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. 534551
doi: 10.1107/97809553602060000569 Chapter 5.1. Dynamical theory of Xray diffraction^{a}Laboratoire de MinéralogieCristallographie, Université P. et M. Curie, 4 Place Jussieu, F75252 Paris CEDEX 05, France This chapter presents the dynamical theory of the diffraction of Xrays by perfect crystals. The most important part is devoted to the case of plane waves (Section 5.1.2). The solutions of the propagation equation of plane waves in crystals are given in Section 5.1.3 using the concept of wavefields introduced by Ewald for Xrays in 1913 and by Bloch for electrons in 1928 (known in solidstate physics as Bloch waves). They are applied to the interpretation of the main properties of dynamical diffraction: anomalous transmission, standing waves and Pendellösung. The expressions for the diffracted intensity are given in both the transmission (Section 5.1.6) and the reflection (Section 5.1.7) cases. The last part (Section 5.1.8) concerns the diffraction of real and spherical waves, which is described in a qualitative way. 
The first experiment on Xray 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 Xrays were developed by Darwin (1914a,b). His dynamical theory took into account the interaction of Xrays 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 Xray diffraction. It gives correct expressions for the reflected intensities and was extended to the absorbingcrystal 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 Xrays 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 Xrays 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 Xrays 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 Xray topographs, accurate measurement of form factors, latticeparameter 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 standingwaves method [see, for instance, the reviews by Authier (1989) and Zegenhagen (1993)], attempts to determine phases using multiple reflections [see, for instance, Chang (1987) and Hümmer & Weckert (1995)], characterization of the crystal perfection of epilayers and superlattices by highresolution diffractometry [see, for instance, Tanner (1990) and Fewster (1993)], etc.
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 & Malgrange (1998), and Authier (2001). Topography is described in Chapter 2.7 of IT C (2004), in Tanner (1976) and in Tanner & Bowen (1992). For the use of Braggangle measurements for accurate latticeparameter mapping, see Hart (1981).
A reminder of some basic concepts in electrodynamics is given in Section A5.1.1.1 of the Appendix.
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 pseudopotential 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 Xrays, 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 Xrays 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 reciprocallattice vector and the summation is extended over all reciprocallattice 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 reciprocallattice 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 Xrays 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 Xrays 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.
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 secondorder partial differential equation with a periodic interaction coefficient, its solution has the same periodicity, where the summation is over all reciprocallattice vectors h. Equation (5.1.2.14) can also be written where
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 solidstate 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 reciprocallattice 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 choice of the `o' component of expansion (5.1.2.15) is arbitrary in an infinite medium. In a semiinfinite 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 nonnegligible amplitude. These wavevectors are associated with reciprocallattice 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 Xrays, there are only two reciprocallattice points on the Ewald sphere. This is the socalled twobeam case to which this treatment is limited. There are, however, many instances where several reciprocallattice points lie simultaneously on the Ewald sphere. This corresponds to the manybeam case which has interesting applications for the determination of phases of reflections [see, for instance, Chang (1987) and Hümmer & Weckert (1995)]. On the other hand, for electrons, there are in general many reciprocallattice points close to the Ewald sphere and many wavefields are excited simultaneously (see Chapter 5.2 ).
In the twobeam case, for reflections that are not highly asymmetric and for Bragg angles that are not close to , the fundamental equations of dynamical theory reduce to where 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 nonabsorbing 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 constantenergy surface and is the equivalent of the index surface in optics. It is the Xray analogue of the constantenergy surfaces known as Fermi surfaces in the electron band theory of solids.
In the twobeam 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 zeroabsorption 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 reciprocallattice points O and H. Given the orientation of the wavevectors, which has been chosen away from the reciprocallattice 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.
The energy of all the waves in a given wavefield propagates in a common direction, which is obtained by calculating either the group velocity or the Poynting vector [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 is where and are unit vectors in the and directions, respectively, c is the velocity of light and is the dielectric permittivity of a vacuum. This result was first shown by von Laue (1952) in the twobeam case and was generalized to the nbeam case by Kato (1958).
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 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.
The wavefields excited in the crystal by the incident wave are determined by applying the boundary condition mentioned above for the continuity of the tangential component of the wavevectors (Section 5.1.2.3). Waves propagating in a vacuum have wavenumber . Depending on whether they propagate in the incident or in the reflected direction, the common extremity, M, of their wavevectors lies on spheres of radius k and centred at O and H, respectively. The intersections of these spheres with the plane of incidence are two circles which can be approximated by their tangents and at their intersection point, , or Laue point (Fig. 5.1.3.1).
