Summary of basic scattering theory

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. 408-410 | 1 | 2 |

Section 4.2.2. Summary of basic scattering theory

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

4.2.2. Summary of basic scattering theory

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Diffuse scattering results from deviations from the identity of translational invariant scattering objects and from long-range correlations in space and time. Fluctuations of scattering amplitudes and/or phase shifts of the scattered wavetrains reduce the maximum capacity of interference (leading to Bragg reflections) and are responsible for the diffuse scattering, i.e. scattering parts which are not located in reciprocal space in distinct spots. Unfortunately, the terms `coherent' and `incoherent' scattering used in this context are not uniquely defined in the literature. Since all scattering processes are correlated in space and time, there is no incoherent scattering at all in its strict sense. (A similar relationship exists for `elastic' and `inelastic' scattering. Here pure inelastic scattering would take place if the momentum and the energy were transferred to a single scatterer; on the other hand, an elastic scattering process would demand a uniform exchange of momentum and energy with the whole crystal.) Obviously, both cases are idealized and the truth lies somewhere in between. In spite of this, a great many authors use the term `incoherent' systematically for the diffuse scattering away from the Bragg peaks, even if some diffuse maxima or minima, other than those due to structure factors of molecules or atoms, are observed. Although this definition is unequivocal as such, it is physically incorrect. Other authors use the term `coherent' for Bragg scattering only; all diffuse contributions are then called `incoherent'. This definition is clear and unique since it considers space and time, but it does not differentiate between incoherent and inelastic. In the case of neutron scattering both terms are essential and cannot be abandoned.

In neutron diffraction the term `incoherent' scattering is generally used in cases where no correlation between spin orientations or between isotopes of the same element exists. Hence, another definition of `incoherence' is proposed for scattering processes that are uncorrelated in space and time. In fact there may be correlations between the spins via their magnetic field, but the correlation length in space (and time) may be very small, such that the scattering process appears to be incoherent. Even in these cases the nuclei contribute to coherent (average structure) and incoherent scattering (diffuse background). Hence, the scattering process cannot really be understood by assuming nuclei which scatter independently. For this reason, it seems to be useful to restrict the term `incoherent' to cases where a random contribution to scattering is realized or, in other words, a continuous function exists in reciprocal space. This corresponds to a δ function in real four-dimensional space. The randomness may be attributed to a nucleus (neutron diffraction) or an atom (molecule). It follows from this definition that the scattering need not be continuous, but may be modulated by structure factors of molecules. In this sense we shall use the term `incoherent', remembering that it is incorrect from a physical point of view.

As mentioned in Chapter 4.1 the theory of thermal neutron scattering must be treated quantum mechanically. (In principle this is true also in the X-ray case.) In the classical limit, however, the final expressions have a simple physical interpretation. Generally, the quantum-mechanical nature of the scattering function of thermal neutrons is negligible at higher temperatures and in those cases where energy or momentum transfers are not too large. In almost all disorder problems this classical interpretation is sufficient for interpretation of diffuse scattering phenomena. This is not quite true in the case of orientational disorder (plastic crystals) where H atoms are involved.

The basic formulae given below are valid in either the X-ray or the neutron case: the atomic form factor f replaces the coherent scattering length $[b_{\rm coh}]$ (abbreviated to b). The formulation in the frame of the van Hove correlation function G(r, t) (classical interpretation, coherent part) corresponds to a treatment by a four-dimensional Patterson function P(r, t).

The basic equations for the differential cross sections are: $[\eqalignno{{\hbox{d}^{2} \sigma_{\rm coh} \over \hbox{d}\Omega \hbox{ d}(\hbar \omega)} &= N {|{\bf k}| \over |{\bf k}_{0}|} {\langle b\rangle^{2} \over 2 \pi \hbar} \int\limits_{{\bf r}} \int\limits_{t} G ({\bf r}, t) &\cr &\quad \times \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \hbox{ d}{\bf r} \; \hbox{d}t &(4.2.2.1a)\cr {\hbox{d}^{2}\sigma_{\rm inc}\over \hbox{d}\Omega \hbox{ d}(\hbar \omega)} &= N {|{\bf k}|\over |{\bf k}_{0}|} {\langle b^{2}\rangle - \langle b\rangle^{2}\over 2 \pi \hbar} \int\limits_{{\bf r}} \int\limits_{t} G_{s} ({\bf r}, t) &\cr &\quad \times \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \hbox{ d}{\bf r}\; \hbox{d}t &(4.2.2.1b)}]$ (N = number of scattering nuclei of same chemical species; $[{\bf k}]$ , $[{\bf k}_{0}]$ = wavevectors after/before scattering).

