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

International Tables for Crystallography (2010). Vol. B, ch. 5.1, pp. 626-630   | 1 | 2 |

Section 5.1.2. Fundamentals of plane-wave dynamical theory

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: AAuthier@wanadoo.fr

5.1.2. Fundamentals of plane-wave dynamical theory

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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_{\rm r} + i\chi_{\rm 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_{\rm r} &= {\textstyle\sum\limits_{{\bf h}}} \chi_{{\rm r}h} \exp (2\pi i{\bf h} \cdot {\bf r})\cr \chi_{\rm i} &= {\textstyle\sum\limits_{{\bf h}}} \chi_{{\rm i}h} \exp (2\pi i{\bf h} \cdot {\bf r}),} \eqno(5.1.2.8)]where[\eqalign{\chi_{{\rm r}h} &= - R \lambda^{2} F_{{\rm r}h} / (\pi V),\cr \chi_{{\rm i}h} &= - R \lambda^{2} F_{{\rm i}h} / (\pi V)} \eqno(5.1.2.9)]and[\eqalignno{F_{{\rm r}h} &= {\textstyle\sum\limits_{j}} (\,f_{j} + f'_{j}) \exp (- M_{j} - 2\pi i{\bf h} \cdot {\bf r}_{j})\cr &= | F_{{\rm r}h} | \exp (i \varphi_{{\rm r}h}), &(5.1.2.10a)\cr F_{{\rm i}h} &= {\textstyle\sum\limits_{j}} (\,f''_{j}) \exp (- M_{j} - 2\pi i{\bf h} \cdot {\bf r}_{j})\cr &= | F_{{\rm i}h} | \exp (i\varphi_{{\rm i}h}). &(5.1.2.10b)}]It is important to note that[F_{{\rm r}h}^{*} = F_{{\rm r}\bar{h}} \hbox{ and } F_{{\rm i}h}^{*} = F_{{\rm 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_{{\rm r}o} / 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_{{\rm r}o})^{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_{{\rm i}o} = 2R\lambda F_{{\rm i}o}/V. \eqno(5.1.2.13)]For example, it is 143.2 cm−1 for silicon and Cu Kα radiation.

5.1.2.2. Wavefields

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The notion of a wavefield, introduced by Ewald (1917)[link], is one of the most fundamental concepts in dynamical theory. It results from the fact that since the propagation equation (5.1.2.2)[link] is a second-order partial differential equation with a periodic interaction coefficient, its solution has the same periodicity,[{\bf D} = \exp (- 2\pi i{\bf K}_{{\bf o}} \cdot {\bf r}) \,{\textstyle\sum\limits_{{\bf h}}} \,{\bf D}_{{\bf h}} \exp (2\pi i{\bf h} \cdot {\bf r}), \eqno(5.1.2.14)]where the summation is over all reciprocal-lattice vectors h. Equation (5.1.2.14)[link] can also be written[{\bf D} = {\textstyle\sum\limits_{\bf h}} \,{\bf D}_{{\bf h}} \exp (- 2\pi i{\bf K}_{{\bf h}} \cdot {\bf r}), \eqno(5.1.2.15)]where[{\bf K}_{{\bf h}} = {\bf K}_{{\bf o}} - {\bf h}. \eqno(5.1.2.16)]

Expression (5.1.2.15)[link] shows that the solution of the propagation equation can be interpreted as an infinite sum of plane waves with amplitudes [{\bf D}_{\bf h}] and wavevectors [{\bf K}_{\bf h}]. This sum is a wavefield, or Ewald wave. The same expression is used to describe the propagation of any wave in a periodic medium, such as phonons or electrons in a solid. Expression (5.1.2.14)[link] was later called a Bloch wave by solid-state physicists.

The wavevectors in a wavefield are deduced from one another by translations of the reciprocal lattice [expression (5.1.2.16)[link]]. They can be represented geometrically as shown in Fig. 5.1.2.1(b)[link]. The wavevectors [{\bf K}_{{\bf o}} = {\bf OP}]; [{\bf K}_{{\bf h}} = {\bf HP}] are drawn away from reciprocal-lattice points. Their common extremity, P, called the tie point by Ewald, characterizes the wavefield.

