General formulation (elastic diffuse scattering)

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

International
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Crystallography
Volume B
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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 429-431 | 1 | 2 |

Section 4.2.4.4.1. General formulation (elastic diffuse scattering)

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

4.2.4.4.1. General formulation (elastic diffuse scattering)

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In this section general formulae for diffuse scattering will be derived which may best be applied to crystals with a well ordered average structure, characterized by (almost) sharp Bragg peaks. Textbooks and review articles concerning defects and local ordering are by Krivoglaz (1969), Dederichs (1973), Peisl (1975), Schwartz & Cohen (1977), Schmatz (1973, 1983), Bauer (1979), and Kitaigorodsky (1984). A series of interesting papers on local order is given by Young (1975) and also by Cowley et al. (1979). Expressions for polycrystalline sample material are given by Warren (1969) and Fender (1973).

Two general methods may be applied:

(a) the average difference cluster method, where a representative cluster of scattering differences between the average structure and the cluster is used; and
(b) the method of short-range-order correlation functions where formal parameters are introduced.

Both methods are equivalent in principle. The cluster method is generally more convenient in cases where a single average cluster is a good approximation. This holds for small concentrations of clusters with sufficient space in between. The method of short-range-order parameters is optimal in cases where isolated clusters are not realized and the correlations do not extend to long distances. Otherwise periodic solutions are more convenient in most cases.

In any case, the first step towards the solution of the diffraction problem is the accurate determination of the average structure. As described in Section 4.2.3.2 important information on fractional occupations, interstitials and displacements (unusual thermal parameters) of atoms may be derived. Unfortunately all defects contribute to diffuse scattering; hence one has to start with the assumption that the disorder to be interpreted is predominant. Fractional occupancy of certain lattice sites by two or more kinds of atoms plays an important role in the literature, especially in metallic or ionic structures. Since vacancies may be treated as atoms with zero scattering amplitude, structures containing vacancies may be formally treated as multi-component systems.

Since the solution of the diffraction problem should not be restricted to metallic systems with a simple (primitive) structure, we have to consider the structure of the unit cell – as given by the average structure – and the propagation of order according to the translation group separately. In simple metallic systems this difference is immaterial. It is well known that the thermodynamic problem of propagation of order in a three-dimensional crystal can hardly be solved analytically in a general way. Some solutions have been published with the aid of the so-called Ising model using next-nearest-neighbour interactions. They are excellent for an understanding of the principles of order–disorder phenomena, but they can scarcely be applied quantitatively in practical problems. Hence, methods have been developed to derive the propagation of order from the diffraction pattern by means of Fourier transformation. This method has been described qualitatively in Section 4.2.3.1, and will be used here for a quantitative application. In a first approximation the assumption of a small number of different configurations of the unit cell is made, represented by the corresponding number of structure factors. Displacements of atoms caused by the configurations of the neighbouring cells are excluded. This problem will be treated subsequently.

The finite number of structures of the unit cell in the disordered crystal is given by $[F_{\nu} ({\bf r}) = \textstyle\sum\limits_{j} \textstyle\sum\limits_{\mu} \pi_{j\mu}^{\nu}\; f_{\mu} ({\bf r - r}_{j}). \eqno(4.2.4.55)]$ Note that $[F_{\nu}({\bf r})]$ is defined in real space, and $[{\bf r}_{j}]$ gives the position vector of site j; $[\pi_{j\mu}^{\nu} = 1]$ if in the νth structure factor the site j is occupied by an atom of kind µ, and 0 elsewhere.

In order to apply the laws of Fourier transformation adequately, it is useful to introduce the distribution function of $[F_{\nu}]$ $[\pi_{\nu} ({\bf r}) = \textstyle\sum\limits_{{\bf n}} \pi_{{\bf n}\nu} \delta ({\bf r} - {\bf n}) \eqno(4.2.4.56)]$ with $[\pi_{{\bf n}\nu} = 1]$ , if the cell $[{\bf n} = n_{1}{\bf a} + n_{2}{\bf b} + n_{3}{\bf c}]$ has the $[F_{\nu}]$ structure, and $[\pi_{{\bf n}\nu} = 0]$ elsewhere.

