and for a change in nuclear screening with electron population, through the parameter κ, which represents an expansion , or a contraction of the radial density distribution.
The atomcentred, spherical harmonic expansion of the electronic part of the charge distribution is defined by where p = ± when m is larger than 0, and is a radial function.
The real spherical harmonic functions and their Fourier transforms have been described in International Tables for Crystallography, Volume B, Chapter 1.2 (Coppens, 2001). They differ from the functions by the normalization condition, defined as . The real spherical harmonic functions are often referred to as multipoles, since each represents the component of the charge distribution ρ(r) that gives a nonzero contribution to the integral for the electrostatic multipole moment where the functions are the Cartesian representations of the real spherical harmonics (Coppens, 2001).
More general models include nonatom centred functions. If the wavefunction ψ in (8.7.2.4) is an antisymmetrized product of molecular orbitals ψ_{i}, expressed in terms of a linear combination of atomic orbitals (LCAO formalism), the integration (8.7.2.4) leads to with R_{μ} and R_{ν} defining the centres of and , respectively, , and the sum is over all molecular orbitals with occupancy . Expression (8.7.3.9) contains products of atomic orbitals, which may have significant values for orbitals centred on adjacent atoms. In the `chargecloud' model (Hellner, 1977), these products are approximated by bondcentred, Gaussianshaped density functions. Such functions can often be projected efficiently into the onecentre terms of the spherical harmonic multipole model, so that large correlations occur if both spherical harmonics and bondcentred functions are adjusted independently in a leastsquares refinement.
According to (8.7.2.4) and (8.7.3.9), the population of the twocentre terms is related to the onecentre occupancies. A molecularorbital based model, which implicitly incorporates such relations, has been used to describe local bonding between transitionmetal and ligand atoms (Becker & Coppens, 1985).
There are several physical constraints that an electrondensity model must satisfy. With the exception of the electroneutrality constraint, they depend strongly on the electron density close to the nucleus, which is poorly determined by the diffraction experiment.
Since a crystal is neutral, the total electron population must equal the sum of the nuclear charges of the constituent atoms. A constraint procedure for linear least squares that does not increase the size of the leastsquares matrix has been described by Hamilton (1964). If the starting point is a neutral crystal, the constraint equation becomes where , g being a general density function, and the are the shifts in the population parameters. For the multipole model, only the monopolar functions integrate to a nonzero value. For normalized monopole functions, this gives If the shifts without constraints are given by the vector y and the constrained shifts by y_{c}, the Hamilton constraint is expressed as where the superscript T indicates transposition, A is the leastsquares matrix of the products of derivatives, and Q is a row vector of the values of for elements representing density functions and zeros otherwise.
Expression (8.7.3.12) cannot be applied if the unconstrained refinement corresponds to a singular matrix. This would be the case if all population parameters, including those of the core functions, were to be refined together with the scale factor. In this case, a new set of independent parameters must be defined, as described in Chapter 8.1 on leastsquares refinements. Alternatively, one may set the scale factor to one and rescale the population parameters to neutrality after completion of the refinement. This will in general give a nonintegral electron population for the core functions. The proper interpretation of such a result is that a corelike function is an appropriate component of the density basis set representing the valence electrons.
The electron density at a nucleus i with nuclear charge must satisfy the electron–nuclear cusp condition given by where is the spherical component of the expansion of the density around nucleus i.
Only 1stype functions have nonzero electron density at the nucleus and contribute to (8.7.3.13). For the hydrogenlike atom or ion described by a single exponent radial function R(r) = N exp (−ζr), (8.7.3.13) gives , where Z is the nuclear charge, and is the Bohr unit. Thus, a modification of ζ for 1s functions, as implied by (8.7.3.6) and (8.7.3.7) if applied to H atoms, leads to a violation of the cusp constraint. In practice, the electron density at the nucleus is not determined by a limited resolution diffraction experiment; the single exponent function R(r) is fitted to the electron density away from, rather than at the nucleus.
Poisson's electrostatic equation gives a relation between the gradient of the electric field ∇^{2}Φ(r) and the electron density at r. As noted by Stewart (1977), this equation imposes a constraint on the radial functions R(r). For , the condition must be obeyed for to be finite at r = 0, which satisfies the requirement of the nondivergence of the electric field ∇V, its gradient ∇^{2}V, the gradient of the field gradient ∇^{3}V, etc.
According to the electrostatic Hellmann–Feynman theorem, which follows from the Born–Oppenheimer approximation and the condition that the forces on the nuclei must vanish when the nuclear configuration is in equilibrium, the nuclear repulsions are balanced by the electron–nucleus attractions (Levine, 1983). The balance of forces is often achieved by a very sharp polarization of the electron density very close to the nuclei (Hirshfeld & Rzotkiewicz, 1974), which may be represented in the Xray model by the introduction of dipolar functions with large values of ζ. The Hellmann–Feynman constraint offers the possibility for obtaining information on such functions even though they may contribute only marginally to the observed Xray scattering (Hirshfeld, 1984).
As the Hellmann–Feynman constraint applies to the static density, its application presumes a proper deconvolution of the thermal motion and the electron density in the scattering model.
8.7.3.4.1. Moments of a charge distribution^{1}
Use of the expectation value expression with the operator gives for the electrostatic moments of a charge distribution ρ(r) in which the are the three components of the vector r (α_{i} = 1, 2, 3), and the integral is over the complete volume of the distribution.
For l = 0, (8.7.3.16) represents the integral over the charge distribution, which is the total charge, a scalar function described as the monopole. The higher moments are, in ascending order of l, the dipole, a vector, the quadrupole, a secondrank tensor, and the octupole, a thirdrank tensor. Successively higher moments are named the hexadecapole (l = 4), the tricontadipole (l = 5), and the hexacontatetrapole (l = 6). An alternative, traceless, definition is often used for moments with . In the traceless definition, the quadrupole moment, , is given by where δ_{αβ} is the Kronecker delta function. The term , which is subtracted from the diagonal elements of the tensor, corresponds to the spherically averaged second moment of the distribution.
Expression (8.7.3.17) is a special case of the following general expression for the lthrank traceless tensor elements.
Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer present in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from (8.7.3.18). For the quadrupole moments, we obtain with (8.7.3.17) and Expressions for the other elements are obtained by simple permutation of the indices.
For a site of point symmetry 1, the electrostatic moment of order l has (l + 1)(l + 2)/2 unique elements. In the traceless definition, not all elements are independent. Because the trace of the tensor has been set to zero, only 2l + 1 independent components remain. For the quadrupole there are 5 independent components of the form (8.7.3.19).
In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics (ITB, 2001) multiplied by , which defines the spherical electrostatic moments
The expressions for are listed in Volume B of International Tables for Crystallography (ITB, 2001); for the l = 2 moment, the have the well known form 3z^{2} − 1, xz, yz, (x^{2} − y^{2})/2, and xy, where x, y and z are the components of a unit vector from the origin to the point being described. The spherical electrostatic moments have (2l + 1) components, which equals the number of independent components in the traceless definition (8.7.3.18), as it should. The linear relationships are
In the multipole model [expression (8.7.3.7)], the charge density is a sum of atomcentred density functions, and the moments of a whole distribution can be written as a sum over the atomic moments plus a contribution due to the shift to a common origin. An atomic moment is obtained by integration over the charge distributions ρ_{total,i}(r) = ρ_{nuclear,i} − ρ_{e,i} of atom i, where the electronic part of the atomic charge distribution is defined by the multipole expansion where p = ± when m , and is a radial function.
