A fast algorithm to find reduced hyperplane unit cells and solve N‐dimensional Bézout's identities
Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplane...
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| Published in | Acta crystallographica. Section A, Foundations and advances Vol. 77; no. 5; pp. 453 - 459 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
5 Abbey Square, Chester, Cheshire CH1 2HU, England
International Union of Crystallography
01.09.2021
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| Subjects | |
| Online Access | Get full text |
| ISSN | 2053-2733 2053-2733 |
| DOI | 10.1107/S2053273321006835 |
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| Abstract | Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N‐dimensional Bézout's identity associated with the Miller indices of the hyperplane.
The paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N‐dimensional Bézout's identity associated with the coordinates of p. |
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| AbstractList | Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the
N
-dimensional Bézout's identity associated with the Miller indices of the hyperplane. The paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N-dimensional Bézout’s identity associated with the coordinates of p. Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout’s identity associated with the Miller indices of the hyperplane. Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N‐dimensional Bézout's identity associated with the Miller indices of the hyperplane. The paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N‐dimensional Bézout's identity associated with the coordinates of p. Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout's identity associated with the Miller indices of the hyperplane.Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout's identity associated with the Miller indices of the hyperplane. |
| Author | Cayron, Cyril |
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| SubjectTerms | hyperplane unit cell integer relation N‐dimensional Bézout's identity Research Papers twinning |
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| Title | A fast algorithm to find reduced hyperplane unit cells and solve N‐dimensional Bézout's identities |
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