Thermal diffuse scattering

Willis, B. T. M.

doi:10.1107/97809553602060000607

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

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International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 653-657

Section 7.4.2. Thermal diffuse scattering

B. T. M. Willis^d

7.4.2. Thermal diffuse scattering

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7.4.2.1. Glossary of symbols

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$[\widehat {e}_{j}]$	Direction cosines of e_j(q)
e _j (q)	Polarization vector of normal mode (jq)
E _j (q)	Energy of mode (jq)
E _meas	Total integrated intensity measured under Bragg peak
E ₀	Integrated intensity from Bragg scattering
E ₁	Integrated intensity from one-phonon scattering
F (h)	Structure factor
$[\hbar]$	Planck's constant h divided by 2π
2πh	Reciprocal-lattice vector
H	Scattering vector
j	Label for branch of dispersion relation
k ₀	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 of IT B (2001)]. 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) 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 LiNbO₃ (Krec, Steiner, Pongratz & Skalicky, 1984), 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, 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).

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).

Let E_meas 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) reduces to the familiar result (James, 1962) $[E_0={{N\lambda ^{3}| F({\bf h}) | ^2}\over{V\omega _{0}\sin 2\theta }}, \eqno(7.4.2.3)]$ 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 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 $[E_{j}\left ({\bf q}\right) ]$ 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 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 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/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) 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 of IT B (2001)], 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) is over the range of measurement, and the summation in (7.4.2.7) is over the three acoustic branches. Only long-wavelength elastic waves, with a linear dispersion relation, equation (7.4.2.5), need be considered.

7.4.2.2.1. Evaluation of J(q)

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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 $[\widehat {e}_1]$ , $[\widehat {e}_2]$ , $[\widehat {e}_3]$ are the direction cosines of the polarization vector with respect to orthogonal axes x, y, z, then the velocity v_j 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, A_km 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 of IT B (2001).

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 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 $[J({\bf q}) ={{k_BT}\over{q^2}}\sum _{m=1}^3\sum _{n=1}^3\left ({\bf A}^{-1}\right) _{mn}H_mH_n, \eqno (7.4.2.8)]$ 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 [E_j(q) = k_BT] . 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 $[c_{11}-c_{12}=c_{44}]$ ), 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) into (7.4.2.6) 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 T_mn, 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) 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 h^T = (h, k, l) . The final expression for α, from (7.4.2.11) and (7.4.2.12), 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) 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 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| 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), 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 $[ T_{mn}={{k_BT}\over {8\pi ^3}}\left \langle \left ({\bf A}^{-1}\right) _{mn}\right \rangle {\int} {1\over q^2}\,{\rm d}q,]$ 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) 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 $[\alpha ={{H^2k_BTq_m}\over {3\pi ^2\rho {\bf v}_L^2}}, \eqno (7.4.2.14)]$ 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 γ-ray 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 $[\exp \alpha \approx 1+\alpha ]$ , 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 $[\Delta B={{8k_BTq_m}\over {3\pi ^2\rho {\bf v}_L^2}}.]$ 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; Stevenson & Harada, 1983).

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).

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. ɛ is either +1 or −1, where ɛ = +1 corresponds to phonon emission (or phonon creation) in the crystal and a loss in energy of the neutrons after scattering, and ɛ = −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) 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) 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).

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; Cooper, 1971):

(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), 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).

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). 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| top | pdf |

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

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International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 653-657