Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 4.3, pp. 445-447   | 1 | 2 |

Section 4.3.4. Dynamical scattering: Bragg scattering effects

J. M. Cowleya and J. K. Gjønnesb

aArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA, and  bInstitute of Physics, University of Oslo, PO Box 1048, N-0316 Oslo 3, Norway

4.3.4. Dynamical scattering: Bragg scattering effects

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The distribution of diffuse scattering is modified by higher-order terms in essentially two ways: Bragg scattering of the incident and diffuse beams or multiple diffuse scattering, or by a combination.

Theoretical treatment of the Bragg scattering effects in diffuse scattering has been given by many authors, starting with Kainuma's (1955[link]) work on Kikuchi-line contrast (Howie, 1963[link]; Fujimoto & Kainuma, 1963[link]; Gjønnes, 1966[link]; Rez et al., 1977[link]; Maslen & Rossouw, 1984a[link],b[link]; Wang, 1995[link]; Allen et al., 1997[link]). Mathematical formalism may vary but the physical pictures and results are essentially the same. They may be discussed with reference to a Born-series expansion, i.e. by introducing the potential [\varphi] in the integral equation, as a sum of a periodic and a nonperiodic part [cf. equation ([link]] and arranging the terms by orders of [\Delta \varphi]. [\eqalignno{ \psi &= \psi_{0} + G\varphi \psi \cr &= [1 + G\varphi + (G\varphi)^{2} + \ldots]\psi_{0} \cr &= [1 + G\bar{\varphi} + (G\bar{\varphi})^{2} + \ldots]\psi_{0} \cr &\quad + [1 + G\bar{\varphi} + (G\bar{\varphi})^{2} + \ldots] \cr &\quad \times G (\Delta \varphi) [1 + G\bar{\varphi} + (G\bar{\varphi})^{2} + \ldots]\psi_{0} \cr &\quad + \hbox{higher-order terms}. &(}]

Some of the higher-order terms contributing to the Bragg scattering can be included by adding the essentially imaginary term [\langle \Delta \varphi G (\Delta \varphi)\rangle] to the static potential [\bar{\varphi}].

Theoretical treatments have mostly been limited to the first-order diffuse scattering. With the usual approximation to forward scattering, the expression for the amplitude of diffuse scattering in a direction [{\bf k}_{0} + {\bf u} + {\bf g}] can be written as [\eqalignno{ \psi ({\bf u} + {\bf g}) &= {\textstyle\sum\limits_{g}} {\textstyle\sum\limits_{f}} {\textstyle\int\limits_{0}^{z}} S_{hg} ({\bf k}_{0} + {\bf u}, z - z_{1}) \cr &\quad \times \Delta \Phi ({\bf u} + {\bf g} - {\bf f}) S_{f0} ({\bf k}_{0}, z_{1})\; \hbox{d}z_{1} &(}] and read (from right to left): [S_{f0}], Bragg scattering of the incident beam above the level [z_{1}]; [\Delta \Phi], diffuse scattering within a thin layer [\hbox{d}z_{1}] through the Fourier components [\Delta \Phi] of the nonperiodic potential [\Delta \Phi]; [S_{hg}], Bragg scattering between diffuse beams in the lower part of the crystal. It is commonly assumed that diffuse scattering at different levels can be treated as independent (Gjønnes, 1966[link]), then the intensity expression becomes [\eqalignno{ I({\bf u} + {\bf g}) &= {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{h'}} {\textstyle\sum\limits_{f}} {\textstyle\sum\limits_{f'}} {\textstyle\int\limits_{0}^{z}} S_{gh} (2) S_{gh'}^{*} (2) \cr &\quad \times \langle \Delta \Phi ({\bf u} + {\bf h} - {\bf f}) \Delta \Phi^{*} ({\bf u} + {\bf h}' - {\bf f}')\rangle \cr &\quad \times S_{f0} (1) S_{f'0}^{*} (1)\; \hbox{d}z_{1}, &(}] where (1) and (2) refer to the regions above and below the diffuse-scattering layer. This expression can be manipulated further, e.g. by introducing Bloch-wave expansion of the scattering matrices, viz [\eqalignno{ I ({\bf u} + {\bf g}) &= {\textstyle\sum\limits_{hh'}} {\textstyle\sum\limits_{ff'}} {\textstyle\sum\limits_{jj'}} {\textstyle\sum\limits_{ii'}} C_{g}^{j} (2) C_{g}^{j'*} (2) C_{h}^{j*} (2) C_{h'}^{j'} (2) \cr &\quad \times {\exp [i(\gamma^{i} - \gamma^{i'})z] - \exp [i(\gamma\hskip 2pt^{j} - \gamma\hskip 2pt^{j'})z]\over \gamma^{i} - \gamma^{i'} - \gamma\hskip 2pt^{j} + \gamma\hskip 2pt^{j'}} \cr &\quad \times \langle f({\bf u} + {\bf h} - {\bf f}) f^{*} ({\bf u} + {\bf h}' - {\bf f}')\rangle \cr &\quad \times C_{f}^{i*} (1) C_{f'}^{i'} (1) C_{0}^{i}(1) C_{0}^{i*} (1), &(}] which may be interpreted as scattering by [\Delta \varphi] between Bloch waves belonging to the same branch (intraband scattering) or different branches (interband scattering). Another alternative is to evaluate the scattering matrices by multislice calculations (Section 4.3.5[link]).

