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Normally supportive sublattices of crystallographic space groups.
Acta Crystallographica Section A: Foundations and Advances ( IF 1.9 ) Pub Date : 2020-01-01 , DOI: 10.1107/s205327331901218x Miles A Clemens 1 , Branton J Campbell 1 , Stephen P Humphries 2
Acta Crystallographica Section A: Foundations and Advances ( IF 1.9 ) Pub Date : 2020-01-01 , DOI: 10.1107/s205327331901218x Miles A Clemens 1 , Branton J Campbell 1 , Stephen P Humphries 2
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The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.
中文翻译：
通常为晶体空间群的支持性子晶格。
将本身是3D晶体空间群（csg）的3D晶体空间群的常规子群制成表格是一个宏伟的目标，但将有多种应用。为了方便起见，这样的亚组被称为“ csg-正态”，而结晶空间群的晶体学点组（cpg）的正态子组被称为“ cpg-正态”。csg-normal子组的点组必须是cpg-normal子组。当前的工作通过列表能够支持csg-normal子组的翻译子组（又称子格），朝着该目标迈出了重要的一步。为了使子格支持csg-normal子组，必须在相对子格的基础上确定两个必要条件：一个取决于空间组的点组的操作，而另一个取决于cpg-normal子组的操作。满足这些条件的子格称为“正常支持”。对于3D空间组的每个算术晶体类的每个cpg-正常子组（不包括标识子组1），所有正常支持子格都以符号形式制成表格，因此表中的大多数条目都包含一个或多个。更多无限范围的整数变量；因此，可以将其更准确地描述为正常支撑子格的无限族的表格。对于给定的一对cpg正常子群和正常支持子格，可以通过组扩展构造父算术晶体类空间群的csg正常子群，
更新日期：2020-01-07
中文翻译：
通常为晶体空间群的支持性子晶格。
将本身是3D晶体空间群（csg）的3D晶体空间群的常规子群制成表格是一个宏伟的目标，但将有多种应用。为了方便起见，这样的亚组被称为“ csg-正态”，而结晶空间群的晶体学点组（cpg）的正态子组被称为“ cpg-正态”。csg-normal子组的点组必须是cpg-normal子组。当前的工作通过列表能够支持csg-normal子组的翻译子组（又称子格），朝着该目标迈出了重要的一步。为了使子格支持csg-normal子组，必须在相对子格的基础上确定两个必要条件：一个取决于空间组的点组的操作，而另一个取决于cpg-normal子组的操作。满足这些条件的子格称为“正常支持”。对于3D空间组的每个算术晶体类的每个cpg-正常子组（不包括标识子组1），所有正常支持子格都以符号形式制成表格，因此表中的大多数条目都包含一个或多个。更多无限范围的整数变量；因此，可以将其更准确地描述为正常支撑子格的无限族的表格。对于给定的一对cpg正常子群和正常支持子格，可以通过组扩展构造父算术晶体类空间群的csg正常子群，
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