Chapter 8.7. Analysis of charge and spin densities

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 least-squares 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 non-zero 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 least-squares 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 least-squares 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 non-integral electron population for the core functions. The proper interpretation of such a result is that a core-like 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 1s-type functions have non-zero electron density at the nucleus and contribute to (8.7.3.13). For the hydrogen-like 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 ∇²Φ(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 non-divergence of the electric field ∇V, its gradient ∇²V, the gradient of the field gradient ∇³V, 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 X-ray 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 X-ray 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.

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 second-rank tensor, and the octupole, a third-rank 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 lth-rank 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 3z² − 1, xz, yz, (x² − y²)/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

8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation

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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 low-order 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).

Table 8.7.3.2| top | pdf | Expressions for the shape factors S for a parallelepiped with edges δx, δy, and δz (from Moss & Coppens, 1981) j0 and j1 are the zero- and first-order spherical Bessel functions: j0(x) = sin x/x, j1(x) = sin x/x2 − cos x/x; VT is volume of integration. Property 1 Charge rα Dipole μα rαrβ Second moment μαβ off-diagonal rαrα Second moment μαα diagonal

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 direct-space 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 reciprocal-lattice 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 self-electronic 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 non-Coulombic 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 exchange-correlation 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 closed-shell 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 closed-shell 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.

In the Born–Oppenheimer approximation, the electrons rearrange instantaneously to the minimum-energy 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 quantum-mechanical 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 pseudo-atom. The convolution approximation thus requires decomposition of the density into atomic fragments. It is related to the thermal-motion formalisms commonly used, and requires that two-centre terms in the theoretical electron density be either projected into the atom-centred density functions, or assigned the thermal motion of a point between the two centres. In the LCAO approximation (8.7.3.9), the two-centre 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 bond-centred 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 rigid-body vibrations and librations of the molecule as a whole. Their frequencies are low because of the weakness of intermolecular interactions. Rigid-body 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 three-dimensional Gaussian probability function, the exponents of which may be obtained by rigid-body 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 Gaussian-type 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_μψ_ν = N_μexp[−α_μ(r − r_A)²] × N_νexp [−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 temperature-factor 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 two-centre density function and is adequate for moderate vibrational amplitudes.

Thermally smeared density functions are obtained from the averaged reciprocal-space function by performing the inverse Fourier transform with phase factors depending on the position coordinates of each orbital product where the orbital product χ_μχ_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.

Let us consider first an isolated open-shell system, whose orbital momentum is quenched: it is a spin-only magnetism case. Let be the spin-magnetization-density 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 spin-magnetization 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 one-particle 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_ijδ_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 spin-magnetization 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 spin-magnetization 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 spin-density 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 magneto-vibrational 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 spin-magnetization 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 open-shell 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 B₀ an applied external field (magneto-crystalline 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 spin-magnetization 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 band-structure description has to be used (Lovesey, 1984). This is the case for transition metals like Ni, where the spin-magnetization 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 spin-orbit 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 L² nor L_z are constants of motion. There is, however, an important exception, when open-shell electrons can be described as localized around atomic centres. This is the case for most rare-earth 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 transition-metal ions. Spin-orbit coupling will be present, and thus only L² will be a constant of motion. One may define the total angular momentum and {J², L², S², 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 scattered-beam intensity is thus . By using a radio-frequency (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.

The experimental set-up is discussed by Forsyth (1980).

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 reciprocal-lattice 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 spin-flip 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 least-squares routine fitting to , minimizing the error function where corresponds to a model and σ²(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).

We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transition-metal 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 open-shell 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 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 centre-term 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.

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 one-centre cases.

When covalency is small, the major aims are the determination of the ground state of the rare-earth 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 angular-momentum algebra. A multipolar expansion of spin and orbital components of the structure factor enables a determination of the expansion coefficient (Schweizer, 1980).

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).

In addition to the usual Thomson scattering (charge scattering), there is a magnetic contribution to the X-ray 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 spin-magnetization 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 X-ray 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 X-ray intensity is therefore Thus, under these conditions, the magnetic effect is typically 10⁻⁶ times the X-ray 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 high-flux 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 two-dimensional 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 off-diagonal 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 diffracted-beam 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.

Figure 8.7.4.1| top | pdf | Some geometrical definitions.

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 diffracted-beam analyser passes only components of the diffracted beam, one can measure , and thus eliminate the charge scattering.

For non-polarized 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 right-polarized photons. If E′ is the field for the diffracted photons, In this case, `mixed-polarization' 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).