In the definitions given above $[\pi_{{\bf n}\nu}]$ are numbers (scalars) assigned to the cell. Since all these are occupied we have $[\textstyle\sum\limits_{\nu} \pi_{\nu} ({\bf r}) = l({\bf r})]$ with $[l({\bf r})]$ = lattice in real space.

The structure of the disordered crystal is given by $[\textstyle\sum\limits_{\nu} \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}). \eqno(4.2.4.57)]$ $[\pi_{\nu} ({\bf r})]$ consists of $[\alpha_{\nu} N]$ points, where $[N = N_{1} N_{2} N_{3}]$ is the total (large) number of unit cells and $[\alpha_{\nu}]$ denotes the a priori probability (concentration) of the νth cell occupation.

It is now useful to introduce $[\Delta \pi_{\nu} ({\bf r}) = \pi_{\nu} ({\bf r}) - \alpha_{\nu} l({\bf r}) \eqno(4.2.4.58)]$ with $[\textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) = \textstyle\sum\limits_{\nu} \pi_{\nu} ({\bf r}) - l({\bf r}) \textstyle\sum\limits_{\nu} \alpha_{\nu} = l({\bf r}) - l({\bf r}) = 0.]$ Introducing (4.2.4.58) into (4.2.4.57): $[\eqalignno{\textstyle\sum\limits_{\nu} \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}) &= \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}) + l({\bf r}) \textstyle\sum\limits_{\nu} \alpha_{\nu} F_{\nu} ({\bf r}) &\cr \noalign{\vskip5pt}&= \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}) + l({\bf r}) \langle F({\bf r})\rangle. &(4.2.4.59)}]$ Similarly: $[\eqalign{&\phantom{\sum_{\nu} \alpha_{\nu} \Delta F_{\nu}} \Delta F_{\nu} ({\bf r}) = F_{\nu} ({\bf r}) - \langle F({\bf r})\rangle\cr &\textstyle\sum\limits_{\nu} \alpha_{\nu} \Delta F_{\nu} ({\bf r}) = \textstyle\sum\limits_{\nu} \alpha_{\nu} F_{\nu} ({\bf r}) - \textstyle\sum\limits_{\nu} \alpha_{\nu} \langle F({\bf r})\rangle = 0.} \eqno(4.2.4.60)]$ Using (4.2.4.60) it follows from (4.2.4.58) that $[\eqalignno{\textstyle\sum\limits_{\nu} \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}) &= \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * \Delta F_{\nu} ({\bf r}) &\cr \noalign{\vskip5pt}&\quad + \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * \langle F({\bf r})\rangle + l({\bf r}) * \langle F({\bf r})\rangle &\cr \noalign{\vskip5pt}&= \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * \Delta F_{\nu} ({\bf r}) + l({\bf r}) * \langle F({\bf r})\rangle. &\cr&&(4.2.4.61)}]$ Comparison with (4.2.4.59) yields $[\textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * F_{\nu} ({\bf r}) = \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * \Delta F_{\nu} ({\bf r}).]$ Fourier transformation of (4.2.4.61) gives $[\textstyle\sum\limits_{\nu} \Pi_{\nu} ({\bf H}) F_{\nu} ({\bf H}) = \textstyle\sum\limits_{\nu} \Delta \Pi_{\nu} ({\bf H}) \Delta F_{\nu} ({\bf H}) + L({\bf H}) \langle F({\bf H})\rangle]$ with $[\textstyle\sum\limits_{\nu} \Delta \Pi_{\nu} ({\bf H}) = 0\hbox{;} \quad \textstyle\sum\limits_{\nu} \Delta F_{\nu} ({\bf H}) = 0.]$ The expression for the scattered intensity is therefore $[\eqalignno{I({\bf H}) &= \left|\textstyle\sum\limits_{\nu} \Delta \Pi_{\nu} ({\bf H}) \Delta F_{\nu} ({\bf H})\right|^{2} + |L({\bf H}) \langle F({\bf H})\rangle |^{2} &\cr \noalign{\vskip5pt}&\quad + L({\bf H}) \left\{\langle F^{+} ({\bf H})\rangle \textstyle\sum\limits_{\nu} \Delta \Pi_{\nu} ({\bf H}) F_{\nu} ({\bf H})\right. &\cr \noalign{\vskip5pt}&\quad + \left.