Transition Probabilities of 4 f-elements and the Judd-Ofelt Theory

5 Transition Probabilities of 4 f-elements and the Judd-Ofelt Theory 

The optical transitions so typical of the spectra of rare earth (RE) within condensed phase usually correspond to intra-1 transitions of predominantly electric-dipole character. For a free ion, electric-dipole transitions between states of the same configuration are strictly parity forbidden, and thus any explanation of the observed spectra of glasses must concern itself with non-centro-symmetric interactions that lead to a mixing of states of opposite parity. This mixing may result from several distinct mechanisms. One of the most obvious mechanisms is simply the coupling of states of opposite parity by way of the odd terms in the crystal field expansion. 

Several selection rules follow in application of the triangular conditions to the 3-j and 6-j symbols contained in the theoretical equations. These are  

The AL = 1 selection rule means that for f configurations the perturbing n figurations may differ only by the single substitution of n d or n g-electrons. Spin-orbit interaction will usually lead to a breakdown of the selection rules on S and L. The selection rule AJ 6 will be valid as long as J is a good quantum number. This selection rule is seen in the usual absence of the transitions Dq 

Eu spectra. In general, the oscillator strength f or a transition may be expressed as 

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