Distortions in binary systems

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 433-435 | 1 | 2 |

Section 4.2.4.4.5. Distortions in binary systems

H. Jagodzinski^a and F. Frey^b

^aInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany, and ^bInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany

4.2.4.4.5. Distortions in binary systems

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In substitutional binary systems (primitive cell with only one sublattice) the Borie–Sparks method is widely used (Sparks & Borie, 1966; Borie & Sparks, 1971). The method is formulated in the short-range-order-parameter formalism. The diffuse scattering may be separated into two parts (a) owing to short-range order and (b) owing to static displacements.

Corresponding to the expansion (4.2.4.75), $[I_{d} = I_{\rm sro} + I_{2} + I_{3}]$ , where $[I_{\rm sro}]$ is given by equation (4.2.4.71b) and the correction terms $[I_{2}]$ and $[I_{3}]$ relate to the linear and the quadratic term in (4.2.4.75). The intensity expression will be split into terms of A–A, A–B, … pairs. More explicitly $[\Delta_{\nu \nu'}{\bf r} = {\bf u}_{{\bf n} \nu} - {\bf u}_{{\bf n}' \nu'}]$ and with the following abbreviations: $[\eqalignno{\delta_{{\bf nn}'|AA} &= {\bf u}_{{\bf n}|A} - {\bf u}_{{\bf n}'|A} = x_{{\bf nn}'|AA} {\bf a} + y_{{\bf nn}'|AA} {\bf b} + z_{{\bf nn}'|AA} {\bf c}\cr \delta_{{\bf nn}'|AB} &= {\bf u}_{{\bf n}|A} - {\bf u}_{{\bf n}'|B} = \ldots \cr F_{{\bf nn}'|AA} &= f_{A}^{2} / (\;f_{A} - f_{B})^{2} \cdot [(c_{A} / c_{B}) + \alpha_{{\bf nn}'}]\cr F_{{\bf nn}'|BB} &= f_{B}^{2} / (\;f_{A} - f_{B})^{2} \cdot [(c_{B} / c_{A}) + \alpha_{{\bf nn}'}]\cr F_{{\bf nn}'|AB} &= 2f_{A} f_{B} / (\;f_{A} - f_{B})^{2} \cdot (1 - \alpha_{{\bf nn}'}) = F_{{\bf nn}'|BA}}]$ one finds (where the short-hand notation is self-explanatory): $[\eqalignno{I_{2} &= 2 \pi ic_{A} c_{B} (\;f_{A} - f_{B})^{2} \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \{H \cdot [F_{{\bf nn}'|AA} \langle x_{{\bf nn}'|AA} \rangle &\cr &\quad + F_{{\bf nn}'|BB} \langle x_{{\bf nn}'|BB} \rangle + F_{{\bf nn}'|AB} \langle x_{{\bf nn}'|AB} \rangle] + K \cdot [\hbox{`}y\hbox{'}] &\cr &\quad + L \cdot [\hbox{`}z\hbox{'}]\} \exp \{2 \pi i{\bf H} \cdot ({\bf n} - {\bf n}')\} &(4.2.4.79)}]$ $[\eqalignno{I_{3} &= c_{A} c_{B} (\;f_{A} - f_{B})^{2} (-2 \pi)^{2} \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \{H^{2} [F_{{\bf nn}'} \langle x_{{\bf nn}'|AA}^{2} \rangle &\cr &\quad + F_{{\bf nn}'|BB} \langle x_{{\bf nn}'|BB}^{2} \rangle + F_{{\bf nn}'|AB} \langle x_{{\bf nn}'|AB}^{2} \rangle] &\cr &\quad + K^{2} \cdot [\hbox{`}y^{2}\hbox{'}] + L^{2} \cdot [\hbox{`}z^{2}\hbox{'}] &\cr &\quad + HK [F_{{\bf nn}'|AA} \langle (xy)_{{\bf