The total energy splitting of ionic eigenstates in the axial crystal fields

Abstract

The relationship between the energy total splitting $ΔE$ of the free-ion electron states in the axial crystal-fields and the second moment of that splitting $σ^{2}$ is thoroughly investigated. The non-Kramers and Kramers states with the quantum number $1\leq J \leq 8$ in the axial crystal-fields of any multipolar composition but fixed $σ^{2}$ are considered. Since the crystal-field Hamiltonian ${\cal H}_{\rm CF}$ is a superposition of the three effective multipoles various $ΔE$ can correspond to a fixed $σ^{2}$ according to the resultant combination of the independent contributions. This $ΔE$ variation range is the subject of the study. For the states under examination $ΔE$ can take the values from $2.00σ$ to $3.75σ$, whereas the difference $ΔE_{max}- ΔE_{min}$, except the states with $J\leq 5/2$, amounts roughly to $σ$. For comparison, the one-multipolar ${\cal H}_{\rm CF}$s yield accurately defined $ΔE$ ranging from $2.50σ$ to $3.00σ$. The limitations of the allowed $ΔE$ values exclude rigorously a number of virtually possible splitting diagrams. The documentary evidence for this restriction has been supplied in the paper collating the nominally admissible total energy splittings $Δ{\cal E}$ (i.e. those preserving the $σ^{2}$) with the $(ΔE_{min}, ΔE_{max})$ ranges occurring in the actual axial crystal-fields. Although the $ΔE$ unlike the $σ^{2}$ is not an essential characteristic and depends on the reference frame orientation, it is useful to know its dispersion range, particularly attempting to assign or verify complex electron spectra.