In an absorbing crystal, wavevectors have an imaginary part,[{\bf K}_{{\bf o}} = {\bf K}_{{\bf o}{\rm r}} + i{\bf K}_{{\bf o}{\rm i}}\semi \,\,{\bf K}_{{\bf h}} = {\bf K}_{{\bf h}{\rm r}} + i{\bf K}_{{\bf h}{\rm i}},]and (5.1.2.16)[link] shows that all wavevectors have the same imaginary part,[{\bf K}_{{\bf o}{\rm i}} = {\bf K}_{{\bf h}{\rm i}}, \eqno(5.1.2.17)]and therefore undergo the same absorption. This is one of the most important properties of wavefields.

The first experimental evidence of the physical existence of Ewald's wavefields is to be found in the light–dark structure of Kossel lines, observed by Borrmann (1936[link]) and explained by von Laue (1937[link]) using the properties of the standing waves formed by the wavefields inside the crystal. It was followed by the discovery of anomalous absorption (Borrmann, 1941[link], 1950[link]) and the theoretical determination of the wavefield trajectories (von Laue, 1952[link]; Kato, 1952[link], 1958; Ewald, 1958[link]), which was confirmed by the experimental observation of spherical wave Pendellösung (Kato & Lang, 1959[link]) and by that of the double refraction of X-rays, predicted by Borrmann (1955[link]) and observed by Authier (1960[link], 1961[link]).

5.1.2.3. Boundary conditions at the entrance surface

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The choice of the `o' component of expansion (5.1.2.15)[link] is arbitrary in an infinite medium. In a semi-infinite medium where the waves are created at the interface with a vacuum or air by an incident plane wave with wavevector [{\bf K}_{{\bf o}}^{(a)}] (using von Laue's notation), the choice of [{\bf K}_{{\bf o}}] is determined by the boundary conditions.

This condition for wavevectors at an interface demands that their tangential components should be continuous across the boundary, in agreement with Descartes–Snell's law. This condition is satisfied when the difference between the wavevectors on each side of the interface is parallel to the normal to the interface. This is shown geometrically in Fig. 5.1.2.2[link] and formally in (5.1.2.18)[link]:[{\bf K}_{{\bf o}} - {\bf K}_{{\bf o}}^{(a)} = {\bf OP} - {\bf OM} = \overline{MP} \cdot {\bf n}, \eqno(5.1.2.18)]where n is a unit vector normal to the crystal surface, oriented towards the inside of the crystal.

[Figure 5.1.2.2]

Figure 5.1.2.2 | top | pdf |

Boundary condition for wavevectors at the entrance surface of the crystal.

There is no absorption in a vacuum and the incident wavevector [{\bf K}_{{\bf o}}^{(a)}] is real. Equation (5.1.2.18)[link] shows that it is the component normal to the interface of wavevector [{\bf K}_{{\bf o}}] which has an imaginary part,[{\bf K}_{{\bf o}{\rm i}} = {\cal I} (\overline{MP}) \cdot {\bf n} = - \mu {\bf n} / (4\pi \gamma_{o}), \eqno(5.1.2.19)]where [{\cal I}(f)] is the imaginary part of f, [\gamma_{o} = \cos ({\bf n} \cdot {\bf s}_{{\bf o}})] and [{\bf s}_{{\bf o}}] is a unit vector in the incident direction. When there is more than one wave in the wavefield, the effective absorption coefficient μ can differ significantly from the normal value, [\mu_{o}], given by (5.1.2.13)[link] – see Section 5.1.5[link].

5.1.2.4. Fundamental equations of dynamical theory

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In order to obtain the solution of dynamical theory, one inserts expansions (5.1.2.15)[link] and (5.1.2.4)[link] into the propagation equation (5.1.2.2)[link]. This leads to an equation with an infinite sum of terms. It is shown to be equivalent to an infinite system of linear equations which are the fundamental equations of dynamical theory. Only those terms in (5.1.2.15)[link] whose wavevector magnitudes [K_{h}] are very close to the vacuum value, k, have a non-negligible amplitude. These wavevectors are associated with reciprocal-lattice points that lie very close to the Ewald sphere. Far from any Bragg reflection, their number is equal to 1 and a single plane wave propagates through the medium. In general, for X-rays, there are only two reciprocal-lattice points on the Ewald sphere. This is the so-called two-beam case to which this treatment is limited. There are, however, many instances where several reciprocal-lattice points lie simultaneously on the Ewald sphere. This corresponds to the many-beam case which has interesting applications for the determination of phases of reflections [see, for instance, Chang (1987[link], 2004[link]), Hümmer & Weckert (1995)[link] and Weckert & Hümmer (1997)[link]]. On the other hand, for electrons, there are in general many reciprocal-lattice points close to the Ewald sphere and many wavefields are excited simultaneously (see Chapter 5.2[link] ).

