A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

Abstract

An algebraic, calculational, non‐fitting method of parametrization of the cubic and hexagonal crystal‐field (CF) Hamiltonians false(HCFfalse) with only two independent CF parameters B40 and B60 is proposed. It is based on the linear relation σ2false(Jfalse)=∑kAk2false(Jfalse)Sk2 between σ2false(Jfalse) the square of the second moment of the electron state false|J〉 CF splitting and the squares of the multipolar CF strengths Sk2=12k+1∑qfalse|Bkq|2, where AkJ=〈Jfalse|false|Cfalse(kfalse)false|false|J〉. For a pair of false|Ji〉 and false|Jj〉 electron states an algebraic solution for S42 and S62, and hence for the B40 and B60 CF parameters can be gained. The necessary conditions for physical correctness of the solutions false(Sk2≥0false) impose limitations on the observed σ2false(Jifalse)/σ2false(Jjfalse) ratios depending on the respective ratios Ak2false(Jifalse)/Ak2false(Jjfalse). These restrictions can be used to verify postulated CF splitting diagrams. In the high‐symmetry crystal fields, the electron states with J=2 or 5/2 undergo splitting only by the k=4 HCF multipole, and from their CF splitting second moments the B40 parameter can be directly obtained. For J=2, false|B40false|=6310Δfalse(Jfalse)false|A4false(Jfalse)false|, whereas for J=5/2, false|B40false|=7Δfalse(Jfalse)false|A4false(Jfalse)false|, where Δ is the energy gap between the two Stark's levels of the false|J〉 state. The proposed method is demonstrated and discussed for several systems of RE3+ ions in some high‐symmetry crystal matrices.

The crystal-field splitting third moment in determination of the crystal-field Hamiltonian parametrization. The cubic symmetry case

“…1 Introduction Our previous paper [1] was concerned with the use of independent linear equations between the squares of crystal-field (CF) splitting second moments σ 2 2 (J) for two free-ion states |J (see Section 2), and the squares of the relevant CF strengths in order to determine the crystalfield parameters (CFP) in the high symmetry systems. The multipolar additivity of the second moment square σ 2 2 (J) was directly applied.…”

“…In the case of the cubic crystal-field Hamiltonians (H CF ), which need only two parameters B 40 and B 60 , our method leads directly to the complete parametrization of the H CF . An anonymous referee of the paper [1] drew our attention to the fact that the third CF splitting moments, as well as higher order ones, can be also employed to this end allowing us to limit the experimental data to only one individual free-ion state [2].…”