scholarly journals FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions

2014 ◽  
Vol 19 (1) ◽  
pp. 95
Author(s):  
Nazife Özdeş Koca ◽  
Mehmet Koca ◽  
Muna Al-Sawafi
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Face Centered Cubic ◽  
Tetrahedral Symmetry ◽  
Orthogonal Projections ◽  
Cubic Lattices ◽  
Dynkin Diagrams ◽  
Octahedral Group ◽  
Face Centered ◽  
Coxeter Plane

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine extended Coxeter groups W(A3) and W(B3)=Aut(A3). It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices which have extensive applications in material science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices.   

scholarly journals Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1082
Author(s):  
Ozdes Koca ◽  
Al-Siyabi ◽  
Koca ◽  
Koc
Keyword(s):  
General Scheme ◽  
Square Lattice ◽  
Face Centered Cubic ◽  
Root Lattice ◽  
Rhombic Dodecahedron ◽  
Orthogonal Projections ◽  
Seitz Cell ◽  
The Face ◽  
Face Centered ◽  
Coxeter Plane

The orthogonal projections of the Voronoi and Delone cells of root lattice An onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2h involving the Coxeter element of order h=n+1 of the Coxeter–Weyl group an play a crucial role for h-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4-fold symmetry.

Equilibration of kinetic temperatures in face-centered cubic lattices

2020 ◽  
Vol 102 (4) ◽  
Author(s):  
Vitaly A. Kuzkin ◽  
Sergei D. Liazhkov
Keyword(s):  
Face Centered Cubic ◽  
Cubic Lattices ◽  
Face Centered

Theory of ordering in ternary alloys with face-centered-cubic lattices

1968 ◽  
Vol 11 (12) ◽  
pp. 56-63
Author(s):  
V. P. Fadin ◽  
Yu. A. Khon ◽  
V. E. Panin
Keyword(s):  
Ternary Alloys ◽  
Face Centered Cubic ◽  
Cubic Lattices ◽  
Face Centered

Zero-Point Effects in Heisenberg Antiferromagnets with Arbitrary Range of Interaction

1972 ◽  
Vol 50 (23) ◽  
pp. 2991-2996 ◽  
Author(s):  
M. F. Collins ◽  
V. K. Tondon
Keyword(s):  
Ground State ◽  
State Energy ◽  
Correction Term ◽  
Zero Point ◽  
Face Centered Cubic ◽  
Heisenberg Antiferromagnet ◽  
Body Centered Cubic ◽  
Cubic Lattices ◽  
Simple Cubic ◽  
Face Centered

The ground state energy, spin-wave energy, and sublattice magnetization have been calculated for a Heisenberg antiferromagnet at the absolute zero of temperature. The treatment extends the earlier work of Anderson, Kubo, and Oguchi to apply for any two-sublattice antiferromagnet with arbitrary range of interaction. It is shown that for each exchange interaction there is a different characteristic correction term to the energies. Explicit calculations are made of these terms for the simple cubic, body-centered cubic, and face-centered cubic lattices, with both first- and second-neighbor interactions. Applications are also made to NiO and MnO. An extra term in the magnetization series beyond that given by earlier workers is derived.

Protein tertiary structure prediction using hidden Markov model based on lattice

2019 ◽  
Vol 17 (02) ◽  
pp. 1950007
Author(s):  
Farzad Peyravi ◽  
Alimohammad Latif ◽  
Seyed Mohammad Moshtaghioun
Keyword(s):  
Amino Acid ◽  
Amino Acid Sequence ◽  
Structure Prediction ◽  
Tertiary Structure ◽  
Hidden Markov ◽  
Face Centered Cubic ◽  
Tertiary Structure Prediction ◽  
Cubic Lattices ◽  
Face Centered ◽  
Protein Tertiary Structure Prediction

The prediction of protein structure from its amino acid sequence is one of the most prominent problems in computational biology. The biological function of a protein depends on its tertiary structure which is determined by its amino acid sequence via the process of protein folding. We propose a novel fold recognition method for protein tertiary structure prediction based on a hidden Markov model and 3D coordinates of amino acid residues. The method introduces states based on the basis vectors in Bravais cubic lattices to learn the path of amino acids of the proteins of each fold. Three hidden Markov models are considered based on simple cubic, body-centered cubic (BCC) and face-centered cubic (FCC) lattices. A 10-fold cross validation was performed on a set of 42 fold SCOP dataset. The proposed composite methodology is compared to fold recognition methods which have HMM as base of their algorithms having approaches on only amino acid sequence or secondary structure. The accuracy of proposed model based on face-centered cubic lattices is quite better in comparison with SAM, 3-HMM optimized and Markov chain optimized in overall experiment. The huge data of 3D space help the model to have greater performance in comparison to methods which use only primary structures or only secondary structures.

scholarly journals Efficient LBM Visual Simulation on Face-Centered Cubic Lattices

2009 ◽  
Vol 15 (5) ◽  
pp. 802-814 ◽  
Author(s):  
K. Petkov ◽  
Feng Qiu ◽  
Zhe Fan ◽  
A.E. Kaufman ◽  
K. Mueller
Keyword(s):  
Visual Simulation ◽  
Face Centered Cubic ◽  
Cubic Lattices ◽  
Face Centered
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