In the two-beam case, for reflections that are not highly asymmetric and for Bragg angles that are not close to [\pi/2], the fundamental equations of dynamical theory reduce to[\eqalign{2X_{o} D_{o} - kC\chi_{\bar{h}} D_{h} &= 0\cr -kC\chi_{h} D_{o} + 2X_{h} D_{h} &= 0,} \eqno(5.1.2.20)]where [C = 1] if [{\bf D}_{{\bf h}}] is normal to the [{\bf K}_{{\bf o}}, {\bf K}_{{\bf h}}] plane and [C = \cos 2\theta] if [{\bf D}_{{\bf h}}] lies in the plane; this is due to the fact that the amplitude with which electromagnetic radiation is scattered is proportional to the sine of the angle between the direction of the electric vector of the incident radiation and the direction of scattering (see, for instance, IT C, Section 6.2.2[link] ). The polarization of an electromagnetic wave is classically related to the orientation of the electric vector; in dynamical theory it is that of the electric displacement which is considered (see Section A5.1.1.3[link] of the Appendix).

The system (5.1.2.20)[link] is therefore a system of four equations which admits four solutions, two for each direction of polarization. In the non-absorbing case, to a very good approximation,[\eqalign{X_{o} &= K_{o} - nk,\cr X_{h} &= K_{h} - nk.} \eqno(5.1.2.21)]In the case of an absorbing crystal, [X_{o}] and [K_{h}] are complex. Equation (5.1.5.2)[link] gives the full expression for [X_{o}].

5.1.2.5. Dispersion surface

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The fundamental equations (5.1.2.20)[link] of dynamical theory are a set of linear homogeneous equations whose unknowns are the amplitudes of the various waves which make up a wavefield. For the solution to be nontrivial, the determinant of the set must be set equal to zero. This provides a secular equation relating the magnitudes of the wavevectors of a given wavefield. This equation is that of the locus of the tie points of all the wavefields that may propagate in the crystal with a given frequency. This locus is called the dispersion surface. It is a constant-energy surface and is the equivalent of the index surface in optics. It is the X-ray analogue of the constant-energy surfaces known as Fermi surfaces in the electron band theory of solids.

In the two-beam case, the dispersion surface is a surface of revolution around the diffraction vector OH. It is made from two spheres and a connecting surface between them. The two spheres are centred at O and H and have the same radius, nk. Fig. 5.1.2.3[link] shows the intersection of the dispersion surface with a plane passing through OH. When the tie point lies on one of the two spheres, far from their intersection, only one wavefield propagates inside the crystal. When it lies on the connecting surface, two waves are excited simultaneously. The equation of this surface is obtained by equating to zero the determinant of system (5.1.2.20)[link]:[X_{o}X_{h} = k^{2}C^{2} \chi_{h} \chi_{\bar{h}}/4. \eqno(5.1.2.22)]

[Figure 5.1.2.3]

Figure 5.1.2.3 | top | pdf |

Intersection of the dispersion surface with the plane of incidence. The dispersion surface is a connecting surface between the two spheres centred at reciprocal-lattice points O and H and with radius nk. [L_{o}] is the Lorentz point.

Equations (5.1.2.21)[link] show that, in the zero-absorption case, [X_{o}] and [X_{h}] are to be interpreted as the distances of the tie point P from the spheres centred at O and H, respectively. From (5.1.2.20)[link] it can be seen that they are of the order of the vacuum wavenumber times the Fourier coefficient of the dielectric susceptibility, that is five or six orders of magnitude smaller than k. The two spheres can therefore be replaced by their tangential planes. Equation (5.1.2.22)[link] shows that the product of the distances of the tie point from these planes is constant. The intersection of the dispersion surface with the plane passing through OH is therefore a hyperbola (Fig. 5.1.2.4[link]) whose diameter [using (5.1.2.5)[link] and (5.1.2.22)[link]] is[\overline{A_{o2}A_{o1}} = |C| R\lambda (F_{h} F_{\bar{h}})^{1/2}/(\pi V \cos \theta). \eqno(5.1.2.23)]

[Figure 5.1.2.4]

Figure 5.1.2.4 | top | pdf |

Intersection of the dispersion surface with the plane of incidence shown in greater detail. The Lorentz point [L_{o}] is far away from the nodes O and H of the reciprocal lattice: [OL_{o} = HL_{o} = 1/\lambda] is about 105 to 106 times larger than the diameter [A_{o1} A_{o2}] of the dispersion surface.

