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. 7.4, pp. 653665
https://doi.org/10.1107/97809553602060000607 Chapter 7.4. Correction of systematic errors^{a}Department of Physics, University of Ioannina, PO Box 1186, Gr45110 Ioannina, Greece,^{b}Department of Physics, University of Warwick, Coventry CV4 7AL, England,^{c}Department of Physics, PO Box 9, University of Helsinki, FIN00014 Helsinki, Finland, and ^{d}Chemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England The theory of the thermal diffuse scattering correction, which is quite different for Xrays and for neutrons, is described. The contribution of Compton scattering to Bragg intensities is explained and tables are presented for the incoherent scattering function for elements up to atomic number 55 and sin θ/λ values up to 2.0 Å^{−1}. Different models for calculating the incoherent scattering are discussed. Background components of Xray scattering from a crystalline sample and from different inelastic components of the scattered radiation are described. 
The positions and intensities of Xray diffraction maxima are affected by absorption, the magnitude of the effect depending on the size and shape of the specimen. Positional effects are treated as they are encountered in the chapters on experimental techniques.
In structure determination, the effect of absorption on intensity may sometimes be negligible, if the crystal is small enough and the radiation penetrating enough. In general, however, this is not the case, and corrections must be applied. They are simplest if the crystal is of a regular geometric shape, produced either through natural growth or through grinding or cutting. Expressions for reflection from and transmission through a flat plate are given in Table 6.3.3.1 , for reflection from cylinders in Table 6.3.3.2 , and for reflection from spheres in Table 6.3.3.3 . The calculation for a crystal bounded by arbitrary plane faces is treated in Subsection 6.3.3.3 .
The values of mass absorption (attenuation) coefficients required for the calculation of corrections are given as a function of the element and of the radiation in Table 4.2.4.3 .
Thermal diffuse scattering (TDS) is a process in which the radiation is scattered inelastically, so that the incident Xray photon (or neutron) exchanges one or more quanta of vibrational energy with the crystal. The vibrational quantum is known as a phonon, and the TDS can be distinguished as onephonon (firstorder), twophonon (secondorder), scattering according to the number of phonons exchanged.
The normal modes of vibration of a crystal are characterized as either acoustic modes, for which the frequency ω(q) goes to zero as the wavevector q approaches zero, or optic modes, for which the frequency remains finite for all values of q [see Section 4.1.1 of IT B (2001)]. The onephonon scattering by the acoustic modes rises to a maximum at the reciprocallattice points and so is not entirely subtracted with the background measured on either side of the reflection. This gives rise to the `TDS error' in estimating Bragg intensities. The remaining contributions to the TDS – the twophonon and multiphonon acoustic mode scattering and all kinds of scattering by the optic modes – are largely removed with the background.
It is not easy in an Xray experiment to separate the elastic (Bragg) and the inelastic thermal scattering by energy analysis, as the energy difference is only a few parts per million. However, this has been achieved by Dorner, Burkel, Illini & Peisl (1987) using extremely high energy resolution. The separation is also possible using Mössbauer spectroscopy. Fig. 7.4.2.1 shows the elastic and inelastic components from the 060 reflection of LiNbO_{3} (Krec, Steiner, Pongratz & Skalicky, 1984), measured with γradiation from a ^{57}Co Mössbauer source. The TDS makes a substantial contribution to the measured integrated intensity; in Fig. 7.4.2.1, it is 10% of the total intensity, but it can be much larger for higherorder reflections. On the other hand, for the extremely sharp Bragg peaks obtained with synchrotron radiation, the TDS error may be reduced to negligible proportions (Bachmann, Kohler, Schulz & Weber, 1985).

