Solutions of plane-wave dynamical theory

Authier, A.

doi:10.1107/97809553602060000569

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

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International Tables for Crystallography (2006). Vol. B. ch. 5.1, pp. 538-541 | 1 | 2 |

Section 5.1.3. Solutions of plane-wave dynamical theory

A. Authier^a ^*

^a Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.3. Solutions of plane-wave dynamical theory

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5.1.3.1. Departure from Bragg's law of the incident wave

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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 $[k = 1/\lambda]$ . Depending on whether they propagate in the incident or in the reflected direction, the common extremity, M, of their wavevectors $[{\bf OM} = {\bf K}_{{\bf o}}^{(a)} \hbox{ and } {\bf HM} = {\bf K}_{{\bf h}}^{(a)}]$ 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 $[T'_{o}]$ and $[T'_{h}]$ at their intersection point, $[L_{a}]$ , or Laue point (Fig. 5.1.3.1).

Figure 5.1.3.1| top | pdf |

Departure from Bragg's law of an incident wave.

Bragg's condition is exactly satisfied according to the geometrical theory of diffraction when M lies at $[L_{a}]$ . The departure Δθ from Bragg's incidence of an incident wave is defined as the angle between the corresponding wavevectors OM and $[{\bf OL}_{a}]$ . As Δθ is very small compared to the Bragg angle in the general case of X-rays or neutrons, one may write $[\eqalign{{\bf K}_{{\bf o}}^{(a)} &= {\bf OM} = {\bf OL}_{a} + {\bf L}_{a}{\bf M},\cr \Delta \theta &= \overline{L_{a}M} / k.} \eqno(5.1.3.1)]$

The tangent $[T'_{o}]$ is oriented in such a way that Δθ is negative when the angle of incidence is smaller than the Bragg angle.

5.1.3.2. Transmission and reflection geometries

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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:

(a) Transmission, or Laue case (Fig. 5.1.3.2). The normal to the crystal surface drawn from M intersects both branches of the dispersion surface (Fig. 5.1.3.2a). The reflected wave is then directed towards the inside of the crystal (Fig. 5.1.3.2b). Let $[\gamma_{o}]$ and $[\gamma_{h}]$ be the cosines of the angles between the normal to the crystal surface, n, and the incident and reflected directions, respectively: $[\gamma_{o} = \cos ({\bf n},{\bf s}_{{\bf o}})\hbox{;} \quad \gamma_{h} = \cos ({\bf n},{\bf s}_{{\bf h}}). \eqno(5.1.3.2)]$ It will be noted that they are both positive, as is their ratio, $[\gamma = \gamma_{h}/\gamma_{o}. \eqno(5.1.3.3)]$ This is the asymmetry ratio, which is very important since the width of the rocking curve is proportional to its square root [equation (5.1.3.6)].

Figure 5.1.3.2| top | pdf |
Transmission, or Laue, geometry. (a) Reciprocal space; (b) direct space.
(b) Reflection, or Bragg case (Fig. 5.1.3.3). In this case there are three possible situations: the normal to the crystal surface drawn from M intersects either branch 1 or branch 2 of the dispersion surface, or the intersection points are imaginary (Fig. 5.1.3.3a). The reflected wave is directed towards the outside of the crystal (Fig. 5.1.3.3b). The cosines defined by (5.1.3.2) are now positive for $[\gamma_{o}]$ and negative for $[\gamma_{h}]$ . The asymmetry factor is therefore also negative.

Figure 5.1.3.3| top | pdf |
Reflection, or Bragg, geometry. (a) Reciprocal space; (b) direct space.