We get for the jth moment of the valence density in which the minus sign arises because of the negative charge of the electrons.
We will use the symbol for the moment operators. We get where, as before, p = ±. The requirement that the integrand be totally symmetric means that only the dipolar terms in the multipole expansion contribute to the dipole moment. If we use the traceless definition of the higher moments, or the equivalent definition of the moments in terms of the spherical harmonic functions, only the quadrupolar terms of the multipole expansion will contribute to the quadrupole moment; more generally, in the traceless definition the lthorder multipoles are the sole contributors to the lth moments. In terms of the spherical moments, we get
Substitution with R_{l} = {(κ′ζ)^{n(l)+3}/[n(l) +2] !}r^{n(l)}exp(−ζr) and and subsequent integration over r gives where the definitions have been used (ITB, 2001). Since the functions are wavefunction normalized, we obtain Application to dipolar terms with n(l) = 2, L_{lm} = 1/π and M_{lm} = (3/4π)^{1/2} gives the x component of the atomic dipole moment as For the atomic quadrupole moments in the spherical definition, we obtain directly, using n(l) = 2, l = 2 in (8.7.3.29), and, for the other elements,
As the traceless quadrupole moments are linear combinations of the spherical quadrupole moments, the corresponding expressions follow directly from (8.7.3.31), (8.7.3.32) and (8.7.3.21). We obtain with n(2) = 2 and and analogously for the other offdiagonal elements.
The moments derived from the total density ρ(r) and from the deformation density Δρ(r) are not identical. To illustrate the relation for the diagonal elements of the secondmoment tensor, we rewrite the xx element as The promolecule is the sum over spherical atom densities, or If R_{i} = (X_{i}, Y_{i}, Z_{i}) is the position vector for atom i, each singleatom contribution can be rewritten as Since the last two integrals are proportional to the atomic dipole moment and its net charge, respectively, they will be zero for neutral spherical atoms. Substitution in (8.7.3.35) gives, with , and and, by substitution in (8.7.3.34), with in which and are the atomic dipole moment and the charge on atom i, respectively.
The last term in (8.7.3.38a) can be derived rapidly from analytical expressions for the atomic wavefunctions. Results for Hartree–Fock wavefunctions have been tabulated by Boyd (1977). Since the offdiagonal elements of the secondmoment tensor vanish for the spherical atom, the second term in (8.7.3.38a) disappears, and the offdiagonal elements are identical for the total and deformation densities.
The relation between the second moments μ_{αβ} and the traceless moments _{αβ} of the deformation density can be illustrated as follows. From (8.7.3.17), we may write Only the spherical density terms contribute to the integral on the right. Assuming for the moment that the spherical deformation is represented by the valenceshell distortion (i.e. neglect of the second monopole in the aspherical atom expansion), we have, with density functions ρ normalized to 1, for each atom and which, on substitution in (8.7.3.39), gives the required relation.
In general, the multipole moments depend on the choice of origin. This can be seen as follows. Substitution of in (8.7.3.16) corresponds to a shift of origin by R_{α}, or X, Y, Z in the original coordinate system. In three dimensions, we get, for the first moment, the charge q, and for the transformed first and second moments
For the traceless quadrupole moments, the corresponding equations are obtained by substitution of and r′ = r − R into (8.7.3.17), which gives Similar expressions for the higher moments are reported in the literature (Buckingham, 1970).
We note that the first nonvanishing moment is originindependent. Thus, the dipole moment of a neutral molecule, but not that of an ion, is independent of origin; the quadrupole moment of a molecule without charge and dipole moment is not dependent on the choice of origin and so on. The molecular electric moments are commonly reported with respect to the centre of mass.
The moments of a molecule or of a molecular fragment are obtained from the sum over the atomic moments, plus a contribution due to the shift to a common origin for all but the monopoles. If individual atomic coordinate systems are used, as is common if chemical constraints are applied in the leastsquares refinement, they must be rotated to have a common orientation. Expressions for coordinate system rotations have been given by Cromer, Larson & Stewart (1976) and by Su & Coppens (1994a).
The transformation to a common coordinate origin requires use of the originshift expressions (8.7.3.42)–(8.7.3.44), with, for an atom at , . The first three moments summed over the atoms i located at become and with α, β = x, y, z; and expressions equivalent to (8.7.3.44) for the traceless components _{αβ}.
Expression (8.7.3.16) for the outer moment of a distribution within a volume element may be written as with , and integration over the volume .
Replacement of ρ(r) by the Fourier summation over the structure factors gives where is the product of l coordinates according to (8.7.3.16), and μ^{l} represents the moment of the static distribution if the F(h) are the structure factors on an absolute scale after deconvolution of thermal motion. Otherwise, the moment of the thermally averaged density is obtained.
The integral is defined as the shape transform S of the volume For regularly shaped volumes, the integral can be evaluated analytically. A volume of complex shape may be subdivided into integrable subvolumes such as parallelepipeds. By choosing the subvolumes sufficiently small, a desired boundary surface can be closely approximated.
If the origin of each subvolume is located at , relative to a coordinate system origin at P, the total electronic moment relative to this origin is given by
Expressions for for and a subvolume parallelepipedal shape are given in Table 8.7.3.2. Since the spherical order Bessel functions that appear in the expressions generally decrease with increasing x, the moments are strongly dependent on the loworder reflections in a data set. An example is the shape transform for the dipole moment. Relative to an origin O, A shift of origin by leads to in agreement with (8.7.3.46).

The electrostatic potential Φ(r′) due to the electronic charge distribution is given by the Coulomb equation, where the constant k is dependent on the units selected, and will here be taken equal to 1. For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain in which Z_{M} is the charge of nucleus M located at R_{M}.
The electric field E at a point in space is the gradient of the electrostatic potential at that point. As E is the negative gradient vector of the potential, the electric force is directed `downhill' and proportional to the slope of the potential function. The explicit expression for E is obtained by differentiation of the operator r − r′^{−1} in (8.7.3.50) towards x, y, z and subsequent addition of the vector components. For the negative slope of the potential in the x direction, one obtains which gives, after addition of the components,
The electric field gradient (EFG) is the tensor product of the gradient operator and the electric field vector E; It follows that in a Cartesian system the EFG tensor is a symmetric tensor with elements
The EFG tensor elements can be obtained by differentiation of the operator in (8.7.3.53) for E_{α} to each of the three directions β. In this way, the traceless result is obtained. We note that according to (8.7.3.57) the electric field gradient can equally well be interpreted as the tensor of the traceless quadrupole moments of the distribution −2ρ_{total}(r)/r − r′^{5} [see equation (8.7.3.17)].
Definition (8.7.3.56) and result (8.7.3.57) differ in that (8.7.3.56) does not correspond to a zerotrace tensor. The situation is analogous to the two definitions of the second moments, discussed above, and is illustrated as follows. The trace of the tensor defined by (8.7.3.56) is given by Poisson's equation relates the divergence of the gradient of the potential to the electron density at that point: Thus, the EFG as defined by (8.7.3.56) is not traceless, unless the electron density at r is zero.
The potential and its derivatives are sometimes referred to as inner moments of the charge distribution, since the operators in (8.7.3.50), (8.7.3.52) and (8.7.3.54) contain the negative power of the position vector. In the same terminology, the electrostatic moments discussed in §8.7.3.4.1 are described as the outer moments.