060 reflection of LiNbO_{3} (Mössbauer diffraction). Inelastic (triangles), elastic (crosses), total (squares) and background (pluses) intensity (after Krec, Steiner, Pongratz & Skalicky, 1984). 
Let E_{meas} represent the total integrated intensity measured in a diffraction experiment, with the contribution from Bragg scattering and that from (onephonon) TDS. Then, where α is the ratio and is known as the `TDS correction factor'. α can be evaluated in terms of the properties of the crystal (elastic constants, temperature) and the experimental conditions of measurement. In the following, it is implied that the intensities are measured using a singlecrystal diffractometer with incident radiation of a fixed wavelength. We shall treat separately the calculation of α for Xrays and for thermal neutrons.
The differential cross section, representing the intensity per unit solid angle for Bragg scattering, is where N is the number of unit cells, each of volume V, and F(h) is the structure factor. H is the scattering vector, defined by with k and k_{0} the wavevectors of the scattered and incident beams, respectively. (The scattering is elastic, so k = k_{0} = 2π /λ, where λ is the wavelength.) 2πh is the reciprocallattice vector and the delta function shows that the scattered intensity is restricted to the reciprocallattice points.
The integrated Bragg intensity is given by where the integration is over the solid angle Ω subtended by the detector at the crystal and over the time t spent in scanning the reflection. Using with dH = H dθ, equation (7.4.2.2) reduces to the familiar result (James, 1962) where is the angular velocity of the crystal and 2θ the scattering angle.
The differential cross section for onephonon scattering by acoustic modes of small wavevector q is [see Section 4.1.1 of IT B (2001)]. Here, e_{j}(q) is the polarization vector of the mode (jq) , where j is an index for labelling the acoustic branches of the dispersion relations, m is the mass of the unit cell and is the mode energy. The delta function in (7.4.2.4) shows that the scattering from the mode (jq) is confined to the points in reciprocal space displaced by ±q from the reciprocallattice point at q = 0. The acoustic modes involved are of small wavenumber, for which the dispersion relation can be written where v_{j} is the velocity of the elastic wave with polarization vector e_{j}(q). Substituting (7.4.2.5) into (7.4.2.4) shows that the intensity from the acoustic modes varies as 1/q^{2}, and so peaks strongly at the reciprocallattice points to give rise to the TDS error.
Integrating the delta function in (7.4.2.4) gives the integrated onephonon intensity with ρ the crystal density. The sum over the wavevectors q is determined by the range of q encompassed in the intensity scan. The density of wavevectors is uniform in reciprocal space [see Section 4.1.1 of IT B (2001)], and so the sum can be replaced by an integral Thus, the correction factor (E_{1}/E_{0}) is given by where The integral in (7.4.2.6) is over the range of measurement, and the summation in (7.4.2.7) is over the three acoustic branches. Only longwavelength elastic waves, with a linear dispersion relation, equation (7.4.2.5), need be considered.
The frequencies ω_{j}(q) and polarization vectors e_{j}(q) of the elastic waves in equation (7.4.2.7) can be calculated from the classical theory of Voigt (1910) [see Wooster (1962)]. If , , are the direction cosines of the polarization vector with respect to orthogonal axes x, y, z, then the velocity v_{j} is determined from the elastic stiffness constants by solving the following equations of motion. Here, A_{km} is the km element of a 3 × 3 symmetric matrix A; if , , are the direction cosines of the wavevector q with reference to x, y, z, the km element is given in terms of the elastic stiffness constants by The four indices klmn can be reduced to two, replacing 11 by 1, 22 by 2, 33 by 3, 23 and 32 by 4, 31 and 13 by 5, and 12 and 21 by 6. The elements of A are then given explicitly by
The setting up of the matrix A is a fundamental first step in calculating the TDS correction factor. This implies a knowledge of the elastic constants, whose number ranges from three for cubic crystals to twenty one for triclinic crystals. The measurement of elastic stiffness constants is described in Section 4.1.6 of IT B (2001).
For each direction of propagation , there are three values of (j = 1, 2, 3), given by the eigenvalues of A. The corresponding eigenvectors of A are the polarization vectors e_{j}(q). These polarization vectors are mutually perpendicular, but are not necessarily parallel or perpendicular to the propagation direction.
The function J(q) in equation (7.4.2.7) is related to the inverse matrix A^{−1} by where , , are the x, y, z components of the scattering vector H, and classical equipartition of energy is assumed [E_{j}(q) = k_{B}T] . Thus A^{−1} determines the anisotropy of the TDS in reciprocal space, arising from the anisotropic elastic properties of the crystal.
Isodiffusion surfaces, giving the locus in reciprocal space for which the intensity J(q) is constant for elastic waves of a given wavelength, were first plotted by Jahn (1942a,b). These surfaces are not spherical even for cubic crystals (unless ), and their shapes vary from one reciprocallattice point to another.
Inserting (7.4.2.8) into (7.4.2.6) gives the TDS correction factor as where T_{mn}, an element of a 3 × 3 symmetric matrix T, is defined by Equation (7.4.2.9) can also be written in the matrix form with representing the transpose of H.
The components of H relate to orthonormal axes, whereas it is more convenient to express them in terms of Miller indices hkl and the axes of the reciprocal lattice. If S is the 3 × 3 matrix that transforms the scattering vector H from orthonormal axes to reciprocallattice axes, then where h^{T} = (h, k, l) . The final expression for α, from (7.4.2.11) and (7.4.2.12), is This is the basic formula for the TDS correction factor.
We have assumed that the entire onephonon TDS under the Bragg peak contributes to the measured integrated intensity, whereas some of it is removed in the background subtraction. This portion can be calculated by taking the range of integration in (7.4.2.10) as that corresponding to the region of reciprocal space covered in the background measurement.
To evaluate T requires the integration of the function A^{−1} over the scanned region in reciprocal space (see Fig. 7.4.2.2). Both the function itself and the scanned region are anisotropic about the reciprocallattice point, and so the TDS correction is anisotropic too, i.e. it depends on the direction of the diffraction vector as well as on .