Bragg's condition is exactly satisfied according to the geometrical theory of diffraction when M lies at . The departure Δθ from Bragg's incidence of an incident wave is defined as the angle between the corresponding wavevectors OM and . As Δθ is very small compared to the Bragg angle in the general case of Xrays or neutrons, one may write
The tangent is oriented in such a way that Δθ is negative when the angle of incidence is smaller than the Bragg angle.
The boundary condition for the continuity of the tangential component of the wavevectors is applied by drawing from M a line, Mz, parallel to the normal n to the crystal surface. The tie points of the wavefields excited in the crystal by the incident wave are at the intersections of this line with the dispersion surface. Two different situations may occur:
It will be apparent from the equations given later that the incident wavevector corresponding to the middle of the reflection domain is, in both cases, OI, where I is the intersection of the normal to the crystal surface drawn from the Lorentz point, , with (Figs. 5.1.3.4 and 5.1.3.5), while, according to Bragg's law, it should be . The angle Δθ between the incident wavevectors and OI, corresponding to the middle of the reflecting domain according to the geometrical and dynamical theories, respectively, is

Boundary conditions at the entrance surface for reflection geometry. (a) Reciprocal space; (b) direct space. 
In the Bragg case, the asymmetry ratio γ is negative and is never equal to zero. This difference in Bragg angle between the two theories is due to the refraction effect, which is neglected in geometrical theory. In the Laue case, is equal to zero for symmetric reflections .
The solutions of dynamical theory are best described by introducing a reduced parameter called the deviation parameter, where whose real part is equal to the half width of the rocking curve (Sections 5.1.6 and 5.1.7). The width 2δ of the rocking curve is sometimes called the Darwin width.
The definition (5.1.3.5) of the deviation parameter is independent of the geometrical situation (reflection or transmission case); this is not followed by some authors. The present convention has the advantage of being quite general.
In an absorbing crystal, and δ are complex, and it is the real part, , of which has the geometrical interpretation given in Section 5.1.3.3. One obtains where β is the phase angle of [or that of ].
This length plays a very important role in the dynamical theory of diffraction by both perfect and deformed crystals. For example, it is 15.3 µm for the 220 reflection of silicon, with Mo Kα radiation and a symmetric reflection.
In transmission geometry, it gives the period of the interference between the two excited wavefields which constitutes the Pendellösung effect first described by Ewald (1917) (see Section 5.1.6.3); in this case is called the Pendellösung distance, denoted hereafter. Its geometrical interpretation, in the zeroabsorption case, is the inverse of the diameter of the dispersion surface in a direction defined by the cosines and with respect to the reflected and incident directions, respectively (Fig. 5.1.3.4). It reduces to the inverse of (5.1.2.23) in the symmetric case.
In reflection geometry, it gives the absorption distance in the totalreflection domain and is called the extinction distance, denoted (see Section 5.1.7.1). Its geometrical interpretation in the zeroabsorption case is the inverse of the length , Fig. 5.1.3.5.
In a deformed crystal, if distortions are of the order of the width of the rocking curve over a distance , the crystal is considered to be slightly deformed, and ray theory (Penning & Polder, 1961; Kato, 1963, 1964a,b) can be used to describe the propagation of wavefields. If the distortions are larger, new wavefields may be generated by interbranch scattering (Authier & Balibar, 1970) and generalized dynamical diffraction theory such as that developed by Takagi (1962, 1969) should be used.
Using (5.1.3.8), expressions (5.1.3.5) and (5.1.3.6) can be rewritten in the very useful form:
The order of magnitude of the Darwin width 2δ ranges from a fraction of a second of an arc to ten or more seconds, and increases with increasing wavelength and increasing structure factor. For example, for the 220 reflection of silicon and Cu Kα radiation, it is 5.2 seconds.
The coordinates of the tie points excited by the incident wave are obtained by looking for the intersection of the dispersion surface, (5.1.2.22), with the normal Mz to the crystal surface (Figs. 5.1.3.4 and 5.1.3.5). The ratio ξ of the amplitudes of the waves of the corresponding wavefields is related to these coordinates by (5.1.2.24) and is found to be where the plus sign corresponds to a tie point on branch 1 (j = 1) and the minus sign to a tie point on branch 2 (j = 2), and is the sign of (+1 in transmission geometry, −1 in reflection geometry).