5.1.3.3. Middle of the reflection domain

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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, $[L_{o}]$ , with $[T'_{o}]$ (Figs. 5.1.3.4 and 5.1.3.5), while, according to Bragg's law, it should be $[{\bf OL}_{a}]$ . The angle Δθ between the incident wavevectors $[{\bf OL}_{a}]$ and OI, corresponding to the middle of the reflecting domain according to the geometrical and dynamical theories, respectively, is $[\Delta \theta_{o} = \overline{L_{a}I}/k = R\lambda^{2} F_{o}(1 - \gamma)/(2\pi V \sin 2\theta). \eqno(5.1.3.4)]$

Figure 5.1.3.4| top | pdf |

Boundary conditions at the entrance surface for transmission geometry.

Figure 5.1.3.5| top | pdf |

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 $[\Delta \theta_{o}]$ 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, $[\Delta \theta_{o}]$ is equal to zero for symmetric reflections $[(\gamma = 1)]$ .

5.1.3.4. Deviation parameter

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The solutions of dynamical theory are best described by introducing a reduced parameter called the deviation parameter, $[\eta = (\Delta \theta - \Delta \theta_{o})/\delta, \eqno(5.1.3.5)]$ where $[\delta = R\lambda^{2}|C|(|\gamma |F_{h}F_{\bar{h}})^{1/2}/(\pi V \sin 2\theta), \eqno(5.1.3.6)]$ 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, $[\eta, \Delta \theta_{o}]$ and δ are complex, and it is the real part, $[\Delta \theta_{or}]$ , of $[\Delta \theta_{o}]$ which has the geometrical interpretation given in Section 5.1.3.3. One obtains $[\eqalign{\eta &= \eta_{r} + i\eta_{i}\cr \eta_{r} &= (\Delta \theta - \Delta \theta_{or})/\delta_{r}\hbox{;} \ \eta_{i} = A\eta_{r} + B\cr A &= - \tan \beta\cr B &= \left\{\chi_{io}/\left[|C|(|\chi_{h} \chi_{\bar{h}}|)^{1/2} \cos \beta\right]\right\} (1 - \gamma)/2(|\gamma |)^{1/2},} \eqno(5.1.3.7)]$ where β is the phase angle of $[(\chi_{h} \chi_{\bar{h}})^{1/2}]$ [or that of $[(F_{h} F_{\bar{h}})^{1/2}]$ ].

5.1.3.5. Pendellösung and extinction distances

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Let $[\Lambda_{o} = \pi V(\gamma_{o} |\gamma_{h}|)^{1/2}\big /[R\lambda |C|(F_{h} F_{\bar{h}})^{1/2}]. \eqno(5.1.3.8)]$

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); $[\Lambda_{o}]$ in this case is called the Pendellösung distance, denoted $[\Lambda_{L}]$ hereafter. Its geometrical interpretation, in the zero-absorption case, is the inverse of the diameter $[A_{2}A_{1}]$ of the dispersion surface in a direction defined by the cosines $[\gamma_{h}]$ and $[\gamma_{o}]$ with respect to the reflected and incident directions, respectively (Fig. 5.1.3.4). It reduces to the inverse of $[A_{o2}A_{o1}]$ (5.1.2.23) in the symmetric case.

In reflection geometry, it gives the absorption distance in the total-reflection domain and is called the extinction distance, denoted $[\Lambda_{B}]$ (see Section 5.1.7.1). Its geometrical interpretation in the zero-absorption case is the inverse of the length $[I_{o1}I_{h1} = I_{h2}I_{o2}]$ , 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 $[\Lambda_{o}]$ , 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: $[\eqalign{\eta &= (\Delta \theta - \Delta \theta_{o}) \Lambda_{o} \sin 2\theta / (\lambda |\gamma_{h}|),\cr \delta &= \lambda |\gamma_{h}| / (\Lambda_{o} \sin 2\theta).} \eqno(5.1.3.9)]$

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.

