# Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3{sup n} grain boundaries

## Abstract

Multiple twinning in cubic crystals is represented geometrically by three-dimensional fractals and algebraically by groupoids. The groupoid composition table can be used to identify the Σ3{sup n} grain boundaries in EBSD maps. Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Σ3{sup n} operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of Σ3{sup n} operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the Σ3{sup n} grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.

- Authors:

- CEA-Grenoble DRT/LITEN/DTH, 17 rue des Martyrs, 38054 Grenoble (France)

- Publication Date:

- OSTI Identifier:
- 22347663

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Acta Crystallographica. Section A, Foundations of Crystallography; Journal Volume: 63; Journal Issue: Pt 1; Other Information: PMCID: PMC2525860; PUBLISHER-ID: au5046; PMID: 17179603; OAI: oai:pubmedcentral.nih.gov:2525860; Copyright (c) International Union of Crystallography 2007; This is an open-access article distributed under the terms described at http://journals.iucr.org/services/termsofuse.html.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- Denmark

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; CRYSTALS; ELECTRONS; FRACTALS; GRAIN BOUNDARIES; MATRICES; TWINNING

### Citation Formats

```
Cayron, Cyril, E-mail: cyril.cayron@cea.fr.
```*Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3{sup n} grain boundaries*. Denmark: N. p., 2007.
Web. doi:10.1107/S0108767306044291.

```
Cayron, Cyril, E-mail: cyril.cayron@cea.fr.
```*Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3{sup n} grain boundaries*. Denmark. doi:10.1107/S0108767306044291.

```
Cayron, Cyril, E-mail: cyril.cayron@cea.fr. Mon .
"Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3{sup n} grain boundaries". Denmark.
doi:10.1107/S0108767306044291.
```

```
@article{osti_22347663,
```

title = {Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3{sup n} grain boundaries},

author = {Cayron, Cyril, E-mail: cyril.cayron@cea.fr},

abstractNote = {Multiple twinning in cubic crystals is represented geometrically by three-dimensional fractals and algebraically by groupoids. The groupoid composition table can be used to identify the Σ3{sup n} grain boundaries in EBSD maps. Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Σ3{sup n} operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of Σ3{sup n} operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the Σ3{sup n} grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.},

doi = {10.1107/S0108767306044291},

journal = {Acta Crystallographica. Section A, Foundations of Crystallography},

number = Pt 1,

volume = 63,

place = {Denmark},

year = {Mon Jan 01 00:00:00 EST 2007},

month = {Mon Jan 01 00:00:00 EST 2007}

}