International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 5.1, p. 626
Section 5.1.1. Introduction^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
The first experiment on X-ray diffraction by a crystal was performed by W. Friedrich, P. Knipping and M. von Laue in 1912 and Bragg's law was derived in 1913 (Bragg, 1913). Geometrical and dynamical theories for the intensities of the diffracted X-rays were developed by Darwin (1914a,b). His dynamical theory took into account the interaction of X-rays with matter by solving recurrence equations that describe the balance of partially transmitted and partially reflected amplitudes at each lattice plane. This is the first form of the dynamical theory of X-ray diffraction. It gives correct expressions for the reflected intensities and was extended to the absorbing-crystal case by Prins (1930). A second form of dynamical theory was introduced by Ewald (1917) as a continuation of his previous work on the diffraction of optical waves by crystals. He took into account the interaction of X-rays with matter by considering the crystal to be a periodic distribution of dipoles which were excited by the incident wave. This theory also gives the correct expressions for the reflected and transmitted intensities, and it introduces the fundamental notion of a wavefield, which is necessary to understand the propagation of X-rays in perfect or deformed crystals. Ewald's theory was later modified by von Laue (1931), who showed that the interaction could be described by solving Maxwell's equations in a medium with a continuous, triply periodic distribution of dielectric susceptibility. It is this form which is most widely used today and which will be presented in this chapter.
The geometrical (or kinematical) theory, on the other hand, considers that each photon is scattered only once and that the interaction of X-rays with matter is so small it can be neglected. It can therefore be assumed that the amplitude incident on every diffraction centre inside the crystal is the same. The total diffracted amplitude is then simply obtained by adding the individual amplitudes diffracted by each diffracting centre, taking into account only the geometrical phase differences between them and neglecting the interaction of the radiation with matter. The result is that the distribution of diffracted amplitudes in reciprocal space is the Fourier transform of the distribution of diffracting centres in physical space. Following von Laue (1960), the expression geometrical theory will be used throughout this chapter when referring to these geometrical phase differences.
The first experimentally measured reflected intensities were not in agreement with the theoretical values obtained with the more rigorous dynamical theory, but rather with the simpler geometrical theory. The integrated reflected intensities calculated using geometrical theory are proportional to the square of the structure factor, while the corresponding expressions calculated using dynamical theory for an infinite perfect crystal are proportional to the modulus of the structure factor. The integrated intensity calculated by geometrical theory is also proportional to the volume of the crystal bathed in the incident beam. This is due to the fact that one neglects the decrease of the incident amplitude as it progresses through the crystal and a fraction of it is scattered away. According to geometrical theory, the diffracted intensity would therefore increase to infinity if the volume of the crystal was increased to infinity, which is of course absurd. The theory only works because the strength of the interaction is very weak and if it is applied to very small crystals. How small will be shown quantitatively in Sections 5.1.6.5 and 5.1.7.2. Darwin (1922) showed that it can also be applied to large imperfect crystals. This is done using the model of mosaic crystals (Bragg et al., 1926). For perfect or nearly perfect crystals, dynamical theory should be used. Geometrical theory presents another drawback: it gives no indication as to the phase of the reflected wave. This is due to the fact that it is based on the Fourier transform of the electron density limited by the external shape of the crystal. This is not important when one is only interested in measuring the reflected intensities. For any problem where the phase is important, as is the case for multiple reflections, interference between coherent blocks, standing waves etc., dynamical theory should be used, even for thin or imperfect crystals.
Until the 1940s, the applications of dynamical theory were essentially intensity measurements. From the 1950s to the 1970s, applications were related to the properties (absorption, interference, propagation) of wavefields in perfect or nearly perfect crystals: anomalous transmission, diffraction of spherical waves, interpretation of images on X-ray topographs, accurate measurement of form factors, lattice-parameter mapping. In recent years, they have been concerned mainly with crystal optics, focusing and the design of monochromators for synchrotron radiation [see, for instance, Batterman & Bilderback (1991)], the location of atoms at crystal surfaces and interfaces using the standing-waves method, determination of phases using multiple reflections [for reviews of n-beam diffraction, see Weckert & Hümmer (1997) and Chang (2004); for recent determinations of phases, see Chang et al. (2002), Mo et al. (2002), Weckert et al. (2002), Shen & Wang (2003)], characterization of the crystal perfection of epilayers and superlattices by high-resolution diffractometry [see, for instance, Tanner (1990) and Fewster (1993)], etc.
