THE JOURNAL OF CHEMICAL PHYSICS VOLUME 37 NUMBER 3 AUGUST I 1962 Intensities of Crystal Spectra of Rare Earth Ions* G. s. OFELT The Johns Hopkins University, Baltimore, Maryland (Received February 26 1962) Magnetic and electric dipole transitions between levels of the 4jx confi gurat ion per turbed by a static c ~ ~ s t a ~ l i n e field are ~ r e a t e d The expression obtained for the pure-electronic electric-dipole transition probability mvolves matnx elements of an even-order unit tensor between the two 4jx states involved in the transition. The contributions to the transition probability from interactions, via the crystalline field, with the nlj,94P-1, 4j<t-lnd, 4jX- l ng configurations are shown to add linearly, in such a manner as to multiply each odd k crystal-field parameter Akq by a constant. f J mixing" in the 4jx configuration is neglected t:J.J between the upper and lower 4jx level s is restricted to six units or less. f L mixing" is neglected tl:en t:J.L is also restricted to six units or less. Application is made to the fluor escen ce s pectra of PrC , and EuC ,. Many of the missing and weak transitions are explained. I INTRODUCTION O BSERVED spectra of rare -earth ions in crystalline fields consists mainly of electric-dipole transitions, although some magnetic-dipole transitions have been noticed. I ,2 The single ion in a static-field model has proved satisfactory in the explanation of much of the experimental data. In some cas es vibrational interaction has been included to account for lines not accounted for by the static model. 3 These extra transitions are usually called "vibronic-electronic transitions" to distinguish them from the "pure-electronic transitions" of the s tati c model. The theory for magnetic-dipole transitions is well known for intermediate coupling. 4 Only the detailed application for the 4fx crystal states need be made. This has been previously hampered by the lack of intermediate coupling and crystal-field wave functions. For electric-dipole transitions we shall consider only those transitions which are purely electronic in nature and therefore assume the static model. The states of the system may be labeled by the usual set of quantum numbers L 5, J, J z but it should be kept in mind that intermediate coupling removes Land 5 as good quantum numbers, and that the crystalline field also removes J and J. but leaves for the latter a weaker quantum number often called the crystal quantum number which corresponds to the irreducible representation of the symmetry group of the crystal field. In many cases the J mixing by the crystalline field is small, such that the level may be assigned a meaningful J value. There is experimental evidence that the electric dipole transitions originate from upper and lower levels which differ in t ,.J by more than one unit, as much as six units in some cases, with no general * This work was carried out with the partial support of the U. S. Air Force Office of Scientific Research. IE. V. Sayer and S. Freed, J. Chern. Phys. 24, 1213 (1956). 2 K. H. Hellwege, U. Johnsen, H. G. Kahle, and G. Schaack, z Physik 148, 112 (1957). 3 R. A. Satten, J. Chern. Phys. 27,286 (1957). 4 S. Pasternack, Astrophys. J. 92,129 (1940). trend of decrease in intensity with increase of t ,.J up to t ,.J=6. In the spectra of many of the rare-earth ions there are lines and groups of lines which, though not forbidden by crystal-symmetry selection rules, have not been observed. One possibility is that the transition probability between the corresponding states is small compared to that for observed lines. However, one cannot rule out the possibi lity of other, as yet unknown, selection rules which might forbid the transitions in question. Since we are concerned only with pureelectronic transitions, we shall consider contributions from configurations of opposite parity to the wave functions of the 4fx configuration by the odd parity terms of the crystalline potential expansion. We then may obtain nonzero matrix elements of the dipolemoment operator between the mixed-parity eigenfunctions. This calculation will be shown to clarify many of these questions. II. jx CONFIGURATION AND LINE STRENGTHS Adopting the static model and expanding the crystalline potential V in a spherical-harmonic series, we obtain the following expression: V = LAkqLrikVkQ(8i, C{ i , 1) kq i where ri is the radial coordinate of the ith electron. V k Q 8i, C{ i is the qth component of the spherical harmonic of order k and 8 i , C{ i are the angular coordinates of the ith electron. Akq are parameters which depend on the specific crystal-symmetry group. This potential may be separated into even parity (even values of k and odd parity (odd values of k parts. We denote these by V even k and Vodd k, respectively. The remainder of the Hamiltonian will be considered to consist of the electrostatic and spin-orbit interactions of the free ion, denoting it by Ho and noticing that Ho is an even parity operator. Thus, the perturbation Hamiltonian for the system is (2) 511 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/t ermsconditions. Downloaded to IP: 200.17.141.103 On: Fri, 16 Jan 2015 12:08:09
