International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 7.4, pp. 653-665
https://doi.org/10.1107/97809553602060000607

Chapter 7.4. Correction of systematic errors

N. G. Alexandropoulos,a M. J. Cooper,b P. Suorttic and B. T. M. Willisd

aDepartment of Physics, University of Ioannina, PO Box 1186, Gr-45110 Ioannina, Greece,bDepartment of Physics, University of Warwick, Coventry CV4 7AL, England,cDepartment of Physics, PO Box 9, University of Helsinki, FIN-00014 Helsinki, Finland, and dChemical 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 X-rays 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 X-ray scattering from a crystalline sample and from different inelastic components of the scattered radiation are described.

7.4.1. Absorption

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The positions and intensities of X-ray 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[link] , for reflection from cylinders in Table 6.3.3.2[link] , and for reflection from spheres in Table 6.3.3.3[link] . The calculation for a crystal bounded by arbitrary plane faces is treated in Subsection 6.3.3.3[link] .

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[link] .

7.4.2. Thermal diffuse scattering

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B. T. M. Willisd

7.4.2.1. Glossary of symbols

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[\widehat {e}_{j}] Direction cosines of ej(q)
ej(q) Polarization vector of normal mode (jq)
Ej(q) Energy of mode (jq)
Emeas Total integrated intensity measured under Bragg peak
E0 Integrated intensity from Bragg scattering
E1 Integrated intensity from one-phonon scattering
F(h) Structure factor
[\hbar] Planck's constant h divided by 2π
h Reciprocal-lattice vector
H Scattering vector
j Label for branch of dispersion relation
k0 Wavevector of incident radiation
k Wavevector of scattered radiation
[k_{B}] Boltzmann's constant
[m_{n}] Neutron mass
m Mass of unit cell
N Number of unit cells in crystal
q Wavevector of normal mode of vibration
[q_{m}] Radius of scanning sphere in reciprocal space
V Volume of unit cell
[{\bf v}_{j}] Elastic wave velocity for branch j
[{\bf v}_{L}] Mean velocity of elastic waves
α TDS correction factor
[2\theta] Scattering angle
[\theta _{B}] Bragg angle
[\left({{{\rm d}\sigma }\over{{\rm d}\Omega}} \right) ^{ (0) }] Differential cross section for Bragg scattering
[\left({{{\rm d}\sigma }\over{{\rm d}\Omega}} \right) ^{ (1) }] Differential cross section for one-phonon scattering
ρ Density of crystal
[\omega _{j} ({\bf q}) ] Frequency of normal mode (jq)

Thermal diffuse scattering (TDS) is a process in which the radiation is scattered inelastically, so that the incident X-ray 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 one-phonon (first-order), two-phonon (second-order), [\ldots] 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[link] of IT B (2001[link])]. The one-phonon scattering by the acoustic modes rises to a maximum at the reciprocal-lattice 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 two-phonon 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 X-ray 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[link]) using extremely high energy resolution. The separation is also possible using Mössbauer spectroscopy. Fig. 7.4.2.1[link] shows the elastic and inelastic components from the 060 reflection of LiNbO3 (Krec, Steiner, Pongratz & Skalicky, 1984[link]), measured with γ-radiation from a 57Co Mössbauer source. The TDS makes a substantial contribution to the measured integrated intensity; in Fig. 7.4.2.1[link], it is 10% of the total intensity, but it can be much larger for higher-order 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[link]).

[Figure 7.4.2.1]

Figure 7.4.2.1 | top | pdf |

060 reflection of LiNbO3 (Mössbauer diffraction). Inelastic (triangles), elastic (crosses), total (squares) and background (pluses) intensity (after Krec, Steiner, Pongratz & Skalicky, 1984[link]).

Let Emeas represent the total integrated intensity measured in a diffraction experiment, with [E_{0}] the contribution from Bragg scattering and [E_{1}] that from (one-phonon) TDS. Then, [E_{\rm meas}=E_{0}+E_{1}=E_{0}\left (1+\alpha \right), \eqno (7.4.2.1)]where α is the ratio [E_{1}/E_{0}] 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 single-crystal diffractometer with incident radiation of a fixed wavelength. We shall treat separately the calculation of α for X-rays and for thermal neutrons.

7.4.2.2. TDS correction factor for X-rays (single crystals)

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The differential cross section, representing the intensity per unit solid angle for Bragg scattering, is [ \left( {{\rm d}\sigma \over{{\rm d}\Omega}}\right) ^{(0)}={N{(2\pi)^3}\over V}| F({\bf h}) | ^2\delta ({\bf H}-2\pi {\bf h}),]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 [{\bf H}={\bf k}-{\bf k}_0, ]with k and k0 the wavevectors of the scattered and incident beams, respectively. (The scattering is elastic, so k = k0 = 2π /λ, where λ is the wavelength.) 2πh is the reciprocal-lattice vector and the delta function shows that the scattered intensity is restricted to the reciprocal-lattice points.

The integrated Bragg intensity is given by [\eqalignno{ E_{0} &=\int\int \left ({{\rm d}\sigma }\over{{\rm d}\Omega}\right) ^{(0) }{\rm d}\Omega{\kern 2pt}{\rm d}t\cr &=N {{(2\pi) ^{3}}\over{V}}| F({\bf h})  | ^{2}\int\int \delta  ({\bf H}-2\pi {\bf h}) {\kern 2pt{\rm d}}\Omega{\kern 2pt{\rm d}}t, & (7.4.2.2)}]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 [\textstyle \int \delta \left ({\bf H}\right)\,{\rm d}{\bf H}=1,]with dH = H dθ, equation (7.4.2.2)[link] reduces to the familiar result (James, 1962[link]) [E_0={{N\lambda ^{3}| F({\bf h}) | ^2}\over{V\omega _{0}\sin 2\theta }}, \eqno(7.4.2.3)]where [\omega _{0}] is the angular velocity of the crystal and 2θ the scattering angle.

