Energies of Crystal Field Split Terms

Estimation of the crystal field energy levels occurs through diagonahsation of the Hamiltonian matrix in Equation 1.33. 

After substituting Equation 1.26 into Equation 1.33, it can be shown that the matrix elements are described by Equation 1.34 [29,42]. 

These eonditions for the 3-j symbols dictate the selection of / = 0, 2,4,6 for the crystal field spfitting. 

The doubly reduced matrix elements(i j 11/ y/fsc) are specific to a given lanthanide ion. It was shown that the crystal field does not affect them substantially. Therefore, instead of ealeulating them for each system, the values tabulated by Nielson and Koster can be used [34]. 

With all of these tabulated values, only the needs to be determined to evaluate the energy level splitting based on the crystal field and its symmetry. 

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