The integrations over space may be replaced by summations in disordered crystals except for cases where structural elements exhibit a liquid-like behaviour. Then the van Hove correlation functions are: $[\eqalignno{G({\bf r}, t) &= {1\over N} \sum\limits_{{\bf r}_{j}, \, {\bf r}_{j'}} \delta \{{\bf r} - [{\bf r}_{j'}(t) - {\bf r}_{j}(0)]\} &(4.2.2.2a)\cr G_{s}({\bf r}, t) &= {1\over N} \sum\limits_{{\bf r}_{j}} \delta \{{\bf r} - [{\bf r}_{j}(t) - {\bf r}_{j}(0)]\}. &(4.2.2.2b)}]$ $[G({\bf r}, t)]$ gives the probability that if there is an atom j at $[{\bf r}_{j}(0)]$ at time zero, there is an arbitrary atom j′ at $[{\bf r}_{j'}(t)]$ at arbitrary time t, while $[G_{s}({\bf r}, t)]$ refers to the same atom j at $[{\bf r}_{j}(t)]$ at time t.

Equations (4.2.2.1) may be rewritten by use of the four-dimensional Fourier transforms of G, and $[G_{s}]$ , respectively: $[\eqalignno{S_{\rm coh}({\bf H}, \omega) &= {1\over 2 \pi} \int\limits_{{\bf r}}\!\! \int\limits_{t} G({\bf r}, t) \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \hbox{ d}{\bf r}\; \hbox{d}t &(4.2.2.3a)\cr S_{\rm inc}({\bf H}, \omega) &= {1\over 2 \pi} \int\limits_{{\bf r}}\!\! \int\limits_{t} G_{s}({\bf r}, t) \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \hbox{ d}{\bf r} \hbox{ d}t &(4.2.2.3b)}]$ $[\eqalignno{{\hbox{d}^{2}\sigma_{\rm coh}\over \hbox{d}\Omega \hbox{ d}(\hbar\omega)} &= N {k\over k_{0}} \langle b\rangle^{2} S_{\rm coh}({\bf H}, \omega) &(4.2.2.4a)\cr {\hbox{d}^{2} \sigma_{\rm inc}\over \hbox{d} \Omega \hbox{ d}(\hbar\omega)} &= N {k\over k_{0}} [\langle b^{2}\rangle - \langle b\rangle^{2}]S_{\rm inc}({\bf H}, \omega). &(4.2.2.4b)}]$ Incoherent scattering cross sections [(4.2.2.3b), (4.2.2.4b)] refer to one and the same particle (at different times). In particular, plastic crystals (see Section 4.2.4.5) may be studied by means of this incoherent scattering. It should be emphasized, however, that for reasons of intensity only disordered crystals with strong incoherent scatterers can be investigated by this technique. In practice, mostly samples with hydrogen atoms were investigated. This topic will not be treated further in this article (see, e.g., Springer, 1972; Lechner & Riekel, 1983). The following considerations are restricted to coherent scattering only.

