International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.1, pp. 634-638   | 1 | 2 |

Section 5.1.6. Intensities of plane waves in transmission geometry

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: AAuthier@wanadoo.fr

5.1.6. Intensities of plane waves in transmission geometry

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5.1.6.1. Absorption coefficient

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In transmission geometry, the imaginary part of [X_{o}] is small and, using a first-order approximation for the expansion of [(\eta^{2} + 1)^{1/2}], (5.1.5.1)[link] and (5.1.5.2)[link], the effective absorption coefficient in the absorption case is[\eqalignno{\mu_{j} &= \mu_{o} \bigg[\textstyle{{1 \over 2}} (1 + \gamma^{-1}) \cr &\quad \mp {(\eta_{\rm r}/2) (1 - \gamma^{-1}) + |C|(\gamma^{-1})^{1/2} |F_{{\rm i}h}/F_{{\rm i}o}| \cos \varphi \over (\eta_{\rm r}^{2} + 1)^{1/2}}\bigg], &(5.1.6.1)}]where [\varphi = \varphi_{{\rm r}h} - \varphi_{{\rm i}h}] is the phase difference between [F_{{\rm r}h}] and [F_{{\rm i}h}] [equation (5.1.2.10)[link]], 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 ([\gamma = 1], reflecting planes normal to the crystal surface), equation (5.1.6.1)[link] reduces to[\mu_{j} = \mu_{o} \left[1 \mp {|C| |F_{{\rm i}h}/F_{{\rm i}o}| \cos \varphi \over (\eta_{\rm r}^{2} + 1)^{1/2}}\right].]

Fig. 5.1.6.1[link] shows the variations of the effective absorption coefficient [\mu_{j}] with [\eta_{\rm r}] 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 [\eta_{\rm r} = 0] the absorption coefficient for branch 1 becomes significantly smaller than the normal absorption coefficient, [\mu_{o}]. The minimum absorption coefficient, [\mu_{o} (1 - |CF_{{\rm i}h}/F_{{\rm i}o}| \cos \varphi)], depends on the nature of the reflection through the structure factor and on the temperature through the Debye–Waller factor included in [F_{{\rm i}h}] [equation (5.1.2.10b)[link]] (Ohtsuki, 1964[link], 1965[link]). For instance, in diamond-type structures, it is smaller for reflections with even indices than for reflections with odd indices. The influence of temperature is very important when [|F_{{\rm i}h}/F_{{\rm i}o}|] 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, [\mu_{o}] (Ludewig, 1969[link]).

[Figure 5.1.6.1]

Figure 5.1.6.1 | top | pdf |

Variation of the effective absorption with the deviation parameter in the transmission case for the 400 reflection of GaAs using Cu Kα radiation. Solid curve: branch 1; broken curve: branch 2.

5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves

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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[link], two wavefields are excited in the crystal, with tie points [P_{1}] and [P_{2}], and amplitudes [D_{o1}, D_{h1}] and [D_{o2}, D_{h2}], respectively. Maxwell's boundary conditions (see Section A5.1.1.2[link] 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[\eqalign{D_{o}^{(a)} &= D_{o1} + D_{o2}\cr 0 &= D_{h1} + D_{h2},} \eqno(5.1.6.2)]where [D_{o}^{(a)}] is the amplitude of the incident wave.

Using (5.1.3.11)[link], (5.1.5.2)[link] and (5.1.6.2)[link], it can be shown that the intensities of the four waves are[\eqalign{|D_{oj}|^{2} &= |D_{o}^{(a)}|^{2} \exp (- \mu_{j}z/\gamma_{o}) \left[(1 + \eta_{\rm r}^{2})^{1/2} \mp \eta_{\rm r}\right]^{2}\cr &\quad \times [4 (1 + \eta_{\rm r}^{2})]^{-1},\cr |D_{hj}|^{2} &= |D_{o}^{(a)}|^{2} \exp (- \mu_{j}z/\gamma_{o}) |F_{h}/F_{\bar{h}}| [4\gamma (1 + \eta_{\rm r}^{2})]^{-1}\semi}\eqno(5.1.6.3)]top sign: [j = 1]; bottom sign: [j = 2].

