*Sander Van Smaalen*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570820
- eISBN:
- 9780191718762
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570820.003.0003
- Subject:
- Physics, Crystallography

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the ...
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This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.Less

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.

*Sander Van Smaalen*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570820
- eISBN:
- 9780191718762
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570820.003.0009
- Subject:
- Physics, Crystallography

This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal ...
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This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.Less

This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.

*Wai-Kee Li, Hung Kay Lee, Dennis Kee Pui Ng, Yu-San Cheung, Kendrew Kin Wah Mak, and Thomas Chung Wai Mak*

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198823902
- eISBN:
- 9780191862724
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198823902.003.0010
- Subject:
- Physics, Crystallography, Condensed Matter Physics / Materials

This chapter presents 47 problems covering the subject of crystal structure, along with their corresponding solutions.

This chapter presents 47 problems covering the subject of crystal structure, along with their corresponding solutions.

*Christopher Hammond*

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198738671
- eISBN:
- 9780191801938
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198738671.003.0012
- Subject:
- Physics, Crystallography, Condensed Matter Physics / Materials

This chapter starts by providing an historical introduction by looking at the origin of the stereographic projection (Claudius Ptolemy and the Planispherium). It looks at great and small circles and ...
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This chapter starts by providing an historical introduction by looking at the origin of the stereographic projection (Claudius Ptolemy and the Planispherium). It looks at great and small circles and stereographic poles. Next, it turns to the construction of the stereographic projection of a cubic crystal, the use of the polar and Wulff nets, non-cubic crystals (distinguishing between plane normals and zone axes), and applications of the stereographic projection (representation of point group symmetry, orientation relationships between crystals, preferred orientation, and trace analysis).Less

This chapter starts by providing an historical introduction by looking at the origin of the stereographic projection (Claudius Ptolemy and the Planispherium). It looks at great and small circles and stereographic poles. Next, it turns to the construction of the stereographic projection of a cubic crystal, the use of the polar and Wulff nets, non-cubic crystals (distinguishing between plane normals and zone axes), and applications of the stereographic projection (representation of point group symmetry, orientation relationships between crystals, preferred orientation, and trace analysis).