Scientific Research

An Academic Publisher

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

**Author(s)**Leave a comment

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

_{k}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_{k}for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4 f^{N}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_{k}in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the A_{k}variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm^{3+}, Er^{3+}, Nd^{3+}, and Pr^{3+}ions. The detailed mechanisms of such A_{k}changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

J. Mulak and M. Mulak, "Capability of the Free-Ion Eigenstates for Crystal-Field Splitting,"

*Journal of Modern Physics*, Vol. 2 No. 11, 2011, pp. 1373-1389. doi: 10.4236/jmp.2011.211170.

[1] | B. G. Wybourne, “Spectroscopic Properties of Rare Earths,” John Wiley, New York, 1965. |

[2] | B. R. Judd, “Operator Techniques in Atomic Spectroscopy,” Mc Graw-Hill, New York, 1963. |

[3] | A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” Princeton University Press, Princeton, New York, 1960. |

[4] | M. Rotenberg, R. Bivins, N. Metropolis and J. K. Wooten, Jr., “The 3-j and 6-j Symbols,” MIT Press, Cambridge, MA, 1963. |

[5] | J. Mulak and Z. Gajek, “The Effective Crystal-Field Potential,” Elsevier, Amsterdam, 2000. |

[6] | 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 |

[7] | M. Reid, “f-Shell Programs,” Private Communication by Courtesy of Z. Gajek, 2010. |

[8] | 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 |

[9] | 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 |

[10] | 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 |

[11] | 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. |

[12] | 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 |

[13] | 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 |

[14] | 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 |

[15] | I. G. Kaplan, “Symmetry of Many Electron Systems,” Academic Press, New York, 1975. |

[16] | S. Hüffner, “Optical Spectra of Transparent Rare Earths Compounds,” Academic Press, New York, San Francisco, London, 1978. |

[17] | S. G. Redsun, “3-j, 6-j, 9-j Symbol Calculators,” Accessed in January 2011. http://www.svengato.com (postware application). |

[18] | C. W. Nielson and G. F. Koster, “Spectroscopic Coefficients for pn, dn and fn Configurations,” MIT Press, Cambridge MA, 1963. |

[19] | J. S. Griffith, “The Theory of Transition-Metal Ions,” Cambridge University Press, London, New York, 1961. |

[20] | J. Mulak and M. Mulak, “A Fundamental Requirement for Crystal-Field Parametrization,” Physica Status Solidi B, Vol. 248, No. 9, 2011, pp. 2159-2164. |

[21] | R. P. Leavitt, J. B. Gruber, N. C. Chang, C. A. Morrison, “Optical Spectra, Energy Levels, and Crystal-Field Analysis of Tripositive Rare-Earth Ions in Y2O3. II. Non- Kramers Ions in C2 Sites,” Journal of Chemical Physics, Vol. 76, No. 10, 1982, pp. 4775-4788. doi:10.1063/1.442796 |

[22] | S. S. Bishton and D. J. Newman, “Parametrization of the Correlation Crystal Field,” Journal of Physics C: Solid State Physics, Vol. 3, No. 8, 1970, pp. 1753-1761. doi:10.1088/0022-3719/3/8/014 |

[23] | D. J. Newman, “Theory of Lanthanide Crystal Field,” Advances in Physics, Vol. 20, No. 84, 1971, pp. 197-256. doi:10.1080/00018737100101241 |

[24] | M. F. Reid, “Correlation Crystal Field Analyses with Orthogonal Operators,” Journal of Chemical Physics, Vol. 87, No. 5, 1987, pp. 2875-2884. doi:10.1063/1.453075 |

Copyright © 2019 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.