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

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_{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.

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

KEYWORDS

Crystal-Field Theory, Crystal-Field Splitting, Rare-Earth Free-Ion Eigenstates, Rare-Earth Ions

Crystal-Field Theory, Crystal-Field Splitting, Rare-Earth Free-Ion Eigenstates, Rare-Earth Ions

Cite this paper

nullJ. 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.

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

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

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[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

[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