Diagrams in reciprocal space illustrating the volume abcd swept out for (a) an ω scan, and (b) a θ/2θ, or ω/2θ, scan. The dimension of ab is determined by the aperture of the detector and of bc by the rocking angle of the crystal. 
Computer programs for calculating the anisotropic TDS correction for crystals of any symmetry have been written by Rouse & Cooper (1969), Stevens (1974), Merisalo & Kurittu (1978), Helmholdt, Braam & Vos (1983), and Sakata, Stevenson & Harada (1983). To simplify the calculation, further approximations can be made, either by removing the anisotropy associated with A^{−1} or that associated with the scanned region. In the first case, the element T_{mn} is expressed as where the angle brackets indicate the average value over all directions. In the second case, where is the radius of the sphere that replaces the anisotropic region (Fig. 7.4.2.2) actually scanned in the experiment, and dS is a surface element of this sphere. q_{m} can be estimated by equating the volume of the sphere to the volume swept out in the scan.
If both approximations are employed, the correction factor is isotropic and reduces to with v_{L} representing the mean velocity of the elastic waves, averaged over all directions of propagation and of polarization.
Experimental values of α have been measured for several crystals by γray diffraction of Mössbauer radiation (Krec & Steiner, 1984). In general, there is good agreement between these values and those calculated by the numerical methods, which take into account anisotropy of the TDS. The correction factors calculated analytically from (7.4.2.14) are less satisfactory.
The principal effect of not correcting for TDS is to underestimate the values of the atomic displacement parameters. Writing , we see from (7.4.2.14) that the overall displacement factor is increased from B to B + ΔB when the correction is made. ΔB is given by Typically, ΔB/B is 10–20%. Smaller errors occur in other parameters, but, for accurate studies of charge densities or bonding effects, a TDS correction of all integrated intensities is advisable (Helmholdt & Vos, 1977; Stevenson & Harada, 1983).
The neutron treatment of the correction factor lies along similar lines to that for Xrays. The principal difference arises from the different topologies of the onephonon `scattering surfaces' for Xrays and neutrons. These surfaces represent the locus in reciprocal space of the endpoints of the phonon wavevectors q (for fixed crystal orientation and fixed incident wavevector k_{0}) when the wavevector k of the scattered radiation is allowed to vary. We shall not discuss the theory for pulsed neutrons, where the incident wavelength varies (see Popa & Willis, 1994).
The scattering surfaces are determined by the conservation laws for momentum transfer, and for energy transfer, where is the neutron mass and is the phonon energy. is either +1 or −1, where = +1 corresponds to phonon emission (or phonon creation) in the crystal and a loss in energy of the neutrons after scattering, and = −1 corresponds to phonon absorption (or phonon annihilation) in the crystal and a gain in neutron energy. In the Xray case, the phonon energy is negligible compared with the energy of the Xray photon, so that (7.4.2.15) reduces to and the scattering surface is the Ewald sphere. For neutron scattering, is comparable with the energy of a thermal neutron, and so the topology of the scattering surface is more complicated. For onephonon scattering by longwavelength acoustic modes with , (7.4.2.15) reduces to where β is the ratio of the sound velocity in the crystal and the neutron velocity. If the Ewald sphere in the neighbourhood of a reciprocallattice point is replaced by its tangent plane, the scattering surface becomes a conic section with eccentricity 1/β. For , the conic section is a hyperboloid of two sheets with the reciprocallattice point P at one focus. The phonon wavevectors on one sheet correspond to scattering with phonon emission and on the other sheet to phonon absorption. For , the conic section is an ellipsoid with P at one focus. Scattering now occurs either by emission or by absorption, but not by both together (Fig. 7.4.2.3).

Scattering surfaces for onephonon scattering of neutrons: (a) for neutrons faster than sound (β < 1); (b) for neutrons slower than sound (β > 1). The scattering surface for Xrays is the Ewald sphere. P_{0}, P_{1}, etc. are different positions of the reciprocallattice point with respect to the Ewald sphere, and the scattering surfaces are numbered to correspond with the appropriate position of P. 
To evaluate the TDS correction, with q restricted to lie along the scattering surfaces, separate treatments are required for fasterthansound and for slowerthansound neutrons. The final results can be summarized as follows (Willis, 1970; Cooper, 1971):

The sharp distinction between cases (a) and (b) has been confirmed experimentally using the neutron Laue technique on singlecrystal silicon (Willis, Carlile & Ward, 1986).
Thermal diffuse scattering in Xray powderdiffraction patterns produces a nonuniform background that peaks sharply at the positions of the Bragg reflections, as in the singlecrystal case (see Fig. 7.4.2.4). For a given value of the scattering vector, the onephonon TDS is contributed by all those wavevectors q joining the reciprocallattice point and any point on the surface of a sphere of radius with its centre at the origin of reciprocal space. These q vectors reach the boundary of the Brillouin zone and are not restricted to those in the neighbourhood of the reciprocallattice point. To calculate α properly, we require a knowledge, therefore, of the lattice dynamics of the crystal and not just its elastic properties. This is one reason why relatively little progress has been made in calculating the Xray correction factor for powders.
In many diffraction studies, it is necessary to correct the intensities of the Bragg peaks for a variety of inelastic scattering processes. Compton scattering is only one of the incoherent processes although the term is often used loosely to include plasmon, Raman, and resonant Raman scattering, all of which may occur in addition to the more familiar fluorescence radiation and thermal diffuse scattering. The various interactions are summarized schematically in Fig. 7.4.3.1, where the dominance of each interaction is characterized by the energy and momentum transfer and the relevant binding energy.