In this case (Fig. 5.1.3.4) is +1 and (5.1.3.10) may be written
Let and be the intersections of the normal to the crystal surface drawn from the Lorentz point with the two branches of the dispersion surface (Fig. 5.1.3.4). From Sections 5.1.3.3 and 5.1.3.4, they are the tie points excited for and correspond to the middle of the reflection domain. Let us further consider the tangents to the dispersion surface at and and let , and , be their intersections with and , respectively. It can be shown that and intersect the dispersion surface at the tie points excited for and , respectively, and that the Pendellösung distance , the width of the rocking curve and the deviation parameter , where and are the intersections of the normal to the crystal surface drawn from the extremity of any incident wavevector OM with and , respectively.
In this case (Fig. 5.1.3.5) is now −1 and (5.1.3.10) may be written
Now, let and be the points of the dispersion surface where the tangent is parallel to the normal to the crystal surface, and further let and be the intersections of these two tangents with and , respectively. For an incident wave of wavevector OM where M lies between I′ and I″, the normal to the crystal surface drawn from M has no real intersection with the dispersion surface and defines the totalreflection domain. The tie points and correspond to and , respectively, the extinction distance , the width of the totalreflection domain and the deviation parameter , where and are the intersections with and of the normal to the crystal surface drawn from the extremity of any incident wavevector OM.
The various waves in a wavefield are coherent and interfere. In the twobeam case, the intensity of the wavefield, using (5.1.2.14) and (5.1.2.24), is where Ψ is the phase of ξ,
Equation (5.1.4.1) shows that the interference between the two waves is the origin of standing waves. The corresponding nodes lie on planes such that is a constant. These planes are therefore parallel to the diffraction planes and their periodicity is equal to (defined in the caption for Fig. 5.1.2.1a). Their position within the unit cell is given by the value of the phase Ψ.
In the Laue case, Ψ is equal to for branch 1 and to for branch 2, where is the phase of the structure factor, (5.1.2.6). This means that the nodes of standing waves lie on the maxima of the hkl Fourier component of the electron density for branch 1 while the antinodes lie on the maxima for branch 2.
In the Bragg case, Ψ varies continuously from to as the angle of incidence is varied from the lowangle side to the highangle side of the reflection domain by rocking the crystal. The nodes lie on the maxima of the hkl Fourier components of the electron density on the lowangle side of the rocking curve. As the crystal is rocked, they are progressively shifted by half a lattice spacing until the antinodes lie on the maxima of the electron density on the highangle side of the rocking curve.
Standing waves are the origin of the phenomenon of anomalous absorption, which is one of the specific properties of wavefields (Section 5.1.5). Anomalous scattering is also used for the location of atoms in the unit cell at the vicinity of the crystal surface: when Xrays are absorbed, fluorescent radiation and photoelectrons are emitted. Detection of this emission for a known angular position of the crystal with respect to the rocking curve and therefore for a known value of the phase Ψ enables the emitting atom within the unit cell to be located. The principle of this method is due to Batterman (1964, 1969). For reviews, see Golovchenko et al. (1982), Materlik & Zegenhagen (1984), Kovalchuk & Kohn (1986), Bedzyk (1988), Authier (1989), and Zegenhagen (1993).
It was shown in Section 5.1.2.2 that the wavevectors of a given wavefield all have the same imaginary part (5.1.2.17) and therefore the same absorption coefficient μ (5.1.2.19). Borrmann (1950, 1954) showed that this coefficient is much smaller than the normal one for wavefields whose tie points lie on branch 1 of the dispersion surface and much larger for wavefields whose tie points lie on branch 2. The former case corresponds to the anomalous transmission effect, or Borrmann effect. As in favourable cases the minimum absorption coefficient may be as low as a few per cent of , this effect is very important from both a fundamental and a practical point of view.
The physical interpretation of the Borrmann effect is to be found in the standing waves described in Section 5.1.4. When the nodes of the electric field lie on the planes corresponding to the maxima of the hkl component of the electron density, the wavefields are absorbed anomalously less than when there is no diffraction. Just the opposite occurs for branch 2 wavefields, whose antinodes lie on the maxima of the electron density and which are absorbed more than normal.
The effective absorption coefficient μ is related to the imaginary part of the wavevectors through (5.1.2.19), and to the imaginary part of the ratio of the amplitude of the reflected to the incident wave through where is the imaginary part of , which, using (5.1.2.24) and (5.1.3.10), is given by Taking the upper sign (+) for the ± term corresponds to tie points on branch 1 and taking the lower sign (−) corresponds to tie points on branch 2.
