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 (10):

$\displaystyle {\hat \mathcal H}= B_l^m \hat O_{lm}({\hat \mathbf J}) - g_... ..._B {\hat \mathbf J} \cdot {\mathbf H} - {\hat \mathbf J} \cdot { \mathbf H^s}$ (10)

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

$\displaystyle \langle \hat \mathbf J \rangle = \sum_{i=1}^{2J+1} p_{i} \langle i \vert\hat \mathbf J \vert i \rangle$ (11)

with

	$\displaystyle p_{i}$	$\textstyle =$	$\displaystyle \frac{\exp(-E_{i}/kT)}{z}$	(12)
	$\displaystyle z$	$\textstyle =$	$\displaystyle \sum_{i=1}^{2J+1} \exp(-E_{i}/kT)$	(13)

Here $z$ is the partition sum, $\vert i\rangle$ the eigenstate corresponding to the eigenvalue $E_{i}$ of the Hamiltonian (10). ⁸