Fig. 5.1.6.2[link] represents the variations of these four intensities with the deviation parameter. Far from the reflection domain, [|D_{h1}|^{2}] and [|D_{h2}|^{2}] tend toward zero, as is normal, while[\eqalign{|D_{o1}|^{2} &\gg |D_{o2}|^{2} \hbox{ for } \eta_{\rm r} \Rightarrow - \infty,\cr |D_{o1}|^{2} &\ll |D_{o2}|^{2} \hbox{ for } \eta_{\rm r} \Rightarrow + \infty.}]

[Figure 5.1.6.2]

Figure 5.1.6.2 | top | pdf |

Variation of the intensities of the reflected and refracted waves in an absorbing crystal for the 220 reflection of Si using Mo Kα radiation, t = 1 mm (μt = 1.42). Solid curve: branch 1; dashed curve: branch 2.

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 so-called interbranch scattering: see, for instance, Authier & Balibar (1970)[link] and Authier & Malgrange (1998)[link]].

5.1.6.3. Boundary conditions at the exit surface

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5.1.6.3.1. Wavevectors

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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, [M_{j}] and [N_{j}], of these wavevectors[{\bf OM}_{j} = {\bf K}_{{\bf o}j}^{(d)}\qquad {\bf HN}_{j} = {\bf K}_{{\bf h}j}^{(d)}]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 [P_{j}] (j = 1 and 2) (Fig. 5.1.6.3[link]).

[Figure 5.1.6.3]

Figure 5.1.6.3 | top | pdf |

Boundary condition for the wavevectors at the exit surface. (a) Reciprocal space. The wavevectors of the emerging waves are determined by the intersections [M_{1}], [M_{2}], [N_{1}] and [N_{2}] of the normals n′ to the exit surface, drawn from the tie points [P_{1}] and [P_{2}] of the wavefields, with the tangents [T'_{o}] and [T'_{h}] to the spheres centred at O and H and of radius k, respectively. (b) Direct space.

If the crystal is wedge-shaped 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.

5.1.6.3.2. Amplitudes – Pendellösung

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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[link]). Their respective amplitudes are given by the boundary conditions[\eqalign{D_{o}^{(d)} \exp (-2\pi i {\bf K}_{{\bf o}}^{(d)} \cdot {\bf r}) &= D_{o1} \exp (-2\pi i {\bf K}_{{{\bf o}1}} \cdot {\bf r})\cr &\quad + D_{o2} \exp (-2\pi i {\bf K}_{{{\bf o}2}} \cdot {\bf r})\cr D_{h}^{(d)} \exp (-2\pi i {\bf K}_{{\bf h}}^{(d)} \cdot {\bf r}) &= D_{h1} \exp (-2\pi i {\bf K}_{{{\bf h}1}} \cdot {\bf r})\cr &\quad + D_{h2} \exp (-2\pi i {\bf K}_{{{\bf h}2}} \cdot {\bf r}),} \eqno(5.1.6.4)]where r is the position vector of a point on the exit surface, the origin of phases being taken at the entrance surface.

[Figure 5.1.6.4]

Figure 5.1.6.4 | top | pdf |

Decomposition of a wavefield into its two components when it reaches the exit surface. [{\bf S}_{1}] and [{\bf S}_{2}] are the Poynting vectors of the two wavefields propagating in the crystal belonging to branches 1 and 2 of the dispersion surface, respectively, and interfering at the exit surface.

In a plane-parallel crystal, (5.1.6.4)[link] reduces to[\eqalign{D_{o}^{(d)} &= D_{o1} \exp \left(-2\pi i \overline{MP_{1}} \cdot t\right) + D_{o2} \exp \left(-2\pi i \overline{MP_{2}} \cdot t\right)\cr D_{h}^{(d)} &= D_{h1} \exp \left(-2\pi i \overline{MP_{1}} \cdot t\right) + D_{h2} \exp \left(-2\pi i \overline{MP_{2}} \cdot t\right),}]where t is the crystal thickness.

In a non-absorbing crystal, the amplitudes squared are of the form[\left|D_{o}^{(d)}\right|^{2} = |D_{o1}|^{2} + |D_{o2}|^{2} + 2 D_{o1} D_{o2} \cos 2\pi \overline{P_{2}P_{1}} t.]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[\Lambda = 1/\overline{P_{2}P_{1}} = \Lambda_{\rm L}/(1 + \eta_{\rm r}^{2})^{1/2}.]

