Chapter 73 Chapter 73
7. Ligand effects
7.6 Correlation crystal fields
host crystal Cs2NaYCI 6 for their analysis of the 4f---, 5d transitions of C e 3 +. They found that the vibronic structure could be explained almost entirely in terms of progressions based on the breathing mode of the CeC16 complex. Schwartz (1975) turned his attention to transitions within the 4f shell by studying the M C D of the vibronic transitions associated with 7FI(Tlg)__,SDI(TIg) and 7Fo(Alg)+ 7FI(Tlg)---~5D2(T2g+E ) of Cs2NaEuC16. The signs of the characteristic M C D parameters proved useful in distinguishing the t~u and t2u vibra- tions. Similar techniques were applied to Cs2NaPrC16 (Schwartz 1976) and to the magnetic-dipole and vibronically induced electric-dipole transitions associated with 7Fj ~ 5D4 of Tb 3 + 4f 8 in Cs2NaTbC16 (Schwartz et al. 1977). A curious feature of the latter is the absence of vibronic lines when A J is odd (as with 5D 4 ~ 7 F 3 and 7F5). The detailed intensity calculation by Faulkner and Richardson (1978b) referred to in section 7.4 accounted rather well for the relative intensities of the dif- ferent kinds of transitions.
A certain amount of M C D work has been carried out for lanthanides in aqueous solution, from which a number of deductions can be made about the local site symmetry (G6rller-Walrand and G o d e m o n t 1977a,b; G6rller-Walrand et al. 1982).
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 147
Bk2k. kq "= (--1)qBk~k2). Bishton and N e w m a n (1970) used time-reversal invariance (which is equivalent to the demand that the total Hamiltonian be real) to further constrain the parameters. Even so, there is an e n o r m o u s increase in their n u m b e r compared to the
Akq.
F o r octahedral symmetry, the two standard single-electron parameters (for k = 4 and 6) are increased by 32 more. Sites with low point sym- metries may require several hundred two-electron parameters FIklk2 ~kq for a complete analysis. It is thus crucial to use physical arguments whenever possible to reduce their number. Bishton and N e w m a n (1970) pointed out that k is necessarily even for C~v point symmetry, a result that can be quickly understood because any tensor T~ k~ for odd k would necessarily split the doublets I+_ML~ that arise in cylindrical symmetry. Since the rotation of such a tensor preserves k, we can deduce that the superposition model, applied to any site symmetry G, limits the tensors (99) that we need consider to those for which k is even. The simplification that this argument affords is comparatively slight, however; for example, the 32 two-electron octa- hedral parameters are only reduced to 31 as the single octahedral scalar belonging to k = 9 drops out (Judd 1977a).The a p p r o a c h via a general two-electron parameterization is not the only one we can take. As early as the mid-1960's, Rajnak and Wybourne (1964a) considered a number of perturbation mechanisms in which the inter-electronic C o u l o m b poten- tial is introduced to mix other configurations into 4f N. Some mechanisms merely lead to overall changes in the Akq, but there are others - such as 4f s ~ 4f N- ~f' - for which the cross-terms of the single-electron crystal-field potential between f and f' lead to effective two-electron operators. Since the C o u l o m b interaction commutes with S and L, all sublevels and levels deriving from a single SL term can be treated by means of the familiar parameters Akq , which are n o w term-dependent. This seemed to offer considerable scope for an enlarged parameterization. However, it is still severely restricted c o m p a r e d to the general treatment based on the operators given by (99); in particular, no tensors for which k > 6 occur. Such freedom as remains was found to be excessive by Rajnak and K r u p k e (1967), who carried out separate parameterizations of the 5G and 5I multiplets of H O 3+ 4f 1° in HoC13 and discovered that they both led to virtually identical sets of parameters Akq.
7.6.1 Spin-correlated crystal fields
The first efforts to construct a simplified parameterization relied on the very slight dependence of the radial wavefunction R(r) of a 4f electron on the associated spin orientation ms. There are strong attractive exchange forces between 4f electrons whose spins are similarly directed, and these forces should lead to less extended radial wavefunctions and hence to different Akq for different 4f orbitals. Such variations in R(r) are excluded, of course, from the central-field approximation on which all of shell theory is based; but it is not difficult to find a species of configuration interaction whose effects, in part, mimic the effects of the exchange forces. In fact, the mechanism f ~ f' mentioned above preserves the f character of the orbitals and can only m a k e itself felt through an equivalent variation in R(r). In his efforts to account for the ground-level splitting of Gd 3 + 4f 78S7/2, Newman (1970)
introduced a spin-dependent operator of the type
S" ~ SiYkq(Oi, ~bi), (100)
i
arguing that the spin-orbit interaction, taken to second order, could produce such terms as corrections to spherical harmonics appearing in the standard crystal-field Hamiltonian. Put slightly differently, each Yk~ in this Hamiltonian can be corrected by making replacements of the type
Ykq(Oi, ~,) ~ Ykq(O,, q~,) + CkS "S, Ykq (0,, q~,). (101) Substitution (101) appears to be the simplest way in which spin-dependent pheno- mena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of V (° a part (proportional to ct) that involves the reduced matrix element of the double tensor V (~t), where v¢lt)= sv "~ (Judd 1977b). Such double tensors are straightforward to evaluate; furthermore the proportionality ( V (°) = A ( V (1°) holds for all terms of maximum multiplicity of a given configuration 4f N, so we can at once understand the consistent fit obtained for the SG and 5I multiplets of Ho 3+ (mentioned in section 7.6).
