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The local structures and the EPR parameters for the trigonal Ru3+ centers in YAG, LuAG, YGG and LuGG are theoretically studied from the perturbation formulas of the EPR parameters for a 4d(5) ion in trigonally distorted octahedra. In these formulas, the contributions from the spin-orbit coupling coefficient and the s- and p-orbitals of the ligands are taken into account from the cluster approach. According to the studies, the local impurity-ligand bonding angles beta(L) related to the C-3 axis in the Ru3+ centers are found to be about 2-4 degrees larger than those (beta) in the host octahedral trivalent cation sites due to size mismatching substitution. This leads to the less elongated ligand octahedra of the impurity centers. The theoretical results based on the above angular distortions Delta beta(=beta(L) - beta) in this work show better agreement than those in the absence of Delta beta in the previous studies with the experimental data. The difficulties of the orbital reduction factor (and the normalization factors) larger than unit and the spin-orbit coupling coefficient larger than the free-ion value in the previous investigations are removed on the basis of the cluster approach in this work. In addition, the theoretical results for the g factors of YGG and LuGG and the hyperfine structure constants of YAG and YGG are also improved. (c) 2006 Elsevier B.V. All rights reserved.