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, pp. 626-630
Section 5.1.2. Fundamentals of plane-wave dynamical theory^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, 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 aswhere 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 bywhere 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 bywhere 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,whereandIt is important to note thatwhere is the imaginary conjugate of f.
The index of refraction of the medium for X-rays iswhere 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 isFor example, it is 143.2 cm^{−1} for silicon and Cu Kα radiation.
The notion of a wavefield, introduced by Ewald (1917), is one of the most fundamental concepts in dynamical theory. It results from the fact that since the propagation equation (5.1.2.2) is a second-order partial differential equation with a periodic interaction coefficient, its solution has the same periodicity,where the summation is over all reciprocal-lattice vectors h. Equation (5.1.2.14) can also be writtenwhere
Expression (5.1.2.15) shows that the solution of the propagation equation can be interpreted as an infinite sum of plane waves with amplitudes and wavevectors . 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) 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)]. They can be represented geometrically as shown in Fig. 5.1.2.1(b). The wavevectors ; 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,and (5.1.2.16) shows that all wavevectors have the same imaginary part,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) and explained by von Laue (1937) 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, 1950) and the theoretical determination of the wavefield trajectories (von Laue, 1952; Kato, 1952, 1958; Ewald, 1958), which was confirmed by the experimental observation of spherical wave Pendellösung (Kato & Lang, 1959) and by that of the double refraction of X-rays, predicted by Borrmann (1955) and observed by Authier (1960, 1961).
The choice of the `o' component of expansion (5.1.2.15) 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 (using von Laue's notation), the choice of 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 and formally in (5.1.2.18):where n is a unit vector normal to the crystal surface, oriented towards the inside of the crystal.
There is no absorption in a vacuum and the incident wavevector is real. Equation (5.1.2.18) shows that it is the component normal to the interface of wavevector which has an imaginary part,where is the imaginary part of f, and 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, , given by (5.1.2.13) – see Section 5.1.5.
In order to obtain the solution of dynamical theory, one inserts expansions (5.1.2.15) and (5.1.2.4) into the propagation equation (5.1.2.2). 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) whose wavevector magnitudes 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, 2004), Hümmer & Weckert (1995) and Weckert & Hümmer (1997)]. 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 ).
In the two-beam case, for reflections that are not highly asymmetric and for Bragg angles that are not close to , the fundamental equations of dynamical theory reduce towhere if is normal to the plane and if 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 ). 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 of the Appendix).
The system (5.1.2.20) 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,In the case of an absorbing crystal, and are complex. Equation (5.1.5.2) gives the full expression for .
The fundamental equations (5.1.2.20) 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 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):
Equations (5.1.2.21) show that, in the zero-absorption case, and 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) 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) 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) whose diameter [using (5.1.2.5) and (5.1.2.22)] is
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 , the interaction is weaker.
The asymptotes and to the hyperbola are tangents to the circles centred at O and H, respectively. Their intersection, , is called the Lorentz point (Fig. 5.1.2.4).
A wavefield propagating in the crystal is characterized by a tie point P on the dispersion surface and two waves with wavevectors and , respectively. The ratio, ξ, of their amplitudes and is given by means of (5.1.2.20):
The hyperbola has two branches, 1 and 2, for each direction of polarization, that is, for or (Fig. 5.1.2.5). 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), the coordinates of the tie point, and , are positive for branch 1 and negative for branch 2. The phase of ξ is therefore equal to and to for the two branches, respectively, where is the phase of the structure factor [equation (5.1.2.6)]. This difference of π between the two branches has important consequences for the properties of the wavefields.
As mentioned above, owing to absorption, wavevectors are actually complex and so is the dispersion surface (see Authier, 2008).
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) [see Section A5.1.1.4, equation (A5.1.1.8) of the Appendix]. It can be shown that, averaged over time and the unit cell, the Poynting vector of a wavefield iswhere and are unit vectors in the and directions, respectively, c is the velocity of light and is the dielectric permittivity of a vacuum.
From (5.1.2.25) and equation (5.1.2.22) 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). This result was first shown by Kato (1952) in the two-beam case for electron diffraction and generalized by him to the n-beam case for X-rays (Kato, 1958). It is also obtained by considering the group velocity of the wavefield (Ewald, 1958; Wagner, 1959). The angle α between the propagation direction and the lattice planes is given by
It should be noted that the propagation direction varies between and for both branches of the dispersion surface.
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