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 σ2 is thoroughly investigated. The non-Kramers and Kramers states with the quantum number 1 ≤ J ≤ 8 in the axial crystal-fields of any multipolar composition but fixed σ2 are considered. Since the crystal-field Hamiltonian ℋ︁CF is a superposition of the three effective multipoles various ΔE can correspond to a fixed σ2 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 ΔEmax – ΔEmin, except the states with J ≤ 5/2, amounts roughly to σ. For comparison, the one-multipolar ℋ︁CFs 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 Δℰ (i.e. those preserving the σ2) with the (ΔEmin, ΔEmax) ranges occurring in the actual axial crystal-fields. Although the ΔE unlike the σ2 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. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)