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

TL;DR

This paper investigates the relationship between total energy splitting and its second moment in ionic eigenstates within axial crystal fields, revealing the range of possible splittings and their restrictions based on multipolar contributions.

Contribution

It provides a detailed analysis of the variation range of energy splittings for different states and multipolar compositions, enhancing understanding of spectral assignments in crystal field theory.

Findings

Energy splitting ranges from 2.00σ to 3.75σ for examined states.

Difference between maximum and minimum splitting is roughly σ for states with J>5/2.

Restrictions exclude certain splitting diagrams, refining spectral analysis.

Abstract

The relationship between the energy total splitting $\Delta E$ of the free-ion electron states in the axial crystal-fields and the second moment of that splitting $\sigma^{2}$ 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 $\sigma^{2}$ are considered. Since the crystal-field Hamiltonian ${\cal H}_{\rm CF}$ is a superposition of the three effective multipoles various $\Delta E$ can correspond to a fixed $\sigma^{2}$ according to the resultant combination of the independent contributions. This $\Delta E$ variation range is the subject of the study. For the states under examination $\Delta E$ can take the values from $2.00\sigma$ to $3.75\sigma$, whereas the difference $\Delta E_{max}- \Delta E_{min}$, except the states with $J\leq 5/2$, amounts roughly to $\sigma$. For comparison, the one-multipolar ${\cal H}_{\rm CF}$s yield accurately defined $\Delta E$ ranging from $2.50\sigma$ to $3.00\sigma$. The limitations of the allowed $\Delta E$ 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 $\Delta {\cal E}$ (i.e. those preserving the $\sigma^{2}$) with the $(\Delta E_{min}, \Delta E_{max})$ ranges occurring in the actual axial crystal-fields. Although the $\Delta E$ unlike the $\sigma^{2}$ 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.