Pub Date : 2024-07-01Epub Date: 2024-06-07DOI: 10.1107/S2053273324003292
Johannes Dallmann, Jonas Graetz, Rainer Hock
Analytical calculations of absorption corrections for X-ray powder diffraction experiments on non-ideal samples with surface roughness, porosity or absorption contrasts from multiple phases require complex mathematical models to represent their material distribution. In a computational approach to this problem, a practicable ray-tracing algorithm is formulated which is capable of simulating angle-dependent absorption corrections in reflection geometry for any given rasterized sample model. Single or multiphase systems with arbitrary surface roughness, porosity and spatial distribution of the phases in any combination can be modeled on a voxel grid by assigning respective values to each voxel. The absorption corrections are calculated by tracing the attenuation of X-rays along their individual paths via a modified shear-warp algorithm. The algorithm is presented in detail and the results of simulated absorption corrections on samples with various surface modulations are discussed in the context of published experimental results.
对表面粗糙度、孔隙率或多相吸收对比的非理想样品进行 X 射线粉末衍射实验的吸收修正分析计算,需要复杂的数学模型来表示其材料分布。针对这一问题的计算方法制定了一种实用的光线跟踪算法,该算法能够模拟任何给定光栅化样品模型在反射几何中与角度相关的吸收修正。单相或多相系统具有任意的表面粗糙度、孔隙率和任意组合的相的空间分布,可以通过给每个象素分配各自的值在象素网格上建模。吸收修正的计算方法是通过改进的剪切-剪切算法追踪 X 射线沿各自路径的衰减情况。本文详细介绍了该算法,并结合已公布的实验结果,讨论了对具有各种表面调制的样品进行吸收修正的模拟结果。
{"title":"Universal simulation of absorption effects for X-ray diffraction in reflection geometry.","authors":"Johannes Dallmann, Jonas Graetz, Rainer Hock","doi":"10.1107/S2053273324003292","DOIUrl":"10.1107/S2053273324003292","url":null,"abstract":"<p><p>Analytical calculations of absorption corrections for X-ray powder diffraction experiments on non-ideal samples with surface roughness, porosity or absorption contrasts from multiple phases require complex mathematical models to represent their material distribution. In a computational approach to this problem, a practicable ray-tracing algorithm is formulated which is capable of simulating angle-dependent absorption corrections in reflection geometry for any given rasterized sample model. Single or multiphase systems with arbitrary surface roughness, porosity and spatial distribution of the phases in any combination can be modeled on a voxel grid by assigning respective values to each voxel. The absorption corrections are calculated by tracing the attenuation of X-rays along their individual paths via a modified shear-warp algorithm. The algorithm is presented in detail and the results of simulated absorption corrections on samples with various surface modulations are discussed in the context of published experimental results.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11216610/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141282318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-01Epub Date: 2024-05-31DOI: 10.1107/S2053273324002730
Felix N Chukhovskii
Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].
描述不完全晶体中 X 射线和电子衍射散射的基本方程是以矩阵 Fredholm-Volterra 第二种积分方程的形式推导出来的。一种理论方法是利用完美晶体的格林函数形式主义。相反,另一种方法则利用了与传统高木-陶平方程和豪伊-惠兰方程的对角化矩阵传播者相关的波场特征函数。利用 Liouville-Neumann 型数列形式建立矩阵 Fredholm-Volterra 积分方程解,得到了 X 射线和电子衍射边界值考奇问题的一般解析函数解。基于解析型解法,目的是揭示衍射散射到晶格缺陷上的特征,包括晶格缺陷核心附近强变形区域的支内和支间波散射机制。利用二阶的两级解析解,通过直接计算厚箔中边缘位错的电子亮场和暗场对比,支持了这一方法。本文讨论了边缘位错的明场和暗场剖面,并与 Howie 和 Whelan [Proc. R. Soc. A (1962), 267, 206] 的类似数值计算结果进行了比较。
{"title":"Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals.","authors":"Felix N Chukhovskii","doi":"10.1107/S2053273324002730","DOIUrl":"10.1107/S2053273324002730","url":null,"abstract":"<p><p>Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141178211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-01Epub Date: 2024-06-27DOI: 10.1107/S2053273324004352
Il Hun Kim, Il Hwan Kim, Kum Ok Jang, Song Won Kim
This paper proposes a new order parameter model which satisfactorily explains complicated symmetry changes, the temperature-pressure (T-P) phase diagram and elastic anomalies observed experimentally with the improper ferroelastic phase transitions in multiferroic KMnF3 single crystal. First, it is shown that the order parameter model is transformed according to the four-dimensional reducible representation of the wavevector star channel group. Second, based on the order parameter model and the singularity theory, the sixth-order structurally stable Landau potential model is constructed. Finally, the theoretical T-P phase diagram is plotted and the elastic anomalies possible for each of the phase transitions are discussed.
