The Er <sub> 3 </ Sub> Ga <sub> 5 </ Sub> O ~ (12) Material Faraday Theoretical Research,

In 1998, M.Guillot et.al reported the magnetic properties and faraday effects in single crystal erbium gallium gallate (ErGG ),which showed some specific properties, and attracted more attentions.In recent years Liu gongqiang and Xu you made the quantitative calculation in magnetic properties and MO effects of the material with doped rare earth ions in theory.Because of the complex microscopic structure of Er3+ ions, so to date, the quantitative calculations by quantum theory reported little.After detail deduction and calculations, we fistly gave the quantitative calculation on faraday rotation angle of the single crystal ErGG under high magnetic field,and gained the magnetic anisotropy.According to traditional theory that faraday rotation angle was mainly due to the electric-dipole transition.Liu gongqiang and Xu you's research showed that thetransition from the ground state(GS) to the excited state(ES) (4fn -4fn-15d) ofCe3+,Pr3+ions was dominant to the faraday rotation angle.But to Er3+ions there wasan energy gap above 60000cm-1 between GS and ES, and the excited spectrum can't be obtained till now.At the same time we considered that when one electron was transited to the 5d ,it can orient its spin in two ways,either parallel with the 7 remaining4f electrons ,giving rise to a high-spin(HS) state,or antiparallel yielding a low-spin(LS) state.According to Hund's rule,the HS state would be lower in energy.Thus the transition from the GS to the lowest 4f105J state would be forbidden and therefore berelatively weak compared to the higher energy 4f105d excitation which was spin allowed.In 1960s' Judd and Ofelt proposed the theory that the 4fn configurationmixed with opposite parity 5d configuration would induce f -f forbidden transition deduction.Based on these consideration,we believed that the faraday rotation angle of ErGG was arisen from the f - f transition. We calculated the transition of4S3/2-4I15/2 when magnetic field was applied in [100] crystallographic direction andgained the result which basically familiar with experiment.In addition we calculated the superficial relationship between the faraday rotation angle and magnetization. We found that (1) the crystal field has a big influence on the magnetization;(2) the indirect exchange of rare earth ion greatly affected on the magnetization.Then we simulated the coefficient relevant to the applied field and gained the better agreement between theory and experiment;(3) finally the effects of temperature on magnetic depended on magnetic direction.