Zeeman energy

When an external magnetic field is applied the term of the form

  $$\hat{\mathcal{H}}_{Z-J}(n) = - \sum_{\alpha=1,2,3} g_n \mu_B \hat{J}^{n}_{\alpha} H_{\alpha}$$ (6)

where $\hat{\mathbf{J}}^n$ is a total angular momentum operator, is included.

Or the term of the form

  $$\hat{\mathcal{H}}_{Z-LS}(n) = - \sum_{\alpha=1,2,3} \mu_B \Bigl( 2 \hat{S}^n_{\alpha} + \hat{L}^n_{\alpha} \Bigr) H_{\alpha}$$ (7)

where $\hat{\mathbf{S}}^n$ and $\hat{\mathbf{L}}^n$ are the (inverse) spin and orbital angular momentum operators of the ion $n$, respectively.

Note that by "external magnetic field $\mathbf{H}$" we refer to the magnetic field in the sample, which is the applied field (e. g. generated by a coil) diminished by the demagnetizing tensor $\overline{n}_{{\rm demag}}$ times the magnetisation $\mathbf{M}$ of the sample, i. e. $\mathbf{H} = \mathbf{H}_{{\rm applied}} - \overline{n}_{{\rm demag}} \mathbf{M}$, e. g. for a spherical sample $\overline{n}_{{\rm demag}} = 1/3\,\overline{I}$, where $\overline{I}$ is the identity matrix. The unit of $\mathbf{H}$ is usually chosen to be Tesla, which actually refers to the quantity $\mu_0 \mathbf{H}$ (in SI units).