We propose a classification of the reflection K-matrices (solutions of the boundary Yang-Baxter equation) for the Uq=Uq vertex-model. The Stokes shift differentiated the emission behaviors of the tetragonal and monoclinic structures. The luminescence properties showed excellent agreement with theoretical predictions based on these factors. The emission and excitation spectra of tetragonal M 2MgSi 2O7 ( M=Ca, Sr, and Ba) changed as a function of the covalency, site symmetry, and crystal field strength. Monoclinic Ba 2MgSi 2O7 was prepared, and the luminescence properties were compared with those of other samples. Ba x Sr 2- x Si 2O7 retained a tetragonal crystal structure similar to the structure of the other compounds up to a stoichiometry of x=1.6, which enabled a systematic study of the common structure. Solid solutions of Ba x Sr 2- x Si 2O7, Ca 2MgSi 2O7, and Sr 2MgSi 2O7 were prepared. The photoluminescence properties of alkali-earth magnesium silicates ( M 2MgSi 2O7, M=Ca, Sr, and Ba) doped with Eu 2+ were investigated. Luminescence properties of Eu 2+ in M 2MgSi 2O7 ( M=Ca, Sr, and Ba) phosphors The analysis was carried out by means of the MODY computer program for vectors k = (0 0 0) and k = (0 0 1) describing the changes of translational symmetry and all positions occupied by the R, M and D atoms.
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For a given transformation it indicates possible displacements of atoms from initial positions in the parent structure, ordering of hydrogen over interstitial sites and also ordering of magnetic moments, described by the smallest possible number of free parameters. The symmetry analysis in the frame of the theory of space groups and their representation gives the opportunity to find all possible transformations from high symmetry parent structure to the structures with symmetry belonging to one of its subgroups. Additionally in the elementary cell, there could be up to 100 atoms of hydrogen (or deuterium) occupying the interstitial positions 4a, 32f3, 96j1 and 96k1. The crystallographic structure of these compounds is described by the Fm3 m space group and contain 116 atoms per unit cell occupying the positions 24e(R), 4b, 24d, 32f1 and 32f 2( M). Symmetry analysis was applied in this work to discuss the behavior of the family R6 M23 compounds upon hydrogenation (deuteration), where different structural transformations and magnetic properties, depending on the type of R and M atoms and hydrogen (deuterium) concentrations, have been found. Symmetry analysis of the behavior of the family R6 M23 compounds upon hydrogenation As an interesting observation, their Hamiltonian operators are linearly combinations of Dx, Dx3, uDx +Dx u and Dx u Dx-1uDx, basic ingredients in the bi-Hamiltonian theory of Korteweg-de Vries and modified Korteweg-de Vries equations. Hence, all cases, having fifth order symmetries, of K( m, n) equation are integrable in the bi-Hamiltonian sense.
For five less studied cases, namely K(0, - 2), K(- 1, - 2), K(- 2, - 2), K(- 1 / 2, - 1 / 2) and K(3 / 2, - 1 / 2), bi-Hamiltonian structures are established through their invertible links with some famous integrable equations. K( m, n) equations with fifth order symmetries and their integrabilityįor K( m, n) equation ut =Dx3(un) + Î☝x(um), all non-degenerate (n ≠0) cases admitting fifth order symmetries are identified, including K( m1, 1), K( m 2, - 1 / 2) and K( m3, - 2), where m1 = 0, 1, 2, 3, m 2 = - 1 / 2, 0, 1, 3 / 2 and m3 = - 2, - 1, 0, 1.
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