Schematic diagram of the inelastic scattering interactions, ΔE = E_{1} − E_{2} is the energy transferred from the photon and K the momentum transfer. The valence electrons are characterized by the Fermi energy, E_{F}, and momentum, k_{F} ( being taken as unity). The core electrons are characterized by their binding energy E_{B}. The dipole approximation is valid when Ka < 1, where a is the orbital radius of the scattering electron. 
With the exception of thermal diffuse scattering, which is known to peak at the reciprocallattice points, the incoherent background varies smoothly through reciprocal space. It can be removed with a linear interpolation under the sharp Bragg peaks and without any energy analysis. On the other hand, in noncrystalline material, the elastic scattering is also diffused throughout reciprocal space; the pointbypoint correction is consequently larger and without energy analysis it cannot be made empirically; it must be calculated. These calculations are imprecise except in the situations where Compton scattering is the dominant process. For this to be the case, there must be an encounter, conserving energy and momentum, between the incoming photon and an individual target electron. This in turn will occur if the energy lost by the photon, ΔE = E_{1} − E_{1}, clearly exceeds the oneelectron binding energy, , of the target electron. Eisenberger & Platzman (1970) have shown that this binary encounter model – alternatively known as the impulse approximation – fails as (E_{B}/ΔE)^{2}.
The likelihood of this failure can be predicted from the Compton shift formula, which for scattering through an angle can be written.
This energy transfer is given as a function of the scattering angle in Table 7.4.3.1 for a set of characteristic Xray energies; it ranges from a few eV for Cr Kα Xradiation at small angles, up to ∼2 keV for backscattered Ag Kα Xradiation. Clearly, in the majority of typical experiments Compton scattering will be inhibited from all but the valence electrons.

For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (dσ/dΩ)_{inc}, can be factorized as follows: where (dσ/dΩ)_{0} is the cross section characterizing the interaction, in this case it is the Thomson cross section, ; and being the initial and final state photon polarization vectors. The dynamics of the target are contained in the incoherent scattering factor S(E_{1}, E_{2}, K, Z), which is usually a function of the energy transfer , the momentum transfer K, and the atomic number Z.
The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian
It produces photoelectric absorption through the term taken in first order, Compton and Raman scattering through the term and resonant Raman scattering through the terms in second order.
If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes where the Born operator is summed over the j target electrons and the matrix element is summed over all final states accessible through energy conservation. In the highenergy limit of , S(E_{1}, E_{2}, K, Z) Z but as Table 7.4.3.1 shows this condition does not hold in the Xray regime.
The evaluation of the matrix elements in equation (7.4.3.4) was simplified by Waller & Hartree (1929) who (i) set E_{2} = E_{1} and (ii) summed over all final states irrespective of energy conservation. Closure relationships were then invoked to reduce the incoherent scattering factor to an expression in terms of form factors : where and the latter term arising from exchange in the manyelectron atom.
According to Currat, DeCicco & Weiss (1971), equation (7.4.3.5) can be improved by inserting the prefactor (E_{2}/E_{1})^{2}, where E_{2} is calculated from equation (7.4.3.1); the factor is an average for the factors inside the summation sign of equation (7.4.3.4) that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974)]. The Waller–Hartree method remains the chosen basis for the most extensive compilations of incoherent scattering factors, including those tabulated here, which were calculated by Cromer & Mann (1967) and Cromer (1969) from nonrelativistic Hartree–Fock selfconsistentfield wavefunctions. Table 7.4.3.2 is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975).

This statistical model of the atomic charge density (Thomas, 1927; Fermi, 1928) considerably simplifies the calculation of coherent and incoherent scattering factors since both can be written as universal functions of K and Z. Numerical values were first calculated by Bewilogua (1931); more recent calculations have been made by Brown (1966) and Veigele (1967). The method is less accurate than Waller–Hartree theory, but it is a much simpler computation.
The matrix elements of (7.4.3.4) can be evaluated exactly for the hydrogen atom. If oneelectron wavefunctions in manyelectron atoms are modelled by hydrogenic orbitals [with a suitable choice of the orbital exponent; see, for example, Slater (1937)], an analytical approach can be used, as was originally proposed by Bloch (1934).
Hydrogenic calculations have been shown to predict accurate K and Lshell photoelectric cross sections (Pratt & Tseng, 1972). The method has been applied in a limited number of cases to Kshell (Eisenberger & Platzman, 1970) and Lshell (Bloch & Mendelsohn, 1974) incoherent scattering factors, where it has served to highlight the deficiencies of the Waller–Hartree approach. In chromium, for example, at an incident energy of ∼17 keV and a Bragg angle of 85°, the Lshell Waller–Hartree cross section is higher than the `exact' calculation by ∼50%. A comparison of Waller–Hartree and exact results for 2s electrons, taken from Bloch & Mendelsohn (1974), is given in Table 7.4.3.3 for illustration. The discrepancy is much reduced when all electrons are considered.
S_{exact} is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. S_{imp} is the incoherent scattering factor calculated by taking the Compton profile derived in the impulse approximation and truncating it for ΔE < E_{B}. S_{W–H} is the Waller–Hartree incoherent scattering factor. Data taken from Bloch & Mendelsohn (1974).

In those instances where the exact method has been used as a yardstick, the comparison favours the `relativistic integrated impulse approximation' outlined below, rather than the Waller–Hartree method.
The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929) theory and the Dirac equation (see Jauch & Rohrlich, 1976). In secondorder relativistic perturbation theory, there is no overt separation of and terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a,b); they are generally small at Xray energies. They are of increasing interest in synchrotronbased experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988).
Somewhat surprisingly, it is the spectral distribution, d^{2}σ/dΩ dE_{2}, rather than the total intensity, dσ/dΩ, which is the better understood. This is a consequence of the exploitation of the Compton scattering technique to determine electron momentum density distributions through the Doppler broadening of the scattered radiation [see Cooper (1985) and Williams (1977) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976) and Ribberfors (1975a,b) have shown that the Compton profile – the projection of the electron momentum density distribution onto the Xray scattering vector – can be isolated from the relativistic differential scattering cross section within the impulse approximation. Several experimental and theoretical investigations have been concerned with understanding the changes in the spectral distribution when electron binding energies cannot be discounted. It has been found (e.g. Pattison & Schneider, 1979; Bloch & Mendelsohn, 1974) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers .
This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from
Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975)] for all elements.
Ribberfors (1983) and Ribberfors & Berggren (1982) have shown that this calculation can be dramatically simplified, without loss of accuracy, by crudely approximating the Compton line shape. Fig. 7.4.3.2 shows the incoherent scattering from aluminium, modelled in this way, and compared with experiment, Waller–Hartree theory, and an exact integral of the truncated impulse Compton profile.

