Two dimentional lattice vibrations from direct product representations of symmetry groups
Arrangements of point masses and ideal harmonic springs are used to model two dimensional crystals. First, the Born cyclic condition is applied to a double chain composed of coupled linear lattices to obtain a cylindrical arrangement. Then the quadratic Lagrangian function for the system is written...
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| Format: | Article |
| Language: | English |
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Wiley
1983-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171283000666 |
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| _version_ | 1832564935993851904 |
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| author | J. N. Boyd P. N. Raychowdhury |
| author_facet | J. N. Boyd P. N. Raychowdhury |
| author_sort | J. N. Boyd |
| collection | DOAJ |
| description | Arrangements of point masses and ideal harmonic springs are used to model
two dimensional crystals. First, the Born cyclic condition is applied to a double chain composed of coupled linear lattices to obtain a cylindrical arrangement. Then the quadratic Lagrangian function for the system is written in matrix notation. The Lagrangian is diagonalized to yield the natural frequencies of the system. The transformation to achieve the diagonalization was obtained from group theorectic considerations. Next, the techniques developed for the double chain are applied to a square lattice. The square lattice is transformed into the toroidal Ising model. The direct product nature of the symmetry group of the torus reveals the transformation to diagonalize the Lagrangian for the Ising model, and the natural frequencies for the principal directions in the model are obtained in closed form. |
| format | Article |
| id | doaj-art-1342ccdc772c469cad47676b87eb6e10 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1983-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-1342ccdc772c469cad47676b87eb6e102025-02-03T01:09:51ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251983-01-016478379410.1155/S0161171283000666Two dimentional lattice vibrations from direct product representations of symmetry groupsJ. N. Boyd0P. N. Raychowdhury1Department of Hathematical Sciences, Virginia Commonwealth University, Richmond 23284, Virginia, USADepartment of Hathematical Sciences, Virginia Commonwealth University, Richmond 23284, Virginia, USAArrangements of point masses and ideal harmonic springs are used to model two dimensional crystals. First, the Born cyclic condition is applied to a double chain composed of coupled linear lattices to obtain a cylindrical arrangement. Then the quadratic Lagrangian function for the system is written in matrix notation. The Lagrangian is diagonalized to yield the natural frequencies of the system. The transformation to achieve the diagonalization was obtained from group theorectic considerations. Next, the techniques developed for the double chain are applied to a square lattice. The square lattice is transformed into the toroidal Ising model. The direct product nature of the symmetry group of the torus reveals the transformation to diagonalize the Lagrangian for the Ising model, and the natural frequencies for the principal directions in the model are obtained in closed form.http://dx.doi.org/10.1155/S0161171283000666group representationsdirect productLagrangian mechanicsBorn cyclic conditionsymmetry coordinatesProjection operators. |
| spellingShingle | J. N. Boyd P. N. Raychowdhury Two dimentional lattice vibrations from direct product representations of symmetry groups International Journal of Mathematics and Mathematical Sciences group representations direct product Lagrangian mechanics Born cyclic condition symmetry coordinates Projection operators. |
| title | Two dimentional lattice vibrations from direct product representations of symmetry groups |
| title_full | Two dimentional lattice vibrations from direct product representations of symmetry groups |
| title_fullStr | Two dimentional lattice vibrations from direct product representations of symmetry groups |
| title_full_unstemmed | Two dimentional lattice vibrations from direct product representations of symmetry groups |
| title_short | Two dimentional lattice vibrations from direct product representations of symmetry groups |
| title_sort | two dimentional lattice vibrations from direct product representations of symmetry groups |
| topic | group representations direct product Lagrangian mechanics Born cyclic condition symmetry coordinates Projection operators. |
| url | http://dx.doi.org/10.1155/S0161171283000666 |
| work_keys_str_mv | AT jnboyd twodimentionallatticevibrationsfromdirectproductrepresentationsofsymmetrygroups AT pnraychowdhury twodimentionallatticevibrationsfromdirectproductrepresentationsofsymmetrygroups |