\langle F^{+} ({\bf H})\rangle \textstyle\sum\limits_{\nu} \Delta \Pi_{\nu}^{+} ({\bf H}) F_{\nu}^{+} ({\bf H})\right\}. &(4.2.4.62)}]$ Because of the multiplication by L(H) the third term in (4.2.4.62) contributes to sharp reflections only. Since they are correctly given by the second term in (4.2.4.62), the third term vanishes, Hence, the diffuse part is given by $[I_{d} ({\bf H}) = \left|\textstyle\sum\limits_{\nu} \Delta \Pi_{\nu} ({\bf H}) \Delta F_{\nu} ({\bf H})\right|^{2}. \eqno(4.2.4.63)]$ For a better understanding of the behaviour of diffuse scattering it is useful to return to real space: $[\eqalignno{i_{d} ({\bf r}) &= \textstyle\sum\limits_{\nu} \Delta \pi_{\nu} ({\bf r}) * \Delta F_{\nu} ({\bf r}) * \textstyle\sum\limits_{\nu'} \Delta \pi_{\nu'} (-{\bf r}) * \Delta F_{\nu'} (-{\bf r}) &\cr \noalign{\vskip5pt}&= \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} \Delta \pi_{\nu} ({\bf r}) * \Delta \pi_{\nu'} (-{\bf r}) * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r}) &\cr&&(4.2.4.64)}]$ and with (4.2.4.58): $[\eqalignno{i_{d} ({\bf r}) &= \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} [\pi_{\nu} ({\bf r}) - \alpha_{\nu} l({\bf r})] * [\pi_{\nu'} (-{\bf r}) - \alpha_{\nu'} l({\bf r})] &\cr \noalign{\vskip5pt}&\quad * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r}). &(4.2.4.65)}]$ Evaluation of this equation for a single term yields $[\eqalignno{&[\pi_{\nu} ({\bf r}) * \pi_{\nu'} (-{\bf r}) - \alpha_{\nu} l({\bf r}) * \pi_{\nu'} (-{\bf r}) &\cr \noalign{\vskip5pt}&\quad - \alpha_{\nu'} l({\bf r}) * \pi_{\nu} ({\bf r}) + \alpha_{\nu} \alpha_{\nu'} l({\bf r}) * l({\bf r})] &\cr \noalign{\vskip5pt}&\quad * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} ({\bf r}). &(4.2.4.66)}]$ Since l(r) is a periodic function of points, all convolution products with l(r) are also periodic. For the final evaluation the decrease of a number of overlapping points (maximum N) in the convolution products with increasing displacements of the functions is neglected (no particle-size effect). Then (4.2.4.66) becomes $[\eqalignno{&[\pi_{\nu} ({\bf r}) * \pi_{\nu'} (-{\bf r}) - N\alpha_{\nu} \alpha_{\nu'} l({\bf r}) - N\alpha_{\nu} \alpha_{\nu'} l({\bf r}) + N\alpha_{\nu} \alpha_{\nu'} l({\bf r})] &\cr \noalign{\vskip5pt}&\quad * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r}) &\cr \noalign{\vskip5pt}&\quad\quad = [\pi_{\nu} ({\bf r}) * \pi_{\nu'} (-{\bf r}) - N\alpha_{\nu} \alpha_{\nu'} l({\bf r})] &\cr \noalign{\vskip5pt}&\quad\quad\quad * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r}). &(4.2.4.67)}]$ If the first term in (4.2.4.67) is considered, the convolution of the two functions for a given distance n counts the number of coincidences of the function $[\pi_{\nu} ({\bf r})]$ with $[\pi_{\nu'} (-{\bf r})]$ . This quantity is given by $[Nl({\bf r}) \alpha_{\nu} p_{\nu \nu'} (-{\bf r})]$ , where $[\alpha_{\nu} p_{\nu \nu'} ({\bf r})]$ is the probability of a pair occupation in the r direction.

