Magnetic dipole-dipole interaction¶
Magnetic dipole-dipole interaction 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 |
|||
|---|---|---|---|
\(\mathcal{H}\) |
\[\mathcal{H}
=
-
\dfrac{1}{2}
\sum_{n\ne n^{\prime},\alpha\beta=1,2,3}
\dfrac{\mu_0\mu_B^2 g_n g_{n^{\prime}}}{2\pi}
\Biggl(
3\dfrac{(R_{n^{\prime}}^{\alpha} - R_{n}^{\alpha})(R_{n^{\prime}}^{\beta} - R_{n}^{\beta})}{|\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}|^5}
-
\dfrac{\delta_{\alpha\beta}}{|\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}|^3}
\Biggr)
\hat{J}_{\alpha}^n
\hat{J}_{\beta}^{n^{\prime}}\]
|
\[\mathcal{H}
=
\frac{1}{2}\frac{\mu_0}{4\pi}
\sum_{i \neq j} \mu_i \mu_j \frac{(\vec{n}_i \cdot \hat{r}_{ij}) (\vec{n}_j\cdot\hat{r}_{ij}) - (\vec{n}_i \cdot \vec{n}_j)}{{r_{ij}}^3}\]
|
\[\mathcal{H}
=
-\dfrac{\mu_0}{4\pi}
\sum_{ij}
\dfrac{
3(\mu_i \cdot \hat{r}_{ij}) (\mu_j \cdot \hat{r}_{ij}) - \mu_i \cdot \mu_j
}{r^3_{ij}}\]
|
Renaming of indices |
\(n \rightarrow (\mu_1, \alpha_1)\), \(n^{\prime} \rightarrow (\mu_2, \alpha_2)\), \(\alpha \rightarrow i_1\), \(\beta \rightarrow i_2\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(C_2\) |
\(-\dfrac{1}{2}\) |
\(\dfrac{1}{2}\) |
\(-1\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\[\dfrac{\mu_0\mu_B^2 g_{\mu_1, \alpha_1} g_{\mu_2, \alpha_2}}{2\pi}
\Biggl(
3\dfrac{(R_{\mu_2, \alpha_2}^{i_1} - R_{\mu_1, \alpha_1}^{i_1})(R_{\mu_2, \alpha_2}^{i_2} - R_{\mu_1, \alpha_1}^{i_2})}{|\mathbf{R}_{\mu_2, \alpha_2} - \mathbf{R}_{\mu_1, \alpha_1}|^5}
-
\dfrac{\delta_{i_1, i_2}}{|\mathbf{R}_{\mu_2, \alpha_2} - \mathbf{R}_{\mu_1, \alpha_1}|^3}
\Biggr)\]
|
\[\frac{\mu_0}{4\pi}\mu_{\mu_1, \alpha_1} \mu_{\mu_2, \alpha_2}
\frac{
\hat{r}_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1}
\hat{r}_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_2}
- \delta_{i_1, i_2}
}
{r_{\mu_1, \alpha_1; \mu_2, \alpha_2}^3}\]
|
\[\dfrac{\mu_0}{4\pi}
\frac{
3\hat{r}_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1}
\hat{r}_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_2}
- \delta_{i_1, i_2}
}
{r_{\mu_1, \alpha_1; \mu_2, \alpha_2}^3}\]
|
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(\hat{J}_{i_1}^{\mu_1; \alpha_1}\) |
\(n_{\mu_1, \alpha_1}^{i_1}\) |
\(\mu_{\mu_1, \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(\hat{J}_{i_2}^{\mu_2; \alpha_2}\) |
\(n_{\mu_2, \alpha_2}^{i_2}\) |
\(\mu_{\mu_2, \alpha_2}^{i_2}\) |