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

International Tables for Crystallography (2006). Vol. B, ch. 4.2, pp. 407-408   | 1 | 2 |

Section 4.2.1.  Scope of this chapter

H. Jagodzinskia and F. Freyb

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

4.2.1. Scope of this chapter

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Diffuse scattering of X-rays, neutrons and other particles is an accompanying effect in all diffraction experiments aimed at structure analysis with the aid of so-called elastic scattering. In this case the momentum exchange of the scattered photon (or particle) includes the crystal as a whole; the energy transfer involved becomes negligibly small and need not be considered in diffraction theory. Inelastic scattering processes, however, are due to excitation processes, such as ionization, phonon scattering etc. Distortions as a consequence of structural changes cause typical elastic or inelastic diffuse scattering. All these processes contribute to scattering, and a general theory has to include all of them. Hence, the exact treatment of diffuse scattering becomes very complex. Fortunately, approximations treating the phenomena independently are possible in most cases, but it should be kept in mind that difficulties may occasionally arise.

A separation of elastic from inelastic diffuse scattering may be made if detectors sensitive to the energy of radiation are used. Difficulties may sometimes result from small energy exchanges, which cannot be resolved for experimental reasons. The latter is true for scattering of X-rays by phonons which have energies of the order of 10−2−10−3 eV, a value which is considerably smaller than 10 keV, a typical value for X-ray quanta. Another equivalent explanation, frequently forwarded in the literature, is the high speed of X-ray photons, such that the rather slow motion of atoms cannot be `observed' by them during diffraction. Hence, all movements appear as static displacement waves of atoms, and temperature diffuse scattering is pseudo-elastic for X-rays. This is not true in the case of thermal neutrons, which have energies comparable to those of phonons. Since thermal diffuse scattering is discussed in Chapter 4.1[link] , this chapter is mainly concerned with the elastic (or pseudo-elastic other than thermal) part of diffuse scattering.

The full treatment of the complicated theoretical background for all other kinds of diffuse scattering lies beyond the scope of this article. It is also impossible to refer to all papers in this wide and complicated field. Different theoretical treatments of one and the same subject are often developed, but only some are given here, in most cases those which may be understood most easily – at least to the authors' feeling. As shown in this chapter, electron-density fluctuations and distribution functions of defects play an important role for the complete interpretation of diffraction patterns. Both quantities may best be studied in the low-angle scattering range, which occasionally represents the only Bragg peak dealing with the full information of the distribution function of the defects. Hence, many problems cannot be solved without a detailed interpretation of low-angle diffraction.

Disorder phenomena in magnetic structures are not specifically discussed here. Magnetic diffuse neutron scattering and special experimental techniques themselves constitute a large subject. Many aspects, however, may be analysed along similar lines as given here. For this particular topic the reader is referred to textbooks of neutron scattering, where the theory of diffraction by magnetic materials is generally included (see, e.g., Lovesey, 1984[link]).

Glasses, liquids or liquid crystals show typical diffuse diffraction phenomena. Particle-size effects and strains have an important influence on the diffuse scattering. The same is true for dislocations and point defects such as interstitials or vacancies. These defects are mainly described by their strain field which influences the intensities of sharp reflections like an artificial temperature factor: the Bragg peaks diminish in intensity, while the diffuse scattering increases predominantly close to them. These phenomena are less important from a structural point of view, at least in the case of metals or other simple structures. This statement is true as long as the structure of the `kernel' of defects may be neglected when compared with the influence of the strain field. Whether dislocations in more complicated structures meet this condition is not yet known.

Radiation damage in crystals represents another field of diffuse scattering which cannot be treated here explicitly. As long as point defects only are generated, the strain field around these defects is the most important factor governing diffuse scattering. Particles with high energy, such as fast neutrons, protons and others, generate complicated defect structures which have to be treated with the aid of the cluster method described below, but no special reference is given here because of the complexity of these phenomena.

Diffuse scattering related to phase transitions, in particular the critical diffuse scattering observed at or close to the transition temperature, cannot be discussed here. In simple cases a satisfactory description may be given with the aid of a `soft phonon', which freezes at the critical temperature, thus generating typical temperature-dependent diffuse scattering. If the geometry of the lattice is maintained during the transformation (no breakdown into crystallites of different cell geometry), the diffuse scattering is very similar to diffraction phenomena described in this article. Sometimes, however, very complicated interim stages (ordered or disordered) are observed demanding a complicated theory for their full explanation (see, e.g., Dorner & Comes, 1977[link]).

Commensurate and incommensurate modulated structures as well as quasicrystals are frequently accompanied by a typical diffuse scattering, demanding an extensive experimental and theoretical study in order to arrive at a satisfactory explanation. A reliable structure determination becomes very difficult in cases where the interpretation of diffuse scattering has not been incorporated. Many erroneous structural conclusions have been published in the past. The solution of problems of this kind needs careful thermodynamical consideration as to whether a plausible explanation of the structural data can be given.

