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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 616-618   | 1 | 2 |

Section 4.6.3.3.3.1. Indexing

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail:  walter.steurer@mat.ethz.ch

4.6.3.3.3.1. Indexing

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There are several indexing schemes in use. The generic one uses a set of six rationally independent reciprocal-basis vectors pointing to the corners of an icosahedron, [{\bf a}_{1}^{*} = a^{*} (0,0,1)], [{\bf a}_{i}^{*} = a^{*} [\sin \theta \cos (2\pi i/5), \sin \theta \sin (2\pi i/5), \cos \theta]], [i = 2, \ldots, 6], [\sin \theta = 2/(5)^{1/2}, \cos \theta = 1/(5)^{1/2}], with [\theta \simeq 63.44^{\circ}], the angle between two neighbouring fivefold axes (setting 1) (Fig. 4.6.3.28)[link]. In this case, the physical-space basis corresponds to a simple projection of the 6D reciprocal basis [{\bf d}_{i}^{*}, i = 1, \ldots, 6]. Sometimes, the same set of six reciprocal-basis vectors is referred to a differently oriented Cartesian reference system (C basis, with basis vectors [{\bf e}_{i}] along the twofold axes) (Bancel et al., 1985[link]). The reciprocal basis is [\pmatrix{{\bf a}_{1}^{*}\cr {\bf a}_{2}^{*}\cr {\bf a}_{3}^{*}\cr {\bf a}_{4}^{*}\cr {\bf a}_{5}^{*}\cr {\bf a}_{6}^{*}\cr} = {a^{*} \over (1 + \tau^{2})^{1/2}} \pmatrix{0 &1 &\tau\cr -1 &\tau &0\cr -\tau &0 &1\cr 0 &-1 &\tau\cr \tau &0 &1\cr 1 &\tau &0\cr}_{C} \pmatrix{{\bf e}_{1}^{C}\cr {\bf e}_{2}^{C}\cr {\bf e}_{3}^{C}\cr}.]

An alternate way of indexing is based on a 3D set of cubic reciprocal-basis vectors [{\bf b}_{i}^{*}, i = 1, \ldots, 3] (setting 2) (Fig. 4.6.3.32)[link]: [\eqalign{\pmatrix{{\bf b}_{1}^{*}\cr {\bf b}_{2}^{*}\cr {\bf b}_{3}^{*}\cr} &= {\textstyle{1 \over 2}} \pmatrix{0 &\bar{1} &0 &0 &0 &1\cr 1 &0 &0 &\bar{1} &0 &0\cr 0 &0 &1 &0 &1 &0\cr}_{D} \pmatrix{{\bf a}_{1}^{*}\cr {\bf a}_{2}^{*}\cr {\bf a}_{3}^{*}\cr {\bf a}_{4}^{*}\cr {\bf a}_{5}^{*}\cr {\bf a}_{6}^{*}\cr}\cr &= {a^{*} \over (1 + \tau^{2})^{1/2}} \pmatrix{{\bf e}_{1}^{C}\cr {\bf e}_{2}^{C}\cr {\bf e}_{3}^{C}\cr}.}]The Cartesian C basis is related to the V basis by a [\theta /2] rotation around [[100]_{C}], yielding [[001]_{V}], followed by a [\pi /10] rotation around [[001]_{C}]: [\displaylines{ \pmatrix{{\bf e}_{1}^{C}\cr {\bf e}_{2}^{C}\cr {\bf e}_{3}^{C}\cr} = \pmatrix{\cos (\pi/10) &\sin (\pi/10) &0\cr - \cos (\theta/2)\sin (\pi/10) &\cos (\theta/2)\cos (\pi/10) &\sin (\theta/2)\cr \sin (\theta/2)\sin (\pi/10) &- \sin (\theta/2)\cos (\pi/10) &\cos (\theta/2)\cr}_{V} \pmatrix{{\bf e}_{1}^{V}\cr {\bf e}_{2}^{V}\cr {\bf e}_{3}^{V}\cr}.}]Thus, indexing the diffraction pattern of an icosahedral phase with integer indices, one obtains for setting 1 [{\bf H} = {\textstyle\sum_{i = 1}^{6}} h_{i}{\bf a}_{i}^{*}, h_{i} \in {\bb Z}]. These indices [(h_{1}\;h_{2}\;h_{3}\;h_{4}\;h_{5}\;h_{6})] transform into the second setting to [(h/h'\;k/k'\;l/l')] with the fractional cubic indices [h_{1}^{\rm c} = h + h' \tau], [h_{2}^{\rm c} = k + k'\tau], [h_{3}^{\rm c} = l + l' \tau]. The transformation matrix is [\pmatrix{h\cr h'\cr k\cr k'\cr l\cr l'\cr}_{C} = \pmatrix{0 &\bar{1} &0 &0 &0 &1\cr 0 &0 &\bar{1} &0 &1 &0\cr 1 &0 &0 &\bar{1} &0 &0\cr 0 &1 &0 &0 &0 &1\cr 0 &0 &1 &0 &1 &0\cr 1 &0 &0 &1 &0 &0\cr} \pmatrix{h_{1}\cr h_{2}\cr h_{3}\cr h_{4}\cr h_{5}\cr h_{6}\cr}_{D} = \pmatrix{h_{6} - h_{2}\cr h_{5} - h_{3}\cr h_{1} - h_{4}\cr h_{6} + h_{2}\cr h_{5} + h_{3}\cr h_{1} + h_{4}\cr}_{D}.]

[Figure 4.6.3.32]

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Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling: the cubic and the icosahedral setting, represented by the bases [{\bf b}_{i}^{*}, i = 1,\ldots, 3], and [{\bf a}_{i}^{*}, i = 1,\ldots, 6], respectively.

References

Bancel, P. A., Heiney, P. A., Stephens, P. W., Goldman, A. I. & Horn, P. M. (1985). Structure of rapidly quenched Al–Mn. Phys. Rev. Lett. 54, 2422–2425.








































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