Second-order crystal field interactions

Energy level diagram for a octahedral complex, ignoring second-order crystal field interaction. 

Let us consider an optical system with two modes at the frequencies oo and 2oo interacting through a nonlinear crystal with second-order susceptibility placed within a Fabry-Perot interferometer. In a general case, both modes are damped and driven with external phase-locked driving fields. The input external fields have the frequencies (0/, and 2(0/,. The classical equations describing second-harmonic generation are [104,105] ... 

Let us consider a quantum optical system with two interacting modes at the frequencies coi and ff>2 = respectively, interacting by way of a nonlinear crystal with second-order susceptibility. Moreover, let us assume that the nonlinear crystal is placed within a Fabry-Perot interferometer. Both modes are damped via a reservoir. The fundamental mode is driven by an external field with the frequency (0/ and amplitude F. The Hamiltonian for our system is given by [169,178] ... 

Consider a dn configuration present in a crystal field that leaves the ground state nondegenerate except for spin. The ground state then consists of (25+ l)-spin states and the effect of the spin-orbit interaction plus the magnetic field can be computed using first- and second-order perturbation theory. If we take as the perturbation operator... 

Two ions are well-known for their highly anisotropic properties. Firstly, in the rare-earth family, Dy3+ which has a 6H15/2 ground state. The spin-orbit interaction is stronger than the crystal field effects. The ratio J /J can be of the order of 100 (Jj. = 0), gjj = 20 and gi = 0 this is practically an ideal case. Secondly, in the transition element series, the ion Co2+ is also characterized by anisotropic interactions (either in the tetrahedral or octahedral coordination), the anisotropy being however lower than in the case of Dy3+. J /J is about 0.5 for this ion. Some Fe2+ compounds also display a behavior approximating to the Ising model. 

Fig. 2 Dependence of the T2g term fine structure for the Cr ion in KMgp3. The curves are the splitting as the functions of the Ham reduction factor y calculated from the first and second order Ham theory. The open circles correspond to the energy of spinors in a static crystal field (y = 1.0), and the filled circles are observed experimental energies. The best fit is obtained for y = 0.31. AH four curves are merged into two (if y = 0 extremely strong Jahn-Teller interaction) with the separation of2 k + p)...

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