Essentially the same formalism as given by equations (4.2.2.1a)–(4.2.2.4a) may be described by the use of a generalized Patterson function , which is more familiar to crystallographers, $[P({\bf r}, t) = \textstyle\int\limits_{{\bf r}'} \textstyle\int\limits_{t'=0}^{\tau} \rho ({\bf r}', t') \rho ({\bf r}' + {\bf r}, t' + t) \;\hbox{d}{\bf r}' \;\hbox{d}t', \eqno(4.2.2.5)]$ where τ denotes the time of observation. The only difference between $[G({\bf r}, t)]$ and $[P({\bf r}, t)]$ is the inclusion of the scattering weight (f or b) in $[P({\bf r}, t)]$ . $[P({\bf r}, t)]$ is an extension of the usual spatial Patterson function $[P({\bf r})]$ . $[\eqalignno{P({\bf r}, t) &\leftrightarrow 2 \pi S({\bf H}, \omega) \equiv |F({\bf H}, \omega)|^{2} &\cr &= \textstyle\int\limits_{{\bf r}} \textstyle\int\limits_{t} P({\bf r}, t) \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \;\hbox{d}{\bf r} \;\hbox{d}t. &(4.2.2.6)}]$ One difficulty arises from neglecting the time of observation. Just as $[S({\bf H})\ (\sim |F({\bf H})|^{2})]$ is always proportional to the scattering volume V, in the frame of a kinematical theory or within Born's first approximation [cf. equation (4.2.2.1a)], so $[S({\bf H}, \omega)\ [\sim |F({\bf H}, \omega)|^{2}]]$ is proportional to volume and observation time. Generally one does not make S proportional to V, but one normalizes S to be independent of τ as $[\tau \rightarrow \infty: 2 \pi S = (1 / \tau)|F|^{2}]$ . Averaging over time τ gives therefore $[\eqalignno{S({\bf H}, \omega) &= {1\over 2 \pi} \int\limits_{{\bf r}} \int\limits_{t} \left\langle \int\limits_{{\bf r}'} \rho ({\bf r}', t') \rho ({\bf r}' + {\bf r}, t' + t) \;\hbox{d}r'\right\rangle_{t'} &\cr &\quad \times \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \;\hbox{d}{\bf r} \;\hbox{d}t. &(4.2.2.7)}]$

Special cases (see, e.g., Cowley, 1981):

(1) Pure elastic measurement $[\eqalignno{I_{e} &\sim S({\bf H}, 0) = \textstyle\int\limits_{{\bf r}} \left[\textstyle\int\limits_{t} P({\bf r}, t) \;\hbox{d}t\right] \exp \{2 \pi i {\bf H} \cdot {\bf r}\} \;\hbox{d}{\bf r} &\cr &= \left|\textstyle\sum\limits_{j} f_{j}\langle \exp \{2 \pi i {\bf H} \cdot {\bf r}_{j}(t)\}\rangle_{t}\right|^{2}. &(4.2.2.8)}]$ In this type of measurement the time-averaged `structure' is determined: $[\langle \rho ({\bf r}, t)\rangle_{t} = \textstyle\int\limits_{{\bf H}} |F({\bf H}, 0)| \exp \{2 \pi i {\bf H} \cdot {\bf r}\} \;\hbox{d}{\bf H}.]$ The projection along the time axis in real (Patterson) space gives a section in Fourier space at $[\omega = 0]$ . True elastic measurement is a domain of neutron scattering. For a determination of the time-averaged structure of a statistically disordered crystal dynamical disorder (phonon scattering) may be separated. For liquids or liquid-like systems this kind of scattering technique is rather ineffective as the time-averaging procedure gives a uniform particle distribution only.
(2) Integration over frequency (or energy) $[\eqalignno{I_{\rm tot} &\sim \textstyle\int\limits_{\omega} |F({\bf H}, \omega)|^{2} \;\hbox{d}\omega = \textstyle\int\limits_{\omega} \textstyle\int\limits_{{\bf r}} \textstyle\int\limits_{t} P({\bf r}, t) &\cr &\quad \times \exp \{2 \pi i ({\bf H} \cdot {\bf r} - \nu t)\} \;\hbox{d}{\bf r} \;\hbox{d}t \;\hbox{d}\omega &\cr &= \textstyle\int\limits_{{\bf r}} P({\bf r}, 0) \exp \{2 \pi i {\bf H} \cdot {\bf r}\} \;\hbox{d}{\bf r} &(4.2.2.9)}]$ (cf. properties of δ functions). In such an experiment one determines the Patterson function for , i.e. the instantaneous structure (`snapshot' of the correlation function): a projection in Fourier space along the energy axis gives a section in direct (Patterson) space at . An energy integration is automatically performed in a conventional X-ray diffraction experiment $[(|{\bf k}| \sim |{\bf k}_{0}|)]$ . One should keep in mind that in a real experiment there is, of course, an average over both the sample volume and the time of observation.

