Crystal Field and Parameter Conventions

For historical reasons, crystal field parameters (effectively the radial matrix elements of the crystal field interactions) may be expressed in two different "normalisation", which we shall call Stevens and Wybourne. Stevens [53,25] initially expressed the radial parts of the crystal field interaction in terms of angular momentum operators $J_x$, $J_y$, $J_z$. He did this by taking the Cartesian expressions for the tesseral harmonic functions (see Appendix F), and replacing all instances of the coordinates $x$, $y$, and $z$ with $J_x$, $J_y$ and $J_z$ and allowing for the commutation relations of the angular momentum operators, but without considering the normalisation condition of these functions and hence are missing the prefactors before the square brackets in the expressions in Appendix F. We denote these prefactors $p_{lm}$. The Stevens crystal field Hamiltonian is thus

\mathcal{H}_{\mathrm{cf}} = \sum_{l,m} A_{lm} \langle r^l \r... J \vert\vert \theta_l \vert\vert J \rangle \hat{O}_l^m (J)

where the product $A_{lm} \langle r^l \rangle$ is commonly taken in the literature as the crystal field parameter, because the factorisation into an intrinsic parameter $A_{lm}$ and the expectation value of the radial wavefunction $\langle r^l \rangle$ is derived from the point charge model and is not generally valid. Alternatively, the product $B_l^m = A_{lm} \langle r^l \rangle \langle J \vert\vert \theta_l \vert\vert J \rangle$ is also commonly used, particularly in the neutron scattering literature. $ \theta_l= \langle J \vert\vert \theta_l \vert\vert J \rangle$ are the Stevens factors: for $l=0,2,4,6$ these correspond to the number of electrons in the unfilled shell $\nu,\alpha_J,\beta_J,\gamma_J$, respectively.

Wybourne [43] and subsequent co-authors on the other hand chose to use the tensor operators $\hat{C}_{lm}$ which transform in the same way as the functions $C_{lm}(\theta,\phi) = \sqrt{4\pi / (2l+1)}
Y_{lm}(\theta,\phi)$, where $Y_{lm}(\theta,\phi)$ are the spherical harmonic functions, to describe the crystal field. Thus the angular-dependent part of the crystal field matrix elements used by Wybourne differed from that of Stevens by the factor $\lambda_{l0} = p_{l0} \sqrt{4\pi / (2l+1)}$ and $\lambda_{lm} = p_{lm} \sqrt{8\pi / (2l+1)}$ for $l\neq 0$. The crystal field Hamiltonian used by Wybourne is thus (in our notation)

\mathcal{H}_{\mathrm{cf}} = \sum_{l,m} D_l^{m} \hat{C}_{lm}

The disadvantage of the Wybourne approach is that one requires imaginary crystal field parameters, because the tensor operators $\hat{C}_{lm}$ are not Hermitian. In McPhase, we have instead chosen to use slightly different tensor operators $\hat{T}_{lm}$, which are the Hermitian combinations of the $\hat{C}_{lm}$,

\hat{T}_{l0} = \hat{C}_{l0}, \qquad \hat{T}_{l,\pm\vert m\v...
...vert} \pm (-1)^{\vert m\vert} \hat{C}_{l,\vert m\vert} \right]

giving the Hamiltonian

\mathcal{H}_{\mathrm{cf}} = \sum_{l,m} L_l^m \hat{T}_{lm}

Our $L_l^m$ parameters therefore have the same normalisation as the Wybourne parameters but will be real.

In summary:

  1. the various types of crystal field parameters are given in table 5.
  2. crystal field parameters may have different units, in McPhase we use meV.
  3. crystal field coordinate systems used in literature may be different from the convention used by single ion modules so1ion,ic1ion,icf1ion, which is the crystal field axes $xyz$ are such that $x\vert\vert a,y\vert\vert b,z\vert\vert c$ in case of orthogonal lattices and in case of non-orthogonal axes the convention is $y\vert\vert\vec b$, $z\vert\vert(\vec a \times \vec b)$ and $x$ perpendicular to $y$ and $z$.

