The Second and Third Crystal Field Splitting Moments in Verification of the Trigonal and Hexagonal Crystal Field Parametrizations

Abstract

The paper, as the third in a series, presents the complete formulas for the crystal field splitting third moment of free ion states as functions of the relevant crystal field parameters in the case of the trigonal and hexagonal crystal fields. The additive structure of the expressions consists of 21 independent components for the trigonal symmetry, 12 for the hexagonal one and only 4 components for the ideal hexagonal coordination. The two scalar characteristics of the crystal field Hamiltonian HCF – the crystal field splitting second and third moments – enable us to verify the HCF parametrizations solely in an algebraic way. Such verification concerns the multipolar structure of the parametrizations as well as the signs of the crystal field parameters. Both the crystal field splitting moments can ensure us in what degree the comprehensive HCF parametrizations are adequate with respect to the individual ionic states of the electron configuration. As an example two chosen HCF parametrizations, one for the hexagonal system Nd3+:LaCl3 of the C3h symmetry, and the second for the trigonal system Tb3+:Y2O2S of the C3v symmetry are subjected to in depth analysis. The presented approach provides a critical insight into the crystal field parametrizations from a different point of view, i.e., omitting the conventional fitting procedure. After the extended verification the studied classical HCF parametrizations withstand, in principle, the test of time.