{"title":"A new order parameter model for the improper ferroelastic phase transitions in KMnF<sub>3</sub> single crystal.","authors":"Il Hun Kim, Il Hwan Kim, Kum Ok Jang, Song Won Kim","doi":"10.1107/S2053273324004352","DOIUrl":"10.1107/S2053273324004352","url":null,"abstract":"<p><p>This paper proposes a new order parameter model which satisfactorily explains complicated symmetry changes, the temperature-pressure (T-P) phase diagram and elastic anomalies observed experimentally with the improper ferroelastic phase transitions in multiferroic KMnF<sub>3</sub> single crystal. First, it is shown that the order parameter model is transformed according to the four-dimensional reducible representation of the wavevector star channel group. Second, based on the order parameter model and the singularity theory, the sixth-order structurally stable Landau potential model is constructed. Finally, the theoretical T-P phase diagram is plotted and the elastic anomalies possible for each of the phase transitions are discussed.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141453791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-01Epub Date: 2024-06-25DOI: 10.1107/S2053273324004418
Ryoko Oishi-Tomiyasu
In ab initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. `Ab initio' means a process performed without any good prior information on the crystal lattice. Newly developed ab initio indexing software is frequently reported in crystallography. However, it is not widely recognized that use of a Bravais lattice determination method, which is tolerant of experimental errors, can simplify indexing algorithms and increase their success rates. One of the goals of this article is to collect information on the lattice-basis reduction theory and its applications. The main result is a Bravais lattice determination algorithm for 2D lattices, along with a mathematical proof that it works even for parameters containing large observational errors. It uses two lattice-basis reduction methods that seem to be optimal for different symmetries, similarly to the algorithm for 3D lattices implemented in the CONOGRAPH software. In indexing, a method for error-stable unit-cell identification is also required to exclude duplicate solutions. Several methods are introduced to measure the difference in unit cells known in crystallography and mathematics.
在 ab initio 索引中,对于给定的衍射/散射图样,要同时确定单位晶胞参数和分配给图案中反射的米勒指数。所谓 "非初始",是指在没有任何关于晶格的可靠信息的情况下进行的过程。晶体学领域经常报道新开发的ab initio 索引软件。然而,人们并没有普遍认识到,使用可容忍实验误差的布拉维晶格确定方法可以简化索引算法并提高其成功率。本文的目的之一是收集有关晶格基础还原理论及其应用的信息。主要成果是二维网格的布拉维网格确定算法,以及该算法即使在参数包含较大观测误差时也有效的数学证明。它使用了两种格子基础还原方法,这两种方法似乎是不同对称性的最佳方法,类似于 CONOGRAPH 软件中实现的三维格子算法。在索引过程中,还需要一种误差稳定的单元格识别方法来排除重复解。本文介绍了几种测量晶体学和数学中已知单元格差异的方法。
{"title":"Ideas of lattice-basis reduction theory for error-stable Bravais lattice determination and abinitio indexing.","authors":"Ryoko Oishi-Tomiyasu","doi":"10.1107/S2053273324004418","DOIUrl":"10.1107/S2053273324004418","url":null,"abstract":"<p><p>In ab initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. `Ab initio' means a process performed without any good prior information on the crystal lattice. Newly developed ab initio indexing software is frequently reported in crystallography. However, it is not widely recognized that use of a Bravais lattice determination method, which is tolerant of experimental errors, can simplify indexing algorithms and increase their success rates. One of the goals of this article is to collect information on the lattice-basis reduction theory and its applications. The main result is a Bravais lattice determination algorithm for 2D lattices, along with a mathematical proof that it works even for parameters containing large observational errors. It uses two lattice-basis reduction methods that seem to be optimal for different symmetries, similarly to the algorithm for 3D lattices implemented in the CONOGRAPH software. In indexing, a method for error-stable unit-cell identification is also required to exclude duplicate solutions. Several methods are introduced to measure the difference in unit cells known in crystallography and mathematics.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141445661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-01Epub Date: 2024-05-28DOI: 10.1107/S2053273324003097
