Module Tasks:

As a single ion module, so1ion provides to mcphas the magnetic properties of a single rare earth ion subject to the crystal field. Its main duty is to calculate the magnetic moment given an effective magnetic field. To be more explicit - given the effective magnetic field $H^s_{eff}$ by mcphas the module so1ion diagonalises the crystal field and Zeeman Hamiltonian (9) of the ion $n$:

$${\mathcal H_n}= B_l^m O_{lm}({\mathbf J}^n) - g_{Jn} \mu_B {\mathbf J}^n {\mathbf H^n_{eff}}$$ (9)

and calculates the expectation value of the angular momentum $\langle \mathbf J^n \rangle$ according to

$$\langle \mathbf J^n \rangle = \sum_{\Gamma} p_{\Gamma} \langle \Gamma \vert \mathbf J^n \vert \Gamma \rangle$$ (10)

with

$$p_{\Gamma} = \frac{\exp(-E_{\Gamma}/kT)}{z}$$ (11)

$$z = \sum_{\Gamma} \exp(-E_{\Gamma}/kT)$$ (12)

Here $z$ is the partition sum, $\vert\Gamma\rangle$ the eigenstate corresponding to the eigenvalue $E_{\Gamma}$ of the Hamiltonian (9). ¹¹