The differential cross section for one-phonon scattering by acoustic modes of small wavevector q is [\eqalignno{ \left ({{\rm d}\sigma }\over {{\rm d}\Omega}\right) ^{(1)}&={(2\pi) ^{3}\over {V}}| F({\bf h}) | ^{2}\cr &\quad\times {\sum _{j=1}^{3}} {{\bf H}\cdot {\bf e}_{j}({\bf q})\over {m\omega _{j}^{2}({\bf q})} }E_{j}({\bf q}) \delta ({\bf H}\pm {\bf q}-2\pi {\bf h}) & (7.4.2.4)}][see Section 4.1.1[link] of IT B (2001[link])]. Here, ej(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 [E_{j}\left ({\bf q}\right) ] is the mode energy. The delta function in (7.4.2.4)[link] shows that the scattering from the mode (jq) is confined to the points in reciprocal space displaced by ±q from the reciprocal-lattice point at q = 0. The acoustic modes involved are of small wavenumber, for which the dispersion relation can be written [\omega _{j}\left ({\bf q}\right) ={\bf v}_{j}\left | {\bf q}\right |, \eqno (7.4.2.5)]where vj is the velocity of the elastic wave with polarization vector ej(q). Substituting (7.4.2.5)[link] into (7.4.2.4)[link] shows that the intensity from the acoustic modes varies as 1/q2, and so peaks strongly at the reciprocal-lattice points to give rise to the TDS error.

Integrating the delta function in (7.4.2.4)[link] gives the integrated one-phonon intensity [\eqalign{ E_1 &={{\lambda ^3}\over {V^2\omega _0\sin 2\theta _B}}H^2 | F({\bf h}) | ^2 \cr &\quad\times {\sum _q}{\sum _j} {{\left [{\bf H}\cdot {\bf e}_j({\bf q}) \right] ^2}\over {\rho \omega _q^2({\bf q}) }}E_j({\bf q}),}]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[link] of IT B (2001[link])], and so the sum can be replaced by an integral [ {\sum _q}\rightarrow {{NV}\over {(2\pi) ^3}}\int {\rm d}{\bf q}.]Thus, the correction factor (E1/E0) is given by [\alpha ={1\over 8\pi ^3}\int J({\bf q})\, {\rm d}{\bf q}, \eqno (7.4.2.6)]where [J({\bf q}) ={\sum _j} {{{\bf H}\cdot {\bf e}_j({\bf q}) ^2}\over{\rho \omega ^2({\bf q}) }}E_j({\bf q}). \eqno (7.4.2.7)]The integral in (7.4.2.6)[link] is over the range of measurement, and the summation in (7.4.2.7)[link] is over the three acoustic branches. Only long-wavelength elastic waves, with a linear dispersion relation, equation (7.4.2.5)[link], need be considered.

7.4.2.2.1. Evaluation of J(q)

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The frequencies ωj(q) and polarization vectors ej(q) of the elastic waves in equation (7.4.2.7)[link] can be calculated from the classical theory of Voigt (1910[link]) [see Wooster (1962[link])]. If [\widehat {e}_1], [\widehat {e}_2], [\widehat {e}_3] are the direction cosines of the polarization vector with respect to orthogonal axes x, y, z, then the velocity vj is determined from the elastic stiffness constants [c_{ijkl}] by solving the following equations of motion. [\eqalignno{ \widehat {e}_1\left (A_{11}-\rho {v}_j^2\right) +\widehat {e}_2A_{12}+\widehat {e}_3A_{13} &=0,\cr \widehat {e}_1A_{12}+\widehat {e}_2\left (A_{22}-\rho {v}_j^2\right) +\widehat {e}_3A_{23} &=0,\cr \widehat {e}_1A_{13}+\widehat {e}_2A_{23}+\widehat {e}_3\left (A_{33}-\rho {v}_j^2\right) &=0.}]Here, Akm is the km element of a 3 × 3 symmetric matrix A; if [\widehat {q}_1], [\widehat {q}_2], [\widehat {q}_3] 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 [ A_{km}=\; {\textstyle\sum\limits _{l=1}^3}\;{\textstyle\sum\limits _{n=1}^3}c_{klmn}\widehat {q}_l\widehat {q}_n.]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 [\eqalignno{ A_{11} &=c_{11}\widehat {q}_1^2+c_{66}\widehat {q}_2^2+c_{55}\widehat {q}_3^2+2c_{56}\widehat {q}_2\widehat {q}_3\cr &\quad +2c_{15}\widehat {q}_3\widehat {q}_1+2c_{16}\widehat {q}_1\widehat {q}_2, \cr A_{22} &= c_{66}\widehat {q}_1^2+c_{22}\widehat {q}_2^2+c_{44}\widehat {q}_3^2+2c_{24}\widehat {q}_2\widehat {q}_3 \cr &\quad +2c_{46}\widehat {q}_3\widehat {q}_1+2c_{26}\widehat {q}_1\widehat {q}_2, \cr A_{33} &= c_{55}\widehat {q}_1^2+c_{44}\widehat {q}_2^2+c_{33}\widehat {q}_3^2+2c_{34}\widehat {q}_2\widehat {q}_3 \cr &\quad+2c_{35}\widehat {q}_3\widehat {q}_1+2c_{45}\widehat {q}_1\widehat {q}_2, \cr A_{12} &=c_{16}\widehat {q}_1^2+c_{26}\widehat {q}_2^2+c_{45}\widehat {q}_3^2+\left (c_{25}+c_{46}\right) \widehat {q}_2\widehat {q}_3 \cr &\quad+\left (c_{14}+c_{56}\right) \widehat {q}_3\widehat {q}_1+\left (c_{12}+c_{66}\right) \widehat {q}_1\widehat {q}_2, \cr A_{13} &=c_{15}\widehat {q}_1^2+c_{46}\widehat {q}_2^2+c_{35}\widehat {q}_3^2+\left (c_{36}+c_{45}\right) \widehat {q}_2\widehat {q}_3 \cr &\quad +\left (c_{13}+c_{55}\right) \widehat {q}_3\widehat {q}_1+\left (c_{14}+c_{56}\right) \widehat {q}_1\widehat {q}_2, \cr A_{23} &= c_{56}\widehat {q}_1^2+c_{24}\widehat {q}_2^2+c_{34}\widehat {q}_3^2+\left (c_{23}+c_{44}\right) \widehat {q}_2\widehat {q}_3 \cr &\quad +\left (c_{36}+c_{45}\right) \widehat {q}_3\widehat {q}_1+\left (c_{25}+c_{46}\right) \widehat {q}_1\widehat {q}_2 .}]

