Magneto-elastic interaction¶
Magneto-elastic interaction involves two entities (\(X\) and \(X\) or \(Y\) and \(X\)). Therefore, it can be expressed via the terms \(\mathcal{H}_{0, 2}\) and \(\mathcal{H}_{1, 1}\).
The summary of the mapping for two terms is given in the tables below.
Magneto-elastic interaction (1)¶
Code |
|
|---|---|
\(\mathcal{H}\) |
\[\hat{\mathcal{H}}_{ME1}
=
-
\sum_n
\sum_{\alpha=1,\dots,6, lm}
G^{\alpha,l,m}_{\text{cfph}}(n)
\epsilon_{\alpha}
O_{lm}(\hat{\mathbf{J}}^n)\]
|
Indices renaming |
\(n \rightarrow (\mu_1; \alpha_1)\), \((l,m) \rightarrow i_1\), \(\alpha \rightarrow j_1\) |
\(C_{1, 1}\) |
\(-1\) |
\(V_{1; \mu_1; \alpha_1}^{j_1,i_1}\) |
\(G^{\alpha, i_1}_{\text{cfph}}(\mu_1,\alpha_1)\) |
\(Y_{1}^{j_1}\) |
\(\epsilon_{j_1}\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(O_{i_1}(\hat{\mathbf{J}}^{\mu_1, \alpha_1})\) |
Magneto-elastic interaction (2)¶
Code |
|
|---|---|
\(\mathcal{H}\) |
\[\begin{split}\hat{\mathcal{H}}_{ME2}
=
-
\sum_{\substack{n<n^{\prime},lm,\\\alpha=1,2,3}}
\Gamma^{\alpha,l,m}(nn^{\prime})
\hat{u}_n^{\alpha}
O_{lm}(\hat{\mathbf{J}}^{n^{\prime}})\end{split}\]
|
Indices renaming |
\(n \rightarrow (\mu_1; \alpha_1)\), \(n^{\prime} \rightarrow (\mu_2; \alpha_2)\), \(\alpha \rightarrow i_1\), \((l,m) \rightarrow i_2\) |
\(C_{0, 2}\) |
\(-1\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\(\Gamma^{i_1, i_2}(\mu_1, \alpha_1, \mu_2, \alpha_2)\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(\hat{u}_{\mu_1, \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(O_{i_2}(\hat{\mathbf{J}}^{\mu_2, \alpha_2})\) |
Note
Pair of indices \((l,m)\) 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 \(i_1\) (\(i_2\)).
In the case of \((2)\), index \(i_1 = x, y, z\), while index \(i_2\) runs over all pairs of integers \((l,m)\) of the original Hamiltonian.