The incoherent scattering function, S(x, Z)/Z, per electron for aluminium shown as a function of x = (sin θ)/λ. The Waller–Hartree theory (dotted line) is compared with the truncated impulse approximation in the tabulated Compton profiles (Biggs, Mendelsohn & Mann, 1975) cut off at E < E_{B} for each electron group (solid line). The third curve (dashed line) shows the simplification introduced by Ribberfors (1983) and Ribberfors & Berggren (1982). The predictions are indistinguishable to within experimental error except at low . Reference to the measurements can be found in Ribberfors & Berggren (1982). 
In typical Xray experiments, as is evident from Table 7.4.3.1, the energy transfer may be so low that Compton scattering will be inhibited from all but the most loosely bound electrons. Indeed, in the situation in metals where K, the momentum transfer, is less than k_{F} (the Fermi momentum), Compton scattering from the conduction electrons may be restricted by exclusion because of the lack of unoccupied final states [see Bushuev & Kuz'min (1977)]. Fortunately, in these uncertain circumstances, the incoherent intensities are low. In this regime, the electron gas may be excited into collective motion. For almost all solids, the plasmon excitation energy is 20–30 eV and, in the random phase approximation, the incoherent scattering factor becomes S(ΔE, K) ∝ (K^{2}/w_{p})δ(ΔE − hω_{p}), where ω_{p} is the plasma frequency.
At slightly higher energies , Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977). The resultant spectrum is often referred to as the Compton–Raman band. In semiclassical radiation theory, Raman scattering is usually differentiated from Compton scattering by dropping the requirement for momentum conservation between the photon and the individual target electron, the recoil being absorbed by the atom. The Raman band corresponds to transitions into the lowest unoccupied levels and these can be calculated within the dipole approximation as long as Ka , where K is the momentum transfer and a the orbital radius of the core electron undergoing the transition. The transition probability in equation (7.4.3.4) becomes which implies that the nearedge structure is similar to the photoelectric absorption spectrum.
Whereas plasmon and Raman scattering are unlikely to make dramatic contributions to the total incoherent intensity, resonant Raman scattering (RRS) may, when . The excitation involves a virtual Kshell vacancy in the intermediate state and a vacancy in the L (or M or N) shell and an electron in the continuum in the final state. It has now been observed in a variety of materials [see, for example, Sparks (1974), Eisenberger, Platzman & Winick (1976), Schaupp et al. (1984)]. It was predicted by Gavrila & Tugulea (1975) and the theory has been treated comprehensively by Åberg & Tulkki (1985). The effect is the exact counterpart, in the inelastic spectrum, of anomalous scattering in the elastic spectrum. It is important because, as the resonance condition is approached, the intensity will exceed that due to Compton scattering and therefore play havoc with any corrections to total intensities based solely on the latter.
Although systematic tabulations of resonance Raman scattering do not exist, Fig. 7.4.3.3, which is based on the calculations of Bannett & Freund (1975), shows how the intensity of RRS clearly exceeds that of the Compton scattering for incident energies just below the absorption edge. However, since the problems posed by anomalous scattering and Xray fluorescence are generally appreciated, the energy range is wisely avoided by crystallographers intent upon absolute intensity measurements.

