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

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.