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[link] of IT B (2001[link]).

For each direction of propagation [\widehat {{\bf q}}], there are three values of [\rho {v}_j^2] (j = 1, 2, 3), given by the eigenvalues of A. The corresponding eigenvectors of A are the polarization vectors ej(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)[link] is related to the inverse matrix A−1 by [J({\bf q}) ={{k_BT}\over{q^2}}\sum _{m=1}^3\sum _{n=1}^3\left ({\bf A}^{-1}\right) _{mn}H_mH_n, \eqno (7.4.2.8)]where [H_1], [H_2], [H_3] are the x, y, z components of the scattering vector H, and classical equipartition of energy is assumed [Ej(q) = kBT] . 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[link],b[link]). These surfaces are not spherical even for cubic crystals (unless [c_{11}-c_{12}=c_{44}]), and their shapes vary from one reciprocal-lattice point to another.

7.4.2.2.2. Calculation of α

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Inserting (7.4.2.8)[link] into (7.4.2.6)[link] gives the TDS correction factor as [\alpha =\textstyle\sum\limits _{m=1}^3\,\textstyle\sum\limits _{n=1}^3T_{mn}H_mH_n, \eqno (7.4.2.9)]where Tmn, an element of a 3 × 3 symmetric matrix T, is defined by [T_{mn}={{k_BT}\over {8\pi ^3}}{\int} {{({\bf A}^{-1}) _{mn}}\over{q^2}}\,{\rm d}{\bf q}. \eqno (7.4.2.10)]Equation (7.4.2.9)[link] can also be written in the matrix form [\alpha ={\bf H}^T{\bf TH}, \eqno (7.4.2.11)]with [{\bf H}^T=\left (H_1,H_2,H_3\right) ] 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 reciprocal-lattice axes, then [ {\bf H}={\bf Sh}, \eqno (7.4.2.12)]where hT = (h, k, l) . The final expression for α, from (7.4.2.11)[link] and (7.4.2.12)[link], is [\alpha ={\bf h}^T{\bf S}^T{\bf TSh}. \eqno (7.4.2.13)]This is the basic formula for the TDS correction factor.

We have assumed that the entire one-phonon 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)[link] 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[link]). Both the function itself and the scanned region are anisotropic about the reciprocal-lattice point, and so the TDS correction is anisotropic too, i.e. it depends on the direction of the diffraction vector as well as on [\sin\theta/\lambda].

[Figure 7.4.2.2]

Figure 7.4.2.2 | top | pdf |

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[link]), Stevens (1974[link]), Merisalo & Kurittu (1978[link]), Helmholdt, Braam & Vos (1983[link]), and Sakata, Stevenson & Harada (1983[link]). 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 Tmn is expressed as [ T_{mn}={{k_BT}\over {8\pi ^3}}\left \langle \left ({\bf A}^{-1}\right) _{mn}\right \rangle {\int} {1\over q^2}\,{\rm d}q,]where the angle brackets indicate the average value over all directions. In the second case, [ T_{mn}={{k_BT}\over {8\pi ^3}}q_m\int\int \left ({\bf A}^{-1}\right) _{mn}{\rm d}S,]where [q_m] is the radius of the sphere that replaces the anisotropic region (Fig. 7.4.2.2[link]) actually scanned in the experiment, and dS is a surface element of this sphere. qm 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 [\alpha ={{H^2k_BTq_m}\over {3\pi ^2\rho {\bf v}_L^2}}, \eqno (7.4.2.14)]with vL 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 γ-ray diffraction of Mössbauer radiation (Krec & Steiner, 1984[link]). 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)[link] are less satisfactory.

The principal effect of not correcting for TDS is to underestimate the values of the atomic displacement parameters. Writing [\exp \alpha \approx 1+\alpha ], we see from (7.4.2.14)[link] that the overall displacement factor is increased from B to B + ΔB when the correction is made. ΔB is given by [\Delta B={{8k_BTq_m}\over {3\pi ^2\rho {\bf v}_L^2}}.]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[link]; Stevenson & Harada, 1983[link]).

7.4.2.3. TDS correction factor for thermal neutrons (single crystals)

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The neutron treatment of the correction factor lies along similar lines to that for X-rays. The principal difference arises from the different topologies of the one-phonon `scattering surfaces' for X-rays and neutrons. These surfaces represent the locus in reciprocal space of the end-points of the phonon wavevectors q (for fixed crystal orientation and fixed incident wavevector k0) 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[link]).