Dmitry G Stepenshchikov, Anton D Pavlushin
The shape of a flat-faceted octahedral crystal can be uniquely defined by the measured distances between pairs of its parallel facets and the length of one of its false edges. In total, only five numerical values are involved in this approach. Some interdependencies of parameters that allow one to control the correctness of measurements were derived. The proposed method is suitable for describing the shape as full-faceted, or as incomplete octahedral crystals (e.g. diamond) with unequally developed facets. This so-called `real crystal form' can be considered as one of the typomorphic features of minerals, connecting the dissymmetry to the anisotropy of the host rock. The measurement results can be used in crystallo-morphological analysis, restoration of the lost crystal shape in the case of man-made damage and in the practice of diamond prospecting.
{"title":"The description of octahedral crystals using five parameters.","authors":"Dmitry G Stepenshchikov, Anton D Pavlushin","doi":"10.1107/S2053273324003097","DOIUrl":"10.1107/S2053273324003097","url":null,"abstract":"<p><p>The shape of a flat-faceted octahedral crystal can be uniquely defined by the measured distances between pairs of its parallel facets and the length of one of its false edges. In total, only five numerical values are involved in this approach. Some interdependencies of parameters that allow one to control the correctness of measurements were derived. The proposed method is suitable for describing the shape as full-faceted, or as incomplete octahedral crystals (e.g. diamond) with unequally developed facets. This so-called `real crystal form' can be considered as one of the typomorphic features of minerals, connecting the dissymmetry to the anisotropy of the host rock. The measurement results can be used in crystallo-morphological analysis, restoration of the lost crystal shape in the case of man-made damage and in the practice of diamond prospecting.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141157116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01Epub Date: 2024-04-29DOI: 10.1107/S2053273324002523
Maxwell Christopher Day, Ali Rostami, Frank Christopher Hawthorne
Following the work of Day & Hawthorne [Acta Cryst. (2022), A78, 212-233] and Day et al. [Acta Cryst. (2024), A80, 258-281], the program GraphT-T has been developed to embed graphical representations of observed and hypothetical chains of (SiO4)4- tetrahedra into 2D and 3D Euclidean space. During embedding, the distance between linked vertices (T-T distances) and the distance between unlinked vertices (T...T separations) in the resultant unit-distance graph are restrained to the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is restrained to be equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. The notional interactions between vertices are described by a 3D spring-force algorithm in which the attractive forces between linked vertices behave according to Hooke's law and the repulsive forces between unlinked vertices behave according to Coulomb's law. Embedding parameters (i.e. spring coefficient, k, and Coulomb's constant, K) are iteratively refined during embedding to determine if it is possible to embed a given graph to produce a unit-distance graph with T-T distances and T...T separations that are compatible with the observed T-T distances and T...T separations in crystal structures. The resultant unit-distance graphs are denoted as compatible and may form crystal structures if and only if all distances between linked vertices (T-T distances) agree with the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. If the unit-distance graph does not satisfy these conditions, it is considered incompatible and the corresponding chain of tetrahedra is unlikely to form crystal structures. Using GraphT-T, Day et al. [Acta Cryst. (2024), A80, 258-281] have shown that several topological properties of chain graphs influence the flexibility (and rigidity) of the corresponding chains of Si tetrahedra and may explain why particular compatible chain arrangements (and the minerals in which they occur) are more common than others and/or why incompatible chain arrangements do not occur in crystals despite being topologically possible.
