Crystal Reduced Motif via the Vectors Exchange Theorem I: Impact of Swapping on two Orthogonalization Processes and the AE Algorithm

IF 1 Q1 MATHEMATICS
Seddik Abdelalim, Ilias Elmouki
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引用次数: 0

Abstract

Crystallographic literature is relying more on observational rules for the determination of the motif that could generate the whole representing Bravais lattice of a crystal. Here, we devise an algebraic method that can serve in this regard at least in cases when the associated unit cell is made of quasi-orthogonal vectors. To let our approach be applicable to other reduction problems, we introduce a concept which is about starting first from any 'bad' crystal cell, not necessarily the primitive elementary cell, in order to find a 'good' crystal cell and that means seeking a motif made of a basis whose vectors are close-to-orthogonal. The orthogonalization loss could happen any time of vectors swapping which represents a very important process in dealing with lattice reduction, but it has insufficiently been discussed in this subject. Thus, through our present version of vectors exchange theorem, and by using examples of two processes, namely the Gram-Schmidt (GS) procedure and its modified version (MGS), we provide formulations for the new reduced unit cell vectors and analyze the impact of the repeated exchange of vectors on the orthogonalization precision. Finally, we give a detailed explanation to our procedure named as AE algorithm. More interestingly, we show that MGS is not only better than GS because of the classical reason related to numerics, but also because its formulation for the new motif vectors in four conditions, has been preserved in three times rather than two for GS, and this may recommend more the introduction of MGS in a harder problem, namely when the crystal dimension is very big.
基于向量交换定理的晶体约简基元I:交换对两个正交化过程的影响及AE算法
晶体学文献更多地依赖于观测规则来确定母基,这些母基可以产生代表晶体的整个布拉维晶格。在这里,我们设计了一种代数方法,至少在相关的单元格由拟正交向量构成的情况下,可以在这方面服务。为了使我们的方法适用于其他约简问题,我们引入了一个概念,即首先从任何“坏”晶体单元开始,不一定是原始的初等细胞,以找到“好”晶体单元,这意味着寻找由向量接近正交的基构成的基序。正交化损失是处理格约简中一个非常重要的过程,在向量交换的任何时候都可能发生,但本课题对其讨论不够。因此,通过我们现有版本的向量交换定理,并通过两个过程的例子,即Gram-Schmidt (GS)过程和它的修改版本(MGS),我们提供了新的简化单位细胞向量的表达式,并分析了向量的重复交换对正交化精度的影响。最后,对AE算法进行了详细的说明。更有趣的是,我们表明MGS不仅比GS好,因为与数字相关的经典原因,而且因为它在四种情况下的新基序向量的公式被保留了三次而不是两次,这可能更适合在更难的问题中引入MGS,即晶体尺寸非常大的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
28.60%
发文量
156
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