The limits of the total crystal-field splittings

TL;DR

This study investigates the maximum and minimum total crystal-field splittings for electron states with fixed second moments, identifying the specific potentials that produce these extreme values across various angular momentum states.

Contribution

The paper numerically determines the admissible ranges of total crystal-field splittings and identifies the specific superpositions of crystal-field components that lead to these extremes.

Findings

Extreme splittings occur in specific superpositions of crystal-field components.

Admissible splitting ranges depend on the total angular momentum J.

Maximum splitting gap is 2.00σ for J=4 states.

Abstract

The crystal-fields causing $|J>$ electron states splittings of the same second moment $\sigma^{2}$ can produce different total splittings $\Delta E$ 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 $\Delta E$ have been identified. For all crystal-fields the admissible ranges $(\Delta E_{min},\Delta E_{max})$ 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 $\Delta E_{min}$ 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 $\Delta E_{max}$ 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 $\Delta E_{min}$ changes with $J$ from $2.00\sigma$ to $2.40\sigma$, whereas the $\Delta E_{max}$ from $2.00\sigma$ to $4.10\sigma$. The maximal gap $\Delta E_{max}-\Delta E_{min}=2.00\sigma$ has been found for the states $|J=4>$. Not all the nominally allowed total splittings, preserving $\sigma^{2}=const$ condition, are physically available, and in consequence not all virtual splittings diagrams can be observed in real crystal-fields.