It can be noted that the larger the diameter of the dispersion surface, the larger the structure factor, that is, the stronger the interaction of the waves with the matter. When the polarization is parallel to the plane of incidence [(C = \cos 2\theta)], the interaction is weaker.

The asymptotes [T_{o}] and [T_{h}] to the hyperbola are tangents to the circles centred at O and H, respectively. Their intersection, [L_{o}], is called the Lorentz point (Fig. 5.1.2.4[link]).

A wavefield propagating in the crystal is characterized by a tie point P on the dispersion surface and two waves with wave­vectors [{\bf K}_{{\bf o}} = {\bf OP}] and [{\bf K}_{{\bf h}} = {\bf HP}], respectively. The ratio, ξ, of their amplitudes [D_{h}] and [D_{o}] is given by means of (5.1.2.20)[link]:[\xi = {D_{h} \over D_{o}} = {2X_{o} \over kC\chi_{\bar{h}}} = {-2\pi VX_{o} \over R\lambda CF_{\bar{h}}}. \eqno(5.1.2.24)]

The hyperbola has two branches, 1 and 2, for each direction of polarization, that is, for [C = 1] or [\cos 2\theta] (Fig. 5.1.2.5)[link]. Branch 2 is the one situated on the same side of the asymptotes as the reciprocal-lattice points O and H. Given the orientation of the wavevectors, which has been chosen away from the reciprocal-lattice points (Fig. 5.1.2.1b[link]), the coordinates of the tie point, [X_{o}] and [X_{h}], are positive for branch 1 and negative for branch 2. The phase of ξ is therefore equal to [\pi + \varphi_{h}] and to [\varphi_{h}] for the two branches, respectively, where [\varphi_{h}] is the phase of the structure factor [equation (5.1.2.6)[link]]. This difference of π between the two branches has important consequences for the properties of the wavefields.

[Figure 5.1.2.5]

Figure 5.1.2.5 | top | pdf |

Dispersion surface for the two states of polarization. Solid curve: polarization normal to the plane of incidence [(C = 1)]; broken curve: polarization parallel to the plane of incidence [(C = \cos 2\theta)]. The direction of propagation of the energy of the wavefields is along the Poynting vector, S, normal to the dispersion surface.

As mentioned above, owing to absorption, wavevectors are actually complex and so is the dispersion surface (see Authier, 2008[link]).

5.1.2.6. Propagation direction

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The energy of all the waves in a given wavefield propagates in a common direction, which is given by that of the Poynting vector (von Laue, 1952[link]) [see Section A5.1.1.4[link], equation (A5.1.1.8)[link] of the Appendix[link]]. It can be shown that, averaged over time and the unit cell, the Poynting vector of a wavefield is[{\bf S} = (c/\varepsilon_{0}) \exp (4\pi {\bf K}_{{\bf o}{\rm i}} \cdot {\bf r}) \left[|D_{o}|^{2} {\bf s}_{{\bf o}} + |D_{h}|^{2} {\bf s}_{{\bf h}}\right], \eqno(5.1.2.25)]where [{\bf s}_{{\bf o}}] and [{\bf s}_{{\bf h}}] are unit vectors in the [{\bf K}_{{\bf o}}] and [{\bf K}_{{\bf h}}] directions, respectively, c is the velocity of light and [\varepsilon_{0}] is the dielectric permittivity of a vacuum.

From (5.1.2.25)[link] and equation (5.1.2.22)[link] of the dispersion surface, it can be shown that the propagation direction of the wavefield lies along the normal to the dispersion surface at the tie point (Fig. 5.1.2.5)[link]. This result was first shown by Kato (1952[link]) in the two-beam case for electron diffraction and generalized by him to the n-beam case for X-rays (Kato, 1958[link]). It is also obtained by considering the group velocity of the wavefield (Ewald, 1958[link]; Wagner, 1959[link]). The angle α between the propagation direction and the lattice planes is given by[\tan \alpha = \left[(1 - |\xi |^{2})/(1 + |\xi |^{2})\right] \tan \theta. \eqno(5.1.2.26)]

It should be noted that the propagation direction varies between [{\bf K}_{{\bf o}}] and [{\bf K}_{{\bf h}}] for both branches of the dispersion surface.

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