It is of interest to evaluate the electric field gradient at the atomic nuclei, which for several types of nuclei can be measured accurately by nuclear quadrupole resonance and Mössbauer spectroscopy. The contribution of the atomic valence shell centred on the nucleus can be obtained by substitution of the multipolar expansion (8.7.3.7) in (8.7.3.57). The quadrupolar (l = 2) terms in the expansion contribute to the integral. For the radial function with n(l) = 2, the following expressions are obtained: with in the case that n_{2} = 4 (Stevens, DeLucia & Coppens, 1980).
The contributions of neighbouring atoms can be subdivided into pointcharge, pointmultipole, and penetration terms, as discussed by Epstein & Swanton (1982) and Su & Coppens (1992, 1994b), where appropriate expressions are given. Such contributions are in particular important when short interatomic distances are involved. For transitionmetal atoms in coordination complexes, the contribution of neighbouring atoms is typically much smaller than the valence contribution.
Hirshfelder, Curtis & Bird (1954) and Buckingham (1959) have given an expression for the potential at a point outside a charge distribution: where summation over repeated indices is implied. The outer moments q, μ_{α}, Φ_{αβ} and Ω_{αβγ} in (8.7.3.61) must include the nuclear contributions, but, for a point outside the distribution, the spherical neutralatom densities and the nuclear contributions cancel, so that the potential outside the charge distribution can be calculated from the deformation density.
The summation in (8.7.3.61) is slowly converging if the charge distribution is represented by a single set of moments. When dealing with experimental charge densities, a multicentre expansion is available from the analysis, and (8.7.3.61) can be replaced by a summation over the distributed moments centred at the nuclear positions, in which case measures the distance from a centre of the expansion to the field point. The result is equivalent to more general expressions given by Su & Coppens (1992), which, for very large values of , reduce to the sum over atomic terms, each expressed as (8.7.3.61). The interaction between two charge distributions, A and B, is given by where ρ_{B} includes the nuclear charge distribution.
The electrostatic properties of a well defined group of atoms can be derived directly from the multipole population coefficients. This method allows the `lifting' of a molecule out of the crystal, and therefore the examination of the electrostatic quantities at the periphery of the molecule, the region of interest for intermolecular interactions. The difficulty related to the origin term, encountered in the reciprocalspace methods, is absent in the directspace analysis.
In order to express the functions as a sum over atomic contributions, we rewrite (8.7.3.51), (8.7.3.54) and (8.7.3.57) for the electrostatic properties at point P as a sum over atomic contributions. in which the exclusion of M = P only applies when the point P coincides with a nucleus, and therefore only occurs for the central contributions. Z_{M} and R_{M} are the nuclear charge and the position vector of atom M, respectively, while r_{P} and r_{M} are, respectively, the vectors from P and from the nucleus M to a point r, such that r_{P} = r − R_{P}, and r_{M} = r − R_{M} = r_{P} + R_{P} − R_{M} = r_{P} − R_{MP}. The subscript M in the second, electronic part of the expressions refers to density functions centred on atom M.
Expressions for the evaluation of (8.7.3.62)–(8.7.3.64) from the chargedensity parameters of the multipole expansion have been given by Su & Coppens (1992). They employ the Fourier convolution theorem, used by Epstein & Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A directspace method based on the Laplace expansion of 1/R_{p} − r was reported by Bentley (1981).
Expression (8.7.3.49) is an example of derivation of electrostatic properties by direct Fourier summations of the structure factors. The electrostatic potential and its derivatives may be obtained in an analogous manner.
In order to obtain the electrostatic properties of the total charge distribution, it is convenient to define the `total' structure factor including the nuclear contribution, where , the summation being over all atoms j with nuclear charge , located at . If Φ(h) is defined as the Fourier transform of the directspace potential, we have and which equals −4πρ_{total} according to the Poisson equation (8.7.3.59). One obtains with and, by inverse Fourier transformation of (8.7.3.65), (Bertaut, 1978; Stewart, 1979). Furthermore, the electric field due to the electrons is given by Thus, with (8.7.3.65), which implies Similarly, the h Fourier component of the electric field gradient tensor with trace is where i : k represents the tensor product of two vectors. This leads to the expression for the electric field gradient in direct space, (The elements of h : h are the products )
The components of E and the elements of the electric field gradient defined by (8.7.3.67a) and (8.7.3.68) are with respect to the reciprocallattice coordinate system. Proper transforms are required to get the values in other coordinate systems. Furthermore, to get the traceless tensor, the quantity −(4π/3)ρ_{e}(r) = −(4π/3V)F(h)exp(−2π ih · r) must be subtracted from each of the diagonal elements
The Coulombic selfelectronic energy of the crystal can be obtained from Since (Parseval's rule), the summation can be performed in reciprocal space, and, for the total Coulombic energy, where the integral has been replaced by a summation.
The summations are rapidly convergent, but suffer from having a singularity at h = 0 (Dahl & Avery, 1984; Becker & Coppens, 1990). The contribution from this term to the potential cannot be ignored if different structures are compared. The term at h = 0 gives a constant contribution to the potential, which, however, has no effect on the energy of a neutral system. For polar crystals, an additional term occurs in (8.7.3.69a,b), which is a function of the dipole moment D of the unit cell (Becker, 1990), To obtain the total energy of the static crystal, electron exchange and correlation as well as electron kinetic energy contributions must be added.
One can write the total energy of a system as where T is the kinetic energy, represents the exchange and electron correlation contributions, and , the Coulomb energy, discussed in the previous section, is given by where is the nuclear charge for an atom at position R_{i}, and R_{ij} = R_{j} − R_{i}.
Because of the theorem of Hohenberg & Kohn (1964), E is a unique functional of the electron density ρ, so that must be a functional of ρ. Approximate density functionals are discussed extensively in the literature (Dahl & Avery, 1984) and are at the centre of active research in the study of electronic structure of various materials. Given an approximate functional, one can estimate nonCoulombic contributions to the energy from the charge density ρ(r).
In the simplest example, the functionals are those applicable to an electronic gas with slow spatial variations (the `nearly free electron gas'). In this approximation, the kinetic energy T is given by with ; and the function . The exchangecorrelation energy is also a functional of ρ,with and .
Any attempt to minimize the energy with respect to ρ in this framework leads to very poor results. However, cohesive energies can be described quite well, assuming that the change in electron density due to cohesive forces is slowly varying in space.
An example is the system AB, with closedshell subsystems A and B. Let ρ_{A} and ρ_{B} be the densities for individual A and B subsystems. The interaction energy is written as This model is known as the Gordon–Kim (1972) model and leads to a qualitatively valid description of potential energy surfaces between closedshell subsystems. Unlike pure Coulombic models, this density functional model can lead to an equilibrium geometry. It has the advantage of depending only on the charge density ρ.
Frequently, the purpose of a charge density analysis is comparison with theory at various levels of sophistication. Though the charge density is a detailed function, the features of which can be compared at several points of interest in space, it is by no means the only level at which comparison can be made. The following sequence represents a progression of functions that are increasingly related to the experimental measurement. where the angle brackets refer to the thermally averaged functions.
The experimental information may be reduced in the opposite sequence:
The crucial step in each sequence is the thermal averaging (top sequence) or the deconvolution of thermal motion (bottom sequence), which in principle requires detailed knowledge of both the internal (molecular) and the external (lattice) modes of the crystal. Even within the generally accepted Born–Oppenheimer approximation, this is a formidable task, which can be simplified for molecular crystals by the assumption that the thermal smearing is dominated by the largeramplitude external modes. The procedure (8.7.3.65) requires an adequate thermalmotion model in the structurefactor formalism applied to the experimental structure amplitudes. The commonly used models may include anharmonicity, as described in Volume B, Chapter 1.2 (ITB, 2001), but assume that a density function centred on an atom can be assigned the thermal motion of that atom, which may be a poor approximation for the more diffuse functions.