The scattering surfaces are determined by the conservation laws for momentum transfer, [{\bf H}={\bf k}-{\bf k}_0=2\pi {\bf h}+{\bf q},]and for energy transfer, [\hbar ^2(k^2-k_0^2) /2m_n=-\varepsilon \hbar \omega _j({\bf q}), \eqno (7.4.2.15)]where [m_n] is the neutron mass and [\hbar \omega _j({\bf q})] is the phonon energy. [epsilon] is either +1 or −1, where [epsilon] = +1 corresponds to phonon emission (or phonon creation) in the crystal and a loss in energy of the neutrons after scattering, and [epsilon] = −1 corresponds to phonon absorption (or phonon annihilation) in the crystal and a gain in neutron energy. In the X-ray case, the phonon energy is negligible compared with the energy of the X-ray photon, so that (7.4.2.15)[link] reduces to [k=k_0,]and the scattering surface is the Ewald sphere. For neutron scattering, [\hbar \omega _j\left ({\bf q}\right) ] is comparable with the energy of a thermal neutron, and so the topology of the scattering surface is more complicated. For one-phonon scattering by long-wavelength acoustic modes with [q\ll k_0], (7.4.2.15)[link] reduces to [ k=k_0-\varepsilon \beta q,]where β [\left (={\bf v}_L/{\bf v}_n\right) ] is the ratio of the sound velocity in the crystal and the neutron velocity. If the Ewald sphere in the neighbourhood of a reciprocal-lattice point is replaced by its tangent plane, the scattering surface becomes a conic section with eccentricity 1/β. For [\beta \lt1], the conic section is a hyperboloid of two sheets with the reciprocal-lattice 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 [\beta \gt1], 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[link]).

[Figure 7.4.2.3]

Figure 7.4.2.3 | top | pdf |

Scattering surfaces for one-phonon scattering of neutrons: (a) for neutrons faster than sound (β < 1); (b) for neutrons slower than sound (β > 1). The scattering surface for X-rays is the Ewald sphere. P0, P1, etc. are different positions of the reciprocal-lattice 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 faster-than-sound [(\beta \lt1)] and for slower-than-sound [(\beta \gt1)] neutrons. The final results can be summarized as follows (Willis, 1970[link]; Cooper, 1971[link]):

  • (a) For faster-than-sound neutrons, the TDS rises to a maximum, just as for X-rays, and the correction factor is given by (7.4.2.13)[link], which applies to the X-ray case. (This is a remarkable result in view of the marked difference in the one-phonon scattering surfaces for X-rays and neutrons.)

  • (b) For slower-than-sound neutrons, the correction factor depends on the velocity (wavelength) of the neutrons and is more difficult to evaluate than in (a). However, α will always be less than that calculated for X-rays of the same wavelength, and under certain conditions the TDS does not rise to a maximum at all so that α is then zero.

The sharp distinction between cases (a) and (b) has been confirmed experimentally using the neutron Laue technique on single-crystal silicon (Willis, Carlile & Ward, 1986[link]).

7.4.2.4. Correction factor for powders

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Thermal diffuse scattering in X-ray powder-diffraction patterns produces a non-uniform background that peaks sharply at the positions of the Bragg reflections, as in the single-crystal case (see Fig. 7.4.2.4[link]). For a given value of the scattering vector, the one-phonon TDS is contributed by all those wavevectors q joining the reciprocal-lattice point and any point on the surface of a sphere of radius [2\sin\theta/\lambda] 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 reciprocal-lattice 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 X-ray correction factor for powders.

[Figure 7.4.2.4]

Figure 7.4.2.4 | top | pdf |

One-phonon scattering calculated for polycrystalline nickel of lattice constant a (after Suortti, 1980a[link]).

7.4.3. Compton scattering

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N. G. Alexandropoulosa and M. J. Cooperb

7.4.3.1. Introduction

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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[link], where the dominance of each interaction is characterized by the energy and momentum transfer and the relevant binding energy.

[Figure 7.4.3.1]

Figure 7.4.3.1 | top | pdf |

Schematic diagram of the inelastic scattering interactions, ΔE = E1E2 is the energy transferred from the photon and K the momentum transfer. The valence electrons are characterized by the Fermi energy, EF, and momentum, kF ([\hbar] being taken as unity). The core electrons are characterized by their binding energy EB. The dipole approximation is valid when |K|a < 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 reciprocal-lattice 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 non-crystalline material, the elastic scattering is also diffused throughout reciprocal space; the point-by-point 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 = E1E1, clearly exceeds the one-electron binding energy, [E_B], of the target electron. Eisenberger & Platzman (1970[link]) have shown that this binary encounter model – alternatively known as the impulse approximation – fails as (EBE)2.

The likelihood of this failure can be predicted from the Compton shift formula, which for scattering through an angle [\varphi] can be written. [\Delta E=E_1-E_2={E^2_1(1-\cos\varphi)\over mc^2[1+(E_1/mc^2)(1-\cos \varphi)]}. \eqno (7.4.3.1)]

This energy transfer is given as a function of the scattering angle in Table 7.4.3.1[link] for a set of characteristic X-ray energies; it ranges from a few eV for Cr Kα X-radiation at small angles, up to ∼2 keV for backscattered Ag Kα X-radiation. Clearly, in the majority of typical experiments Compton scattering will be inhibited from all but the valence electrons.

Table 7.4.3.1| top | pdf |
The energy transfer, in eV, in the Compton scattering process for selected X-ray energies

Scattering angle ϕ (°)Cr KαCu KαMo KαAg Kα
5411 eV8040 eV17 443 eV22 104 eV
0 0 0 0 0
30 8 17 79 127
60 29 63 292 467
90 57 124 575 915
120 85 185 849 1344
150 105 229 1043 1648
180 112 245 1113 1757

Data calculated from equation (7.4.3.1)[link].