Table 5: Different Crystal Field Parameter Notation Schemes in Literature. Here half($m$) = $\frac {1}{2}$ when $m\neq 0$ and half($m$)=1 when $m=0$. Note that lm should be replaced by numbers - e.g. an example of Alm is A20, and that the operator equivalent factor $\langle J \vert\vert \theta _l \vert\vert J \rangle $ [25] should be $\langle L \vert\vert \theta _l \vert\vert L \rangle $ for an $L$-manifold which is the default for $d$-electron ions [46]. The constants $\lambda _{lm}$ are given in table 6.
 Literature Parameter Relation to $L_l^m$ Normalisation
   name in    
 $A_l^m$ in Hutchings 1964 [25] Alm = $ \lambda_{lm} L_l^m /\langle r^l \rangle$ Stevens
 $A_l^m$ in Kassman 1970 [54]      
 $A_l^m$ in Abragam 1970 [46]      
 $B_l^m$ in Hutchings 1964 [25] Blm = $ \left \{ \begin{array}{lr} \lambda_{lm} L_l^m \langle J \vert\vert \theta_l \v...
...rt \theta_l \vert\vert L \rangle & {\rm for~trans~metals}
\end{array} \right .$ Stevens
 $B_l^m$ in Jensen 1991 [1]      
 $B^l_m$ in Mulak 2000 [55]      
 $B^l_m$ in FOCUS      
 $B^l_m$ in AMOS      
 $V_l^m$ in Elliott 1953 [56]   = $ \lambda_{lm} L_l^m $ Stevens
   Wlm = half($m$) $ \lambda_{lm} L_l^m /\langle r^l \rangle$ Stevens
   Vlm = half($m$) $ \lambda_{lm} L_l^m \langle J \vert\vert \theta_l \vert\vert J \rangle$ Stevens
   Llm = $L_l^m$ Wybourne
  $\begin{array}{l}B^l_0 \\ B^l_m(c) \\ B^l_m(s) \end{array} $ in SPECTRE   = $\left \{ \begin{array}{l}L_l^0 \\ L_l^m \\ -L_l^{-m} \end{array} \right .$ with $m>0$ Wybourne
 $B_m^l$ in Wybourne 1965 [43]24 Dlm = $ \left \{ \begin{array}{lr} L_l^0 & m=0 \\
(-1)^m(L_l^m-iL_l^{-m}) & m >0 \\
L_l^{-m}+iL_l^{m} &m<0 \\
\end{array} \right . $ Wybourne
 $B_m^l$ in Kassman 1970 [54]      
 $B_{lm}$ in Mulak 2000 [55]      
 $Ak(),CAk()$ in XTLS      
 $B_m^l$ in Newman 2000 [57]   = $(-1)^m L_l^m$ Wybourne

Table 6: Ratios $\lambda _{lm}$ of the Stevens to the real valued Wybourne normalised parameters. $\lambda _{lm} \equiv B_l^m/(\theta _lL_l^m)= A_{lm}\langle r^l \rangle / L_l^m$. $\lambda _{lm}$ are related to the prefactors in the tesseral harmonics (see appendix F) $p_{lm}$ by $\lambda _{l0} =\sqrt {\frac {4\pi }{2l+1}}\vert p_{l0}\vert$ and $\lambda _{lm} =\sqrt {\frac {8\pi }{2l+1}}\vert p_{lm}\vert$ for $m\neq 0$ [57, note that Newman on p.30, equ (2.7) defines his real valued Wybourne parameters $B_m^l({\rm Newman})\equiv (-1)^m L_l^m$].
$\vert m\vert$
$l$ $0$ $1$ $2$ $3$ $4$ $5$ $6$
$0$ $1$            
$2$ $\frac {1}{2}$ $\sqrt{6}$ $\frac{1}{2}\sqrt{6}$        
$4$ $\frac{1}{8}$ $\frac{1}{2}\sqrt{5}$ $\frac{1}{4}\sqrt{10}$ $\frac{1}{2}\sqrt{35}$ $\frac{1}{8}\sqrt{70}$    
$6$ $\frac{1}{16}$ $\frac{1}{8}\sqrt{42}$ $\frac{1}{16}\sqrt{105}$ $\frac{1}{8}\sqrt{105}$ $\frac{3}{16}\sqrt{14}$ $\frac{3}{8}\sqrt{77}$ $\frac{1}{16}\sqrt{231}$

Martin Rotter 2022-03-11