Fault vectors of twin boundaries in merohedral twins

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 433-434

Section 3.3.10.4.2. Fault vectors of twin boundaries in merohedral twins

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.10.4.2. Fault vectors of twin boundaries in merohedral twins

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Twin displacement vectors can occur in twin boundaries of both non-merohedral (see above) and merohedral twins. For merohedral twins, the displacement vector is usually called the `fault vector', because of the close similarity of these twin boundaries with antiphase boundaries and stacking faults (cf. Section 3.3.2.4, Note 7). In contrast to non-merohedral twins, for merohedral twins these displacement vectors can be determined by imaging the twin boundaries by means of electron or X-ray diffraction methods. The essential reason for this possibility is the exact parallelism of the lattices of the two twin partners 1 and 2, so that for any reflection hkl the electron and X-ray diffraction conditions are always simultaneously fulfilled for both partners. Thus, in transmission electron microscopy and X-ray topography, both domains 1 and 2 are simultaneously imaged under the same excitation conditions. By a proper choice of imaging reflections, both twin domains exhibit the same diffracted intensity (no `domain contrast'), and the twin boundary is imaged by fringe contrast analogously to the imaging of stacking faults and antiphase boundaries (`stacking fault contrast').

This contrast results from the `phase jump' of the structure factor upon crossing the boundary. For stacking faults and antiphase boundaries this phase jump is $[2\pi {\bf f} \cdot {\bf g}_{hkl}]$ , with f the fault vector of the boundary and $[{\bf g}_{hkl}]$ the diffraction vector of the reflection used for imaging. For (merohedral) twin boundaries the total phase jump $[\Psi_{hkl}]$ is composed of two parts, $[\Psi_{hkl} = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl},]$ with $[\phi_{hkl}]$ the phase change due to the twin operation and $[2\pi {\bf f }\cdot {\bf g}_{hkl}]$ the phase change resulting from the lattice displacement vector f. The boundary contrast is strongest if the phase jump $[\Psi_{hkl}]$ is an odd integer multiple of $[\pi]$ , and it is zero if $[\Psi_{hkl}]$ is an integer multiple of $[2\pi]$ . By imaging the boundary in various reflections hkl and analysing the boundary contrast, taking into account the known phase change $[\phi_{hkl}]$ (calculated from the structure-factor phases of the reflections [hkl_1] and [hkl_2] related by the twin operation), the fault vector f can be determined (see the examples below). This procedure has been introduced into transmission electron microscopy by McLaren & Phakey (1966, 1969) and into X-ray topography by Lang (1967 a,b) and McLaren & Phakey (1969).

In the equation given above, for each reflection hkl the total phase jump $[\Psi_{hkl}]$ is independent of the origin of the unit cell. The individual quantities $[\phi_{hkl}]$ and $[2\pi {\bf f} \cdot {\bf g}_{hkl}]$ , however, vary with the choice of the origin but are coupled in such a way that $[\Psi_{hkl}]$ (which alone has a physical meaning) remains constant. This is illustrated by the following simple example of an inversion twin.

The twin operation relates reflections [hkl_1] and [hkl_2 =] $[{\bar h}{\bar k}{\bar l}_1]$ . Their structure factors are (assuming Friedel's rule to be valid) $[F_1 = \vert F\vert \exp(-{i}\varphi)\ \ {\rm and}\ \ F_2 = \vert F \vert \exp(+ {i}\varphi).]$

The phase difference of the two structure factors is $[\phi_{hkl} = 2 \varphi]$ and depends on the choice of the origin. If the origin is chosen at the twin inversion centre (superscript ^o), the phase jump $[\Psi_{hkl}]$ at the boundary is given by $[ \Psi_{hkl} = \phi^o_{hkl} = 2\varphi^o.]$

This is the total phase jump occurring for reflection pairs $[hkl_1/hkl_2 = hkl/{\bar h}{\bar k}{\bar l}]$ at the twin boundary.

If the origin is not located at the twin inversion centre but is displaced from it by a vector $[{\textstyle{1\over 2}} {\bf f}]$ , the phases of the structure factors of reflections [hkl_1] and [hkl_2] are $[\eqalign{\varphi_1 &= \varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot{\bf g}_{hkl} \ \ {\rm and}\cr \varphi_2 &= -\varphi_1 = - [\varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot {\bf g}_{hkl}].}]$ From these equations the phase difference of the structure factors is calculated as $[\phi_{hkl} = \varphi_1 - \varphi_2 = 2\varphi^o - 2\pi {\bf f} \cdot {\bf g}_{hkl},]$ and the total phase jump at the boundary is $[\Psi_{hkl} = 2\varphi^o = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl}.]$

This shows that here the fault vector f has no physical meaning. It merely compensates for the phase contributions that result from an `improper' choice of the origin. If, by the procedures outlined above, a fault vector f is determined, the true twin inversion centre is located at the endpoint of the vector $[{\textstyle{1\over 2}}{\bf f}]$ attached to the chosen origin.

Similar considerations apply to reflection and twofold rotation twins. In these cases, the components of the fault vectors normal to the twin plane or to the twin axis can also be eliminated by a proper choice of the origin. The parallel components, however, cannot be modified by changes of the origin and have a real physical significance for the structure of the boundary.

Particularly characteristic fault vectors occur in (merohedral) `antiphase domains ' (APD). Often the fault vector is the lattice-translation vector lost in a phase transition.

References

Lang, A. R. (1967a). Some recent applications of X-ray topography. Adv. X-ray Anal. 10, 91–107.Google Scholar

Lang, A. R. (1967b). Fault surfaces in alpha quartz: their analysis by X-ray diffraction contrast and their bearing on growth history and impurity distribution. In Crystal growth, edited by H. S. Peiser, pp. 833–838. (Supplement to Phys. Chem. Solids.) Oxford: Pergamon Press.Google Scholar

McLaren, A. C. & Phakey, P. P. (1966). Electron microscope study of Brazil twin boundaries in amethyst quartz. Phys. Status Solidi, 13, 413–422.Google Scholar

McLaren, A. C. & Phakey, P. P. (1969). Diffraction contrast from Dauphiné twin boundaries in quartz. Phys. Status Solidi, 31, 723–737.Google Scholar

Phakey, P. P. (1969). X-ray topographic study of defects in quartz. I. Brazil twin boundaries. Phys. Status Solidi, 34, 105–119.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 433-434