Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 545-548   | 1 | 2 |

Section 5.1.7. Intensity of plane waves in reflection geometry

A. Authiera*

aLaboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail:

5.1.7. Intensity of plane waves in reflection geometry

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Rocking curve. The geometrical construction in Fig.[link] 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, [P_{1}, P'_{1}] , for branch 1, [P_{2}, P'_{2}] for branch 2, or at imaginary points. It was shown in Section[link] that the propagation of the wavefields inside the crystal is along the normal to the dispersion surface at the corresponding tie points. Fig.[link] shows that this direction is oriented towards the outside of the crystal for tie points [P'_{1}] and [P'_{2}]. In a very thick crystal, these wavefields cannot exist because there is always a small amount of absorption. One concludes that in the thick-crystal case and in reflection geometry, only one wavefield is excited inside the crystal. It corresponds to branch 1 on the low-angle side of the rocking curve and to branch 2 on the high-angle side. Using the same approximations as in Section[link], the amplitude [{\bf D}_{{\bf h}}^{(a)}] of the wave reflected at the crystal surface is obtained by applying the boundary conditions, which are particularly simple in this case: [{\bf D}_{{\bf o}} = {\bf D}_{{\bf o}}^{(a)}, \ {\bf D}_{{\bf h}}^{(a)} = {\bf D}_{{\bf h}}.]

The reflecting power is given by an expression similar to ([link]: [I_{h} = |\gamma| |\xi_{j}|^{2},] where the expression for [\xi_{j}] is given by ([link], 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 [-1 \;\lt\; \eta \;\lt\; +1], [|\xi|^{2} = |\gamma|^{-1}, \ I_{h} = 1, \eqno(] and there is total reflection. Outside the total-reflection domain, the reflecting power is given by [I_{h} = \left[|\eta| - (\eta^{2} - 1)^{1/2}\right]^{2}. \eqno(]

The rocking curve has the well known top-hat shape (Fig.[link]). Far from the total-reflection domain, the curve can be approximated by the function [I_{h} \approx 1/(4\eta^{2}).]


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Theoretical rocking curve in the reflection case for a non-absorbing thick crystal in terms of the deviation parameter.

Width of the total-reflection domain. The width of the total-reflection domain is equal to [\Delta \eta = 2] and its angular width is therefore equal, using ([link], to 2δ, where δ is given by ([link]. It is proportional to the structure factor, the polarization factor C and the square root of the asymmetry factor [|\gamma |]. 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)[link]]. It is possible to deduce the value of the form factor from very accurate measurements of the rocking curve; see, for instance, Kikuta (1971)[link].

Integrated intensity. The integrated intensity is defined by ([link]: [I_{hi} = 8\delta / 3. \eqno(]

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 [\mu = -4\pi K_{oi} \gamma_{o} = 2\pi \gamma_{o} (1 - \eta^{2})^{1/2} / \Lambda_{B}, \eqno(] where [\Lambda_{B}] is the value taken by [\Lambda_{o}] [equation ([link]] 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 [\Lambda_{B}/2\pi]. This attenuation effect is called extinction, and [\Lambda_{B}] 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)[link]]. Absorbing crystals

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Rocking curve. Since the sign of γ is negative, [[\eta^{2} + \hbox{S}(\gamma h)]^{1/2}] in ([link] has a very large imaginary part when [|\eta_{r}| \leq 1]. It cannot be calculated using the same approximations as in the Laue case. Let us set [Z \exp (i\Psi') = \eta \mp (\eta^{2} - 1)^{1/2}. \eqno(] The reflecting power is [I_{h} = (F_{h}/F_{\bar{h}})^{1/2} Z^{2}, \eqno(] where [Z = [L - (L^{2} - 1)^{1/2}]^{1/2}, L = |\eta|^{2} + \rho^{2}] and [\rho = |\eta^{2} - 1|] is the modulus of expression ([link] 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)[link]. The way of representing it given here was first used by Hirsch & Ramachandran (1950)[link]. The properties of the rocking curve have been described by Fingerland (1971)[link].