The calculation of the imaginary part is different in the Laue and in the Bragg cases. In the former case, the imaginary part of is small and can be approximated while in the latter, the imaginary part of is large when the real part of the deviation parameter, , lies between 1 and −1, and cannot be calculated using the same approximation.
In transmission geometry, the imaginary part of is small and, using a firstorder approximation for the expansion of , (5.1.5.1) and (5.1.5.2), the effective absorption coefficient in the absorption case is where is the phase difference between and [equation (5.1.2.10)], the upper sign (−) for the ∓ term corresponds to branch 1 and the lower sign (+) corresponds to branch 2 of the dispersion surface. In the symmetric Laue case (, reflecting planes normal to the crystal surface), equation (5.1.6.1) reduces to
Fig. 5.1.6.1 shows the variations of the effective absorption coefficient with for wavefields belonging to branches 1 and 2 in the case of the 400 reflection of GaAs with Cu Kα radiation. It can be seen that for the absorption coefficient for branch 1 becomes significantly smaller than the normal absorption coefficient, . The minimum absorption coefficient, , depends on the nature of the reflection through the structure factor and on the temperature through the Debye–Waller factor included in [equation (5.1.2.10b)] (Ohtsuki, 1964, 1965). For instance, in diamondtype structures, it is smaller for reflections with even indices than for reflections with odd indices. The influence of temperature is very important when is close to one; for example, for germanium 220 and Mo Kα radiation, the minimum absorption coefficient at 5 K is reduced to about 1% of its normal value, (Ludewig, 1969).
5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves
Let us consider an infinite plane wave incident on a crystal plane surface of infinite lateral extension. As has been shown in Section 5.1.3, two wavefields are excited in the crystal, with tie points and , and amplitudes and , respectively. Maxwell's boundary conditions (see Section A5.1.1.2 of the Appendix) imply continuity of the tangential component of the electric field and of the normal component of the electric displacement across the boundary. Because the index of refraction is so close to unity, one can assume to a very good approximation that there is continuity of the three components of both the electric field and the electric displacement. As a consequence, it can easily be shown that, along the entrance surface, for all components of the electric displacement where is the amplitude of the incident wave.
Using (5.1.3.11), (5.1.5.2) and (5.1.6.2), it can be shown that the intensities of the four waves are top sign: ; bottom sign: .
Fig. 5.1.6.2 represents the variations of these four intensities with the deviation parameter. Far from the reflection domain, and tend toward zero, as is normal, while
This result shows that the wavefield of highest intensity `jumps' from one branch of the dispersion surface to the other across the reflection domain. This is an important property of dynamical theory which also holds in the Bragg case and when a wavefield crosses a highly distorted region in a deformed crystal [the socalled interbranch scattering: see, for instance, Authier & Balibar (1970) and Authier & Malgrange (1998)].
When a wavefield reaches the exit surface, it breaks up into its two constituent waves. Their wavevectors are obtained by applying again the condition of the continuity of their tangential components along the crystal surface. The extremities, and , of these wavevectors lie at the intersections of the spheres of radius k centred at O and H, respectively, with the normal n′ to the crystal exit surface drawn from (j = 1 and 2) (Fig. 5.1.6.3).
If the crystal is wedgeshaped and the normals n and n′ to the entrance and exit surfaces are not parallel, the wavevectors of the waves generated by the two wavefields are not parallel. This effect is due to the refraction properties associated with the dispersion surface.
We shall assume from now on that the crystal is plane parallel. Two wavefields arrive at any point of the exit surface. Their constituent waves interfere and generate emerging waves in the refracted and reflected directions (Fig. 5.1.6.4). Their respective amplitudes are given by the boundary conditions where r is the position vector of a point on the exit surface, the origin of phases being taken at the entrance surface.
In a planeparallel crystal, (5.1.6.4) reduces to where t is the crystal thickness.
In a nonabsorbing crystal, the amplitudes squared are of the form This expression shows that the intensities of the refracted and reflected beams are oscillating functions of crystal thickness. The period of the oscillations is called the Pendellösung distance and is
For an absorbing crystal, the intensities of the reflected and refracted waves are where and is given by equation (5.1.6.1).
Depending on the absorption coefficient, the cosine terms are more or less important relative to the hyperbolic cosine term and the oscillations due to Pendellösung have more or less contrast.
For a nonabsorbing crystal, these expressions reduce to
What is actually measured in a counter receiving the reflected or the refracted beam is the reflecting power, namely the ratio of the energy of the reflected or refracted beam on the one hand and the energy of the incident beam on the other. The energy of a beam is obtained by multiplying its intensity by its cross section. If l is the width of the trace of the beam on the crystal surface, the cross sections of the incident (or refracted) and reflected beams are proportional to (Fig. 5.1.6.5) and , respectively.
