Crystal-Field Symmetry Method For The Study Of The Spectroscopic And Magnetic Properties Of Rare-Earth Doped Laser Crystals

Based on the group-chain scheme introduced by Butler, the electronic level structures of Rare-earth (RE) ions doped in laser crystals are analyzed. The energy level analysis starts from a parametrized, static point-charge model Hamiltonian H_cf to represent the components of one-electron crystal-field (CF) interactions. The low site-symmetry that the activator ions occupy in the lattice restricts the terms of the CF expansion, which are determined by the Branching-rule of the group-subgroup chain. By using the group-chain, the wavefunctions of the 4fⁿ configuration for RE ions are also expressed as linear combinations of group-chain basis functions in Butler's irreducible representation notation. The matrix elements of the CF Hamiltonian then are calculated by means of the Wigner-Eckart theorem and factorization lemma of the jm factors. Before the CF level fitting, a point-charge calculation of the conventional CF parameters Bkq has been performed by means of the lattice-sum technique from the known crystal structure data. The corresponding parameters in the group-chain are calculated by the conversion relationship between two schemes that proved identical in essence, which are usually used as initial values in the fit to the experimental levels. The ratios of the same k-value CF parameters are suggested as the constraint conditions in the fit, in order to utilize fully the symmetry distortion degree of the studied system. The subsequent energy levels fitting is performed by the proposed two-step method: (1) The intermediate-coupling wavefunctions in a Russell-Saunders basis of J states are computed by diagonalizing a free-ion Hamiltonian, and thus the reduced matrix elements (RME) of U^(k)(k=2,4,6) can be calculated; (2) Matrices representing the CF interaction are diagonalized simultaneously for all 2S+1LJ states whose experimental levels have been measured, and the CF parameters, as well as the group-chain basis wavefunctions of the terms of interest, are determined in a least-squares fit to these data. The eigen functions of the CF levels of all the manifolds are obtained and expressed as linear combinations of the group-chain basis functions. By utilizing these wavefunctions, the g-factors of the terms of interest are calculated by the group-chain scheme treatment of the Zeeman interaction. The results confirm the partial g-sum rule raised by Karayianis. Likewise, the magnetic properties such as Schottky specific heat and magnetic susceptibility can be theoretically treated in consideration of the CF effect of the ground state, which show basically agreement with the experiment. Another important application of the group-chain scheme is to predict the relative intensities and the polarizations of the transitions between the CF levels by the Judd-Ofelt formalism in the group-chain scheme developed in this thesis. The odd CF parameters responsible for the ED transition can be obtained by the fit to the experimental line-strengths or transition rates for a series of J-J'transitions selected. Once the spectral line-shapes are known, we may further theoretically calculate RE absorption and/or emission spectra. In order to fully utilize the group-chain approach we apply the proposed method to several representative examples, in which RE ions occupy different low site-symmetry, such as the case of S4 site-symmetry (Er3+:LiYF4 and Nd3+:LiYF4), D2d group(Tm3+:LiYF4, Tm3+:YVO4 and Nd3+:YVO4) and D3 group [Nd3+:YAl3(BO3)4]. The method has been proved to be effective and useful in the study of both the spectroscopic characteristics and the magnetic properties of RE ions in crystals.