5.1.6.4. Reflecting power

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For an absorbing crystal, the intensities of the reflected and refracted waves are[\eqalign{\left|D_{o}^{(d)}\right|^{2} &= \left|D_{o}^{(a)}\right|^{2} A_{\eta} \Big\{\cosh (2v + \mu_{a} t)\cr &\quad + \cos \left[2\pi t \Lambda^{-1} - 2\eta_{\rm i} (1 + \eta_{\rm r}^{2})^{-1/2}\right]\Big\}\cr \left|D_{h}^{(d)}\right|^{2} &= \left|D_{o}^{(a)}\right|^{2} |F_{h}/F_{\bar{h}}| \gamma^{-1} A_{\eta} \left[\cosh (\mu_{a} t) - \cos (2\pi t\Lambda^{-1})\right],} \eqno(5.1.6.5)]where[\eqalign{ A_{\eta} &= \left[\exp - \mu_{o} t (\gamma_{o}^{-1} + \gamma_{h}^{-1})\right] / 2 (1 + \eta_{\rm r}^{2}),\cr \mu_{a} &= \mu_{j} \Big[1/2 (\gamma_{o}^{-1} - \gamma_{h}^{-1}) \eta_{\rm r}\cr&\quad +\, |C| |F_{{\rm i}h}/F_{{\rm i}o}| \cos \varphi / (\gamma_{o} \gamma_{h})^{1/2}\Big] (1 + \eta_{\rm r}^{2})^{-1/2}, \cr v &= \arg \sinh \eta_{\rm r}}]and [\mu_{j}] is given by equation (5.1.6.1)[link].

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 non-absorbing crystal, these expressions reduce to[\eqalign{\left|D_{o}^{(d)}\right|^{2} &= \left|D_{o}^{(a)}\right|^{2} \left[{1 + 2\eta^{2} + \cos (2\pi t \Lambda^{-1}) \over 2 (1 + \eta_{\rm r}^{2})}\right],\cr \left|D_{h}^{(d)}\right|^{2} &= \left|D_{o}^{(a)}\right|^{2} \left[{1 - \cos (2\pi t\Lambda^{-1}) \over 2\gamma (1 + \eta_{\rm r}^{2})}\right].} \eqno(5.1.6.6)]

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)[link] [l_{o} = l\gamma_{o}] and [l_{h} = l\gamma_{h}], respectively.

[Figure 5.1.6.5]

Figure 5.1.6.5 | top | pdf |

Cross sections of the incident, [{\bf K}_{{\bf o}}^{(a)}], refracted, [{\bf K}_{{\bf o}}], and reflected, [{\bf K}_{{\bf h}}], waves.

The reflecting powers are therefore:[\eqalign{\hbox{Refracted beam: } I_{o} &= l_{o} \left|D_{o}^{(d)}\right|^{2}\Big/l_{o} \left|D_{o}^{(a)}\right|^{2} = \phantom{\gamma} \left|D_{o}^{(d)}\right|^{2}\Big/\left|D_{o}^{(a)}\right|^{2},\cr \hbox{Reflected beam: } I_{h} &= l_{h} \left|D_{h}^{(d)}\right|^{2}\Big/l_{o} \left|D_{o}^{(a)}\right|^{2} = \gamma \left|D_{h}^{(d)}\right|^{2}\Big/\left|D_{o}^{(a)}\right|^{2}.} \eqno(5.1.6.7)]Using (5.1.6.6)[link], it is easy to check that [I_{o} + I_{h} = 1] in the non-absorbing case; that is, that conservation of energy is satisfied. Equations (5.1.6.6)[link] 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 Lefeld-Sosnowska & Malgrange (1968[link], 1969[link]). Their periodicity can be used for accurate measurements of the form factor [see, for instance, Bonse & Teworte (1980)[link]]. Fig. 5.1.6.6[link] shows the shape of the rocking curve for various values of [t/\Lambda_{\rm L}].

[Figure 5.1.6.6]

Figure 5.1.6.6 | top | pdf |

Theoretical rocking curves in the transmission case for non-absorbing crystals and for various values of [t/\Lambda_{\rm L}]: (a) [t/\Lambda_{\rm L} = 1.25]; (b) [t/\Lambda_{\rm L} = 1.5]; (c) [t/\Lambda_{\rm L} = 1.75]; (d) [t/\Lambda_{\rm L} = 2.0].