The reflecting powers are therefore: Using (5.1.6.6), it is easy to check that in the nonabsorbing case; that is, that conservation of energy is satisfied. Equations (5.1.6.6) show that there is a periodic exchange of energy between the refracted and the reflected waves as the beam penetrates the crystal; this is why Ewald introduced the expression Pendellösung.
The oscillations in the rocking curve were first observed by LefeldSosnowska & Malgrange (1968, 1969). Their periodicity can be used for accurate measurements of the form factor [see, for instance, Bonse & Teworte (1980)]. Fig. 5.1.6.6 shows the shape of the rocking curve for various values of .

Theoretical rocking curves in the transmission case for nonabsorbing crystals and for various values of : (a) ; (b) ; (c) ; (d) . 
The width at halfheight of the rocking curve, averaged over the Pendellösung oscillations, corresponds in the nonabsorbing case to , that is, to , where δ is given by (5.1.3.6).
The integrated intensity is the ratio of the total energy recorded in the counter when the crystal is rocked to the intensity of the incident beam. It is proportional to the area under the line profile:
The integration was performed by von Laue (1960). Using (5.1.3.5), (5.1.6.6) and (5.1.6.7) gives where is the zerothorder Bessel function and Fig. 5.1.6.7 shows the variations of the integrated intensity with .
Using (5.1.6.6) and (5.1.6.7), the reflecting power of the reflected beam may also be written where and When is very small, tends asymptotically towards the function and towards the value given by geometrical theory. The condition for geometrical theory to apply is, therefore, that the crystal thickness be much smaller than the Pendellösung distance. In practice, the two theories agree to within a few per cent for a crystal thickness smaller than or equal to a third of the Pendellösung distance [see Authier & Malgrange (1998)].
Rocking curve. The geometrical construction in Fig. 5.1.3.5 shows that, in the Bragg case, the normal to the crystal surface drawn from the extremity of the incident wavevector intersects the dispersion surface either at two points of the same branch, , for branch 1, for branch 2, or at imaginary points. It was shown in Section 5.1.2.6 that the propagation of the wavefields inside the crystal is along the normal to the dispersion surface at the corresponding tie points. Fig. 5.1.3.5 shows that this direction is oriented towards the outside of the crystal for tie points and . In a very thick crystal, these wavefields cannot exist because there is always a small amount of absorption. One concludes that in the thickcrystal case and in reflection geometry, only one wavefield is excited inside the crystal. It corresponds to branch 1 on the lowangle side of the rocking curve and to branch 2 on the highangle side. Using the same approximations as in Section 5.1.6.2, the amplitude of the wave reflected at the crystal surface is obtained by applying the boundary conditions, which are particularly simple in this case:
The reflecting power is given by an expression similar to (5.1.6.7): where the expression for is given by (5.1.3.12), and j = 1 or 2 depending on which wavefield propagates towards the inside of the crystal. When the normal to the entrance surface intersects the dispersion surface at imaginary points, i.e. when , and there is total reflection. Outside the totalreflection domain, the reflecting power is given by
The rocking curve has the well known tophat shape (Fig. 5.1.7.1). Far from the totalreflection domain, the curve can be approximated by the function

Theoretical rocking curve in the reflection case for a nonabsorbing thick crystal in terms of the deviation parameter. 
Width of the totalreflection domain. The width of the totalreflection domain is equal to and its angular width is therefore equal, using (5.1.3.5), to 2δ, where δ is given by (5.1.3.6). It is proportional to the structure factor, the polarization factor C and the square root of the asymmetry factor . Using an asymmetric reflection, it is therefore possible to decrease the width at wish. This is used in monochromators to produce a pseudo plane wave [see, for instance, Kikuta & Kohra (1970)]. It is possible to deduce the value of the form factor from very accurate measurements of the rocking curve; see, for instance, Kikuta (1971).
Integrated intensity. The integrated intensity is defined by (5.1.6.8):
Penetration depth. Within the domain of total reflection, there are two wavefields propagating inside the crystal with imaginary wavevectors, one towards the inside of the crystal and the other one in the opposite direction, so that they cancel out and, globally, no energy penetrates the crystal. The absorption coefficient of the waves penetrating the crystal is where is the value taken by [equation (5.1.3.8)] in the Bragg case.