Obviously, there is a close relationship between thermodynamics and diffuse scattering in disordered systems, representing a stable or metastable thermal equilibrium. From the thermodynamical point of view the system is then characterized by its grand partition function, which is intimately related to the correlation functions used in the interpretation of diffuse scattering. The latter is nothing other than a kind of `partial partition function' where two atoms, or two cell occupations, are fixed such that the sum of all partial partition functions represents the grand partition function. This fact yields the useful correlation between thermodynamics and diffuse scattering mentioned above, which may well be used for a determination of thermodynamical properties of the crystal. This subject could not be included here for the following reason: real three-dimensional crystals generally exhibit diffuse scattering by defects and/or disordering effects which are not in thermal equilibrium. They are created during crystal growth, or are frozen-in defects formed at higher temperatures. Hence, a thermodynamical interpretation of diffraction data needs a careful study of diffuse scattering as a function of temperature or some other thermodynamical parameters. This can be done in very rare cases only, so the omission of this subject seems justified.

For all of the reasons mentioned above, this article cannot be complete. It is hoped, however, that it will provide a useful guide for those who need the information for the full understanding of the crystal chemistry of a given structure.

There is no comprehensive treatment of all aspects of diffuse scattering. Essential parts are treated in the textbooks of James (1954)[link], Wilson (1962)[link], Wooster (1962)[link] and Schwartz & Cohen (1977)[link]; handbook articles are written by Jagodzinski (1963[link], 1964a[link],b[link], 1987[link]), Schulz (1982)[link], Welberry (1985)[link]; and a series of interesting papers is collected by Collongues et al. (1977)[link].

Many differences are caused by different symbols and by different `languages' used in the various diffraction methods. Quite a few of the new symbols in use are not really necessary, but some are caused by differences in the experimental techniques. For example, the neutron scattering length b may usually be equated with the atomic form factor f in X-ray diffraction. The differential cross section introduced in neutron diffraction represents the intensity scattered into an angular range dΩ and an energy range dE. The famous scattering law in neutron work corresponds to the square of an (extended) structure factor; the `static structure factor', a term used by neutron diffractionists, is nothing other than the conventional Patterson function. The complicated resolution functions in neutron work correspond to the well known Lorentz factors in X-ray diffraction. These have to be derived in order to include all techniques used in diffuse-scattering work.

References

Collongues, R., Fayard, M. & Gautier, F. (1977). Ordre et désordre dans les solides. J. Phys. (Paris) Colloq. 38(C7), Suppl.
Dorner, B. & Comes, R. (1977). Phonons and structural phase transformation. In Dynamics of solids and liquids by neutron scattering, ch. 3, edited by S. Lovesey & T. Springer. Topics in current physics, Vol. 3. Berlin: Springer.
Jagodzinski, H. (1963). On disorder phenomena in crystals. In Crystallography and crystal perfection, edited by G. N. Ramachandran, pp. 177–188. London: Academic Press.
Jagodzinski, H. (1964a). Allgemeine Gesichtspunkte für die Deutung diffuser Interferenzen von fehlgeordneten Kristallen. In Advances in structure research by diffraction methods, Vol. 1, edited by R. Brill & R. Mason, pp. 167–198. Braunschweig: Vieweg.
Jagodzinski, H. (1964b). Diffuse disorder scattering by crystals. In Advanced methods of crystallography, edited by G. N. Ramachandran, pp. 181–219. London: Academic Press.
Jagodzinski, H. (1987). Diffuse X-ray scattering from crystals. In Progress in crystal growth and characterization, edited by P. Krishna, pp. 47–102. Oxford: Pergamon Press.
James, R. W. (1954). The optical principles of diffraction of X-rays, ch. X. London: Bell.
Lovesey, S. W. (1984). Theory of neutron scattering from condensed matter, Vols. 1, 2. Oxford: Clarendon Press.
Schulz, H. (1982). Diffuse X-ray diffraction and its application to materials research. In Current topics in materials science, ch. 4, edited by E. Kaldis. Amsterdam: North-Holland.
Schwartz, L. H. & Cohen, J. B. (1977). Diffraction from materials. New York: Academic Press.
Welberry, T. R. (1985). Diffuse X-ray scattering and models of disorder. Rep. Prog. Phys. 48, 1543–1593.
Wilson, A. J. C. (1962). X-ray optics, 2nd ed., chs. V, VI, VIII. London: Methuen.
Wooster, W. A. (1962). Diffuse X-ray reflections from crystals, chs. IV, V. Oxford: Clarendon Press.








































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