7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section

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7.4.3.2.1. Semi-classical radiation theory

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For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (dσ/dΩ)inc, can be factorized as follows: [\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{\rm inc}=\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{0}S(E_1,E_2,{\bf K},Z), \eqno (7.4.3.2)]where (dσ/dΩ)0 is the cross section characterizing the interaction, in this case it is the Thomson cross section, [(e^2/mc^2)^2\boldvarepsilon_1\cdot\boldvarepsilon_2]; [\boldvarepsilon_1] and [\boldvarepsilon_2] being the initial and final state photon polarization vectors. The dynamics of the target are contained in the incoherent scattering factor S(E1, E2, K, Z), which is usually a function of the energy transfer [\Delta E=E_1-E_2], the momentum transfer K, and the atomic number Z.

The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian [H={e\over me}{\bf p}\cdot {\bf A}+{e^2\over 2me^2}{\bf A}\cdot {\bf A}. \eqno (7.4.3.3)]

It produces photoelectric absorption through the [{\bf p}\cdot{\bf A}] term taken in first order, Compton and Raman scattering through the [{\bf A}\cdot{\bf A}] term and resonant Raman scattering through the [{\bf p}\cdot{\bf A}] terms in second order.

If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes [S=\textstyle\sum\limits_f (E_2/E_1)^2\bigg| \langle\psi_f| \textstyle\sum\limits_j\exp (i{\bf K}\cdot {\bf r}_j)|\psi_i\rangle\bigg|^2\delta(E_f-E_i-\Delta E),\eqno (7.4.3.4)]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 high-energy limit of [\Delta E\gg E_B], S(E1, E2, K, Z) [\rightarrow] Z but as Table 7.4.3.1[link] shows this condition does not hold in the X-ray regime.

The evaluation of the matrix elements in equation (7.4.3.4)[link] was simplified by Waller & Hartree (1929[link]) who (i) set E2 = E1 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 [f_{jk}]: [S=\textstyle\sum\limits_j[1-|\,f_j({\bf K})|^2]-\textstyle\sum\limits_j^{\phantom{X}j}\textstyle\sum\limits^{\ne \,\,k\phantom{XX}}_k|\, f_{jk}({\bf K})|^2,\eqno (7.4.3.5)]where [f_j({\bf K})=\langle\psi_j|\!\exp (i{\bf K}\cdot {\bf r}_j)|\psi_j\rangle]and [f_{jk}=\langle\psi_k|\!\exp [i{\bf K}\cdot ({\bf r}_k-{\bf r}_j)]|\psi_j\rangle,]the latter term arising from exchange in the many-electron atom.

According to Currat, DeCicco & Weiss (1971[link]), equation (7.4.3.5)[link] can be improved by inserting the prefactor (E2/E1)2, where E2 is calculated from equation (7.4.3.1)[link]; the factor is an average for the factors inside the summation sign of equation (7.4.3.4)[link] that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974[link])]. 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[link]) and Cromer (1969[link]) from non-relativistic Hartree–Fock self-consistent-field wavefunctions. Table 7.4.3.2[link] is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975[link]).

Table 7.4.3.2| top | pdf |
The incoherent scattering function for elements up to Z = 55

