Module Tasks:

The 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 external magnetic field $\mathbf H$ and an exchange field $\mathbf H^s$. To be more explicit - given the external magnetic and the exchange field by mcphas the module so1ion diagonalises the crystal field and Zeeman Hamiltonian (9):

$${\hat \mathcal H}= B_l^m \hat O_{lm}({\hat \mathbf J}) ... ... J} \cdot {\mathbf H} - {\hat \mathbf J} \cdot { \mathbf H^s}$$ (9)

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

$$\langle \hat \mathbf J \rangle = \sum_{i=1}^{2J+1} p_{i} \langle i \vert\hat \mathbf J \vert i \rangle$$ (10)

with

$$p_{i} = \frac{\exp(-\epsilon_{i}/kT)}{z}$$ (11)

$$z = \sum_{i=1}^{2J+1} \exp(-\epsilon_{i}/kT)$$ (12)

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