The cross section for resonant Raman scattering (RRS) and fluorescence (F) as a function of the ratio of the incident energy, E, and the Kbinding energy, E_{B}. The units of dσ/dΩ are (e^{2}/mc^{2})^{2} and the data are taken from Bannett & Freund (1975). For comparison, the intensity of Compton scattering (C) from copper through an angle of 30° is also shown [data taken from Hubbell et al. (1975)]. 
Finally, and for completeness, it should be noted that the intensity of Compton scattering from a magnetic material with a net spin moment will, in principle, differ from that from a nonmagnetic material. For unpolarized radiation, the effects are only discernible at photon energies greatly in excess of the electron rest mass energy, mc^{2} = 511 keV, but for circularly polarized radiation effects at the 1% level can be found in Compton scattering experiments carried out at on ferromagnets such as iron. See Lipps & Tolhoek (1954a,b) for a comprehensive description of polarization phenomena in magnetic scattering and Lovesey (1993) for an account of the scattering theory.
By definition, the background includes everything except the signal. In an Xray diffraction measurement, the signal is the pattern of Bragg reflections. The profiles of the reflections should be determined by the structure of the sample, and so the broadening due to the instrument should be considered as background. In the ideal angledispersive experiment, a well collimated beam of Xrays having a well defined energy (and a polarization, perhaps) falls on the sample, and only the radiation scattered by the sample is detected. Furthermore, the detector should be able to resolve all the components of scattering by energy, so that each scattering process could be studied separately. It is obvious that only after this kind of analysis are the Bragg reflections (plus the possible disorder scattering) unequivocally separated from the background arising from other processes. In most cases, however, this analysis is not feasible, and the reflections are separated by using certain assumptions concerning their profile, and the success of this procedure depends on the peaktobackground ratio.
The ideal situation described above is all too often not encountered, and experimenters are satisfied with too low a level of resolution. The aim of the present article is to point out the sources of the unwanted and unresolved components of the registered radiation and to suggest how these may be eliminated or resolved, so that the quality of the diffraction pattern is as high as possible. The article can cover only a few of the possible experimental situations, and only the `almost ideal' angledispersive instrument is considered. It is assumed that the beam incident on the sample is monochromatized by reflection from a crystal and that the scattered radiation is registered by a lownoise quantum detector, which is the standard arrangement for modern diffractometers. Filtered radiation and photographic recording are used in certain applications, but these are excluded from the following discussion. The wavelengthdispersive or Laue methods are becoming popular at the synchrotronradiation laboratories, and a short comment on these techniques will be included. Other sections of this volume deal with the components of scattering that are present even in the ideal experiment: thermal diffuse scattering (TDS), Compton and plasmon scattering, fluorescence and resonant Raman scattering, multiple scattering (coherent and incoherent), and disorder scattering.
The rest of the background may be termed `parasitic' scattering, and it arises from three sources:

Parasitic scattering is occasionally mentioned in the literature, but it has hardly ever been the subject of a detailed study. Therefore, the present article will discuss the general principles of the minimization of the background and then illustrate these ideas with examples. Most of the discussion will be directed to the first of the three sources of parasitic scattering, because the other two depend on the details of the experiment.
An ideal diffraction experiment should be viewed as an Xray optical system where all the parts are properly matched for the desired resolution and efficiency. The impurities of the incident beam are the wavelengths and divergent rays that do not contribute to the signal but scatter from the sample through the various processes mentioned above. The propagation of the Xray beam through the instrument is perhaps best illustrated by the socalled phasespace analysis. The threedimensional version, which will be used in the following, was introduced by Matsushita & Kaminaga (1980a,b) and was elaborated further by Matsushita & Hashizume (1983). The width, divergence and wavelength range of the beam are given as a contour diagram, which originates in the Xray source, and is modified by slits, monochromator, sample, and the detection system. The actual fivedimensional diagram is usually given as threedimensional projections on the plane of diffraction and on the plane perpendicular to it and the beam axis, and in most cases the first projection is sufficient for an adequate description of the geometry of the experiment.
The limitations of the actual experiments are best studied through a comparison with the ideal situation. A close approximation to the ideal experimental arrangement is shown in Fig. 7.4.4.1 as a series of phasespace diagrams. The characteristic radiation from a conventional Xray tube is almost uniformly distributed over the solid angle of 2π, and the relative width of the Kα_{1} or Kα_{2} emission line is typically Δλ/λ = 5 × 10^{−4}. The acceptance and emittance windows of a flat perfect crystal are given in Fig. 7.4.4.1(b). The angular acceptance of the crystal (Darwin width) is typically less than 10^{−4} rad, and, if the width of the slit s or that of the crystal is small enough, none of the Kα_{2} distribution falls within the window. Therefore, it is sufficient to study the size and divergence distributions of the beam in the λ(Kα_{1}) plane only, as shown in Fig. 7.4.4.1(c). The beam transmitted by the flat monochromator and a slit is shown as the hatched area, and the part reflected by a small crystal by the crosshatched area. The reflectivity curve of the crystal is probed when the crystal is rotated. In this schematic case, almost 100% of the beam contributes to the signal. The typical reflection profile shown in Fig. 7.4.4.2 reveals the details of the crystallite distribution of the sample (Suortti, 1985). The broken curve shows the calculated profile of the same reflection if the incident beam from a mosaic crystal monochromator had been used (see below).

Equatorial phasespace diagrams for a conventional Xray source and parallelbeam geometry; x is the size and x′ = dx/dz the divergence of the Xrays. (a) Radiation distributions for two wavelengths, λ_{1} and λ_{2}, at the source of width Δx, and downstream at a slit of width ±s_{1}. (b) Acceptance and emittance windows of a flat perfect crystal, where the phasespace volume remains constant, Aw_{a}Δλ = Ew_{e}Δλ, and the (x′, λ) section shows the reflection of a polychromatic beam (Laue diffraction). (c) Distributions for one wavelength at the source, flat perfectcrystal monochromator, sample (marked with the broken line), and the receiving slit (RS); z is the distance from the source. 

Reflection 400 of LiH measured with a parallel beam of Mo Kα_{1} radiation (solid curve). The broken curve shows the reflection as convoluted by a Gaussian instrumentation function of 2σ = 0.1° and θ(α_{2}) − θ(α_{1}) = 0.13°, which values are comparable with those in Fig. 7.4.4.4. 
The window of acceptance of a flat mosaic crystal is determined by the width of the mosaic distribution, which may be 100 times larger than the Darwin width of the reflection in question. This means that a convergent beam is reflected in the same way as from a bent perfect crystal in Johann or Johansson geometry. Usually, the window is wide enough to transmit an energy band that includes both and components of the incident beam. The distributions of these components are projected on the (x, x′, λ_{1}) plane in Fig. 7.4.4.3. The sample is placed in the (para)focus of the beam, and often the divergence of the beam is much larger than the width of the rocking curve of the sample crystal. This means that at any given time the signal comes from a small part of the beam, but the whole beam contributes to the background. The profile of the reflection is a convolution of the actual rocking curve with the divergence and wavelength distributions of the beam. The calculated profile in Fig. 7.4.4.2 demonstrates that in a typical case the profile is determined by the instrument, and the peaktobackground ratio is much worse than with a perfectcrystal monochromator.

