Two-ion crystal field¶
Two-ion crystal field involves two entities. Therefore, it can be expressed as a \(\mathcal{H}_2\) term.
The summary of the mapping for each code is given in the table below.
Code |
McPhase (v1) |
McPhase (v2) |
|---|---|---|
\(\mathcal{H}\) |
\[\hat{\mathcal{H}}
=
-\dfrac{1}{2}
\sum_{nn^{\prime}}
\sum_{ll^{\prime}}
\sum_{mm^{\prime}}
K_{ll^{\prime}}^{mm^{\prime}}(nn^{\prime})
\hat{O}_{lm}(\mathbf{J}^n)
\hat{O}_{l^{\prime}m^{\prime}}(\mathbf{J}^{n^{\prime}})\]
|
\[\hat{\mathcal{H}}
=
-
\dfrac{1}{2}
\sum_{nn^{\prime}}
\Biggl[
\sum_{kk^{\prime}}
\sum_{qq^{\prime}}
\mathcal{K}_{kk^{\prime}}^{qq^{\prime}}(nn^{\prime})
\hat{T}_{kq}^n
\hat{T}_{k^{\prime}q^{\prime}}^{n^{\prime}}
\Biggr]\]
|
Renaming of indices |
\((l,m) \rightarrow i_1\), \((l^{\prime},m^{\prime}) \rightarrow i_2\), \(n \rightarrow (\mu_1, \alpha_1)\), \(n^{\prime} \rightarrow (\mu_2, \alpha_2)\) |
\((k,q) \rightarrow i_1\), \((k^{\prime},q^{\prime}) \rightarrow i_2\), \(n \rightarrow (\mu_1, \alpha_1)\), \(n^{\prime} \rightarrow (\mu_2, \alpha_2)\) |
\(C_2\) |
\(-\dfrac{1}{2}\) |
\(-\dfrac{1}{2}\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\(K_{i_1, i_2}(\mu_1, \alpha_1; \mu_2, \alpha_2)\) |
\(\mathcal{K}_{i_1, i_2}(\mu_1, \alpha_1; \mu_2, \alpha_2)\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(\hat{O}_{i_1}(\mathbf{J}^{\mu_1, \alpha_1})\) |
\(\hat{T}_{i_1}^{\mu_1, \alpha_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(\hat{O}_{i_2}(\mathbf{J}^{\mu_2, \alpha_2})\) |
\(\hat{T}_{i_2}^{\mu_2, \alpha_2}\) |
Note
Pairs of indices \((l,m)\) (\((l^{\prime},m^{\prime})\)) or \((k,q)\) (\((k^{\prime},q^{\prime})\)) run over the finite set of the integer pairs. Therefore, they can be enumerated with a single integer and mapped to the index \(i_1\) (\(i_2\)).