The limits of the total crystal-field splittings

Abstract

The crystal-fields causing |J electron states splittings of the same second moment σ 2 can produce different total splittings ∆E magnitudes. Based on the numerical data on crystal-field splittings for the representative sets of crystal-field Hamiltonians H CF = k q B kq C Mixing the H CF components with different q yields a secondary effect and does not determine the extreme splittings. The admissible ∆E min changes with J from 2.00σ to 2.40σ, whereas the ∆E max from 2.00σ to 4.10σ. The maximal gap ∆E max − ∆E min = 2.00σ has been found for the states |J = 4 . Not all the nominally allowed total splittings, preserving σ 2 = const condition, are physically available, and in consequence not all virtual splittings diagrams can be observed in real crystal-fields.

Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

“…The electron eigenstates asphericities A k allows us to verify the fitted CFP sets comparing the calculated (Eq. ( 3)) and experimental second moments of the splitting [20]. The fitted CFP sets that well reproduce the experimental spectrum of energy levels for intentionally approximated initial eigenfunctions have by definition an effective character.…”

Terminological confusions and problems at the interface between the crystal field Hamiltonians and the zero-field splitting Hamiltonians—Survey of the CF=ZFS confusion in recent literature

“…Moreover, the symbols [63] B k q are confusingly akin to the CFPs in the ESO notation B q k . As an aside, we note that concerning the physical validity of a particular CF parameterization, Mulak and Mulak have proposed the method based on the second moments [64] and the criterion whether CF parameterization reproduces or not the actual multipolar structure of the surrounding CF [65]. It is worthwhile to note that the orthorhombic standardization may also be employed for the electric quadrupole 'interaction' Hamiltonians of the type H Q ¼ I Á Q Á I for systems with the nuclear spin quantum number IZ 1.…”

Verification of the fitted crystal‐field Hamiltonian parametrization by the crystal‐field splitting second moment

“…The second moment is an invariant of the R 3 rotation group and may be treated as a more reliable measure of CF interaction than the total CF splitting Δ E 24. There are two ways to obtain $\sigma {}^{2} $ , either from spectroscopic (or other experimental) data, or by calculations using Eq.…”

“…The total splitting Δ E of any electron state can serve as an auxiliary (although uncertain) measure of the relevant $\sigma {}^{2} $ 24. As a rule, the bigger $\sigma {}^{2} $ , the bigger Δ E .…”