There is no longer a total-reflection domain and energy penetrates the crystal at all incidence angles, although with a very high absorption coefficient within the domain [|\eta_{r}| \leq 1]. Fig.[link] gives an example of a rocking curve for a thick absorbing crystal. It was first observed by Renninger (1955)[link]. The shape is asymmetric and is due to the anomalous-absorption effect: it is lower than normal on the low-angle side, which is associated with wavefields belonging to branch 1 of the dispersion surface, and larger than normal on the high-angle side, which is associated with branch 2 wavefields. The amount of asymmetry depends on the value of the ratio [A/B] of the coefficients in the expression for the imaginary part of the deviation parameter in ([link]: the smaller this ratio, the more important the asymmetry.


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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[link]) [\mu = \mu_{o} + 2(|F_{h}F_{\bar{h}}|)^{1/2} {R \lambda \over V(|\gamma|)^{1/2}} Z \sin (\beta + \Psi'),] where β is defined in equation ([link] and Ψ′ in equation ([link], and where the sign is chosen in such a way that Z converges. Fig.[link] shows the variation of the penetration depth [z_{o} = \gamma_{o} / \mu] with the deviation parameter.


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Bragg case: thick crystals. Variation of the penetration depth with incidence angle (represented here by the dimensionless deviation parameter η). Thin curve: without absorption; thick curve: with absorption for the 400 reflection of GaAs using Cu Kα radiation. Thin crystals

| top | pdf | Non-absorbing crystals

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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.[link]). 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:


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Bragg case: thin crystals. Two wavefields propagate in the crystal. (a) Direct space; (b) reciprocal space.

  • (i) a partially transmitted wave outside the crystal, [{\bf D}_{{\bf o}}^{(d)} \exp (-2\pi i {\bf K}_{{\bf o}}^{(d)} \cdot {\bf r})];

  • (ii) a partially reflected wavefield inside the crystal.

The corresponding tiepoints are obtained by applying the usual condition of the continuity of the tangential components of wavevectors (Fig.[link]). If the crystal is a plane-parallel slab, these points are M and [P_{2}], respectively, and [{\bf K}_{{\bf o}}^{(d)} = {\bf K}_{{\bf o}}^{(a)}].

The boundary conditions are then written:

  • (i) entrance surface: [D_{o1} + D_{o2} = D_{o}^{(a)}, \;\; D_{h1} + D_{h2} = D_{h}^{(a)}\hbox{;}]

  • (ii) back surface: [\eqalign{D_{o1} \exp (-2\pi i {\bf K}_{{\bf o}1} \cdot {\bf r}) + D_{o2} \exp (-2\pi i {\bf K}_{{\bf o}2} \cdot {\bf r}) &= D_{o}^{(a)} \exp \big(-2\pi i {\bf K}_{{\bf o}}^{(d)} \cdot {\bf r}\big),\cr D_{h1} \exp (-2\pi i {\bf K}_{{\bf h}1} \cdot {\bf r}) + D_{h2} \exp (-2\pi i {\bf K}_{{\bf h}2} \cdot {\bf r}) &= 0.}]

Rocking curve. Using ([link], it can be shown that the expressions for the intensities reflected at the entrance surface and transmitted at the back surface, [I_{h}] and [I_{o}], respectively, are given by different expressions within total reflection and outside it:

  • (i) [|\eta| \;\lt\; 1]: [\eqalign{I_{h} &= |\gamma | {|D_{h}^{(a)}|^{2} \over |D_{o}^{(a)}|^{2}} = {\cosh^{2} [\pi (t/\Lambda_{B})(1 - \eta^{2})^{1/2}] - 1 \over \cosh^{2} [\pi (t/\Lambda_{B})(1 - \eta^{2})^{1/2}] - \eta^{2}}\cr I_{o} &= {|D_{o}^{(d)}|^{2} \over |D_{o}^{(a)}|^{2}} = {1 - \eta^{2} \over \cosh^{2} [\pi (t/\Lambda_{B})(1 - \eta^{2})^{1/2}] - \eta^{2}},} \eqno(] where [\Lambda_{B}] is the value taken by [\Lambda_{o}] [equation ([link]] in the Bragg case.

    There is no longer a total-reflection domain but the extinction effect still exists, as is shown by the hyperbolic cosine term. The maximum height of the rocking curve decreases as the thickness of the crystal decreases.

