Crystal field¶
Crystal field terms involve either one or \(k\) entities (depending on the mapping procedure choice). Consider case with one entity as a preferred mapping scheme. Then, it can be expressed via the term \(\mathcal{H}_1\).
The summary of the mapping for each code is given in the table below.
Code |
McPhase (v1) |
McPhase (v2) |
|
|---|---|---|---|
\(\mathcal{H}\) |
\[\hat{\mathcal{H}}
=
\sum_n
\sum_{lm}
B_l^m
\hat{O}_{lm}(\mathbf{J}^n)\]
|
\[\hat{\mathcal{H}}
=
\sum_n
\sum_{lm}
L_l^m(n)
\hat{T}_{lm}^{n}\]
|
\[\mathcal{H}
=
\sum_{i}\sum_{k,q}
c_{i,k,q}
\mathcal{O}_{k,q}
(\mathbf{S}_{i})\]
|
Indices renaming |
\(n \rightarrow (\mu_1, \alpha_1)\), \((l,m) \rightarrow i_1\) |
\(n \rightarrow (\mu_1, \alpha_1)\), \((l,m) \rightarrow i_1\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \((k,q) \rightarrow i_1\) |
\(C_1\) |
1 |
1 |
1 |
\(V_{\mu_1; \alpha_1}^{i_1}\) |
\(B^{i_1}\) |
\(L^{i_1}(\mu_1, \alpha_1)\) |
\(c_{\mu_1, \alpha_1, i_1}\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(\hat{O}_{i_1}(\mathbf{J}^{\mu_1, \alpha_1})\) |
\(\hat{T}_{i_1}^{\mu_1, \alpha_1}\) |
\(\mathcal{O}_{i_1}(\mathbf{S}_{\mu_1, \alpha_1})\) |
Note
Pairs of indices \((l,m)\) or \((k,q)\) run over the finite set of the sets of two integers. Therefore, they can be enumerated with a single integer and mapped to the index \(i_1\).
Crystal field (ambiguity)¶
Alternatively, the crystal field term can be mapped to the terms with one site and various amount of entities if one is to express the Stevens operators via the angular momentum/spin operators.
Here is an example that illustrate it. Consider the term (in McPhase's notation)
One can express the Stevens operators through angular momentum operator as
By renaming indices as \(n \rightarrow (\mu_1, \alpha_1)\) one can express the non-constant part as
where