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Original Articles

Linear algebraic proof and examples of composite-lattice-based liquid crystalline phases

Pages 13-31 | Published online: 08 Mar 2018
 

ABSTRACT

A novel concept of composite-lattice-based liquid crystalline phases is presented in this article. The author explains and proves this concept from linear algebra and X-ray crystallography. When the ab faces in a 3D crystal continuously rotate and/or slip between each other by heating, the 3D vector space is disintegrated at the first step into the 2D subspace and the 1D subspace. This state can be described as [2D⊕1D] composite vector subspaces in linear algebra. [2D⊕1D] is not equal to 3D and the mathematical sign ⊕ means direct sum in linear algebra. The [2D⊕1D] liquid crystalline phase is composite-lattice-based. Besides the [2D⊕1D] phase, two other composite-lattice-based liquid crystalline phases having [1D⊕1D] and [1D⊕1D⊕1D] can be also theoretically considered from linear algebra. In this article, these real-life examples are also demonstrated at the first time.

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