Equatorial phasespace diagrams for two wavelengths, λ_{1} (solid lines) and λ_{2} (broken lines), projected on the plane λ = λ_{1}. The monochromator at z = 200 mm is a flat mosaic crystal, and a small sample is located at z = 400 mm, as shown by the shaded area. The reflected beams at the receiving slit are shown for the (+, +) and (+, −) configurations of the monochromator and the sample. 
An alternative arrangement, which has become quite popular in recent years, is one where the plane of diffraction at the monochromator is perpendicular to that at the sample. The beam is limited by slits only in the latter plane, and the wavelength varies in the perpendicular plane. An example of rocking curves measured by this kind of diffractometer is given in Fig. 7.4.4.4. The and components are seen separately plus a long tail due to continuum radiation, and the profile is that of the divergence of the beam.

Two reflections of beryllium acetate measured with Mo Kα. The graphite (002) monochromator reflects in the vertical plane, while the crystal reflects in the horizontal plane. The equatorial divergence of the beam is 0.8°, FWHM. 
In the Laue method, a well collimated beam of white radiation is reflected by a stationary crystal. The wavelength band reflected by a perfect crystal is indicated in Fig. 7.4.4.1(b). The mosaic blocks select a band of wavelengths from the incident beam and the wavelength deviation is related to the angular deviation by . The angular resolution is determined by the divergences of the incident beam and the spatial resolution of the detector. The detector is not energy dispersive, so that the background arises from all scattering that reaches the detector. An estimate of the background level involves integrations over the incident spectrum at a fixed scattering angle, weighted by the cross sections of inelastic scattering and the attenuation factors. This calculation is very complicated, but at any rate the background level is far higher than that in a diffraction measurement with a monochromatic incident beam.
The detecting system is an integral part of the Xray optics of a diffraction experiment, and it can be included in the phasespace diagrams. In singlecrystal diffraction, the detecting system is usually a rectangular slit followed by a photon counter, and the slit is large enough to accept all the reflected beam. The slit can be stationary during the scan (ω scan) or follow the rotation of the sample (ω/2θ scan). The included TDS depends on these choices, but otherwise the amount of background is proportional to the area of the receiving slit. It is obvious from a comparison between Fig. 7.4.4.1 and Fig. 7.4.4.3 that a much smaller receiving slit is sufficient in the parallelbeam geometry than in the conventional divergentbeam geometry. Mathieson (1985) has given a thorough analysis of various monochromator–sample–detector combinations and has suggested the use of a twodimensional ω/2θ scan with a narrow receiving slit. This provides a deconvolution of the reflection profile measured with a divergent beam, but the same result with better intensity and resolution is obtained by the parallelbeam techniques.
The above discussion has concentrated on improving the signaltobackground ratio by optimization of the diffraction geometry. This ratio can be improved substantially by an energydispersive detector, but, on the other hand, all detectors have some noise, which increases the background. There have been marked developments in recent years, and traditional technology has been replaced by new constructions. Much of this work has been carried out in synchrotronradiation laboratories (for references, see Thomlinson & Williams, 1984; Brown & Lindau, 1986).
A positionsensitive detector can replace the receiving slit when a reciprocal space is scanned. TV area detectors with an Xraytovisible light converter and twodimensional CCD arrays have moderate resolution and efficiency, but they work in the current mode and do not provide pulse discrimination on the basis of the photon energy. One and twodimensional proportional chambers have a spatial resolution of the order of 0.1 mm, and the relative energy resolution, , is sufficient for rejection of some of the parasitic scattering.
The NaI(Tl) scintillation counter is used most frequently as the Xray detector in crystallography. It has 100% efficiency for the commonly used wavelengths, and the energy resolution is comparable to that of a proportional counter. The detector has a long life, and the level of the lowenergy noise can be reduced to about 0.1 counts s^{−1}.
The Ge and Si(Li) solidstate detectors (SSD) have an energy resolution ΔE/E = 0.01 to 0.03 for the wavelengths used in crystallography. The relative Compton shift, Δλ/λ, is , where 2θ is the scattering angle, so that even this component can be eliminated to some extent by a SSD. These detectors have been bulky and expensive, but new constructions that are suitable for Xray diffraction have become available recently. The effects of the detector resolution are shown schematically in Fig. 7.4.4.5 for a scintillation counter and a SSD.