Expressions such as ([link] contain a large number of terms. Unless very detailed calculations relating to a precisely defined model are to be carried out, attention should be focused on the most important terms.

In Kikuchi-line contrast, the scattering in the upper part of the crystal is usually not considered and frequently the angular variation of the [\Phi ({\bf u})] is also neglected. In diffraction contrast from small-angle inelastic scattering, it may be sufficient to consider the intraband terms [[i = i',\; j = j'] in ([link]].

In studies of diffuse-scattering distribution, the factor [\langle \Phi ({\bf u} + {\bf h}) \Phi ({\bf u} + {\bf h'})\rangle] will produce two types of terms: Those with [{\bf h} = {\bf h'}] result only in a redistribution of intensity between corresponding points in the Brillouin zones, with the same total intensity. Those with [{\bf h} \neq {\bf h'}] lead to enhancement or reduction of the total diffuse intensity and hence absorption from the Bragg beam and enhanced/reduced intensity of secondary radiation, i.e. anomalous absorption and channelling effects. They arise through interference between different Fourier components of the diffuse scattering and carry information about position of the sources of diffuse scattering, referred to the projected unit cell. This is exploited in channelling experiments, where beam direction is used to determine atom reaction (Taftø & Spence, 1982[link]; Taftø & Lehmpfuhl, 1982[link]).

Gjønnes & Høier (1971[link]) expressed this information in terms of the Fourier transform R of the generalized or Kikuchi-line form factor; [{\langle \Delta \Phi ({\bf u}) \Delta \Phi^{*} ({\bf u} + {\bf h})\rangle {\bf u}^{2} ({\bf u} + {\bf h})^{2} = Q ({\bf u} + {\bf h}) = {\scr F} \{R ({\bf r},{\bf g})\}\eqno (}] includes information both about correlations between sources of diffuse scattering and about their position in the projected unit cell. It is seen that [Q({\bf u}, 0)] represents the kinematical intensity, hence [\int R ({\bf r},{\bf g})] dg is the Patterson function. The integral of [Q({\bf u},{\bf h})] in the plane gives the anomalous absorption (Yoshioka, 1957[link]) which is related to the distribution [R ({\bf r}, 0)] of scattering centres across the unit cell.

The scattering factor [\langle \Delta \Phi ({\bf u}) \Delta \Phi^{*} ({\bf u} - {\bf h})\rangle] can be calculated for different modes. For one-electron excitations as an extension of the Waller–Hartree expression (Gjønnes, 1962[link]; Whelan, 1965[link]): [\eqalignno{ \langle \Delta \Phi ({\bf u}) \Delta \Phi^{*} ({\bf u} - {\bf h})\rangle &= {\sum\limits_{i}}\; {f_{ii} ({\bf h}) - f_{ii} ({\bf u}) f_{ii} (|{\bf h} - {\bf u}|)\over u^{2}({\bf h} - {\bf u})^{2}} \cr &\quad - {\sum\limits_{i}} {\sum\limits_{i \neq j}}\; {f_{ij} ({\bf u}) f_{ij} (|{\bf h} - {\bf u}|)\over u^{2} ({\bf h} - {\bf u})^{2}}, &(}] where [f_{ij}] are the one-electron amplitudes (Freeman, 1959[link]).

A similar expression for scattering by phonons is obtained in terms of the scattering factors [G_{j} ({\bf u},{\bf g})] for the branch j, wavevector g and a polarization vector [{\bf l}_{j,\, q}] (see Chapter 4.1[link] ): [\langle \Delta \Phi ({\bf u}) \Delta \Phi^{*} ({\bf u} - {\bf h})\rangle = G_{j} ({\bf u},{\bf q}) G_{j}^{*} ({\bf u} - {\bf h},{\bf q}), \eqno(] independent phonons being assumed.

