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. 634638
Section 5.1.6. Intensities of plane waves in transmission geometry^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France 
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 iswhere 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 displacementwhere 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 aretop 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 wavevectorslie 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 conditionswhere 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 towhere t is the crystal thickness.
In a nonabsorbing crystal, the amplitudes squared are of the formThis 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 arewhereand 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) giveswhere is the zerothorder Bessel function andFig. 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 writtenwhereandWhen is very small, tends asymptotically towards the functionand 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)].
References
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Bonse, U. & Teworte, R. (1980). Measurement of Xray scattering factors of Si from the fine structure of Laue case rocking curves. J. Appl. Cryst. 13, 410–416.
Kato, N. (1955). Integrated intensities of the diffracted and transmitted Xrays due to ideally perfect crystal. J. Phys. Soc. Jpn, 10, 46–55.
Laue, M. von (1960). RöntgenstrahlInterferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.
LefeldSosnowska, M. & Malgrange, C. (1968). Observation of oscillations in rocking curves of the Laue reflected and refracted beams from thin Si single crystals. Phys. Status Solidi, 30, K23–K25.
LefeldSosnowska, M. & Malgrange, C. (1969). Experimental evidence of planewave rocking curve oscillations. Phys. Status Solidi, 34, 635–647.
Ludewig, J. (1969). Debye–Waller factor and anomalous absorption (Ge; 293–5 K). Acta Cryst. A25, 116–118.
Ohtsuki, Y. H. (1964). Temperature dependence of Xray absorption by crystals. I. Photoelectric absorption. J. Phys. Soc. Jpn, 19, 2285–2292.
Ohtsuki, Y. H. (1965). Temperature dependence of Xray absorption by crystals. II. Direct phonon absorption. J. Phys. Soc. Jpn, 20, 374–380.