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  1. Article: Generating random unit cells.

    Andrews, Lawrence C / Bernstein, Herbert J

    Journal of applied crystallography

    2022  Volume 55, Issue Pt 4, Page(s) 782–786

    Abstract: Methods of generating random unit-cell data for testing software are discussed. Working within the ... ...

    Abstract Methods of generating random unit-cell data for testing software are discussed. Working within the space
    Language English
    Publishing date 2022-06-23
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020879-0
    ISSN 1600-5767 ; 0021-8898
    ISSN (online) 1600-5767
    ISSN 0021-8898
    DOI 10.1107/S1600576722004423
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  2. Article ; Online: An invertible seven-dimensional Dirichlet cell characterization of lattices.

    Bernstein, Herbert J / Andrews, Lawrence C / Xerri, Mario

    Acta crystallographica. Section A, Foundations and advances

    2023  Volume 79, Issue Pt 4, Page(s) 369–380

    Abstract: Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on ... ...

    Abstract Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice vectors) or by Delaunay-reduced cells (based on four non-coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point as they are to any other lattice point. The three non-coplanar lattice vectors chosen are here called the Niggli-reduced cell edges. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by 13 lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face-diagonals and the four body-diagonals, but seven of the lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths, and the shortest body-diagonal length. These seven are sufficient to recover the Niggli-reduced cell.
    Language English
    Publishing date 2023-06-20
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020844-3
    ISSN 2053-2733 ; 1600-5724 ; 0108-7673
    ISSN (online) 2053-2733 ; 1600-5724
    ISSN 0108-7673
    DOI 10.1107/S2053273323003121
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  3. Article ; Online: Approximating lattice similarity.

    Andrews, Lawrence C / Bernstein, Herbert J / Sauter, Nicholas K

    Acta crystallographica. Section A, Foundations and advances

    2023  Volume 79, Issue Pt 5, Page(s) 480–484

    Abstract: A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated. ...

    Abstract A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.
    Language English
    Publishing date 2023-07-24
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020844-3
    ISSN 2053-2733 ; 1600-5724 ; 0108-7673
    ISSN (online) 2053-2733 ; 1600-5724
    ISSN 0108-7673
    DOI 10.1107/S2053273323003200
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  4. Book ; Online: Measuring Lattices

    Andrews, Lawrence C. / Bernstein, Herbert J.

    2023  

    Abstract: Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a ... ...

    Abstract Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. We present a review of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Niggli-reduced unit cells G6. Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths to boundary-interrupted paths, looking for a path of minimal length.

    Comment: 45 pages, 24 figures
    Keywords Condensed Matter - Materials Science
    Subject code 514
    Publishing date 2023-01-24
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  5. Book ; Online: Delone lattice studies in C3, the space of three complex variables

    Andrews, Lawrence C. / Bernstein, Herbert J.

    2023  

    Abstract: The Delone (Selling) scalars, which are used in unit cell reduction and in lattice type determination, are studied in $\mathbb{C}^3$, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars. ... ...

    Abstract The Delone (Selling) scalars, which are used in unit cell reduction and in lattice type determination, are studied in $\mathbb{C}^3$, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars.

    Comment: 8 pages, 5 figures, revision added the 5 figures and section 5. Note processing added copy of a figure as decoration on 9th page
    Keywords Condensed Matter - Materials Science
    Publishing date 2023-01-30
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: SELLA -- A Program for Determining Bravais Lattice Types

    Andrews, Lawrence C. / Bernstein, Herbert J. / Sauter, Nicholas K.

    2023  

    Abstract: We introduce a new Bravais lattice determination algorithm. SELLA is a straight-forward algorithm and a program for determining Bravais lattice type based on Selling (Delone) reduction. It is a complete, closed solution, and it provides a clear metric of ...

    Abstract We introduce a new Bravais lattice determination algorithm. SELLA is a straight-forward algorithm and a program for determining Bravais lattice type based on Selling (Delone) reduction. It is a complete, closed solution, and it provides a clear metric of fit to each type.

    Comment: 20 pages, 7 figures
    Keywords Condensed Matter - Materials Science
    Publishing date 2023-01-30
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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    Inter-library loan at ZB MED

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  7. Book ; Online: An Invertible Seven-Dimensional Dirichlet Cell Characterization of Lattices

    Bernstein, Herbert J. / Andrews, Lawrence C. / Xerri, Mario

    2022  

    Abstract: Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on ... ...

    Abstract Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice edge vectors) or by Delaunay-reduced cells (based on four edge vectors summing to zero and all meeting at obtuse or right angles) are commonly used. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point than they are to any other lattice point. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by thirteen lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face diagonals and the four body-diagonals, but seven of the edge lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths and the shortest body-diagonal length, from which the Niggli-reduced cell may be recovered.

    Comment: 37 pages, 2 figures, general copy-edit, update matrices to full 2D display
    Keywords Condensed Matter - Materials Science
    Subject code 612
    Publishing date 2022-07-15
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  8. Article ; Online: Converting three-space matrices to equivalent six-space matrices for Delone scalars in S

    Andrews, Lawrence C / Bernstein, Herbert J / Sauter, Nicholas K

    Acta crystallographica. Section A, Foundations and advances

    2020  Volume 76, Issue Pt 1, Page(s) 79–83

    Abstract: The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices ... ...

    Abstract The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S
    Language English
    Publishing date 2020-01-01
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020844-3
    ISSN 2053-2733 ; 1600-5724 ; 0108-7673
    ISSN (online) 2053-2733 ; 1600-5724
    ISSN 0108-7673
    DOI 10.1107/S2053273319014542
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  9. Article ; Online: Selling reduction versus Niggli reduction for crystallographic lattices.

    Andrews, Lawrence C / Bernstein, Herbert J / Sauter, Nicholas K

    Acta crystallographica. Section A, Foundations and advances

    2019  Volume 75, Issue Pt 1, Page(s) 115–120

    Abstract: The unit-cell reduction described by Selling and used by Delone (whose early publications were under the spelling Delaunay) is explained in a simple form. The transformations needed to implement the reduction are listed. The simplicity of this reduction ... ...

    Abstract The unit-cell reduction described by Selling and used by Delone (whose early publications were under the spelling Delaunay) is explained in a simple form. The transformations needed to implement the reduction are listed. The simplicity of this reduction contrasts with the complexity of Niggli reduction.
    Language English
    Publishing date 2019-01-01
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020844-3
    ISSN 2053-2733 ; 1600-5724 ; 0108-7673
    ISSN (online) 2053-2733 ; 1600-5724
    ISSN 0108-7673
    DOI 10.1107/S2053273318015413
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  10. Article ; Online: A space for lattice representation and clustering.

    Andrews, Lawrence C / Bernstein, Herbert J / Sauter, Nicholas K

    Acta crystallographica. Section A, Foundations and advances

    2019  Volume 75, Issue Pt 3, Page(s) 593–599

    Abstract: Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial ... ...

    Abstract Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation-stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree-based nearest-neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation-stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six-dimensional real vectors or equivalently as three-dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.).
    Language English
    Publishing date 2019-04-30
    Publishing country United States
    Document type Journal Article
    ZDB-ID 2020844-3
    ISSN 2053-2733 ; 1600-5724 ; 0108-7673
    ISSN (online) 2053-2733 ; 1600-5724
    ISSN 0108-7673
    DOI 10.1107/S2053273319002729
    Database MEDical Literature Analysis and Retrieval System OnLINE

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