The penetration depth is a minimum at the middle of the reflection domain and at this point it is equal to . This attenuation effect is called extinction, and is called the extinction length. It is a specific property owing to the existence of wavefields. The resulting propagation direction of energy is parallel to the crystal surface, but with a cross section equal to zero: it is an evanescent wave [see, for instance, Cowan et al. (1986)].
Rocking curve. Since the sign of γ is negative, in (5.1.3.10) has a very large imaginary part when . It cannot be calculated using the same approximations as in the Laue case. Let us set The reflecting power is where and is the modulus of expression (5.1.7.5) where the sign is chosen in such a way that Z is smaller than 1.
The expression for the reflected intensity in the absorbing Bragg case was first given by Prins (1930). The way of representing it given here was first used by Hirsch & Ramachandran (1950). The properties of the rocking curve have been described by Fingerland (1971).
There is no longer a totalreflection domain and energy penetrates the crystal at all incidence angles, although with a very high absorption coefficient within the domain . Fig. 5.1.7.2 gives an example of a rocking curve for a thick absorbing crystal. It was first observed by Renninger (1955). The shape is asymmetric and is due to the anomalousabsorption effect: it is lower than normal on the lowangle side, which is associated with wavefields belonging to branch 1 of the dispersion surface, and larger than normal on the highangle side, which is associated with branch 2 wavefields. The amount of asymmetry depends on the value of the ratio of the coefficients in the expression for the imaginary part of the deviation parameter in (5.1.3.7): the smaller this ratio, the more important the asymmetry.

Theoretical rocking curve in the reflection case for a thick absorbing crystal. The 400 reflection of GaAs using Cu Kα radiation is shown. 
Absorption coefficient. The effective absorption coefficient, taking into account both the Borrmann effect and extinction, is given by (Authier, 1986) where β is defined in equation (5.1.3.7) and Ψ′ in equation (5.1.7.5), and where the sign is chosen in such a way that Z converges. Fig. 5.1.7.3 shows the variation of the penetration depth with the deviation parameter.
Boundary conditions. If the crystal is thin, the wavefield created at the reflecting surface at A and penetrating inside can reach the back surface at B (Fig. 5.1.7.4a). The incident direction there points towards the outside of the crystal, while the reflected direction points towards the inside. The wavefield propagating along AB will therefore generate at B:

Bragg case: thin crystals. Two wavefields propagate in the crystal. (a) Direct space; (b) reciprocal space. 
The corresponding tiepoints are obtained by applying the usual condition of the continuity of the tangential components of wavevectors (Fig. 5.1.7.4b). If the crystal is a planeparallel slab, these points are M and , respectively, and .
The boundary conditions are then written:
Rocking curve. Using (5.1.3.10), it can be shown that the expressions for the intensities reflected at the entrance surface and transmitted at the back surface, and , respectively, are given by different expressions within total reflection and outside it:
Integrated intensity. The integrated intensity is where t is the crystal thickness. When this thickness becomes very large, the integrated intensity tends towards
This expression differs from (5.1.7.3) by the factor π, which appears here in place of . von Laue (1960) pointed out that because of the differences between the two expressions for the reflecting power, (5.1.7.2) and (5.1.7.7b), perfect agreement could not be expected. Since some absorption is always present, expression (5.1.7.3), which includes the factor , should be used for very thick crystals. In the presence of absorption, however, expression (5.1.7.8) for the reflected intensity for thin crystals does tend towards that for thick crystals as the crystal thickness increases.
Comparison with geometrical theory. When is very small (thin crystals or weak reflections), (5.1.7.8) tends towards which is the expression given by geometrical theory. If we call this intensity (geom.), comparison of expressions (5.1.7.8) and (5.1.7.10) shows that the integrated intensity for crystals of intermediate thickness can be written which is the expression given by Darwin (1922) for primary extinction.
Reflected intensity. The intensity of the reflected wave for a thin absorbing crystal is where L, ρ and ψ′ are defined in (5.1.7.5), β is defined in (5.1.3.7) and ω is the phase angle of .
Comparison with geometrical theory. When decreases towards zero, expression (5.1.7.12) tends towards ; using (5.1.3.5) and (5.1.3.8), it can be shown that expression (5.1.7.12) can be written, in the nonabsorbing symmetric case, as where d is the lattice spacing and Δθ is the difference between the angle of incidence and the middle of the reflection domain. This expression is the classical expression given by geometrical theory [see, for instance, James (1950)].
