I am a bit confused by some of the points of operator equivalent methods in crystal field theory. I would appreciate any help I could get with any of my questions.

First, I am confused as to the difference in the operator equivalent method for the single electron case and the multiple electron case. I have read, for example, in Hutchings 1964 and Bleaney & Stevens 1953 that one considers $\mathcal{H}=\sum_{n,m}B_n^mO_n^m(L)$ where $L=\sum \ell_z$. However, this confuses me because for example, in $(3d)^7$, by Hund's rules, $L=3$. Therefore, in this operator equivalent method, one would expect a splitting into two triplets and singlet state. However, the d orbital has $\ell=2$, so one should have a five dimensional representation which splits into a doublet and triplet by group theory (as in Dresselhaus or many others). Indeed, I have seen conflicting diagrams from different sources. For example, in Section 6.2 here, it has $\ce{Co^{2+}}$ splitting from seven levels. However, in this paper, $\ce{Co^{2+}}$ splits from five levels, which makes sense to me because even if $L=3$ it is still the $d$ orbital. I think it may be that I'm not understanding what the levels mean in the operator equivalent method, but I'm not sure where to look.

Second, even if one determines the Hamiltonian using operator equivalent methods, I am not sure how to make good use of it. The operators tend to be written as some complicated function of $J$ and $J_z$ for rare earths (or $L$ and $L_z$ for the iron group), so I'm not sure how to do calculations familiar from more basic quantum mechanics, like finding expectation values of angular momentum or energies.


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