5.1.3.6. Solution of the dynamical theory

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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 $[\eqalign{\xi_{j} &= D_{hj}/D_{oj}\cr &= -\hbox{S}(C)\hbox{S} (\gamma_{h}) [(F_{h} F_{\bar{h}})^{1/2} / F_{\bar{h}}]\cr &\quad \times \left\{\eta \pm \left[\eta^{2} + \hbox{S} (\gamma_{h})\right]^{1/2}\right\} / (|\gamma |)^{1/2},} \eqno(5.1.3.10)]$ 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 $[\hbox{S}(\gamma_{h})]$ is the sign of $[\gamma_{h}]$ (+1 in transmission geometry, −1 in reflection geometry).

5.1.3.7. Geometrical interpretation of the solution in the zero-absorption case

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5.1.3.7.1. Transmission geometry

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In this case (Fig. 5.1.3.4) $[\hbox{S}(\gamma_{h})]$ is +1 and (5.1.3.10) may be written $[\xi_{j} = -\hbox{S}(C) \left[\eta \pm (\eta^{2} + 1)^{1/2}\right] / \gamma^{1/2}. \eqno(5.1.3.11)]$

Let $[A_{1}]$ and $[A_{2}]$ be the intersections of the normal to the crystal surface drawn from the Lorentz point $[L_{o}]$ 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 $[\eta = 0]$ and correspond to the middle of the reflection domain. Let us further consider the tangents to the dispersion surface at $[A_{1}]$ and $[A_{2}]$ and let $[I_{o1}]$ , $[I_{o2}]$ and $[I_{h1}]$ , $[I_{h2}]$ be their intersections with $[T_{o}]$ and $[T_{h}]$ , respectively. It can be shown that $[\overline{I_{o1}I_{h2}}]$ and $[\overline{I_{o2}I_{h1}}]$ intersect the dispersion surface at the tie points excited for $[\eta = -1]$ and $[\eta = +1]$ , respectively, and that the Pendellösung distance $[\Lambda_{L} = 1/\overline{A_{2}A_{1}}]$ , the width of the rocking curve $[2\delta = \overline{I_{o1}I_{o2}}/k]$ and the deviation parameter $[\eta = \overline{M_{o}M_{h}}/\overline{A_{2}A_{1}}]$ , where $[M_{o}]$ and $[M_{h}]$ are the intersections of the normal to the crystal surface drawn from the extremity of any incident wavevector OM with $[T_{o}]$ and $[T_{h}]$ , respectively.

5.1.3.7.2. Reflection geometry

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In this case (Fig. 5.1.3.5) $[\hbox{S}(\gamma_{h})]$ is now −1 and (5.1.3.10) may be written $[\xi_{j} = \hbox{S}(C) \left[\eta \pm (\eta^{2} - 1)^{1/2}\right] / |\gamma|^{1/2}. \eqno(5.1.3.12)]$

Now, let $[I_{1}]$ and $[I_{2}]$ be the points of the dispersion surface where the tangent is parallel to the normal to the crystal surface, and further let $[I_{o1}, I_{h1}, I']$ and $[I_{o2}, I_{h2}, I'']$ be the intersections of these two tangents with $[T_{o}, T_{h}]$ and $[T'_{o}]$ , 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 $[\overline{I'I''}]$ defines the total-reflection domain. The tie points $[I_{1}]$ and $[I_{2}]$ correspond to $[\eta = -1]$ and $[\eta = +1]$ , respectively, the extinction distance $[\Lambda_{L} = 1/\overline{I_{o1}I_{h1}}]$ , the width of the total-reflection domain $[2\delta = \overline{I_{o1}I_{o2}}/k = \overline{I'I''}/k]$ and the deviation parameter $[\eta = - \overline{M_{o}M_{h}} / \overline{I_{o1}I_{h1}}]$ , where $[M_{o}]$ and $[M_{h}]$ are the intersections with $[T_{o}]$ and $[T_{h}]$ of the normal to the crystal surface drawn from the extremity of any incident wavevector OM.

References

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International Tables for Crystallography (2006). Vol. B. ch. 5.1, pp. 538-541