International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 534-535
Section 5.1.2.1. Propagation equation^{a}Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France |
The wavefunction Ψ associated with an electron or a neutron beam is scalar while an electromagnetic wave is a vector wave. When propagating in a medium, these waves are solutions of a propagation equation. For electrons and neutrons, this is Schrödinger's equation, which can be rewritten as where is the wavenumber in a vacuum, (ϕ is the potential in the crystal and W is the accelerating voltage) in the case of electron diffraction and [ is the Fermi pseudo-potential and h is Planck's constant] in the case of neutron diffraction. The dynamical theory of electron diffraction is treated in Chapter 5.2 [note that a different convention is used in Chapter 5.2 for the scalar wavenumber: ; compare, for example, equation (5.2.2.1 ) and its equivalent, equation (5.1.2.1)] and the dynamical theory of neutron diffraction is treated in Chapter 5.3 .
In the case of X-rays, the propagation equation is deduced from Maxwell's equations after neglecting the interaction with protons. Following von Laue (1931, 1960), it is assumed that the positive charge of the nuclei is distributed in such a way that the medium is everywhere locally neutral and that there is no current. As a first approximation, magnetic interaction, which is very weak, is not taken into account in this review. The propagation equation is derived in Section A5.1.1.2 of the Appendix. Expressed in terms of the local electric displacement, , it is given for monochromatic waves by
The interaction of X-rays with matter is characterized in equation (5.1.2.2) by the parameter χ, which is the dielectric susceptibility. It is classically related to the electron density by where nm is the classical radius of the electron [see equation (A5.1.1.2) in Section A5.1.1.2 of the Appendix].
The dielectric susceptibility, being proportional to the electron density, is triply periodic in a crystal. It can therefore be expanded in Fourier series: where h is a reciprocal-lattice vector and the summation is extended over all reciprocal-lattice vectors. The sign convention adopted here for Fourier expansions of periodic functions is the standard crystallographic sign convention defined in Section 2.5.2.3 . The relative orientations of wavevectors and reciprocal-lattice vectors are defined in Fig. 5.1.2.1, which represents schematically a Bragg reflection in direct and reciprocal space (Figs. 5.1.2.1a and 5.1.2.1b, respectively).
The coefficients of the Fourier expansion of the dielectric susceptibility are related to the usual structure factor by where V is the volume of the unit cell and the structure factor is given by is the form factor of atom j, and are the dispersion corrections [see, for instance, IT C, Section 4.2.6 ] and is the Debye–Waller factor. The summation is over all the atoms in the unit cell. The phase of the structure factor depends of course on the choice of origin of the unit cell. The Fourier coefficients are dimensionless. Their order of magnitude varies from to depending on the wavelength and the structure factor. For example, is for the 220 reflection of silicon for Cu radiation.
In an absorbing crystal, absorption is taken into account phenomenologically through the imaginary parts of the index of refraction and of the wavevectors. The dielectric susceptibility is written
The real and imaginary parts of the susceptibility are triply periodic in a crystalline medium and can be expanded in a Fourier series, where and It is important to note that where is the imaginary conjugate of f.
The index of refraction of the medium for X-rays is where is the number of electrons per unit volume. This index is very slightly smaller than one. It is for this reason that specular reflection of X-rays takes place at grazing angles. From the value of the critical angle, , the electron density of a material can be determined.
The linear absorption coefficient is For example, it is 143.2 cm^{−1} for silicon and Cu Kα radiation.
References
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.