The limits of the total crystal-field splittings

Abstract

The crystal-fields causing $|J>$ electron states splittings of the same second moment $σ^{2}$ can produce different total splittings $ΔE$ magnitudes. Based on the numerical data on crystal-field splittings for the representative sets of crystal-field Hamiltonians ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$ with fixed indexes either $k$ or $q$, the potentials leading to the extreme $ΔE$ have been identified. For all crystal-fields the admissible ranges $(ΔE_{min},ΔE_{max})$ have been found numerically for $1\leq J\leq 8$. The extreme splittings are reached in the crystal-fields for which ${\cal H}_{\rm CF}s$ are the definite superpositions of the $C_{q}^{(k)}$ components with different rank $k=2,4$ and 6 and the same index $q$. Apart from few exceptions, the lower limits $ΔE_{min}$ occur in the axial fields of ${\cal H}_{\rm CF}(q=0)=B_{20}C_{0}^{(2)}+B_{40}C_{0}^{(4)}+B_{60}C_{0}^{(6)}$, whereas the upper limits $ΔE_{max}$ in the low symmetry fields of ${\cal H}_{\rm CF}(q=1)=B_{21}C_{1}^{(2)}+B_{41}C_{1}^{(4)}+B_{61}C_{1}^{(6)}$. Mixing the ${\cal H}_{\rm CF}$ components with different $q$ yields a secondary effect and does not determine the extreme splittings. The admissible $ΔE_{min}$ changes with $J$ from $2.00σ$ to $2.40σ$, whereas the $ΔE_{max}$ from $2.00σ$ to $4.10σ$. The maximal gap $ΔE_{max}-ΔE_{min}=2.00σ$ has been found for the states $|J=4>$. Not all the nominally allowed total splittings, preserving $σ^{2}=const$ condition, are physically available, and in consequence not all virtual splittings diagrams can be observed in real crystal-fields.