The missing link in scheme (8.7.3.75) is the sequence . In order to describe the wavefunction analytically, a basis set is required. The number of coefficients in the wavefunction is minimized by the use of a minimal basis for the molecular orbitals, but calculations in general lead to poorquality electron densities.
If the additional approximation is made that the wavefunction is a single Slater determinant, the idempotency condition can be used in the derivation of the wavefunction from the electron density. A simplified twovalenceelectron twoorbital system has been treated in this manner (Massa, Goldberg, Frishberg, Boehme & La Placa, 1985), and further developments may be expected.
A special case occurs if the overlap between the orbitals on an atom and its neighbours is very small. In this case, a direct relation can be derived between the populations of a minimal basis set of valence orbitals and the multipole coefficients, as described in the following sections.
In general, the atomcentred density model functions describe both the valence and the twocentre overlap density. In the case of transition metals, the latter is often small, so that to a good approximation the atomic density can be expressed in terms of an atomic orbital basis set , as well as in terms of the multipolar expansion. Thus, in which are the density functions.
The orbital products can be expressed as linear combinations of spherical harmonic functions, with coefficients listed in Volume B, Chapter 1.2 (ITB, 2001), which leads to relations between the and . In matrix notation, where is a vector containing the coefficients of the 15 spherical harmonic functions with l = 0, 2, or 4 that are generated by the products of d orbitals. The matrix M is also a function of the ratio of orbital and densityfunction normalization coefficients, given in Volume B, Chapter 1.2 (ITB, 2001).
The dorbital occupancies can be derived from the experimental multipole populations by the inverse expression, (Holladay, Leung & Coppens, 1983).
The matrix M^{−1} is given in Table 8.7.3.3. Pointgroupspecific expressions can be derived by omission of symmetryforbidden terms. Matrices for point group 4/mmm (square planar) and for trigonal point groups (3, , 32, 3m, ) are listed in Tables 8.7.3.4 and 8.7.3.5, respectively. Point groups with and without vertical mirror planes are distinguished by the occurrence of both and functions in the latter case, and only in the former, with m being restricted to n, the order of the rotation axis. The functions can be eliminated by rotation of the coordinate system around a vertical axis through an angle given by .



In the Born–Oppenheimer approximation, the electrons rearrange instantaneously to the minimumenergy state for each nuclear configuration. This approximation is generally valid, except when very low lying excited electronic states exist. The thermally smeared electron density is then given bywhere R represents the 3N nuclear space coordinates and P(R) is the probability of the configuration R. Evaluation of (8.7.3.79) is possible if the vibrational spectrum is known, but requires a large number of quantummechanical calculations at points along the vibrational path. A further approximation is the convolution approximation, which assumes that the charge density near each nucleus can be convoluted with the vibrational motion of that nucleus, where ρ_{n} stands for the density of the nth pseudoatom. The convolution approximation thus requires decomposition of the density into atomic fragments. It is related to the thermalmotion formalisms commonly used, and requires that twocentre terms in the theoretical electron density be either projected into the atomcentred density functions, or assigned the thermal motion of a point between the two centres. In the LCAO approximation (8.7.3.9), the twocentre terms are represented by where χ_{μ} and χ_{ν} are basis functions centred at R_{μ} and R_{ν}, respectively. As the motion of a point between the two vibrating atoms depends on their relative phase, further assumptions must be made. The simplest is to assume a gradual variation of the thermal motion along the bond, which gives at a point r_{i} on the internuclear vector of length R_{μν} This expression may be used to assign thermal parameters to a bondcentred function.
Thermal averaging of the electron density is considerably simplified for modes in which adjacent atoms move in phase. In molecular crystals, such modes correspond to rigidbody vibrations and librations of the molecule as a whole. Their frequencies are low because of the weakness of intermolecular interactions. Rigidbody motions therefore tend to dominate thermal motion, in particular at temperatures for which kT (k = 0.7 cm^{−1}) is large compared with the spacing of the vibrational energy levels of the external modes (internal modes are typically not excited to any extent at or below room temperature).
For a translational displacement (u), the dynamic density is given by with ρ(r) defined by (8.7.3.81) (Stevens, Rees & Coppens, 1977). In the harmonic approximation, P(u) is a normalized threedimensional Gaussian probability function, the exponents of which may be obtained by rigidbody analysis of the experimental data. In general, for a translational displacement (u) and a librational oscillation (ω), If correlation between u and ω can be ignored (neglect of the screw tensor S), P(u, ω) = P(u)P(ω), and both types of modes can be treated independently. For the translations where F^{−1} is the inverse Fourier transform operator, and T_{tr}(h) is the translational temperature factor.
If R is an orthogonal rotation matrix corresponding to a rotation ω, we obtain for the librations in which f(h) has been averaged over the distribution of orientations of h with respect to the molecule;
Evaluation of (8.7.3.85) and (8.7.3.86) is most readily performed if the basis functions ψ have a Gaussiantype radial dependence, or are expressed as a linear combination of Gaussian radial functions.
For Gaussian products of s orbitals, the molecular scattering factor of the product ψ_{μ}ψ_{ν} = N_{μ}exp[−α_{μ}(r − r_{A})^{2}] × N_{ν}exp [−α_{ν} (r − r_{B})^{2}], where N_{μ} and N_{ν} are the normalization factors of the orbitals μ and ν centred on atoms A and B, is given by where the centre of density is defined by r_{c} = (α_{μ} r_{A} + α_{ν} r_{B})/(α_{μ} + α_{ν}).
For the translational modes, the temperaturefactor exponent is simply added to the Gaussian exponent in (8.7.3.88) to give For librations, we may write As , for a function centred at r, which shows that for ss orbital products the librational temperature factor can be factored out, or Expressions for P(ω) are described elsewhere (Pawley & Willis, 1970).
For general Cartesian Gaussian basis functions of the type the scattering factors are more complicated (Miller & Krauss, 1967; Stevens, Rees & Coppens, 1977), and the librational temperature factor can no longer be factored out. However, it may be shown that, to a first approximation, (8.7.3.90) can again be used. This `pseudotranslation' approximation corresponds to a neglect of the change in orientation (but not of position) of the twocentre density function and is adequate for moderate vibrational amplitudes.
Thermally smeared density functions are obtained from the averaged reciprocalspace function by performing the inverse Fourier transform with phase factors depending on the position coordinates of each orbital product where the orbital product χ_{μ}χ_{ν} is centred at . If the summation is truncated at the experimental limit of (sin θ)/λ, both thermal vibrations and truncation effects are properly introduced in the theoretical densities.
It is often important to obtain an estimate of the uncertainty in the deformation densities in Table 8.7.3.1. If it is assumed that the density of the static atoms or fragments that are subtracted out are precisely known, three sources of uncertainty affect the deformation densities: (1) the uncertainties in the experimental structure factors; (2) the uncertainties in the positional and thermal parameters of the density functions that influence ρ_{calc}; and (3) the uncertainty in the scale factor k.
If we assume that the uncertainties in the observed structure factors are not correlated with the uncertainties in the refined parameters, the variance of the electron density is given by where , u_{p} is a positional or thermal parameter, and the γ(u_{p}, k) are correlation coefficients between the scale factor and the other parameters (Rees, 1976, 1978; Stevens & Coppens, 1976).
Similarly, for the covariance between the deformation densities at points A and B, where it is implied that the second term includes the effect of the scale factor/parameter correlation.