The preceding sections have dealt with the diffraction of a plane wave by a semiinfinite perfect crystal. This situation is actually never encountered in practice, although with various devices, in particular using synchrotron radiation, it is possible to produce highly collimated monochromated waves which behave like pseudo plane waves. The wave from an Xray tube is best represented by a spherical wave. The first experimental proof of this fact is due to Kato & Lang (1959) in the transmission case. Kato extended the dynamical theory to spherical waves for nonabsorbing (Kato, 1961a,b) and absorbing crystals (Kato, 1968a,b). He expanded the incident spherical wave into plane waves by a Fourier transform, applied planewave dynamical theory to each of these components and took the Fourier transform of the result again in order to obtain the final solution. Another method for treating the problem was used by Takagi (1962, 1969), who solved the propagation equation in a medium where the lateral extension of the incident wave is limited and where the wave amplitudes depend on the lateral coordinates. He showed that in this case the set of fundamental linear equations (5.1.2.20) should be replaced by a set of partial differential equations. This treatment can be applied equally well to a perfect or to an imperfect crystal. In the case of a perfect crystal, Takagi showed that these equations have an analytical solution that is identical with Kato's result. Uragami (1969, 1970) observed the spherical wave in the Bragg (reflection) case, interpreting the observed intensity distribution using Takagi's theory. Saka et al. (1973) subsequently extended Kato's theory to the Bragg case.
Without using any mathematical treatment, it is possible to make some elementary remarks by considering the fact that the divergence of the incident beam falling on the crystal from the source is much larger than the angular width of the reflection domain. Fig. 5.1.8.1(a) shows a spatially collimated beam falling on a crystal in the transmission case and Fig. 5.1.8.1(b) represents the corresponding situation in reciprocal space. Since the divergence of the incident beam is larger than the angular width of the dispersion surface, the plane waves of its Fourier expansion will excite every point of both branches of the dispersion surface. The propagation directions of the corresponding wavefields will cover the angular range between those of the incident and reflected beams (Fig. 5.1.8.1a) and fill what is called the Borrmann triangle. The intensity distribution within this triangle has interesting properties, as described in the next two sections.
The first property of the Borrmann triangle is that the angular density of the wavefield paths is inversely proportional to the curvature of the dispersion surface around their tie points. Let us consider an incident wavepacket of angular width . It will generate a packet of wavefields propagating within the Borrmann triangle. The angular width (Fig. 5.1.8.2) between the paths of the corresponding wavefields is related to the radius of curvature of the dispersion surface by where α is the angle between the wavefield path and the lattice planes [equation (5.1.2.26)] and is called the amplification ratio. In the middle of the reflecting domain, the radius of curvature of the dispersion surface is very much shorter than its value, k, far from it (about 10^{4} times shorter) and the amplification ratio is therefore very large. As a consequence, the energy of a wavepacket of width in reciprocal space is spread in direct space over an angle given by (5.1.8.1). The intensity distribution on the exit surface BC (Fig. 5.1.8.1a) is therefore proportional to . It is represented in Fig. 5.1.8.3 for several values of the absorption coefficient:
Fig. 5.1.8.4 shows that along any path Ap inside the Borrmann triangle two wavefields propagate, one with tie point , on branch 1, the other with the point , on branch 2. These two points lie on the extremities of a diameter of the dispersion surface. The two wavefields interfere, giving rise to Pendellösung fringes, which were first observed by Kato & Lang (1959), and calculated by Kato (1961b). These fringes are of course quite different from the planewave Pendellösung fringes predicted by Ewald (Section 5.1.6.3) because the tie points of the interfering wavefields are different and their period is also different, but they have in common the fact that they result from interference between wavefields belonging to different branches of the dispersion surface.

Interference at the origin of the Pendellösung fringes in the case of an incident spherical wave. (a) Direct space; (b) reciprocal space. 
Kato has shown that the intensity distribution at any point at the base of the Borrmann triangle is proportional to where and and are the distances of p from the sides AB and AC of the Borrmann triangle (Fig. 5.1.8.4). The equalintensity fringes are therefore located along the locus of the points in the triangle for which the product of the distances to the sides is constant, that is hyperbolas having AB and AC as asymptotes (Fig. 5.1.8.4b). These fringes can be observed on a section topograph of a wedgeshaped crystal (Fig. 5.1.8.5). The technique of section topography is described in IT C, Section 2.7.2.2 . The Pendellösung distance depends on the polarization state [see equation (5.1.3.8)]. If the incident wave is unpolarized, one observes the superposition of the Pendellösung fringes corresponding to the two states of polarization, parallel and perpendicular to the plane of incidence. This results in a beat effect, which is clearly visible in Fig. 5.1.8.5.