Element(sin θ)/λ (Å−1)
0.100.200.300.400.500.600.700.800.901.001.502.00
1 H 0.343 0.769 0.937 0.983 0.995 0.998 0.994 0.999 1.000 1.000 1.000 1.000
2 He 0.296 0.881 1.362 1.657 1.817 1.902 1.947 1.970 1.983 1.990 1.999 2.000
3 Li 1.033 1.418 1.795 2.143 2.417 2.613 2.746 2.834 2.891 2.928 2.989 2.998
4 Be 1.170 2.121 2.471 2.744 3.005 3.237 3.429 3.579 3.693 3.777 3.954 3.989
5 B 1.147 2.531 3.190 3.499 3.732 3.948 4.146 4.320 4.469 4.590 4.895 4.973
6 C 1.039 2.604 3.643 4.184 4.478 4.690 4.878 5.051 5.208 5.348 5.781 5.930
7 N 1.08 2.858 4.097 4.792 5.182 5.437 5.635 5.809 5.968 6.113 6.630 6.860
8 O 0.977 2.799 4.293 5.257 5.828 6.175 6.411 6.596 6.755 6.901 7.462 7.764
9 F 0.880 2.691 4.347 5.552 6.339 6.832 7.151 7.376 7.552 7.703 8.288 8.648
10 Ne 0.812 2.547 4.269 5.644 6.640 7.320 7.774 8.085 8.312 8.490 9.113 9.517
11 Na 1.503 2.891 4.431 5.804 6.903 7.724 8.313 8.729 9.028 9.252 9.939 10.376
12 Mg 2.066 3.444 4.771 6.064 7.181 8.086 8.784 9.304 9.689 9.975 10.766 11.229
13 Al 2.264 4.047 5.250 6.435 7.523 8.459 9.225 9.830 10.296 10.652 11.592 12.083
14 Si 2.293 4.520 5.808 6.903 7.937 8.867 9.667 10.330 10.864 11.286 12.408 12.937
15 P 2.206 4.732 6.312 7.435 8.419 9.323 10.131 10.827 11.411 11.888 13.209 13.790
16 S 2.151 4.960 6.795 8.002 8.960 9.829 10.626 11.336 11.952 12.472 13.990 14.641
17 Cl 2.065 5.074 7.182 8.553 9.539 10.382 11.158 11.867 12.499 13.050 14.750 15.487
18 Ar 1.956 5.033 7.377 8.998 10.106 10.967 11.726 12.424 13.061 13.629 15.489 16.324
19 K 2.500 5.301 7.652 9.405 10.650 11.568 12.329 13.014 13.645 14.220 16.212 17.152
20 Ca 3.105 5.690 7.981 9.790 11.157 12.163 12.953 13.635 14.256 14.830 16.921 17.970
21 Sc 3.136 5.801 8.169 10.071 11.561 12.648 13.545 14.256 14.885 15.460 17.630 18.782
22 Ti 3.114 5.860 8.312 10.304 11.901 13.140 14.093 14.856 15.509 16.095 18.334 19.585
23 V 3.067 5.858 8.375 10.454 12.156 13.514 14.574 15.413 16.111 16.721 19.032 20.379
24 Cr 2.609 5.577 8.206 10.415 12.264 13.770 14.960 15.902 16.670 17.323 19.730 21.168
25 Mn 2.949 5.791 8.380 10.604 12.486 14.062 15.346 16.376 17.211 17.910 20.411 21.938
26 Fe 2.891 5.781 8.432 10.733 12.687 14.343 15.716 16.831 17.737 18.488 21.097 22.704
27 Co 2.832 5.764 8.469 10.844 12.867 14.596 16.050 17.249 18.229 19.039 21.777 23.462
28 Ni 2.772 5.726 8.461 10.894 12.980 14.780 16.317 17.602 18.664 19.543 22.445 24.211
29 Cu 2.348 5.455 8.310 10.778 12.942 14.847 16.494 17.885 19.043 20.002 23.107 24.957
30 Zn 2.654 5.631 8.388 10.901 13.094 15.020 16.709 18.163 19.395 20.427 23.745 25.683
31 Ga 2.791 5.939 8.599 11.082 13.290 15.233 16.947 18.445 19.734 20.831 24.370 26.400
32 Ge 2.839 6.229 8.912 11.338 13.536 15.486 17.215 18.741 20.074 21.224 24.983 27.109
33 As 2.793 6.365 9.236 11.658 13.828 15.775 17.511 19.056 20.420 21.612 25.583 27.810
34 Se 2.799 6.589 9.601 12.033 14.168 16.098 17.835 19.391 20.778 22.003 26.171 28.504
35 Br 2.771 6.748 9.940 12.440 14.552 16.456 18.185 19.747 21.149 22.399 26.747 29.190
36 Kr 2.703 6.760 10.157 12.828 14.969 16.849 18.562 20.123 21.535 22.804 27.313 29.870
37 Rb 3.225 7.062 10.431 13.206 15.410 17.282 18.974 20.526 21.940 23.221 27.871 30.543
38 Sr 3.831 7.464 10.746 13.576 15.860 17.745 19.420 20.956 22.367 23.654 28.423 31.210
39 Y 3.999 7.700 11.010 13.899 16.279 18.215 19.891 21.416 22.820 24.110 28.970 31.870
40 Zr 4.064 7.879 11.236 14.176 16.658 18.672 20.373 21.895 23.294 24.583 29.517 32.522
41 Nb 3.672 7.684 11.213 14.317 16.949 19.081 20.844 22.386 23.787 25.077 30.067 33.167
42 Mo 3.625 7.690 11.260 14.444 17.196 19.455 21.300 22.877 24.288 25.581 30.620 33.808
43 Tc 3.987 7.984 11.512 14.653 17.456 19.816 21.748 23.370 24.797 26.093 31.173 34.447
44 Ru 3.559 7.857 11.531 14.782 17.685 20.150 22.172 23.855 25.312 26.621 31.740 35.081
45 Rh 3.499 7.863 11.591 14.883 17.858 20.428 22.557 24.318 25.819 27.148 32.309 35.715
46 Pd 3.103 7.725 11.441 14.824 17.943 26.653 22.904 24.756 26.316 27.677 32.888 36.349
47 Ag 3.362 7.785 11.598 14.969 18.082 20.858 23.212 25.162 26.792 28.195 33.465 36.983
48 Cd 3.700 7.980 11.812 15.185 18.263 21.064 23.501 25.546 27.252 28.705 34.046 37.618
49 In 3.852 8.297 12.083 15.444 18.489 21.288 23.779 25.906 27.691 29.203 34.634 38.255
50 Sn 3.917 8.615 12.415 15.746 18.760 21.541 24.059 26.252 28.113 29.687 35.226 38.894
51 Sb 3.871 8.811 12.777 16.088 19.067 21.823 24.349 26.590 28.518 30.157 35.822 39.536
52 Te 3.097 9.076 13.171 16.466 19.407 22.134 25.655 26.927 28.912 30.613 36.422 40.181
53 I 3.903 9.287 13.564 16.876 19.227 22.471 24.980 27.269 29.298 31.056 37.024 40.827
54 Xe 3.841 9.340 13.892 17.307 20.175 22.833 25.324 27.619 29.680 31.488 37.628 41.477
55 Cs 4.320 9.615 14.217 17.753 20.612 23.228 25.691 27.981 30.064 31.914 38.232 42.129

7.4.3.2.2. Thomas–Fermi model

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This statistical model of the atomic charge density (Thomas, 1927[link]; Fermi, 1928[link]) 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[link]); more recent calculations have been made by Brown (1966[link]) and Veigele (1967[link]). The method is less accurate than Waller–Hartree theory, but it is a much simpler computation.

7.4.3.2.3. Exact calculations

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The matrix elements of (7.4.3.4)[link] can be evaluated exactly for the hydrogen atom. If one-electron wavefunctions in many-electron atoms are modelled by hydrogenic orbitals [with a suitable choice of the orbital exponent; see, for example, Slater (1937[link])], an analytical approach can be used, as was originally proposed by Bloch (1934[link]).