Following Rees (1976), we may derive a simplified expression for the covariance valid for the space group . Since the structure factors are not correlated with each other, where the latter equality is specific for , and indicates summation over a hemisphere in reciprocal space. In general, the second term rapidly averages to zero as increases, while the first term may be replaced by its average with u = 2π r_{A} − r_{B}h_{max}, or cov(ρ_{obs, A}ρ_{obs, B}) (2/V^{2})C(u) × and σ^{2}(ρ_{obs}) , a relation derived earlier by Cruickshank (1949). A discussion of the applicability of this expression in other centrosymmetric space groups is given by Rees (1976).
The electrostatic moments are functions of the scale factor, the positional parameters x, y and z of the atoms, and their chargedensity parameters κ and . The standard uncertainties in the derived moments are therefore dependent on the variances and covariances of these parameters.
If M_{p} represents the m × m variance–covariance matrix of the parameters , and T is an n × m matrix defined by for the lth moment with n independent elements, the variances and covariances of the elements of m^{l} are obtained from If a moment of an assembly of pseudoatoms is evaluated, the elements of T include the effects of coordinate rotations required to transfer atomic moments into a common coordinate system.
A frequently occurring case of interest is the evaluation of the magnitude of a molecular dipole moment and its standard deviation. Defining where is the dipolemoment vector and G is the directspace metric tensor of the appropriate coordinate system. If Y is the 1 × 3 matrix of the derivatives , where M_{μ} is defined by (8.7.3.96a). The standard uncertainty in μ may be obtained from σ(μ) = σ(y)/2μ. Significant contributions often result from uncertainties in the positional and chargedensity parameters of the H atoms.
Magnetism and magnetic ordering are among the central problems in condensedmatter research. One of the main issues in macroscopic studies of magnetism is a description of the magnetization density as a function of temperature and applied field: phase diagrams can be explained from such studies.
Diffraction techniques allow determination of the same information, but at a microscopic level. Let m(r) be the microscopic magnetization density, a function of the position r in the unit cell (for crystalline materials). Macroscopic and microscopic magnetization densities are related by the simple expression where V is the volume of the unit cell, is the sum of two contributions: originating from the spins of the electrons, and originating from their orbital motion.
The scattering process is discussed in Section 6.1.3 and only the features that are essential to the present chapter will be summarized here.
For neutrons, the nuclear structure factor is given by , , are the coherent scattering length, the temperature factor, and the equilibrium position of the jth atom in the unit cell.
Let σ be the spin of the neutron (in units of ). There is a dipolar interaction of the neutron spin with the electron spins and the currents associated with their motion. The magnetic structure factor can be written as the scalar product of the neutron spin and an `interaction vector' Q(h): Q(h) is the sum of a spin and an orbital term: and , respectively. If r_{0} = γr_{e} ~ 0.54 × 10^{−12} cm, where γ is the gyromagnetic factor (= 1.913) of the neutron and the classical Thomson radius of the electron, the spin term is given by being the unit vector along h, while M_{s}(h) is defined as where is the spin of the electron at position , and angle brackets denote the ensemble average over the scattering sample. is the Fourier transform of the spinmagnetization density , given by This is the spindensity vector field in units of 2μ_{B}.
The orbital part of Q(h) is given by where is the momentum of the electrons. If the current density vector field is defined by Q_{L}(h) can be expressed as where J(h) is the Fourier transform of the current density j(r).
The electrodynamic properties of j(r) allow it to be written as the sum of a rotational and a nonrotational part: where is a `conduction' component and is an `orbitalmagnetization' density vector field.
Substitution of the Fourier transform of (8.7.4.11) into (8.7.4.10) leads in analogy to (8.7.4.5) to where M_{L}(h) is the Fourier transform of m_{L}(r). The rotational component of j(r) does not contribute to the neutron scattering process. It is therefore possible to write Q(h) as with being the Fourier transform of the `total' magnetization density vector field, and As Q(h) is the projection of M(h) onto the plane perpendicular to h, there is no magnetic scattering when M is parallel to h. It is clear from (8.7.4.13) that M(h) can be defined to any vector field V(h) parallel to h, i.e. such that h × V(h) = 0.
This means that in real space m(r) is defined to any vector field v(r) such that × v(r) = 0. Therefore, m(r) is defined to an arbitrary gradient.
As a result, magnetic neutron scattering cannot lead to a uniquely defined orbital magnetization density. However, the definition (8.7.4.7) for the spin component is unambiguous.
However, the integrated magnetic moment is determined unambiguously and must thus be identical to the magnetic moment defined from the principles of quantum mechanics, as discussed in §8.7.4.5.1.3.
Before discussing the analysis of magnetic neutron scattering in terms of spindensity distributions, it is necessary to give a brief description of the quantummechanical aspects of magnetization densities.
Let us consider first an isolated openshell system, whose orbital momentum is quenched: it is a spinonly magnetism case. Let be the spinmagnetizationdensity operator (in units of ): and are, respectively, the position and the spin operator (in units) of the jth electron. This definition is consistent with (8.7.4.7).
The system is assumed to be at zero temperature, under an applied field, the quantization axis being Oz. The ground state is an eigenstate of and , where is the total spin: Let S and M_{s} be the eigenvalues of and . (M_{s} will in general be fixed by Hund's rule: M_{S} = S.) where and are the numbers of electrons with and spin, respectively.
The spinmagnetization density is along z, and is given by is proportional to the normalized spin density that was defined for a pure state in (8.7.2.10). If and are the charge densities of electrons of a given spin, the normalized spin density is defined as compared with the total charge density ρ(r) given by A strong complementarity is thus expected from joint studies of ρ(r) and s(r).
In the particular case of an independent electron model, where is an occupied orbital for a given spin state of the electron.
If the ground state is described by a correlated electron model (mixture of different configurations), the oneparticle reduced density matrix can still be analysed in terms of its eigenvectors and eigenvalues (natural spin orbitals and natural occupancies), as described by the expression where <ψ_{iα}ψ_{jβ}> = δ_{ij}δ_{αβ}, since the natural spin orbitals form an orthonormal set, and
As the quantization axis is arbitrary, (8.7.4.20) can be generalized to Equation (8.7.4.26) expresses the proportionality of the spinmagnetization density to the normalized spin density function.
The system is now assumed to be at a given temperature T. S remains a good quantum number, but all states are now populated according to Boltzmann statistics. We are interested in the thermal equilibrium spinmagnetization density: where is the population of the state. The operator fulfils the requirements to satisfy the Wigner–Eckart theorem (Condon & Shortley, 1935), which states that, within the S manifold, all matrix elements of are proportional to . The consequence of this remarkable property is that where is a function that depends on S, but not on . Comparison with (8.7.4.26) shows that is the normalized spindensity function s(r), which therefore is an invariant for the S manifold [s(r) is calculated as the normalized spin density for any . Expression (8.7.4.27) can thus be written as where <S> is the expected value for the total spin, at a given temperature and under a given external field. As s(r) is normalized, the total moment of the system is The behaviour of <S> is governed by the usual laws of magnetism: it can be measured by macroscopic techniques. In paramagnetic species, it will vary as T^{−1} to a first approximation; unless the system is studied at very low temperatures, the value of <S> will be very small. The dependence of <S> on temperature and orienting field is crucial.
Finally, (8.7.4.29) has to be averaged over vibrational modes. Except for the case where there is strong magnetovibrational interaction, only s(r) is affected by thermal atomic motion. This effect can be described in terms similar to those used for the charge density (Subsection 8.7.3.7).