nn}'|AA} \rangle + F_{{\bf nn}'|BB} \langle (xy)_{{\bf nn}'|BB} \rangle &\cr &\quad + F_{{\bf nn}'|AB} \langle (xy)_{{\bf nn}'|AB} \rangle] &\cr &\quad + KL [\hbox{`}(yz)\hbox{'}] + LH [\hbox{`}(zx)\hbox{'}]\} &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf n} - {\bf n}')\}. &(4.2.4.80)}]$ With further abbreviations $[\eqalign{\gamma_{{\bf nn}'|x} &= 2 \pi (F_{{\bf nn}'|AA} \langle x_{{\bf nn}'|AA} \rangle + F_{{\bf nn}'|BB} \langle x_{{\bf nn}'|BB} \rangle \cr &\quad + F_{{\bf nn}'|AB} \langle x_{{\bf nn}'|AB} \rangle) \cr \gamma_{{\bf nn}'|y} &= \ldots \cr \gamma_{{\bf nn}'|z} &= \ldots \cr \delta_{{\bf nn}'|x} &= (-2 \pi^{2}) (F_{{\bf nn}'|AA} \langle x_{{\bf nn}'|AA}^{2} \rangle + F_{{\bf nn}'|BB} \langle x_{{\bf nn}'|BB}^{2} \rangle \cr &\quad + F_{{\bf nn}'|AB} \langle x_{{\bf nn}'|AB}^{2} \rangle) \cr \delta_{{\bf nn}'|y} &= \ldots \cr \delta_{{\bf nn}'|z} &= \ldots \cr \varepsilon_{{\bf nn}'|xy} &= (-4 \pi^{2}) (F_{{\bf nn}'|AA} \langle (xy)_{{\bf nn}'|AA} \rangle + F_{{\bf nn}'|BB} \langle (xy)_{{\bf nn}'|BB} \rangle \cr &\quad + F_{{\bf nn}'|AB} \langle (xy)_{{\bf nn}'|AB} \rangle) \cr \varepsilon_{{\bf nn}'|yz} &= \ldots \cr \varepsilon_{{\bf nn}'|zx} &= \ldots \cr {I}_{2} &= c_{A} c_{B} (\;f_{A} - f_{B})^{2} \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} i (\gamma_{{\bf nn}'|x} + \gamma_{{\bf nn}'|y} + \gamma_{{\bf nn}'|z}) \cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf n} - {\bf n}')\}\cr I_{3} &= c_{A} c_{B} (\;f_{A} - f_{B})^{2} \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} (\delta_{{\bf nn}'|x} H^{2} + \delta_{{\bf nn}'|y} K^{2}\cr &\quad + \delta_{{\bf nn}'|z} L^{2} + \varepsilon_{{\bf nn}'|xy} HK + \varepsilon_{{\bf nn}'|yz} KL\cr &\quad + \varepsilon_{{\bf nn}'|zx} LH \exp \{2 \pi i {\bf H} \cdot ({\bf n} - {\bf n}')\}.}]$ If the $[F_{{\bf nn}'|AA}, \ldots]$ are independent of $[|{\bf H}|]$ in the range of measurement which is better fulfilled with neutrons than with X-rays (see below), γ, δ, ɛ are the coefficients of the Fourier series: $[\eqalign{Q_{x} &= \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} i\gamma_{{\bf nn}'|x} \exp \{2 \pi i {\bf H} \cdot ({\bf n} - {\bf n}')\}\hbox{;}\cr Q_{y} &= \ldots\hbox{;} \qquad Q_{z} = \ldots\hbox{;}\cr R_{x} &= \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \delta_{{\bf nn}'|x} \exp \{2 \pi i {\bf H} \cdot ({\bf n} - {\bf n}')\}\hbox{;}\cr R_{y} &= \ldots\hbox{;} \qquad R_{z} = \ldots\hbox{;}\cr S_{xy} &= \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \varepsilon_{{\bf nn}'|xy} \exp \{2 \pi i {\bf H} \cdot ({\bf n} - {\bf n}')\}\hbox{;}\cr S_{yz} &= \ldots\hbox{;} \qquad S_{zx} = \ldots.}]$ The functions Q, R, S are then periodic in reciprocal space.