Modern developments include the extension of dynamical theory to time-dependent phenomena (Chukhovskii & Förster, 1995; Shastri et al., 2001; Graeff, 2002a,b, 2004; Malgrange & Graeff, 2003; Sondhauss & Wark, 2003; Adams, 2004) and the study of the influence of the coherence of the source (Yamazaki & Ishikawa, 2002, 2004).
For reviews of dynamical theory, see Zachariasen (1945), von Laue (1960), James (1963), Batterman & Cole (1964), Authier (1970), Kato (1974), Brümmer & Stephanik (1976), Pinsker (1978), Authier et al. (1996), Authier & Malgrange (1998), and Authier (2005). Topography is described in Chapter 2.7 of IT C (2004), in Tanner (1976) and in Tanner & Bowen (1992). For the use of Bragg-angle measurements for accurate lattice-parameter mapping, see Hart (1981). For online calculations in the case of multiple diffraction, grazing incidence or for strained crystals, see http://sergey.gmca.aps.anl.gov .
A reminder of some basic concepts in electrodynamics is given in Section A5.1.1.1 of the Appendix.
References
International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Adams, B. W. (2004). Time-dependent Takagi–Taupin eikonal theory of X-ray diffraction in rapidly changing crystal structures. Acta Cryst. A60, 120–133.
Authier, A. (1970). Ewald waves in theory and experiment. Adv. Struct. Res. Diffr. Methods, 3, 1–51.
Authier, A. (2005). Dynamical Theory of X-ray Diffraction. (First printed 2001, revised 2003, 2005.) IUCr Monographs on Crystallography. Oxford University Press.
Authier, A., Lagomarsino, S. & Tanner, B. K. (1996). Editors. X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357. New York, London: Plenum Press.
Authier, A. & Malgrange, C. (1998). Diffraction physics. Acta Cryst. A54, 806–819.
Batterman, B. W. & Bilderback, D. H. (1991). X-ray monochromators and mirrors. In Handbook on Synchrotron Radiation, Vol. 3, edited by G. Brown & D. E. Moncton, pp. 105–153. Amsterdam: Elsevier Science Publishers BV.
Batterman, B. W. & Cole, H. (1964). Dynamical diffraction of X-rays by perfect crystals. Rev. Mod. Phys. 36, 681–717.
Bragg, W. L. (1913). The diffraction of short electromagnetic waves by a crystal. Proc. Cambridge Philos. Soc. 17, 43–57.
Bragg, W. L., Darwin, C. G. & James, R. W. (1926). The intensity of reflection of X-rays by crystals. Philos. Mag. 1, 897–922.
Brümmer, O. & Stephanik, H. (1976). Dynamische Interferenztheorie. Leipzig: Akademische Verlagsgesellshaft.
Chang, S.-L. (2004). X-ray Multiple-Wave Diffraction: Theory and Application. Springer Series in Solid-State Sciences, Vol. 143. Berlin: Springer-Verlag.
Chang, S.-L., Stetsko, Yu. P. & Lee, Y.-R. (2002). Quantitative determination of phase for macromolecular crystals using multiple diffraction methods and dynamical theory. Z. Kristallogr. 217, 662–667.
Chukhovskii, F. N. & Förster, E. (1995). Time-dependent X-ray Bragg diffraction. Acta Cryst. A51, 668–672.
Darwin, C. G. (1914a). The theory of X-ray reflection. Philos. Mag. 27, 315–333.
Darwin, C. G. (1914b). The theory of X-ray reflection. Part II. Philos. Mag. 27, 675–690.
Darwin, C. G. (1922). The reflection of X-rays from imperfect crystals. Philos. Mag. 43, 800–829.