The expression (8.7.4.29) is very important and shows that the microscopic spinmagnetization density carries two types of information: the nature of spin ordering in the system, described by <S>, and the delocalized nature of the electronic ground state, represented by s(r).
A complex magnetic system can generally be described as an ensemble of well defined interacting openshell subsystems (ions or radicals), where each subsystem has a spin , and is assumed to be a good quantum number. The magnetic interaction occurs essentially through exchange mechanisms that can be described by the Heisenberg Hamiltonian: where is the exchange coupling between two subsystems, and B_{0} an applied external field (magnetocrystalline anisotropic effects may have to be added). Expression (8.7.4.30) is the basis for the understanding of magnetic ordering and phase diagrams. The interactions lead to a local field B_{n}, which is the effective orienting field for the spin S_{n}.
The expression for the spinmagnetization density is The relative arrangement of <S_{n}> describes the magnetic structure; is the normalized spin density of the nth subsystem.
In some metallic systems, at least part of the unpaired electron system cannot be described within a localized model: a bandstructure description has to be used (Lovesey, 1984). This is the case for transition metals like Ni, where the spinmagnetization density is written as the sum of a localized part [described by (8.7.4.31)] and a delocalized part [described by (8.7.4.29)].
We must now address the case where the orbital moment is not quenched. In that case, there is some spinorbit coupling, and the description of the magnetization density becomes less straightforward.
The magnetic moment due to the angular momentum of the electron is (in units of 2μ_{B}). As does not commute with the position , orbital magnetization density is defined as If L is the total orbital moment, Only open shells contribute to the orbital moment. But, in general, neither L^{2} nor L_{z} are constants of motion. There is, however, an important exception, when openshell electrons can be described as localized around atomic centres. This is the case for most rareearth compounds, for which the 4f electrons are too close to the nuclei to lead to a significant interatomic overlap. It can also be a first approximation for the d electrons in transitionmetal ions. Spinorbit coupling will be present, and thus only L^{2} will be a constant of motion. One may define the total angular momentum and {J^{2}, L^{2}, S^{2}, J_{z}} become the four constants of motion.
Within the J manifold of the ground state, and do not, in general, fulfil the conditions for the Wigner–Eckart theorem (Condon & Shortley, 1935), which leads to a very complex description of m(r) in practical cases.
However, the Wigner–Eckart theorem can be applied to the magnetic moments themselves, leading to with the Lande factor and, equivalently, The influence of spin–orbit coupling on the scattering will be discussed in Subsection 8.7.4.5.
The magnetic structure factor [equation (8.7.4.4)] depends on the spin state of the neutron. Let λ be the unit vector defining a quantization axis for the neutron, which can be either parallel or antiparallel to . If stands for the cross section where the incident neutron has the polarization σ and the scattered neutron the polarization σ′, one obtains the following basic expressions: If no analysis of the spin state of the scattered beam is made, the two measurable cross sections are which depend only on the polarization of the incident neutron.
If the incident neutron beam is not polarized, the scattering cross section is given by Magnetic and nuclear contributions are simply additive. With , one obtains Owing to its definition, x can be of the order of 1 if and only if the atomic moments are ordered close to saturation (as in the ferro or antiferromagnets). In many situations of structural and chemical interest, x is small.
If, for example, x , the magnetic contribution in (8.7.4.41) is only 0.002 of the total intensity. Weak magnetic effects, such as occur for instance in paramagnets, are thus hardly detectable with unpolarized neutron scattering.
However, if the magnetic structure does not have the same periodicity as the crystalline structure, magnetic components in (8.7.4.40) occur at scattering vectors for which the nuclear contribution is zero. In this case, the unpolarized technique is of unique interest. Most phase diagrams involving antiferromagnetic or helimagnetic order and modulations of such ordering are obtained by this method.
It is generally possible to polarize the incident beam by using as a monochromator a ferromagnetic alloy, for which at a given Bragg angle , because of a cancellation of nuclear and magnetic scattering components. The scatteredbeam intensity is thus . By using a radiofrequency (r.f.) coil tuned to the Larmor frequency of the neutron, the neutron spin can be flipped into the state for which the scattered beam intensity is . This allows measurement of the `flipping ratio' R(h): As the two measurements are made under similar conditions, most systematic effects are eliminated by this technique, which is only applicable to cases where both and occur at the same scattering vectors. This excludes any antiferromagnetic type of ordering.
If is assumed to be in the vertical Oz direction, M(h) will in most situations be aligned along Oz by an external orienting field. If α is the angle between M and h, and with expressed in the same units as , one obtains, for centrosymmetric crystals, If , For and α = π/2, R now departs from 1 by as much as 20%, which proves the enormous advantage of polarized neutron scattering in the case of low magnetism.
Equation (8.7.4.44) can be inverted, and x and its sign can be obtained directly from the observation. However, in order to obtain M(h), the nuclear structure factor must be known, either from nuclear scattering or from a calculation. All systematic errors that affect are transferred to M(h).
For two reasons, it is not in general feasible to access all reciprocallattice vectors. First, in order to have reasonable statistical accuracy, only reflections for which both and are large enough are measured; i.e. reflections having a strong nuclear structure factor. Secondly, should be as close to 1 as possible, which may prevent one from accessing all directions in reciprocal space. If M is oriented along the vertical axis, the simplest experiment consists of recording reflections with h in the horizontal plane, which leads to a projection of m(r) in real space. When possible, the sample is rotated so that other planes in the reciprocal space can be recorded.
Finally, if α = π/2, vanishes, and neutron spin is conserved in the experiment.
If the space group is noncentrosymmetric, both and M have a phase, and , respectively.
If for simplicity one assumes α = π/2, and, defining δ = ϕ_{M} − ϕ_{N}, which shows that x and δ cannot both be obtained from the experiment.
The noncentrosymmetric case can only be solved by a careful modelling of the magnetic structure factor as described in Subsection 8.7.4.5.
In practice, neither the polarization of the incident beam nor the efficiency of the r.f. flipping coil is perfect. This leads to a modification in the expression for the flipping ratios [see Section 6.1.3 or Forsyth (1980)].
Since most measurements correspond to strong nuclear structure factors, extinction severely affects the observed data. To a first approximation, one may assume that both and will be affected by this process, though the spinflip processes and are not. If and are the associated extinction factors, the observed flipping ratio is where the expressions for are given elsewhere (Bonnet, Delapalme, Becker & Fuess, 1976).
It should be emphasized that, even in the case of small magnetic structure factors, extinction remains a serious problem since, even though and may be very close to each other, so are and . An incorrect treatment of extinction may entirely bias the estimate of x.
In the most general case, it is not possible to obtain x, and thus M(h) directly from R. Moreover, it is unlikely that all Bragg spots within the reflection sphere could be measured. Modelling of M(h) is thus of crucial importance. The analysis of data must proceed through a leastsquares routine fitting to , minimizing the error function where corresponds to a model and σ^{2}(R) is the standard uncertainty for R.
If the same counting time for and for is assumed, only the counting statistical error may be considered important in the estimate of R, as most systematic effects cancel. In the simple case where α = π/2, and the structure is centrosymmetric, a straightforward calculation leads to with one obtains the result In the common case where , this reduces to In addition to this estimate, care should be taken of extinction effects.
The real interest is in M(h), rather than x: If is obtained by a nuclear neutron scattering experiment, where a accounts for counting statistics and b for systematic effects.
The first term in (8.7.4.53) is the leading one in many situations. Any systematic error in can have a dramatic effect on the estimate of M(h).
