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. 638640
Section 5.1.7. Intensity of plane waves in reflection geometry^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France 
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 iswhere 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 setThe reflecting power iswhere 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 iswhere 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 towardswhich 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 writtenwhich is the expression given by Darwin (1922) for primary extinction.
Reflected intensity. The intensity of the reflected wave for a thin absorbing crystal iswhereL, 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, aswhere 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)].
References
Authier, A. (1986). Angular dependence of the absorption induced nodal plane shifts of Xray stationary waves. Acta Cryst. A42, 414–426.Batterman, B. W. & Hildebrandt, G. (1967). Observation of Xray Pendellösung fringes in Darwin reflection. Phys. Status Solidi, 23, K147–K149.
Batterman, B. W. & Hildebrandt, G. (1968). Xray Pendellösung fringes in Darwin reflection. Acta Cryst. A24, 150–157.
Cowan, P. L., Brennan, S., Jach, T., Bedzyk, M. J. & Materlik, G. (1986). Observations of the diffraction of evanescent Xrays at a crystal surface. Phys. Rev. Lett. 57, 2399–2402.
Darwin, C. G. (1922). The reflection of Xrays from imperfect crystals. Philos. Mag. 43, 800–829.
Fingerland, A. (1971). Some properties of the single crystal rocking curve in the Bragg case. Acta Cryst. A27, 280–284.
Hirsch, P. B. & Ramachandran, G. N. (1950). Intensity of Xray reflection from perfect and mosaic absorbing crystals. Acta Cryst. 3, 187–194.
James, R. W. (1950). The Optical Principles of the Diffraction of Xrays. London: G. Bell and Sons Ltd.
Kikuta, S. (1971). Determination of structure factors of Xrays using halfwidths of the Bragg diffraction curves from perfect single crystals. Phys. Status Solidi B, 45, 333–341.
Kikuta, S. & Kohra, K. (1970). Xray collimators using successive asymmetric diffractions and their applications to measurements of diffraction curves. I. General considerations on collimators. J. Phys. Soc. Jpn, 29, 1322–1328.
Laue, M. von (1960). RöntgenstrahlInterferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.
Prins, J. A. (1930). Die Reflexion von Röntgenstrahlen an absorbierenden idealen Kristallen. Z. Phys. 63, 477–493.
Renninger, M. (1955). Messungen zur RöntgenstrahlOptik des Idealkristalls. I. Bestätigung der Darwin–Ewald–Prins–KohlerKurve. Acta Cryst. 8, 597–606.