Crystal field

Single-ion crystal field is written with Stevens operators as

  $$\hat{\mathcal{H}}_{CF1-S}(n) = \sum_{lm} B_l^m \hat{O}_{lm}(\mathbf{J}^n)$$ (13)

or with Wybourne operators $\hat{T}_{lm}^n$ and Wybourne parameters $L_l^m$ as

  $$\hat{\mathcal{H}}_{CF1-W}(n) = \sum_{lm} L_l^m(n) \hat{T}_{lm}^{n}$$ (14)

where $\hat{T}_{lm}^n$ is an operator equivalents of real valued spherical harmonic functions

	$$\hat{T}_{l0}$$	$=$	$$\sqrt{4\pi/(2l+1)} \sum_i Y_{l0}(\Omega_{i_n}),$$
	$$\hat{T}_{l,\pm\vert m\vert}$$	$=$	$$\sqrt{4\pi/(2l+1)} \sum_i \sqrt{\pm 1} [Y_{l,-\vert m\vert}(\Omega_{i_n}) \pm (-1)^m Y_{l,\vert m\vert}(\Omega_{i_n})]$$	(15)

for the ion $n$, where index $i_n$ runs over all electrons in the open shell of an ion (d, f). Two-ion crystal field can be written with the Stevens operators as

  $$\hat{\mathcal{H}}_{CF2-S} = -\frac{1}{2} \sum_{nn^{\prime}}... ..._{lm}(\mathbf{J}^n) \hat{O}_{l^{\prime}m^{\prime}}(\mathbf{J}^{n^{\prime}})$$ (16)

or with the Wybourne operators as

  $$\hat{\mathcal{H}}_{CF2-W} = - \frac{1}{2} \sum_{nn^{\prim... ...me}) \hat{T}_{kq}^n \hat{T}_{k^{\prime}q^{\prime}}^{n^{\prime}} \Biggr]$$ (17)

where operators are defined as $\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{O}_{lm}(\mathbf{J}^n)$ or $\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{T}_{kq}^n$ and index $\alpha$ runs over $m$ pairs of $(l,m)$ or $(k,q)$ respectively. Interaction parameters are either $\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{ll^{\prime}}^{mm^{\prime}}(nn^{\prime})$ or $\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow \mathcal{K}_{kk^{\prime}}^{qq^{\prime}}(nn^{\prime})$, where index $\beta$ runs over $m$ pairs of $(l^{\prime},m^{\prime})$ or $(k^{\prime},q^{\prime})$ respectively.