Polyhedron Distortion and Site-Specific Rigidity in Garnet-Type Compounds
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Rigidity of a crystal structure has a simple and imaginable meaning, although its quantitative expression has not been established. A widely accepted approach is to select the Debye temperature as a measurement of rigidity, yet it overlooks the site differences within a unit cell. In this work, the correlation between polyhedron distortion and rigidity of the corresponding crystallographic site is explored in a series of garnet-type compounds. The polyhedron distortion is characterized by determining to what extent the real polyhedron differs from its best fitted ideal one. The smaller octahedron and tetrahedron are much closer to the ideal shapes than the larger dodecahedron. Nevertheless, the dodecahedron has the least variation percentage change, indicating that the 24c site has the smallest deformation on component changing. This observation could explain the excellent thermal luminescence resistance in Ce3+-doped garnet phosphors, in which Ce3+ is accommodated in the dodecahedral site. This new interpretation sheds light on understanding the rigidity of crystal structure and will promote its quantitative characterization for wider applications.Keywords:
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Polyhedra are widely used in the verification of numerical programs. Specially, in the field of static analysis by abstract interpretation to express the program invariants. Polyhedra make the analysis very expressive but also very time consuming. That cost is mostly due to the minimization function, which is used to maintain polyhedra in their minimal representation without redundant constraints or generators. In this article, we propose method to over-approximate a polyhedron by minimizing the loss of accuracy. The idea is to find a good trade off between accuracy and execution time. The proposed method is applied as an alternative to the minimization function for the template polyhedra abstract domain.
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So, just what is a polyhedron? we could say that a polyhedron is a simple closed surface in space whose boundary is composed of polygonal regions, but most middle school students would not understand this definition. Instead, involving students in an activity in which they actually create polyhedra enable them to understand the characteristics of different polyhedra and develop a clear definition for themselves. Once students gain a hands-on understanding of polyhedra, the shapes can be used to reveal a plethora of other geometric concepts.
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