Ewald, P. P. (1917). Zur Begründung der Kristalloptik. III. Röntgenstrahlen. Ann. Phys. (Leipzig), 54, 519–597.
Fewster, P. F. (1993). X-ray diffraction from low-dimensional structures. Semicond. Sci. Technol. 8, 1915–1934.
Graeff, W. (2002a). Short X-ray pulses in a Laue-case crystal. J. Synchrotron Rad. 9, 82–85.
Graeff, W. (2002b). Time dependence of the polarization of short X-ray pulses after crystal reflection. J. Synchrotron Rad. 9, 293–297.
Graeff, W. (2004). Tailoring the time response of a Bragg reflection to short X-ray pulses. J. Synchrotron Rad. 11, 261–265.
Hart, M. (1981). Bragg angle measurement and mapping. J. Cryst. Growth, 55, 409–427.
James, R. W. (1963). The dynamical theory of X-ray diffraction. Solid State Phys. 15, 53.
Kato, N. (1974). X-ray diffraction. In X-ray Diffraction, edited by L. V. Azaroff, R. Kaplow, N. Kato, R. J. Weiss, A. J. C. Wilson & R. A. Young, pp. 176–438. New York: McGraw-Hill.
Laue, M. von (1931). Die dynamische Theorie der Röntgenstrahl interferenzen in neuer Form. Ergeb. Exakten Naturwiss. 10, 133–158.
Laue, M. von (1960). Röntgenstrahl-Interferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.
Malgrange, C. & Graeff, W. (2003). Diffraction of short X-ray pulses in the general asymmetric Laue case – an analytic treatment. J. Synchrotron Rad. 10, 248–254.
Mo, F., Mathiesen, R. H., Alzari, P. M., Lescar, J. & Rasmussen, B. (2002). Physical estimation of triplet phases from two new proteins. Acta Cryst. D58, 1780–1786.
Pinsker, Z. G. (1978). Dynamical scattering of X-rays in crystals. Springer Series in Solid-State Sciences. Berlin: Springer-Verlag.
Prins, J. A. (1930). Die Reflexion von Röntgenstrahlen an absorbierenden idealen Kristallen. Z. Phys. 63, 477–493.
Shastri, S. D., Zambianchi, P. & Mills, D. M. (2001). Dynamical diffraction of ultrashort X-ray free-electron laser pulses. J. Synchrotron Rad. 8, 1131–1135.
Shen, Q. & Wang, J. (2003). Recursive direct phasing with reference-beam diffraction. Acta Cryst. D59, 809–814.
Sondhauss, P. & Wark, J. S. (2003). Extension of the time-dependent dynamical diffraction theory to `optical phonon'-type distortions: application to diffraction from coherent acoustic and optical phonons. Acta Cryst. A59, 7–13.
Tanner, B. K. (1976). X-ray Diffraction Topography. Oxford: Pergamon Press.
Tanner, B. K. (1990). High resolution X-ray diffraction and topography for crystal characterization. J. Cryst. Growth, 99, 1315–1323.
Tanner, B. K. & Bowen, D. K. (1992). Synchrotron X-radiation topography. Mater. Sci. Rep. 8, 369–407.
Weckert, E. & Hümmer, K. (1997). Multiple-beam X-ray diffraction for physical determination of reflection phases and its applications. Acta Cryst. A53, 108–143.
Weckert, E., Müller, R., Zellner, J., Zegers, I. & Loris, R. (2002). Physical measurement of triplet invariants: present state of the experiment, data evaluation and future perspectives. Z. Kristallogr. 217, 651–661.
Yamazaki, H. & Ishikawa, T. (2002). Propagation of X-ray coherence for diffraction of perfect crystals. J. Appl. Cryst. 35, 314–318.
Yamazaki, H. & Ishikawa, T. (2004). Analysis of the mutual coherence function of X-rays using dynamical diffraction. J. Appl. Cryst. 37, 48–51.
Zachariasen, W. H. (1945). Theory of X-ray Diffraction in Crystals. New York: John Wiley.