The double sums over n, n′ may be replaced by $[N \sum_{m, \, n, \, p}]$ where m, n, p are the coordinates of the interatomic vectors $[({\bf n} - {\bf n}')]$ and $[I_{2}]$ becomes $[\eqalignno{I_{2} &= - Nc_{A} c_{B} (\;f_{A} - f_{B})^{2} \textstyle\sum\limits_{m} \textstyle\sum\limits_{n} \textstyle\sum\limits_{p} (H\gamma_{lmn|x} + \ldots + \ldots) &\cr &\quad \times \sin 2 \pi (Hm + Kn + Lp). &(4.2.4.81)}]$ The intensity is therefore modulated sinusoidally and increases with scattering angle. The modulation gives rise to an asymmetry in the intensity around a Bragg peak. Similar considerations for $[I_{3}]$ reveal an intensity contribution $[h_{i}^{2}]$ times a sum over cosine terms which is symmetric around the Bragg peaks. This term shows quite an analogous influence of local static displacements and thermal movements: an increase of diffuse intensity around the Bragg peaks and a reduction of Bragg intensities, which is not discussed here. The second contribution $[I_{2}]$ has no analogue owing to the non-vanishing average displacement. The various diffuse intensity contributions may be separated by symmetry considerations. Once they are separated, the single coefficients may be determined by Fourier inversion. Owing to the symmetry constraints there are relations between the displacements $[\langle x \ldots \rangle]$ and, in turn, between the γ and Q components. The same is true for the δ, ɛ, and R, S components. Consequently, there are symmetry conditions for the individual contributions of the diffuse intensity which may be used to distinguish them. Generally the total diffuse intensity may be split into only a few independent terms. The single components of Q, R, S may be expressed separately by combinations of diffuse intensities which are measured in definite selected volumes in reciprocal space. Only a minimum volume must be explored in order to reveal the behaviour over the whole reciprocal space. This minimum repeat volume is different for the single components: $[I_{\rm sro}]$ , Q, R, S or combinations of them.

The Borie–Sparks method has been applied very frequently to binary and even ternary systems; some improvements have been communicated by Bardhan & Cohen (1976). The diffuse scattering of the historically important metallic compound Cu₃Au has been studied by Cowley (1950a,b), and the pair correlation parameters could be determined. The typical fourfold splitting was found by Moss (1966) and explained in terms of atomic displacements. The same splitting has been found for many similar compounds such as Cu₃Pd (Ohshima et al., 1976), Au₃Cu (Bessière et al., 1983), and Ag_1−xMg_x (x = 0.15–0.20) (Ohshima & Harada, 1986). Similar pair correlation functions have been determined. In order to demonstrate the disorder parameters in terms of structural models, computer programs were used (e.g. Gehlen & Cohen, 1965). A similar microdomain model was proposed by Hashimoto (1974, 1981, 1983, 1987). According to approximations made in the theoretical derivation the evaluation of diffuse scattering is generally restricted to an area in reciprocal space where the influence of displacements is of the same order of magnitude as that of the pair correlation function. The agreement between calculation and measurement is fairly good but it should be remembered that the amount and quality of the experimental information used is low. No residual factors are so far available; these would give an idea of the reliability of the results.

The more general case of a multi-component system with several atoms per lattice point was treated similarly by Hayakawa & Cohen (1975). Sources of error in the determination of the short-range-order coefficients are discussed by Gragg et al. (1973). In general the assumption of constant $[F_{{\bf nn}'|AA},\ldots]$ produces an incomplete separation of the order- and displacement-dependent components of diffuse scattering. By an alternative method, by separation of the form factors from the Q, R, S functions and solving a large array of linear relationships by least-squares methods, the accuracy of the separation of the various contributions is improved (Tibbals, 1975; Georgopoulos & Cohen, 1977; Wu et al., 1983). The method does not work for neutron diffraction. Also, the case of planar short-range order with corresponding diffuse intensity along rods in reciprocal space may be treated along the Borie & Sparks method (Ohshima & Moss, 1983).

Multi-wavelength methods taking advantage of the variation of the structure factor near an absorption edge (anomalous dispersion) are discussed by Cenedese et al. (1984). The same authors show that in some cases the neutron method allows for a contrast variation by using samples with different isotope substitution.

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 433-435