The width at half-height of the rocking curve, averaged over the Pendellösung oscillations, corresponds in the non-absorbing case to [\Delta \eta = 2], that is, to [\Delta \theta = 2\delta], where δ is given by (5.1.3.6)[link].

5.1.6.5. Integrated intensity

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5.1.6.5.1. Non-absorbing crystals

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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:[I_{h{\rm i}} = {\textstyle\int\limits_{-\infty}^{+\infty}} I_{h} \hbox{ d} (\Delta \theta). \eqno(5.1.6.8)]

The integration was performed by von Laue (1960)[link]. Using (5.1.3.5)[link], (5.1.6.6)[link] and (5.1.6.7)[link] gives[I_{h{\rm i}} = A {\textstyle\int\limits_{0}^{2\pi t\Lambda_{\rm L}^{-1}}} J_{0} (z) \hbox{ d}z,]where [J_{0}(z)] is the zeroth-order Bessel function and[A = {R\lambda^{2} |C F_{h}| (\gamma)^{1/2} \over 2V \sin 2\theta}.]Fig. 5.1.6.7[link] shows the variations of the integrated intensity with [t/\Lambda_{\rm L}].

[Figure 5.1.6.7]

Figure 5.1.6.7 | top | pdf |

Variations with crystal thickness of the integrated intensity in the transmission case (no absorption) (arbitrary units). The expression for A is given in the text.

5.1.6.5.2. Absorbing crystals

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The integration was performed for absorbing crystals by Kato (1955)[link]. The integrated intensity in this case is given by[\eqalign{I_{h{\rm i}} &= A |F_{h}/F_{\bar{h}}| \exp \left[-1/2 \mu_{o} t (\gamma_{o}^{-1} + \gamma_{h}^{-1})\right]\cr &\quad \times \left[\textstyle\int\limits_{0}^{2\pi t\Lambda_{\rm L}^{-1}} J_{0}(z) \hbox{ d}z - 1 + I_{0} (\zeta)\right],}]where[\zeta = \mu_{o} t \left\{\left[\left|C\right|^{2} |F_{{\rm i}h}/F_{{\rm i}o}|^{2} \cos^{2} \varphi + (\gamma_{h} - \gamma_{o})/(4\gamma_{o} \gamma_{h})\right] / (\gamma_{o} \gamma_{h})\right\}^{1/2}]and [I_{0} (\zeta)] is a modified Bessel function of zeroth order.

5.1.6.6. Thin crystals – comparison with geometrical theory

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Using (5.1.6.6)[link] and (5.1.6.7)[link], the reflecting power of the reflected beam may also be written[I_{h} = \pi^{2} t^{2} \Lambda_{o}^{-2} f(\eta),]where[f(\eta) = \left[{\sin U(1 + \eta^{2})^{1/2} \over U(1 + \eta^{2})^{1/2}}\right]^{2}]and[U = \pi t \Lambda_{o}^{-1}.]When [t \Lambda_{o}^{-1}] is very small, [f(\eta)] tends asymptotically towards the function[f_{1}(\eta) = \left[{\sin U\eta \over U\eta}\right]^{2}]and [I_{h}] 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[link])].

References

Authier, A. & Balibar, F. (1970). Création de nouveaux champs d'onde généralisés dus à la présence d'un objet diffractant. II. Cas d'un défaut isolé. Acta Cryst. A26, 647–654.
Authier, A. & Malgrange, C. (1998). Diffraction physics. Acta Cryst. A54, 806–819.
Bonse, U. & Teworte, R. (1980). Measurement of X-ray 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 X-rays due to ideally perfect crystal. J. Phys. Soc. Jpn, 10, 46–55.
Laue, M. von (1960). Röntgenstrahl-Interferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.
Lefeld-Sosnowska, 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.
Lefeld-Sosnowska, M. & Malgrange, C. (1969). Experimental evidence of plane-wave 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 X-ray absorption by crystals. I. Photo-electric absorption. J. Phys. Soc. Jpn, 19, 2285–2292.
Ohtsuki, Y. H. (1965). Temperature dependence of X-ray absorption by crystals. II. Direct phonon absorption. J. Phys. Soc. Jpn, 20, 374–380.








































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