Capability of the Free-Ion Eigenstates for Crystal-Field Splitting
- 1
- 2
Abstract
Any electronic eigenstate of the paramagnetic ion open-shell is characterized by the three independent multipole asphericities for and 6 related to the second moments of the relevant crystal-field splittings by , where . The A<sub>k</sub> as the reduced matrix elements can serve as a reliable measure of the state capability for the splitting produced by the k-rank component of the crystal-field Hamiltonian. These multipolar characteristics allow one to verify any fitted crystal-field parameter set by comparing the calculated second moments and the experimental ones of the relevant crystal-field splittings. We present the multipole characteristics A<sub>k</sub> for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4 f <sup>N</sup> ions applying in the calculations the improved eigenfunctions of the free lanthanide ions obtained based on the M. Reid f-shell programs. Such amended asphericities are compared with those achieved for the simplified Russell-Saunders states. Next, they are classified with respect to the absolute or relative weight of A<sub>k</sub> in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the A<sub>k</sub> variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm<sup>3+</sup>, Er<sup>3+</sup>, Nd<sup>3+</sup>, and Pr<sup>3+</sup> ions. The detailed mechanisms of such A<sub>k</sub> changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.
- B. G. Wybourne, “Spectroscopic Properties of Rare Earths,” John Wiley, New York, 1965.
- B. R. Judd, “Operator Techniques in Atomic Spectroscopy,” Mc Graw-Hill, New York, 1963.
- A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” Princeton University Press, Princeton, New York, 1960.
- M. Rotenberg, R. Bivins, N. Metropolis and J. K. Wooten, Jr., “The 3-j and 6-j Symbols,” MIT Press, Cambridge, MA, 1963.
- J. Mulak and Z. Gajek, “The Effective Crystal-Field Potential,” Elsevier, Amsterdam, 2000.
- J. Mulak and M. Mulak, “Multipole Characteristic of the Open-Shell Electron Eigenstates,” Physica Status Solidi B, Vol. 245, No. 6, 2008, pp. 1156-1164. doi:10.1002/pssb.200743527
- M. Reid, “f-Shell Programs,” Private Communication by Courtesy of Z. Gajek, 2010.
- W.T. Carnall, G.L. Goodman, K. Rajnak and R.S. Rana, “A Systematic Analysis of the Spectra of the Lanthanides Doped into Single Crystal LaF3,” Journal of Chemical Physics, Vol. 90, No. 7, 1989, pp. 3443-3457. doi:10.1063/1.455853
- F. Auzel and O. L. Malta, “A Scalar Crystal Field Strength Parameter for Rare Earth Ions: Meaning and Usefulness,” Journal of Physique, Vol. 44, No. 2, 1983, pp. 201-206. doi:10.1051/jphys:01983004402020100
- R. P. Leavitt, “On the Role of Certain Rational Invariants in Crystal-Field Theory,” Journal of Chemical Physics, Vol. 77, No. 4, 1982, pp. 1661-1663. doi:10.1063/1.444088
- C. Rudowicz and J. Qin, “Noether’s Theorem and Conserved Quantities for the Crystal- and Ligand-Field Hamiltonians Invariant under Continuous Rotational Symmetry,” Physical Review B, Vol. 67, No. 17, 2003, pp. 174420+14.
- C. Rudowicz and J. Qin, “Can the Low Symmetry Crystal (Ligand) Field Parameters Be Considered Compatible and Reliable,” Journal of Luminescence, Vol. 110, No. 1-2, 2004, pp. 39-64. doi:10.1016/j.jlumin.2004.04.003
- Y. Y. Yeung, “Invariants and Moments,” In: D. J. Newman and B. Ng, Ed., Crystal Field Handbook, Cambridge University Press, Cambridge, MA, 2000, pp. 160-175. doi:10.1017/CBO9780511524295.010
- J. Mulak and M. Mulak, “On a Complementary Scale of Crystal-Field Parametrization,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 9, 2007, pp. 2063-2076. doi:10.1088/1751-8113/40/9/012
- I. G. Kaplan, “Symmetry of Many Electron Systems,” Academic Press, New York, 1975.