{"title":"GraphT-T (V1.0Beta), a program for embedding and visualizing periodic graphs in 3D Euclidean space.","authors":"Maxwell Christopher Day, Ali Rostami, Frank Christopher Hawthorne","doi":"10.1107/S2053273324002523","DOIUrl":"https://doi.org/10.1107/S2053273324002523","url":null,"abstract":"<p><p>Following the work of Day & Hawthorne [Acta Cryst. (2022), A78, 212-233] and Day et al. [Acta Cryst. (2024), A80, 258-281], the program GraphT-T has been developed to embed graphical representations of observed and hypothetical chains of (SiO<sub>4</sub>)<sup>4-</sup> tetrahedra into 2D and 3D Euclidean space. During embedding, the distance between linked vertices (T-T distances) and the distance between unlinked vertices (T...T separations) in the resultant unit-distance graph are restrained to the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is restrained to be equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. The notional interactions between vertices are described by a 3D spring-force algorithm in which the attractive forces between linked vertices behave according to Hooke's law and the repulsive forces between unlinked vertices behave according to Coulomb's law. Embedding parameters (i.e. spring coefficient, k, and Coulomb's constant, K) are iteratively refined during embedding to determine if it is possible to embed a given graph to produce a unit-distance graph with T-T distances and T...T separations that are compatible with the observed T-T distances and T...T separations in crystal structures. The resultant unit-distance graphs are denoted as compatible and may form crystal structures if and only if all distances between linked vertices (T-T distances) agree with the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. If the unit-distance graph does not satisfy these conditions, it is considered incompatible and the corresponding chain of tetrahedra is unlikely to form crystal structures. Using GraphT-T, Day et al. [Acta Cryst. (2024), A80, 258-281] have shown that several topological properties of chain graphs influence the flexibility (and rigidity) of the corresponding chains of Si tetrahedra and may explain why particular compatible chain arrangements (and the minerals in which they occur) are more common than others and/or why incompatible chain arrangements do not occur in crystals despite being topologically possible.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11067947/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140846642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01Epub Date: 2024-03-18DOI: 10.1107/S2053273324001554
Xiaodong Zhang, James P Donahue
A crystal structure with N atoms in its unit cell can be solved starting from a model with atoms 1 to j - 1 being located. To locate the next atom j, the method uses a modified definition of the traditional R1 factor where its dependencies on the locations of atoms j + 1 to N are removed. This modified R1 is called the single-atom R1 (sR1), because the locations of atoms 1 to j - 1 in sR1 are the known parameters, and only the location of atom j is unknown. Finding the correct position of atom j translates thus into the optimization of the sR1 function, with respect to its fractional coordinates, xj, yj, zj. Using experimental data, it has been verified that an sR1 has a hole near each missing atom. Further, it has been verified that an algorithm based on sR1, hereby called the sR1 method, can solve crystal structures (with up to 156 non-hydrogen atoms in the unit cell). The strategy to carry out this calculation has also been optimized. The main feature of the sR1 method is that, starting from a single arbitrarily positioned atom, the structure is gradually revealed. With the user's help to delete poorly determined parts of the structure, the sR1 method can build the model to a high final quality. Thus, sR1 is a viable and useful tool for solving crystal structures.
{"title":"The single-atom R1: a new optimization method to solve crystal structures.","authors":"Xiaodong Zhang, James P Donahue","doi":"10.1107/S2053273324001554","DOIUrl":"10.1107/S2053273324001554","url":null,"abstract":"<p><p>A crystal structure with N atoms in its unit cell can be solved starting from a model with atoms 1 to j - 1 being located. To locate the next atom j, the method uses a modified definition of the traditional R1 factor where its dependencies on the locations of atoms j + 1 to N are removed. This modified R1 is called the single-atom R1 (sR1), because the locations of atoms 1 to j - 1 in sR1 are the known parameters, and only the location of atom j is unknown. Finding the correct position of atom j translates thus into the optimization of the sR1 function, with respect to its fractional coordinates, x<sub>j</sub>, y<sub>j</sub>, z<sub>j</sub>. Using experimental data, it has been verified that an sR1 has a hole near each missing atom. Further, it has been verified that an algorithm based on sR1, hereby called the sR1 method, can solve crystal structures (with up to 156 non-hydrogen atoms in the unit cell). The strategy to carry out this calculation has also been optimized. The main feature of the sR1 method is that, starting from a single arbitrarily positioned atom, the structure is gradually revealed. With the user's help to delete poorly determined parts of the structure, the sR1 method can build the model to a high final quality. Thus, sR1 is a viable and useful tool for solving crystal structures.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11067948/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140142350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01Epub Date: 2024-04-29DOI: 10.1107/S2053273324002432
Maxwell Christopher Day, Frank Christopher Hawthorne, Ali Rostami
In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6-8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si4+ (plus P5+, V5+, As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+) have a grand nearest-neighbour ⟨T-T⟩ distance of 3.06±0.15 Å and a minimum T...T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (mv); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (mi). The parameters mv and mi are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT-T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T-T and T...T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T...T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO<2.5 do not occur in crystal structures.