Equation (4.2.4.67) then reads: $[\eqalignno{&[Nl({\bf r}) \alpha_{\nu} p_{\nu \nu'} (-{\bf r}) - N\alpha_{\nu} \alpha_{\nu'} l({\bf r})] &\cr \noalign{\vskip5pt}&\quad * \Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r}) &\cr \noalign{\vskip5pt}&\quad\quad = N\alpha_{\nu} [p'_{\nu \nu'} (-{\bf r}) l({\bf r})] &\cr \noalign{\vskip5pt}&\quad\quad\quad * [\Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r})] &(4.2.4.68)}]$ with $[p'_{\nu \nu'} ({\bf r}) = p_{\nu \nu'} ({\bf r}) - \alpha_{\nu'}]$ . The function $[\alpha_{\nu} p'_{\nu \nu'} ({\bf r})]$ is usually called the pair-correlation function $[g\leftrightarrow \alpha_{\nu \nu'|{\bf n n}'}]$ in the physical literature.

The following relations hold: $[\displaylines{\hfill \textstyle\sum\limits_{\nu} \alpha_{\nu} = 1 \hfill (4.2.4.69a)\cr \noalign{\vskip5pt}\hfill \textstyle\sum\limits_{\nu'} \alpha_{\nu} p_{\nu \nu'} ({\bf r}) = \alpha_{\nu'}\hbox{;}\quad \textstyle\sum\limits_{\nu'} \alpha_{\nu} p'_{\nu \nu'} ({\bf r}) = 0 \hfill (4.2.4.69b)\cr \noalign{\vskip5pt}\hfill \textstyle\sum\limits_{\nu} \alpha_{\nu} p_{\nu \nu'} ({\bf r}) = \alpha_{\nu}\hbox{;}\quad \textstyle\sum\limits_{\nu} \alpha_{\nu} p'_{\nu \nu'} ({\bf r}) = 0 \hfill (4.2.4.69c)\cr \noalign{\vskip5pt}\hfill\alpha_{\nu} p_{\nu \nu'} ({\bf r}) = \alpha_{\nu'} p_{\nu' \nu} (-{\bf r}). \hfill (4.2.4.69d)}]$ Also, functions normalized to unity are in use. Obviously the following relation is valid: $[p'_{\nu \nu'} (0) = \delta_{\nu \nu'} - \alpha_{\nu'}]$ .

Hence: $[\alpha_{\nu \nu'|{\bf n n}'} = \alpha_{\nu'} p'_{\nu \nu'} ({\bf r}) / (\delta_{\nu \nu'} - \alpha_{\nu'})]$ is unity for $[{\bf r} = 0\; ({\bf n} = {\bf n}')]$ . This property is especially convenient in binary systems.