Components of scattering at small scattering angles when the incident energy is just below the K absorption edge of the sample [upper part, (a)], and at large scattering angles when the incident energy is about twice the Kedge energy [upper part, (b)]. The abbreviations indicate resonant Raman scattering (RRS), plasmon (P) and Compton (C) scattering, coherent scattering (Coh) and sample fluorescence (Kα and Kβ). The lower part shows these components as convoluted by the resolution function of the detector: (a) a SSD and (b) a scintillation counter (Suortti, 1980b). 
Crystal monochromators placed in front of the detector eliminate all inelastic scattering but the TDS. The monochromator must be matched with the preceding Xray optical system, the sample included, and therefore diffractedbeam monochromators are used in powder diffraction only (see Subsection 7.4.4.4).
The signaltobackground ratio is much worse in powder diffraction than in singlecrystal diffraction, because the background is proportional to the irradiated volume in both cases, but the powder reflection is distributed over a ring of which only the order of 1% is recorded. The phasespace diagrams of a typical measurement are shown in Fig. 7.4.4.6. The Johansson monochromator is matched to the incident beam to provide maximum flux and good energy resolution. The Bragg–Brentano geometry is parafocusing, and, if the geometrical aberrations are ignored, the reflected beam is a convolution between the angular width of the monochromator focus (as seen from the sample) and the reflectivity curve of an average crystallite of the powder sample. The profile of this function is scanned by a narrow slit, as shown in the last diagram. The slit can be followed with a Johann or Johansson monochromator that has a narrow wavelength passband. In this case, there is no primarybeam monochromator, so that the incident beam at the sample is that given at z = 100 mm. The slit RS is the `source' for the monochromator, which focuses the beam at the detector. The obvious advantages of this arrangement are counterbalanced by certain limitations such as that the effective receiving slit is determined by the reflectivity curve of the monochromator, and this may vary over the effective area.

Equatorial phasespace diagrams for powder diffraction in the Bragg–Brentano geometry. (a) The acceptance and emittance windows of a Johannson monochromator; (b) the beam in the λ = λ_{1} plane: the exit beam from the Johansson monochromator is shown by the hatched area (z = 100 mm), the beam on the sample by two closely spaced lines, the reflectivity range of powder particles in a small area of the sample by the hatched area (z = 300 mm, note the change of scales), and the scan of the reflected beam by a slit RS by broken lines (z = 400 mm, at the parafocus). 
Examples of a powder reflection measured with different instruments and 1.5 Å radiation are given in Fig. 7.4.4.7. It should be noticed that scattering from the impurities of the sample and from the sample environment is negligible in all three cases. The width of the mosaic distribution of the 00.2 reflection of the pyrolytic graphite monochromator is 0.3°, which corresponds to a 180 eV (0.034 Å) wide transmitted beam. This is almost 10 times the separation between and , and 70 times the natural width of the line. The width of the focal line is about 0.2 mm, or 0.07°, and is seen as broadening of the reflection profile. The quartz (10.1) monochromator reflects a band that is determined by the projected width of the Xray source. In the present case, the band is 15 eV wide, so that the monochromator can be tuned to transmit the component only. The focal line is very sharp, 0.05 mm wide, and so the reflection is much narrower than in the preceding case. The third measurement was made with synchrotron radiation, and the receiving slit was replaced by a perfectcrystal analyser. The divergences of the incident and diffracted beams are about 0.1 mrad (less than 0.01°) in the plane of diffraction, so that the ideal parallelbeam geometry should prevail. However, the reflection is clearly broader than that measured with the conventional diffractometer. The reason is a wavelength gradient across the beam, which was monochromatized by a flat perfect crystal. On the other hand, the Ge (111) analyser crystal transmits elastic scattering and TDS only, and 2° away from the peak the background is 0.5% of the maximum intensity.

Three measurements of the 220 reflection of Ni at λ = 1.541 Å scaled to the same peak value; (a) in linear scale, (b) in logarithmic scale. Dotted curve: graphite (00.2) Johann monochromator, conventional 0.1 mm wide Xray source (Suortti & Jennings, 1977); solid curve: quartz (10.1) Johansson monochromator, conventional 0.05 mm wide Xray source; broken curve: synchrotron radiation monochromatized by a (+, −) pair of Si (111) crystals, where the second crystal is sagittally bent for horizontal focusing (Suortti, Hastings & Cox, 1985). The horizontal line indicates the halfmaximum value. In all cases, the effective slit width is much less than the FWHM of the reflection. 
The above considerations may seem to have little relevance to everyday crystallographic practices. Unfortunately, many standard methods yield diffraction patterns of poor quality. The quest for maximum integrated intensity has led to designs that make reflections broad and background high. It should be realized that not the flux but the brilliance of the incident beam is important in a diffraction measurement. The other aspect is that the information should not be lost in the experiment, and a divergent wide wavelength band is quite an ignorant probe of a reflection from a single crystal.
A situation where even small departures from the ideal diffraction geometry may cause large effects is measurement at an energy just below an absorption edge. Even a small tail of the energy band of the incident beam may excite radiation that becomes the dominant component of background. Similar effects are due to the harmonic energy bands reflected by most monochromators, particularly when the continuous spectrum of synchrotron radiation is used.
Scattering from the surroundings of the sample can be eliminated almost totally by shielding and beam tunnels. The general idea of the construction should be that an optical element of the instrument `sees' the preceding element only. Inevitably, the detector sees some of the environment of the sample. The density of air is about 1/1000 of that of a solid sample, so that 10 mm^{3} of irradiated air contributes to the background as much as a spherical crystal of 0.3 mm diameter. Strong spurious peaks may arise from slit edges and entrance windows of the specimen chamber, which should never be seen by the detector. A complete measurement without the sample is always a good starting point for an experiment.
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