Hydrogenic calculations have been shown to predict accurate K- and L-shell photoelectric cross sections (Pratt & Tseng, 1972[link]). The method has been applied in a limited number of cases to K-shell (Eisenberger & Platzman, 1970[link]) and L-shell (Bloch & Mendelsohn, 1974[link]) 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 L-shell 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[link]), is given in Table 7.4.3.3[link] for illustration. The discrepancy is much reduced when all electrons are considered.

Table 7.4.3.3| top | pdf |
Compton scattering of Mo Kα X-radiation through 170° from 2s electrons

ElementSexactSimpSW–H
Li 0.879 0.878 0.877
B 0.879 0.878 0.877
O 0.878 0.877 0.876
Ne 0.875 0.875 0.875
Mg 0.863 0.863 0.872
Si 0.851 0.850 0.868
Ar 0.843 0.826 0.877
V 0.663 0.716 0.875
Cr 0.568 0.636 0.875

Sexact is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. Simp is the incoherent scattering factor calculated by taking the Compton profile derived in the impulse approximation and truncating it for ΔE < EB. SW–H is the Waller–Hartree incoherent scattering factor. Data taken from Bloch & Mendelsohn (1974[link]).

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.

7.4.3.3. Relativistic treatment of incoherent scattering

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The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929[link]) theory and the Dirac equation (see Jauch & Rohrlich, 1976[link]). In second-order relativistic perturbation theory, there is no overt separation of [{\bf p} \cdot {\bf A}] and [{\bf A} \cdot {\bf A}] terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a[link],b[link]); they are generally small at X-ray energies. They are of increasing interest in synchrotron-based experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988[link]).

Somewhat surprisingly, it is the spectral distribution, d2σ/dΩ dE2, 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[link]) and Williams (1977[link]) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976[link]) and Ribberfors (1975a[link],b[link]) have shown that the Compton profile – the projection of the electron momentum density distribution onto the X-ray 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[link]; Bloch & Mendelsohn, 1974[link]) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers [E\le E_B].

This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from [{{\rm d}\sigma \over {\rm d}\Omega}=\int\limits^\infty_{E_1-E_B}{{\rm d}^2\sigma \over {\rm d}\Omega\,{\rm d}E_2}\,{\rm d}E_2. \eqno (7.4.3.6)]

Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975[link])] for all elements.

Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]) 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[link] 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.

[Figure 7.4.3.2]

Figure 7.4.3.2 | top | pdf |

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[link]) cut off at E < EB for each electron group (solid line). The third curve (dashed line) shows the simplification introduced by Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]). The predictions are indistinguishable to within experimental error except at low [(\sin\theta/\lambda)]. Reference to the measurements can be found in Ribberfors & Berggren (1982[link]).

7.4.3.4. Plasmon, Raman, and resonant Raman scattering

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In typical X-ray experiments, as is evident from Table 7.4.3.1[link], 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 kF (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[link])]. 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 SE, K) ∝ (K2/wp)δ(ΔEhωp), where ωp is the plasma frequency.

At slightly higher energies [(\Delta E \ge E_B)], Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977[link]). The resultant spectrum is often referred to as the Compton–Raman band. In semi-classical 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 |K|a [\lt1], 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)[link] becomes [\sum_f|\langle \psi_f|{\bf r}|\psi_i\rangle|^2\delta (E_f-E_i-\Delta E), \eqno (7.4.3.7)]which implies that the near-edge 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 [E_1\le E_B]. The excitation involves a virtual K-shell 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[link]), Eisenberger, Platzman & Winick (1976[link]), Schaupp et al. (1984[link])]. It was predicted by Gavrila & Tugulea (1975[link]) and the theory has been treated comprehensively by Åberg & Tulkki (1985[link]). 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[link], which is based on the calculations of Bannett & Freund (1975[link]), 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 X-ray fluorescence are generally appreciated, the energy range [0.9\lt E_1 /E_B \lt 1.1] is wisely avoided by crystallographers intent upon absolute intensity measurements.

[Figure 7.4.3.3]

Figure 7.4.3.3 | top | pdf |

The cross section for resonant Raman scattering (RRS) and fluorescence (F) as a function of the ratio of the incident energy, E, and the K-binding energy, EB. The units of dσ/dΩ are (e2/mc2)2 and the data are taken from Bannett & Freund (1975[link]). 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[link])].

7.4.3.5. Magnetic scattering

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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 non-magnetic material. For unpolarized radiation, the effects are only discernible at photon energies greatly in excess of the electron rest mass energy, mc2 = 511 keV, but for circularly polarized radiation effects at the 1% level can be found in Compton scattering experiments carried out at [E_1\simeq1/10\,mc^2] on ferromagnets such as iron. See Lipps & Tolhoek (1954a[link],b[link]) for a comprehensive description of polarization phenomena in magnetic scattering and Lovesey (1993[link]) for an account of the scattering theory.

7.4.4. White radiation and other sources of background

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P. Suorttic

7.4.4.1. Introduction

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By definition, the background includes everything except the signal. In an X-ray 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 angle-dispersive experiment, a well collimated beam of X-rays 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 peak-to-background 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' angle-dispersive 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 low-noise 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 wavelength-dispersive or Laue methods are becoming popular at the synchrotron-radiation 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:

  • (1) impurities of the incident beam;

  • (2) impurities of the sample;

  • (3) surroundings of the sample.

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.