With (4.2.4.68), equation (4.2.4.64) becomes $[{i_{d} ({\bf r}) = N \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} [\alpha_{\nu} p'_{\nu \nu'} (-{\bf r}) l({\bf r})] * [\Delta F_{\nu} ({\bf r}) * \Delta F_{\nu'} (-{\bf r})]} \eqno(4.2.4.70)]$ and Fourier transformation yields $[{I_{d} ({\bf H}) = N \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} [\alpha_{\nu} P'_{\nu \nu'} (-{\bf H}) * L({\bf H})] \Delta F_{\nu} ({\bf H}) \Delta F_{\nu'}^{+} ({\bf H}).} \eqno(4.2.4.71)]$ It may be concluded from equations (4.2.4.69) that all functions $[p'_{\nu \nu'} ({\bf r})]$ may be expressed by $[p'_{11} ({\bf r})]$ in the case of two structure factors $[F_{1}]$ , $[F_{2}]$ . Then all $[p'_{\nu \nu'} ({\bf r})]$ are symmetric in r; the same is true for the $[P'_{\nu \nu'} ({\bf H})]$ . Consequently, the diffuse reflections described by (4.2.4.71) are all symmetric. The position of the diffuse peak depends strongly on the behaviour of $[p'_{\nu \nu'} ({\bf r})]$ ; in the case of cluster formation Bragg peaks and diffuse peaks coincide. Diffuse superstructure reflections are observed if the $[p'_{\nu \nu'} ({\bf r})]$ show some damped periodicities.

It should be emphasized that the condition $[p'_{\nu \nu'} ({\bf r}) = p'_{\nu \nu'} (-{\bf r})]$ may be violated for $[\nu \neq \nu']$ if more than two cell occupations are involved. As shown below, the possibly asymmetric functions may be split into symmetric and antisymmetric parts. From equation (4.2.3.8) it follows that the Fourier transform of the antisymmetric part of $[p'_{\nu \nu'} ({\bf r})]$ is also antisymmetric. Hence, the convolution in the two terms in square brackets in (4.2.4.71) yields an antisymmetric contribution to each diffuse peak, generated by the convolution with the reciprocal lattice L(h).

Obviously, equation (4.2.4.71) may also be applied to primitive lattices, occupied by two or more kinds of atoms. Then the structure factors $[F_{\nu}]$ are merely replaced by the atomic scattering factors $[f_{\nu}]$ and the $[\alpha_{\nu}]$ are equivalent to the concentrations of atoms $[c_{\nu}]$ . In terms of the $[\alpha_{\nu \nu'|{\bf n n}'}]$ (Warren short-range-order parameters ) equation (4.2.4.71) reads $[{I_{d} ({\bf H}) = N(\;\overline{f^{2}} - \bar{f}^{2}) \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \alpha_{\nu \nu'|{\bf n n}'} \exp \{2 \pi i {\bf H} ({\bf n} - {{\bf n}'})\}.} \eqno(4.2.4.71a)]$ In the simplest case of a binary system A, B $[\displaylines{\alpha_{{\bf nn}'} = (1 - p_{AB|{\bf n n}'}) / c_{B} = (1 - p_{BA|{\bf n n'}}) / c_{A}\hbox{;} \cr \noalign{\vskip5pt} c_{A} p_{AB} = c_{B} p_{BA}\hbox{;} \quad p_{AA} = 1 - p_{AB}\hbox{;}}]$ $[\eqalignno{I_{d} ({\bf H}) &= Nc_{A} c_{B} (\;f_{A} - f_{B})^{2} \textstyle\sum\limits_{\bf n} \textstyle\sum\limits_{{\bf n}'} \alpha_{{\bf nn}'} \cr \noalign{\vskip5pt} &\quad \times \exp \{2\pi i {\bf H} ({\bf n - n}')\}. &(4.2.4.71b)}]$ [The exponential in (4.2.4.71b) may even be replaced by a cosine term owing to the centrosymmetry of this particular case.]

It should be mentioned that the formulations of the problem in terms of pair probabilities, pair correlation functions, short-range-order parameters or concentration waves (Krivoglaz, 1969) are equivalent. Using continuous electron (or nuclear) density functions where site occupancies are implied, the Patterson function may be used, too (Cowley, 1981).

References

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 429-431