{"title":"Bond topology of chain, ribbon and tube silicates. Part II. Geometrical analysis of infinite 1D arrangements of (TO<sub>4</sub>)<sup>n-</sup> tetrahedra.","authors":"Maxwell Christopher Day, Frank Christopher Hawthorne, Ali Rostami","doi":"10.1107/S2053273324002432","DOIUrl":"https://doi.org/10.1107/S2053273324002432","url":null,"abstract":"<p><p>In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6-8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si<sup>4+</sup> (plus P<sup>5+</sup>, V<sup>5+</sup>, As<sup>5+</sup>, Al<sup>3+</sup>, Fe<sup>3+</sup>, B<sup>3+</sup>, Be<sup>2+</sup>, Zn<sup>2+</sup> and Mg<sup>2+</sup>) have a grand nearest-neighbour ⟨T-T⟩ distance of 3.06±0.15 Å and a minimum T...T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (m<sub>v</sub>); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (m<sub>i</sub>). The parameters m<sub>v</sub> and m<sub>i</sub> are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT-T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T-T and T...T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T...T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO<sub><2.5</sub> do not occur in crystal structures.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11067949/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140846471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01Epub Date: 2024-03-11DOI: 10.1107/S2053273323010628
Fatma Kablan, Béla Vizvári, Benedek Nagy
The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.
{"title":"A digital distance on the kisrhombille tiling.","authors":"Fatma Kablan, Béla Vizvári, Benedek Nagy","doi":"10.1107/S2053273323010628","DOIUrl":"10.1107/S2053273323010628","url":null,"abstract":"<p><p>The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140092997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01Epub Date: 2024-03-21DOI: 10.1107/S2053273324001645
Sizhuo Yu, Jean Michel Gillet
Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO2). In this work, a new method is assessed on crystalline urea [CO(NH2)2] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.
量子晶体学的最新进展表明,除了传统的电荷密度细化外,还可以利用联合实验 X 射线结构因子和定向康普顿剖面(DCP),通过半定量编程重建满足 N 代表性条件的单电子还原密度矩阵(1-RDM)。迄今为止,这种不受限于幂等性的 1-RDM 重建方法只在一个玩具模型系统(CO2)上进行过测试。在这项工作中,利用静态(0 K)和动态(50 K)人工实验数据,对结晶脲[CO(NH2)2]的新方法进行了评估。为了更好地处理不断增加的系统复杂性,引入了一个改进的模型,包括对称性约束和冻结的核心电子贡献。对重建的 1-RDM、形变密度和 DCP 各向异性进行了分析,结果表明,即使在信息不足和数据损坏的情况下,模型的变化也能显著提高重建质量。因此,模型的稳健性和策略被证明非常适合解决实际实验散射数据的重建问题。
{"title":"N-representable one-electron reduced density matrix reconstruction with frozen core electrons.","authors":"Sizhuo Yu, Jean Michel Gillet","doi":"10.1107/S2053273324001645","DOIUrl":"10.1107/S2053273324001645","url":null,"abstract":"<p><p>Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO<sub>2</sub>). In this work, a new method is assessed on crystalline urea [CO(NH<sub>2</sub>)<sub>2</sub>] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.</p>","PeriodicalId":106,"journal":{"name":"Acta Crystallographica Section A: Foundations and Advances","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11067946/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140173648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}