In this subsection, the case of spinonly magnetization is considered. The modelling of is very similar to that of the charge density.
We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transitionmetal atoms. The magnetization density is expanded as where is the spin at site j, and the thermally averaged normalized spin density , the Fourier transform of , is known as the `magnetic form factor'. Thus, where and are the Debye–Waller factor and the equilibrium position of the jth site, respectively.
Most measurements are performed at temperatures low enough to ensure a fair description of at the harmonic level (Coppens, 2001). represents the vibrational relaxation of the openshell electrons and may, in some situations, be different from the Debye–Waller factor of the total charge density, though at present no experimental evidence to this effect is available.
In the crudest model, is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function U(r), the magnetic form factor is given by where is the zeroorder spherical Bessel function. For free atoms and ions, these form factors can be found in IT IV (1974).
One of the important features of magnetic neutron scattering is the fact that, to a first approximation, closed shells do not contribute to the form factor. Thus, it is a unique probe of the electronic structure of heavy elements, for which theoretical calculations even at the atomic level are questionable. Relativistic effects are important. Theoretical relativistic form factors can be used (Freeman & Desclaux, 1972; Desclaux & Freeman, 1978). It is also possible to parametrize the radial behaviour of U. A single contractionexpansion model [κ refinement, expression (8.7.3.6)] is easy to incorporate.
Crystalfield effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the groundstate configuration of the system. Thus, they have to be incorporated in the model.
Taking the case of a transitionmetal compound, and neglecting small contributions that may arise from spin polarization in the closed shells (see Subsections 8.7.4.9 and 8.7.4.10), the normalized spin density can be written by analogy with (8.7.3.76) as where is the normalized spin population matrix. If and are the densities of a given spin, the dtype charge density is and is expanded in a similar way to s(r) [see (8.7.3.76)], writing with σ = and , one obtains Similarly to , can be expanded as where U(r) describes the radial dependence. Spin polarization leads to a further modification of this expression. Since the numbers of electrons of a given spin are different, the exchange interaction is different for the two spin states, and a spindependent effective screening occurs. This leads to with σ = or , where two κ parameters are needed.
The complementarity between charge and spin density in the crystal field approximation is obvious. At this particular level of approximation, expansions are exact and it is possible to estimate dorbital populations for each spin state.
The magnetic structure factor is scaled to . Whether the nuclear structure factors are calculated from refined structural parameters or obtained directly from a measurement, their scale factor is not rigorously fixed.
As a result, it is not possible to obtain absolute values of the effective spins <S_{n}> from a magnetic neutron scattering experiment. It is necessary to scale them through the sum rule where is the macroscopic magnetization of the sample.
The practical consequences of this constraint for a refinement are similar to the electroneutrality constraint in chargedensity analysis (§8.7.3.3.1).
In this subsection, the localized magnetism picture is assumed to be valid. However, each subunit can now be a complex ion or a radical. Covalent interactions must be incorporated.
If are atomic basis functions, the spin density s(r) can always be written as an expansion that is similar to (8.7.3.9) for the charge density.
To a first approximation, only the basis functions that are required to describe the open shell have to be incorporated, which makes the expansion simple and more flexible than for the charge density.
As for the charge density, it is possible to project (8.7.4.66) onto the various sites of the ion or molecule by a multipolar expansion: P(R_{j}) is the vibrational p.d.f. of the jth atom, which implies use of the convolution approximation.
The monopolar terms D_{j00} give an estimate of the amount of spin transferred from a central metal ion to the ligands, or of the way spins are shared among the atoms in a radical.
The constraint (8.7.4.65) becomes where (n) refers to the various subunits in the unit cell. Expansion (8.7.4.67) is the key for solving noncentrosymmetric magnetic structure (Boucherle, Gillon, Maruani & Schweizer, 1982).
One may wish to take advantage of the fact that, to a good approximation, only a few molecular orbitals are involved in s(r). In an independent particle model, one expands the relevant orbitals in terms of atomic basis functions (LCAO): with σ = or , and the spin density is expanded according to (8.7.4.21) and (8.7.4.23). Fourier transform of two centreterm products is required. Details can be found in Forsyth (1980) and Tofield (1975).
In the case of extended solids, expansion (8.7.4.69) must refer to the total crystal, and therefore incorporate translational symmetry (Brown, 1986).
Finally, in the simple systems such as transition metals, like Ni, there is a d–s type of interaction, leading to some contribution to the spin density from delocalized electrons (Mook, 1966). If and are the localized and delocalized parts of the density, respectively, where a is the fraction of localized spins. s_{d}(r) can be modelled as being either constant or a function with a very small number of Fourier adjustable coefficients.
Q_{L}(h) is given by (8.7.4.10) and (8.7.4.12). Since in (8.7.4.11) does not play any role in the scattering cross section, we can use the restriction where is defined to an arbitrary gradient. It is possible to constrain to have the form since any radial component could be considered as the radial component of a gradient. With spherical coordinates, (8.7.4.71) becomes which can be integrated as one finally obtains With f(x) defined by the expression for M_{L}(h) is obtained by Fourier transformation of (8.7.4.72): which leads to by using the definition (8.7.4.9) of j.
This expression clearly shows the connection between orbital magnetism and the orbital angular momentum of the electrons. It is of general validity, whatever the origin of orbital magnetism.
The simplest approximation involves decomposing j(r) into atomic contributions: One obtains is the atomic magnetic orbital structure factor.
We notice that as defined in (8.7.4.73) can be expanded as where is a spherical Bessel function of order l, and are the complex spherical harmonic functions.
If one considers only the spherically symmetric term in (8.7.4.78), one obtains the `dipolar approximation', which gives with and is the radial function of the atomic electrons whose orbital momentum is unquenched. Thus, in the dipolar approximation, the atomic orbital scattering is proportional to the effective orbital angular momentum and therefore to the orbital part of the magnetic dipole moment of the atom.
Within the same level of approximation, the spin structure factor is with and Finally, the atomic contribution to the total magnetic structure factor is If <J_{n}> is the total angular momentum of atom n and its gyromagnetic ratio, (8.7.4.35)–(8.7.4.37) lead to:
Another approach, which is applicable only to the atomic case, is often used, which is based on Racah's algebra (Marshall & Lovesey, 1971). At the dipolar approximation level, it leads to a slightly different result, according to which <γ_{0n}> is replaced by The two results are very close for small h where the dipolar approximation is correct. With (8.7.4.35)–(8.7.4.37), (8.7.4.84a) can also be written as where the second term is the `orbital correction'. Its magnitude clearly depends on the difference between and 2, which is small in 3d elements but can become important for rare earths.
Expressions (8.7.4.74) and (8.7.4.78) are valid in any situation where orbital scattering occurs. They can in principle be used to estimate from the diffraction experiment the contribution of a few configurations that interact due to the operator. In delocalized situations, (8.7.4.74) is the most suitable approach, while Racah's algebra can only be applied to onecentre cases.
When covalency is small, the major aims are the determination of the ground state of the rareearth ion, and the amount of delocalized magnetization density via the conduction electrons.
The ground state ψ> of the ion is written as which is well suited for the Johnston (1966) and Marshall & Lovesey (1971) formulation in terms of general angularmomentum algebra. A multipolar expansion of spin and orbital components of the structure factor enables a determination of the expansion coefficient (Schweizer, 1980).
The derivation of electrostatic properties from the charge density was treated in Subsection 8.7.3.4. Magnetostatic properties can be derived from the spinmagnetization density m_{s}(r) using parallel expressions.
