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

International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 534-535   | 1 | 2 |

Section 5.1.2.1. Propagation equation

A. Authiera*

aLaboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.2.1. Propagation equation

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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 [\Delta \Psi + 4\pi^{2} k^{2} (1 + \chi) \Psi = 0, \eqno(5.1.2.1)] where [k = 1/\lambda] is the wavenumber in a vacuum, [\chi = \varphi/W] (ϕ is the potential in the crystal and W is the accelerating voltage) in the case of electron diffraction and [\chi = -2mV ({\bf r})/h^{2} k^{2}] [[V({\bf r})] 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[link] [note that a different convention is used in Chapter 5.2[link] for the scalar wavenumber: [k = 2\pi/\lambda]; compare, for example, equation (5.2.2.1[link] ) and its equivalent, equation (5.1.2.1)[link]] and the dynamical theory of neutron diffraction is treated in Chapter 5.3[link] .

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[link], 1960[link]), 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[link] of the Appendix. Expressed in terms of the local electric displacement, [{\bf D}({\bf r})], it is given for monochromatic waves by [\Delta {\bf D(r)} + \hbox{ curl curl } \chi {\bf D(r)} + 4\pi^{2} k^{2} {\bf D(r)} = 0. \eqno(5.1.2.2)]

The interaction of X-rays with matter is characterized in equation (5.1.2.2)[link] by the parameter χ, which is the dielectric susceptibility. It is classically related to the electron density [\rho ({\bf r})] by [\chi ({\bf r}) = -R \lambda^{2} \rho ({\bf r})/\pi, \eqno(5.1.2.3)] where [R = 2.81794 \times 10^{-6}] nm is the classical radius of the electron [see equation (A5.1.1.2)[link] in Section A5.1.1.2[link] 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: [\chi = {\textstyle\sum\limits_{{\bf h}}} \chi_{h} \exp (2\pi i {\bf h} \cdot {\bf r}), \eqno(5.1.2.4)] 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[link] . The relative orientations of wavevectors and reciprocal-lattice vectors are defined in Fig. 5.1.2.1[link], which represents schematically a Bragg reflection in direct and reciprocal space (Figs. 5.1.2.1a[link] and 5.1.2.1b[link], respectively).

[Figure 5.1.2.1]

Figure 5.1.2.1 | top | pdf |

Bragg reflection. (a) Direct space. Bragg reflection of a wave of wavevector [{\bf K}_{{\bf o}}] incident on a set of lattice planes of spacing d. The reflected wavevector is [{\bf K}_{{\bf h}}]. Bragg's law [2d\sin \theta = n\lambda] can also be written [2d_{hkl} \sin \theta = \lambda], where [d_{hkl} = d/n = 1/OH = 1/h] is the inverse of the length of the corresponding reciprocal-lattice vector [{\bf OH} = {\bf h}] (see part b). (b) Reciprocal space. P is the tie point of the wavefield consisting of the incident wave [{\bf K}_{{\bf o}} = {\bf OP}] and the reflected wave [{\bf K}_{{\bf h}} = {\bf HP}]. Note that the wavevectors are oriented towards the tie point.

The coefficients [\chi_{h}] of the Fourier expansion of the dielectric susceptibility are related to the usual structure factor [F_{h}] by [\chi_{h} = - R \lambda^{2} F_{h}/(\pi V), \eqno(5.1.2.5)] where V is the volume of the unit cell and the structure factor is given by [\eqalignno{F_{h} &= {\textstyle\sum\limits_{j}} (\;f_{j} + f'_{j} + if''_{j}) \exp (- M_{j} - 2 \pi i {\bf h} \cdot {\bf r}_{j}) &\cr &= | F_{h} | \exp (i \varphi_{h}). &(5.1.2.6)}] [f_{j}] is the form factor of atom j, [f'_{j}] and [f''_{j}] are the dispersion corrections [see, for instance, IT C, Section 4.2.6[link] ] and [\exp (- M_{j})] is the Debye–Waller factor. The summation is over all the atoms in the unit cell. The phase [\varphi_{h}] of the structure factor depends of course on the choice of origin of the unit cell. The Fourier coefficients [\chi_{h}] are dimensionless. Their order of magnitude varies from [10^{-5}] to [10^{-7}] depending on the wavelength and the structure factor. For example, [\chi_{h}] is [-9.24 \times 10^{-6}] for the 220 reflection of silicon for Cu [K\alpha] 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 [\chi = \chi_{r} + i\chi_{i}. \eqno(5.1.2.7)]

The real and imaginary parts of the susceptibility are triply periodic in a crystalline medium and can be expanded in a Fourier series, [\eqalign{\chi_{r} &= {\textstyle\sum\limits_{{\bf h}}} \chi_{rh} \exp (2\pi i{\bf h} \cdot {\bf r})\cr \chi_{i} &= {\textstyle\sum\limits_{{\bf h}}} \chi_{ih} \exp (2\pi i{\bf h} \cdot {\bf r}),} \eqno(5.1.2.8)] where [\eqalign{\chi_{rh} &= - R \lambda^{2} F_{rh} / (\pi V),\cr \chi_{ih} &= - R \lambda^{2} F_{ih} / (\pi V)} \eqno(5.1.2.9)] and [\eqalignno{F_{rh} &= {\textstyle\sum\limits_{j}} (\;f_{j} + f'_{j}) \exp (- M_{j} - 2\pi i{\bf h} \cdot {\bf r}_{j})\cr &= | F_{rh} | \exp (i \varphi_{rh}), &(5.1.2.10a)\cr F_{ih} &= {\textstyle\sum\limits_{j}} (\;f''_{j}) \exp (- M_{j} - 2\pi i{\bf h} \cdot {\bf r}_{j})\cr &= | F_{ih} | \exp (i\varphi_{ih}). &(5.1.2.10b)}] It is important to note that [F_{rh}^{*} = F_{r\bar{h}} \hbox{ and } F_{ih}^{*} = F_{i\bar{h}} \hbox{ but that } F_{h}^{*} \neq F_{\bar{h}}, \eqno(5.1.2.11)] where [f^{*}] is the imaginary conjugate of f.

The index of refraction of the medium for X-rays is [n = 1 + \chi_{ro} / 2 = 1 - R \lambda^{2} F_{o}/(2\pi V), \eqno(5.1.2.12)] where [F_{o}/V] 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, [(- \chi_{ro})^{1/2}], the electron density [F_{o}/V] of a material can be determined.

The linear absorption coefficient is [\mu_{o} = -2\pi k \chi_{io} = 2R\lambda F_{io}/V. \eqno(5.1.2.13)] 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.








































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