For scattering from substitutional order in a binary alloy with ordering on one site only, we obtain simply [{\Delta \Phi ({\bf u}) \Delta \Phi^{*} ({\bf u} - {\bf h}) = |\Delta \varphi ({\bf u})|^{2}\; {f_{A} (|{\bf u} - {\bf h}|) - f_{B} (|{\bf u} - {\bf h}|)\over f_{A}({\bf u}) - f_{B}({\bf u})},} \eqno(] where [f_{A, \, B}] are atomic scattering factors. It is seen that [Q({\bf u})] then does not contain any new information; the location of the site involved in the ordering is known.

When several sites are involved in the ordering, the dynamical scattering factor becomes less trivial, since scattering factors for the different ordering parameters (for different sites) will include a factor [\exp(2\pi i{\bf r}_{m} \cdot {\bf h})] (see Andersson et al., 1974[link]).

From the above expressions, it is found that the Bragg scattering will affect diffuse scattering from different sources differently: Diffuse scattering from substitutional order will usually be enhanced at low and intermediate angles, whereas scattering from thermal and electronic fluctuations will be reduced at low angles and enhanced at higher angles. This may be used to study substitutional order and displacement order (size effect) separately (Andersson, 1979[link]).

The use of such expressions for quantitative or semiquantitative interpretation raises several problems. The Bragg scattering effects occur in all diffuse components, in particular the inelastic scattering, which thus may no longer be represented by a smooth, monotonic background. It is best to eliminate this experimentally. When this cannot be done, the experiment should be arranged so as to minimize Kikuchi-line excess/deficient terms, by aligning the incident beam along a not too dense zone. In this way, one may optimize the diffuse-scattering information and minimize the dynamical corrections, which then are used partly as guides to conditions, partly as refinement in calculations.

The multiple scattering of the background remains as the most serious problem. Theoretical expressions for multiple scattering in the absence of Bragg scattering have been available for some time (Moliere, 1948[link]), as a sum of convolution integrals [I ({\bf u}) = [(\mu t) I_{1} ({\bf u}) + (1/2) (\mu t)^{2} I_{2} ({\bf u}) + \ldots] \exp (-\mu t), \eqno (] where [I_{2} ({\bf u}) = I_{1} ({\bf u}) * I_{1} ({\bf u})\ldots] etc., and [I_{1} ({\bf u})] is normalized.

A complete description of multiple scattering in the presence of Bragg scattering should include Bragg scattering between diffuse scattering at all levels [z_{1}, z_{2}], etc. This quickly becomes unwieldy. Fortunately, the experimental patterns seem to indicate that this is not necessary: The Kikuchi-line contrast does not appear to be very sensitive to the exact Bragg condition of the incident beam. Høier (1973[link]) therefore introduced Bragg scattering only in the last part of the crystal, i.e. between the level [z_{n}] and the final thickness z for n-times scattering. He thus obtained the formula: [\eqalignno{ I ({\bf u} + {\bf h}) &= {\textstyle\sum\limits_{j}} |C_{h}^{j}|^{2} \left\{A_{1}^{j} {\textstyle\sum\limits_{g}} {\textstyle\sum\limits_{\neq g'}} F_{1} ({\bf u},{\bf g},{\bf g}') C_{g}^{j} C_{g'}^{j*} \right.\cr &\quad\left. + {\textstyle\sum\limits_{n}} {\textstyle\sum\limits_{g}} A_{ng}^{j} F_{n} ({\bf u},{\bf g})|C_{g}^{\;j}|^{2}\right\}, &(}] where [F_{n}] are normalized scattering factors for nth-order multiple diffuse scattering and [A_{n}^{j}] are multiple-scattering coefficients which include absorption.

When the thickness is increased, the variation of [F_{n} ({\bf u},{\bf g})] with angle becomes slower, and an expression for intensity of the channelling pattern is obtained (Gjønnes & Taftø, 1976[link]): [\eqalignno{ I ({\bf u} + {\bf h}) &= {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{g}} {\textstyle\sum\limits_{n}} |C_{h}^{j}|^{2}|C_{g}^{j}|^{2} A_{n}^{j} \cr &= {\textstyle\sum\limits_{j}} |C_{h}^{j}|^{2} {\textstyle\sum\limits_{n}} A_{n}^{j} \rightarrow |C_{h}^{j}|^{2} / \mu\hskip 2pt^{j} ({\bf u}). &(}] Another approach is the use of a modified diffusion equation (Ohtsuki et al., 1976[link]).

These expressions seem to reproduce the development of the general background with thickness over a wide range of thicknesses. It may thus appear that the contribution to the diffuse background from known sources can be treated adequately – and that such a procedure must be included together with adequate filtering of the inelastic component in order to improve the quantitative interpretation of diffuse scattering.


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