7.4.4.2. Incident beam and sample

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An ideal diffraction experiment should be viewed as an X-ray 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 X-ray beam through the instrument is perhaps best illustrated by the so-called phase-space analysis. The three-dimensional version, which will be used in the following, was introduced by Matsushita & Kaminaga (1980a[link],b[link]) and was elaborated further by Matsushita & Hashizume (1983[link]). The width, divergence and wavelength range of the beam are given as a contour diagram, which originates in the X-ray source, and is modified by slits, monochromator, sample, and the detection system. The actual five-dimensional diagram is usually given as three-dimensional 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[link] as a series of phase-space diagrams. The characteristic radiation from a conventional X-ray tube is almost uniformly distributed over the solid angle of 2π, and the relative width of the Kα1 or Kα2 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)[link]. 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 Kα2 distribution falls within the window. Therefore, it is sufficient to study the size and divergence distributions of the beam in the λ(Kα1) plane only, as shown in Fig. 7.4.4.1(c)[link]. 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 cross-hatched 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[link] reveals the details of the crystallite distribution of the sample (Suortti, 1985[link]). 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).

[Figure 7.4.4.1]

Figure 7.4.4.1 | top | pdf |

Equatorial phase-space diagrams for a conventional X-ray source and parallel-beam geometry; x is the size and x′ = dx/dz the divergence of the X-rays. (a) Radiation distributions for two wavelengths, λ1 and λ2, at the source of width Δx, and downstream at a slit of width ±s1. (b) Acceptance and emittance windows of a flat perfect crystal, where the phase-space volume remains constant, AwaΔλ = EweΔλ, and the (x′, λ) section shows the reflection of a polychromatic beam (Laue diffraction). (c) Distributions for one wavelength at the source, flat perfect-crystal monochromator, sample (marked with the broken line), and the receiving slit (RS); z is the distance from the source.

[Figure 7.4.4.2]

Figure 7.4.4.2 | top | pdf |

Reflection 400 of LiH measured with a parallel beam of Mo Kα1 radiation (solid curve). The broken curve shows the reflection as convoluted by a Gaussian instrumentation function of 2σ = 0.1° and θ(α2) − θ(α1) = 0.13°, which values are comparable with those in Fig. 7.4.4.4[link].

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 [K\alpha_1] and [K\alpha_2] components of the incident beam. The distributions of these components are projected on the (x, x′, λ1) plane in Fig. 7.4.4.3[link]. 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[link] demonstrates that in a typical case the profile is determined by the instrument, and the peak-to-background ratio is much worse than with a perfect-crystal monochromator.

[Figure 7.4.4.3]

Figure 7.4.4.3 | top | pdf |

Equatorial phase-space diagrams for two wavelengths, λ1 (solid lines) and λ2 (broken lines), projected on the plane λ = λ1. 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[link]. The [K\alpha_1] and [K\alpha_2] components are seen separately plus a long tail due to continuum radiation, and the profile is that of the divergence of the beam.

[Figure 7.4.4.4]

Figure 7.4.4.4 | top | pdf |

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)[link]. The mosaic blocks select a band of wavelengths from the incident beam and the wavelength deviation is related to the angular deviation by [\Delta\lambda/\lambda=\cot\theta\Delta\theta]. 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.

7.4.4.3. Detecting system

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The detecting system is an integral part of the X-ray optics of a diffraction experiment, and it can be included in the phase-space diagrams. In single-crystal 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[link] and Fig. 7.4.4.3[link] that a much smaller receiving slit is sufficient in the parallel-beam geometry than in the conventional divergent-beam geometry. Mathieson (1985[link]) has given a thorough analysis of various monochromator–sample–detector combinations and has suggested the use of a two-dimensional 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 parallel-beam techniques.

The above discussion has concentrated on improving the signal-to-background ratio by optimization of the diffraction geometry. This ratio can be improved substantially by an energy-dispersive 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 synchrotron-radiation laboratories (for references, see Thomlinson & Williams, 1984[link]; Brown & Lindau, 1986[link]).

A position-sensitive detector can replace the receiving slit when a reciprocal space is scanned. TV area detectors with an X-ray-to-visible light converter and two-dimensional 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 two-dimensional proportional chambers have a spatial resolution of the order of 0.1 mm, and the relative energy resolution, [\Delta E/E\simeq 0.2], is sufficient for rejection of some of the parasitic scattering.

The NaI(Tl) scintillation counter is used most frequently as the X-ray 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 low-energy noise can be reduced to about 0.1 counts s−1.

The Ge and Si(Li) solid-state 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 [(0.024\,{\rm \AA}/\lambda)\times(1-\cos2\theta)], 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 X-ray diffraction have become available recently. The effects of the detector resolution are shown schematically in Fig. 7.4.4.5[link] for a scintillation counter and a SSD.

[Figure 7.4.4.5]

Figure 7.4.4.5 | top | pdf |

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 K-edge 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[link]).

Crystal monochromators placed in front of the detector eliminate all inelastic scattering but the TDS. The monochromator must be matched with the preceding X-ray optical system, the sample included, and therefore diffracted-beam monochromators are used in powder diffraction only (see Subsection 7.4.4.4[link]).

7.4.4.4. Powder diffraction

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The signal-to-background ratio is much worse in powder diffraction than in single-crystal 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 phase-space diagrams of a typical measurement are shown in Fig. 7.4.4.6[link]. 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 pass-band. In this case, there is no primary-beam 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.

[Figure 7.4.4.6]

Figure 7.4.4.6 | top | pdf |

Equatorial phase-space 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 = λ1 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[link]. 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 [K\alpha _1] and [K\alpha _2], and 70 times the natural width of the [K\alpha _1] 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 X-ray source. In the present case, the band is 15 eV wide, so that the monochromator can be tuned to transmit the [K\alpha_1] 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 perfect-crystal 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 parallel-beam 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.

[Figure 7.4.4.7]

Figure 7.4.4.7 | top | pdf |

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 X-ray source (Suortti & Jennings, 1977[link]); solid curve: quartz (10.1) Johansson monochromator, conventional 0.05 mm wide X-ray 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[link]). The horizontal line indicates the half-maximum 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 mm3 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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