  • (ii) [|\eta | \;\gt\; 1]: [\eqalign{I_{h} &= {1 - \cos^{2} [\pi (t/\Lambda_{B})(\eta^{2} - 1)^{1/2}] \over \eta^{2} - \cos^{2} [\pi (t/\Lambda_{B})(\eta^{2} - 1)^{1/2}]},\cr I_{o} &= {\eta^{2} - 1 \over \eta^{2} - \cos^{2} [\pi (t/\Lambda_{B})(\eta^{2} - 1)^{1/2}]}.} \eqno(]

    The cosine terms show that the two wavefields propagating within the crystal interfere, giving rise to Pendellösung fringes in the rocking curve. These fringes were observed for the first time by Batterman & Hildebrandt (1967[link], 1968[link]). The angular positions of the minima of the reflected beam are given by [\eta = \mp (K^{2}\Lambda_{B}^{2}t^{-2} + 1)^{1/2},] where K is an integer.

Integrated intensity. The integrated intensity is [I_{hi} = \pi \delta \tanh [\pi t/\Lambda_{B}], \eqno(] where t is the crystal thickness. When this thickness becomes very large, the integrated intensity tends towards [I_{hi} = \pi \delta. \eqno(]

This expression differs from ([link] by the factor π, which appears here in place of [8/3]. von Laue (1960)[link] pointed out that because of the differences between the two expressions for the reflecting power, ([link] and ([link], perfect agreement could not be expected. Since some absorption is always present, expression ([link], which includes the factor [8/3], should be used for very thick crystals. In the presence of absorption, however, expression ([link] for the reflected intensity for thin crystals does tend towards that for thick crystals as the crystal thickness increases.

Comparison with geometrical theory. When [t/\Lambda_{B}] is very small (thin crystals or weak reflections), ([link] tends towards [I_{hi} = R^{2}\lambda^{2}t |F_{h}|^{2}\big/(V^{2} \gamma_{o} \sin 2\theta), \eqno(] which is the expression given by geometrical theory. If we call this intensity [I_{hi}](geom.), comparison of expressions ([link] and ([link] shows that the integrated intensity for crystals of intermediate thickness can be written [I_{hi} = I_{hi}\hbox{(geom.)} {\tanh (\pi t/\Lambda_{B}) \over (\pi t/\Lambda_{B})}, \eqno(] which is the expression given by Darwin (1922)[link] for primary extinction. Absorbing crystals

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Reflected intensity. The intensity of the reflected wave for a thin absorbing crystal is [\eqalign{I_{h} &= |\gamma | \left|{D_{h}^{(a)} \over D_{o}^{(a)}}\right|^{2}\cr &= \left|{F_{h} \over F_{\bar{h}}}\right| {\cosh 2b - \cos 2a \over L\cosh 2b + (L^{2} - 1)^{1/2}\sinh 2b - \cos (2a + 2\Psi')},} \eqno(] where [\eqalign{2a &= [\pi t/\Lambda_{B}] \rho \cos (\beta + \omega),\cr 2b &= [\pi t/\Lambda_{B}] \rho \sin (\beta + \omega)}.] L, ρ and ψ′ are defined in ([link], β is defined in ([link] and ω is the phase angle of [(\eta^{2} - 1)^{1/2}].

Comparison with geometrical theory. When [t / \Lambda_{B}] decreases towards zero, expression ([link] tends towards [[\sin (\pi t \eta / \Lambda_{B}) / \eta]^{2}]; using ([link] and ([link], it can be shown that expression ([link] can be written, in the non-absorbing symmetric case, as [I_{h} = {R^{2} \lambda^{2} C^{2} |F_{h}|^{2} t^{2} \over V^{2} \sin^{2} \theta} \left\{{\sin [2\pi k \cos (\theta) t \Delta \theta] \over [2\pi k \cos (\theta) t \Delta \theta]}\right\}^{2}, \eqno(] 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)[link]].


Authier, A. (1986). Angular dependence of the absorption induced nodal plane shifts of X-ray stationary waves. Acta Cryst. A42, 414–426.
Batterman, B. W. & Hildebrandt, G. (1967). Observation of X-ray Pendellösung fringes in Darwin reflection. Phys. Status Solidi, 23, K147–K149.
Batterman, B. W. & Hildebrandt, G. (1968). X-ray Pendellösung fringes in Darwin reflection. Acta Cryst. A24, 150–157.
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Darwin, C. G. (1922). The reflection of X-rays from imperfect crystals. Philos. Mag. 43, 800–829.
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