Appendix A5.1.1
Under the influence of the incident electromagnetic radiation, the medium becomes polarized. The dielectric susceptibility, which relates this polarization to the electric field, thus characterizes the interaction of the medium and the electromagnetic wave. The classical derivation of the dielectric susceptibility, χ, which is summarized here is only valid for a very high frequency which is also far from an absorption edge. Let us consider an electromagnetic wave, incident on a bound electron. The electron behaves as if it were held by a spring with a linear restoring force and is an oscillator with a resonant frequency . The equation of its motion is written in the following way: where the driving force F is due to the electric field of the wave and is equal to . The magnetic interaction is neglected here.
The solution of the equation of motion is The resonant frequencies of the electrons in atoms are of the order of the ultraviolet frequencies and are therefore much smaller than Xray frequencies. They can be neglected and the expression of the amplitude of the electron reduces to The dipolar moment is therefore
von Laue assumes that the negative charge is distributed continuously all over space and that the charge of a volume element is , where ρ is the electronic density. The electric moment of the volume element is The polarization is equal to the moment per unit volume: It is related to the electric field and electric displacement through We finally obtain the expression of the dielectric susceptibility, where is the classical radius of the electron.
The electromagnetic field is represented by two vectors, E and B, which are the electric field and the magnetic induction, respectively. To describe the interaction of the field with matter, three other vectors must be taken into account, the electrical current density, j, the electric displacement, D, and the magnetic field, H. The space and time derivatives of these vectors are related in a continuous medium by Maxwell's equations: where ρ is the electric charge density.
The electric field and the electric displacement on the one hand, and the magnetic field and the magnetic induction on the other hand, are related by the socalled material relations, which describe the reaction of the medium to the electromagnetic field: where and μ are the dielectric constant and the magnetic permeability, respectively.
These equations are complemented by the following boundary conditions at the surface between two neighbouring media:
From the second and the third equations of (A5.1.1.3), and using the identity it follows that
In a vacuum, ρ and j are equal to zero, and the first two equations of (A5.1.1.3) can be written where and are the dielectric constant and the magnetic permeability of a vacuum, respectively.
By taking the curl of both sides of the second equation, it follows that Using the identity , the relation , where c is the velocity of light, and noting that , one finally obtains the equation of propagation of an electromagnetic wave in a vacuum: Its simplest solution is a plane wave: of which the wavenumber and the frequency ν are related by The basic properties of the electromagnetic field are described, for instance, in Born & Wolf (1983).
The propagation equation of Xrays in a crystalline medium is derived following von Laue (1960). The interaction of Xrays with charged particles is inversely proportional to the mass of the particle and the interaction with the nuclei can be neglected. As a first approximation, it is also assumed that the magnetic interaction of Xrays with matter is neglected, and that the magnetic permeability μ can be taken as equal to the magnetic permeability of a vacuum, . It is further assumed that the negative and positive charges are both continuously distributed and compensate each other in such a way that there is neutrality and no current everywhere: ρ and j are equal to zero and div D is therefore also equal to zero. The electric displacement is related to the electric field by (A5.1.1.1) and the electric part of the interaction of Xrays with matter is expressed through the dielectric susceptibility χ, which is given by (A5.1.1.2). This quantity is proportional to the electron density and varies with the space coordinates. It is therefore concluded that div E is different from zero, as opposed to what happens in a vacuum. For this reason, the propagation equation of Xrays in a crystalline medium is expressed in terms of the electric displacement rather than in terms of the electric field. It is obtained by eliminating H, B and E in Maxwell's equations and taking into account the above assumptions: Only coherent scattering is taken into account here, that is, scattering without frequency change. The solution is therefore a wave of the form By replacing D with this expression in equation (A5.1.1.7), one finally obtains the propagation equation (5.1.2.2) (Section 5.1.2.1).
In a crystalline medium, χ is a triply periodic function of the space coordinates and the solutions of this equation are given in terms of Fourier series which can be interpreted as sums of electromagnetic plane waves. Each of these waves is characterized by its wavevector, , its electric displacement, , its electric field, , and its magnetic field, . It can be shown that, as a consequence of the fact that and is a transverse wave ( and and are mutually orthogonal) while is not. The electric displacement is therefore a more suitable vector for describing the state of the field inside the crystal than the electric field.
The propagation direction of the energy of an electromagnetic wave is given by that of the Poynting vector defined by (see Born & Wolf, 1983) where means real part of ().
The intensity I of the radiation is equal to the energy crossing unit area per second in the direction normal to that area. It is given by the value of the Poynting vector averaged over a period of time long compared with :
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