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

Abstract

A physically valid parametrization of the crystal-field (CF) Hamiltonian H CF has to reproduce not only the energy of Stark sublevels but also the multipolar structure of the central ion surroundings. The square of the second moment s 2 J j i ð Þ of J j i electronic state CF splitting must be a superposition of squares of the second moments of the partial splittings produced separately by the individual multipoles:is the k-rank CF strength. The relationship between s 2 J j i ð Þ known from experiment and S 2 k obtained from fitting procedure allows us to verify whether the parametrization is physically valid. It is demonstrated by several examples of CF splittings of electronic states of trivalent lanthanide ions in different crystal matrices. The approach is confined only to the states with good quantum numbers J. Nevertheless, by finding such states in any analyzed spectrum, the method can provide the actual multipole characteristics of H CF , and plays the role of a preparametrization. In general, the stepwise fitting procedure can never lead to parametrizations with a correct multipolar structure due to the limitation of the initial eigenfunctions. In consequence, the pertinent crystalfield parameters (CFPs) are burdened with a methodical inherent fault. The loss of physical meaning of fitted CFPs, its source and consequences, constitute the main theme of the paper.

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

“…The B 40 calculated in this way for five such states of various RE 3+ ions in the hexachloroelpasolite crystal matrix are presented in Table 1 (column 5). Taking into account a noticeable diversity in the B 40 found in the literature [11,[14][15][16][17], the calculated magnitudes seem to be acceptable. There are at least two reasons of potential weaknesses of the method:…”

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

“…The procedure of the verification loop [21] for the second and third moment, separately, was applied for individual freeion states |J in order to obtain two independent equations for the two unknowns B 40 and B 60 . Therefore, the method boils down to a comparison of the σ 2 2 and σ 3 3 found from the experimental CF splitting diagrams (Eq.…”

“…B 40 : 1938 [15], 2050 [21], 2059 [16], 2171 [17], 2290 [19], 2326 [18] B 60 : 236 [19], 247 [18], 250 [17], 290 [15], 290 [ [22], 1492 [15], 1494 [19], 1497 [21], 1531 [16], 1555 [20] B 60 : 140 [21], 153 [22], 154 [20], 156 [19], 160 [16], 163 [15], 174 [17]…”

Comment on the Crystal‐Field Analysis Underlying “Breakdown of Crystallographic Site Symmetry in Lanthanide‐Doped NaYF4 Crystals”

“…To distinguish between such sets and the sets that likely correspond to global minima, other criteria must be utilized. An important criterion for a correct CF parameterization is agreement of the second moments s 2 (SLJ) [9,10] determined using the experimental values of the splitting of a given SLJ-multiplet (s 2 (expt)), and the CFP values B kq (s 2 (calc)). Appropriate expressions are provided in Equations (S4)-(S6) in the Supporting Information.…”