The vector potential field is defined as In the case of a crystal, it can be expanded in Fourier series: the magnetic field is simply One notices that there is no convergence problem for the h = 0 term in the B(r) expansion.
The magnetostatic energy, i.e. the amount of energy that is required to obtain the magnetization m_{s}, is
It is often interesting to look at the magnetostatics of a given subunit: for instance, in the case of paramagnetic species.
For example, the vector potential outside the magnetized system can be obtained in a similar way to the electrostatic potential (8.7.3.30):
If can be easily expanded in powers of 1/r′, and A(r′) can thus be obtained in powers of 1/r′. If m_{s}(r) = <S>s(r),
Among the various properties that are derivable from the delocalized spin density function, the dipole coupling tensor is of particular importance: where R_{n} is a nuclear position and r_{n} = r − R_{n}. This dipolar tensor is involved directly in the hyperfine interaction between a nucleus with spin and an electronic system with spin s, through the interaction energy This tensor is measurable by electron spin resonance for either crystals or paramagnetic species trapped in matrices. The complementarity with scattering is thus of strong importance (Gillon, Becker & Ellinger, 1983).
Computational aspects are the same as in the electric field gradient calculation, ρ(r) being simply replaced by s(r) (see Subsection 8.7.3.4).
Since it is a measure of the imbalance between the densities associated with the two spin states of the electron, the spindensity function is a probe that is very sensitive to the exchange forces in the system. In an independentparticle model (Hartree–Fock approximation), the exchange mean field potential involves exchange between orbitals with the same spin. Therefore, if the numbers of and spins are different, one expects to be different from . The main consequence of this is the necessity to solve two different Fock equations, one for each spin state. This is known as the spinpolarization effect: starting from a paired orbital, a slight spatial decoupling arises from this effect, and closed shells do have a participation in the spin density.
It can be shown that this effect is hardly visible in the charge density, but is enhanced in the spin density.
Spin densities are a very good probe for calculations involving this spinpolarization effect: The unrestricted Hartree–Fock approximation (Gillon, Becker & Ellinger, 1983).
From a common spinrestricted approach, spin polarization can be accounted for by a mixture of Slater determinants (configuration interaction), where the configuration interaction is only among electrons with the same spin. There is also a correlation among electrons with different spins, which is more difficult to describe theoretically. There seems to be evidence for such effects from comparison of experimental and theoretical spin densities in radicals (Delley, Becker & Gillon, 1984), where the unrestricted Hartree–Fock approximation is not sufficient to reproduce experimental facts. In such cases, localspindensity functional theory has revealed itself very satisfactorily. It seems to offer the most efficient way to include correlation effects in spindensity functions.
As noted earlier, analysis of the spindensity function depends more on modelling than that of the charge density. Therefore, in general, `experimental' spin densities at static densities and the problem of theoretical averaging is minor here. Since spin density involves essentially outerelectron states, resolution in reciprocal space is less important, except for analysis of the polarization of the core electrons.
Combined charge and spindensity analysis requires performing Xray and neutron diffraction experiments at the same temperature. Magnetic neutron experiments are often only feasible around 4 K, and such conditions are more difficult to achieve by Xray diffraction. Even if the two experiments are to be performed at different temperatures, it is often difficult to identify compounds suitable for both experiments.
Owing to the common parametrization of ρ(r) and s(r), a combined leastsquaresrefinement procedure can be implemented, leading to a description of and , the spindependent electron densities. Covalency parameters are obtainable together with spin polarization effects in the closed shells, by allowing and to have different radial behaviour.
Spinpolarization effects would be difficult to model from the spin density alone. But the arbitrariness of the modelling is strongly reduced if both ρ and s are analysed at the same time (Becker & Coppens, 1985; Coppens, Koritsanszky & Becker, 1986).
In addition to the usual Thomson scattering (charge scattering), there is a magnetic contribution to the Xray amplitude (de Bergevin & Brunel, 1981; Blume, 1985; Brunel & de Bergevin, 1981; Blume & Gibbs, 1988). In units of the chemical radius of the electron, the total scattering amplitude is where F_{C} is the charge contribution, and F_{M} the magnetic part.
Let and be the unit vectors along the electric field in the incident and diffracted direction, respectively. k and k′ denote the wavevectors for the incident and diffracted beams. With these notations, where F(h) is the usual structure factor, which was discussed in Section 8.7.3 [see also Coppens (2001)]. and are the orbital and spinmagnetization vectors in reciprocal space, and A and B are vectors that depend in a rather complicated way on the polarization and the scattering geometry: For comparison, the magnetic neutron scattering amplitude can be written in the form with .
From (8.7.4.99), it is clear that spin and orbital contributions cannot be separated by neutron scattering. In contrast, the polarization dependencies of and are different in the Xray case. Therefore, owing to the well defined polarization of synchrotron radiation, it is in principle possible to separate experimentally spin and orbital magnetization.
However, the prefactor makes the magnetic contributions weak relative to charge scattering. Moreover, F_{C} is roughly proportional to the total number of electrons, and F_{M} to the number of unpaired electrons. As a result, one expects to be about 10^{−3}.
It should also be pointed out that F_{M} is in quadrature with F_{C}. In many situations, the total Xray intensity is therefore Thus, under these conditions, the magnetic effect is typically 10^{−6} times the Xray intensity.
Magnetic contributions can be detected if magnetic and charge scattering occur at different positions (antiferromagnetic type of ordering). Furthermore, Blume (1985) has pointed out that the photon counting rate for at synchrotron sources is of the same order as the neutron rate at highflux reactors.
Finally, situations where the `interference' term is present in the intensity are very interesting, since the magnetic contribution becomes 10^{−3} times the charge scattering.
The polarization dependence will now be discussed in more detail.
Some geometrical definitions are summarized in Fig. 8.7.4.1, where parallel and perpendicular polarizations will be chosen in order to describe the electric field of the incident and diffracted beams. In this twodimensional basis, vectors A and B of (8.7.4.98) can be written as (2 × 2) matrices: (i and f refer to the incident and diffracted beams, respectively); By comparison, for the Thomson scattering, The major difference with Thomson scattering is the occurrence of offdiagonal terms, which correspond to scattering processes with a change of polarization. We obtain for the structure factors : For a linear polarization, the measured intensity in the absence of diffractedbeam polarization analysis is where α is the angle between E and . In the centrosymmetric system, without anomalous scattering, no interference term occurs in (8.7.4.103). However, if anomalous scattering is present, F = F′ + iF′′, and terms involving or appear in the intensity expression.
The radiation emitted in the plane of the electron or positron orbit is linearly polarized. The experimental geometry is generally such that is a vertical plane. Therefore, the polarization of the incident beam is along (α = 0). If a diffractedbeam analyser passes only components of the diffracted beam, one can measure , and thus eliminate the charge scattering.
For nonpolarized radiation (with a rotating anode, for example), the intensity is The radiation emitted out of the plane of the orbit contains an increasing amount of circularly polarized radiation. There also exist experimental devices that can produce circularly polarized radiation. For such incident radiation, for left or rightpolarized photons. If E′ is the field for the diffracted photons, In this case, `mixedpolarization' contributions are in phase with F, leading to a strong interference between charge and magnetic scattering.
The case of radiation with a general type of polarization is more difficult to analyse. The most elegant formulation involves Stokes vectors to represent the state of polarization of the incident and scattered radiation (see Blume & Gibbs, 1988).
Acknowledgements
Support of this work by the National Science Foundation (CHE8711736 and CHE9615586) is gratefully acknowledged.
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