In most practical cases averaging over time is equivalent to averaging over space: the total diffracted intensity may be regarded as the sum of intensities from a large number of independent regions due to the limited coherence of a beam. At any time these regions take all possible configurations. Therefore, this sum of intensities is equivalent to the sum of intensities from any one region at different times: $[\eqalignno{\langle I_{\rm tot}\rangle_{t} &= \left\langle \textstyle\sum\limits_{j} \textstyle\sum\limits_{j'} f_{j}\;f_{j'} \exp \{2 \pi i {\bf H} \cdot ({\bf r}_{j} - {\bf r}_{j'})\}\right\rangle_{t} &\cr &\quad = \textstyle\sum\limits_{j, \, j'} f_{j}\;f_{j'} \langle \exp \{2 \pi i {\bf H} \cdot ({\bf r}_{j} - {\bf r}_{j'})\}\rangle_{t}. &(4.2.2.9a)}]$ From the basic formulae one also derives the well known results of X-ray or neutron scattering by a periodic arrangement of particles in space [cf. equation (4.1.3.2) of Chapter 4.1 ]: $[\eqalignno{{\hbox{d}\sigma\over \hbox{d}\Omega} &= N {(2 \pi)^{3}\over V_{c}} \sum\limits_{{\bf h}} |F({\bf H})|^{2} \delta ({\bf H} - {\bf h}) &(4.2.2.10)\cr F({\bf H}) &= \sum\limits_{j} f_{j}({\bf H}) \exp \{- W_{j}\} \exp \{2 \pi i {\bf H} \cdot {\bf r}_{j}\}. &(4.2.2.11)}%(4.2.2.11)]$ $[F({\bf H})]$ denotes the Fourier transform of one cell (structure factor); the f's are assumed to be real.

The evaluation of the intensity expressions (4.2.2.6), (4.2.2.8) or (4.2.2.9), (4.2.2.9a) for a disordered crystal must be performed in terms of statistical relationships between scattering factors and/or atomic positions.

From these basic concepts the generally adopted method in a disorder problem is to try to separate the scattering intensity into two parts, namely one part $[\langle \rho \rangle]$ from an average periodic structure where formulae (4.2.2.10), (4.2.2.11) apply and a second part $[\Delta \rho]$ resulting from fluctuations from this average (see, e.g., Schwartz & Cohen, 1977). One may write formally: $[\rho = \langle \rho \rangle + \Delta \rho, \eqno(4.2.2.12a)]$ where $[\langle \rho \rangle]$ is defined to be time independent and periodic in space and $[\langle \Delta \rho \rangle = 0]$ . Because cross terms $[\langle \rho \rangle * \Delta \rho]$ vanish by definition, the Patterson function is $[\eqalignno{&[\langle \rho ({\bf r})\rangle + \Delta \rho ({\bf r})] * [\langle \rho (-{\bf r})\rangle + \Delta \rho (-{\bf r})] &\cr &\qquad =[\langle \rho ({\bf r})\rangle * \langle \rho (-{\bf r})\rangle] + [\Delta \rho ({\bf r}) * \Delta \rho (-{\bf r})]. &(4.2.2.12b)}]$ Fourier transformation gives $[\eqalignno{I &\approx |\langle F\rangle |^{2} + |\Delta F|^{2} &(4.2.2.13a)\cr |\Delta F|^{2} &= \langle |F|^{2}\rangle - |\langle F\rangle |^{2}. &(4.2.2.13b)}]$ Since $[\langle \rho \rangle]$ is periodic, the first term in (4.2.2.13) describes Bragg scattering where $[\langle F\rangle]$ plays the normal role of a structure factor of one cell of the averaged structure. The second term corresponds to diffuse scattering. In many cases diffuse interferences are centred exactly at the positions of the Bragg reflections. It is then a serious experimental problem to decide whether the observed intensity distribution is Bragg scattering obscured by crystal-size limitations or other scattering phenomena.

If disordering is time dependent exclusively, $[\langle \rho \rangle]$ represents the time average, whereas $[\langle F\rangle]$ gives the pure elastic scattering part [cf. equation (4.2.2.8)] and ΔF refers to inelastic scattering only.

References

Cowley, J. M. (1981). Diffraction physics, 2nd ed. Amsterdam: North-Holland.Google Scholar

Lechner, R. E. & Riekel, C. (1983). Application of neutron scattering in chemistry. In Neutron scattering and muon spin rotation. Springer tracts in modern physics, Vol. 101, edited by G. Höhler, pp. 1–84. Berlin: Springer.Google Scholar

Schwartz, L. H. & Cohen, J. B. (1977). Diffraction from materials. New York: Academic Press.Google Scholar

Springer, T. (1972). Quasielastic neutron scattering for the investigation of diffuse motions in solid and liquids. Springer tracts in modern physics, Vol. 64. Berlin: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 408-410