The signs of the parameters Ck are not so easily accounted for. If the exchange forces are to contract the orbitals whose c o m m o n ms value predominates, leading in turn to reduced parameters Akq for those orbitals, then it transpires that we must have c, < 0 (k = 2, 4, 6). Some of the evidence adduced to support negative values of the Ck has turned out to be less weighty and more ambiguous than was once thought (Judd 1977b). A further complication seemed to arise when it was realized that a positive value of c6 would account for the remarkable drop in the magnitudes of the parameters A6q in passing from EuCI3 to TbCI 3 (Judd 1980). However, a positive c6 was found to be produced by a charge-transfer mechanism of the type Eu 3 + + e - ~ Eu z +, as well as by the effects of covalency (Newman et al. 1982). In recent years, work on this topic has culminated in the analysis of Crosswhite and Newman (1984), which was referred to in section 1. Positive values of c6 were found to give excellent fits for the crystal spectra of Gd 3 + and Ho 3 + in LaCl3, and the especially troublesome deviations of the 3K 8 level of Ho 3+ 4f 1° were largely eliminated. This interpretation of the discrepancies left by the standard crystal-field theory has received support from the work of Reid and Richardson (1985b) on several crystals of the type Cs2NaYC16, where Sm 3÷, Dy 3+ and H o 3 ÷ w e r e
substituted for y3 + Their data sets are less extensive than those used by Crosswhite and Newman (1984); nevertheless, positive values of c6 were obtained for all three systems, and the numerical values of around 0.1 are comparable to those obtained for the trichlorides.
Although the argument that led to a negative co cannot be the dominant one, it was noticed by Siu and Newman (1983) that the variation of R(r) as a result of exchange forces leads to slightly different matrix elements of the Coulomb in-
A T O M I C T H E O R Y A N D O P T I C A L S P E C T R O S C O P Y 149
teraction. They showed that these effects could be accommodated by changes in the standard two-electron and three-electron parameters of section 6.1.2, but that, of the latter, only T 2 (and no other T i) is involved. The reason for this is rather.interesting.
Of all the six ti (i = 2, 3, 4, 6, 7, 8), only t2 has group labels W U that are identical to those of a two-electron operator, namely e3. The effective operators in 4f 2 that reproduce the energy shifts coming from having two radial functions Rj (where j = 2 5
(Kk) (~'tk,) (~:-k,t)O
or ~) must necessarily be formed from operators of the type (vi vj ) , where the spin ranks x and K' can be 0 or 1. These operators could be carried forward to 4f3; but it would clearly be preferable to represent their effects in 4f 3 by incre- ments to operators already present in the analysis wherever possible. The constraint that the symmetries represented by W and U be preserved limits the possible ti to t z alone.
7.6.2. Spin-correlated intensities
Attempts to include correlation effects in the intensity theory of section 5.4.2 have been made by Jankowski and Smentek-Mielczarek (1979, 1981). The appearance of two-electron operators leads necessarily to a large increase in the number of parameters. In analogy to the substitution (101), Siu and Newman (1986) have replaced the even-rank tensor V It) appearing in eq. (90) according to the scheme
V I° ~ V ~'~ + d, ~ S'si r~ '~,
i
thereby introducing three new parameters (d2, d4 and d6) into the theory. Some improvement in the fit to the solution data occurs, and the values of the dt are reasonably stable (though unnervingly large in a few cases). It is probably too soon to judge how successful this approach will turn out to be in the long run.
7.7. Spectra of divalent lanthanides
The allowed transitions 4f N --~4f s 15d occur in the ultraviolet for ions of the type R 3+. The contraction in the energy scale that takes place when the effective nuclear charge is decreased in passing from R 3+ to R E+ brings a number of transitions within the optical region of the spectrum. Unlike the narrow lines characteristic of the transitions 4f N -~ 4f N, broad bands are common. As has been mentioned in section 7.5.2, M C D analyses can be helpful here. An early survey of the absorption spectra of the divalent lanthanides in CaF2 was that of McClure and Kiss (1963). Just how to produce the R 2+ species is a problem in its own right, for which some ingenious solutions were found (see Kiss and Yocom 1964). The appreciable interaction of the 5d electron with the crystal lattice makes it impractic- able to attempt calculations on systems other than those of octahedral symmetry.
For these, a single-electron parameterization requires only one parameter (the one traditionally named Dq) for the d electron in addition to the two for the 4f electrons.
The complexities of as elementary a system as Yb 2 + 4 f - 1 5 d in SrC12 is apparent in the work of Piper et al. (1967), where all transitions are of the type Alg ~ T l u . Reasonably good agreement with experiment was obtained, though Piper et al.
(1967) pointed out that setting the f-electron crystal-field parameters equal to zero
hardly affected the quality of the fit to the energies, for which discrepancies of around 5 0 0 c m - 1 were common.
Alig et al. (1969) interpreted the absorption spectrum of CaFE:Ce 2+ in terms of transitions of the type 4 f 5 d - ~ 4f 2. That 4f5d lies lower than 4f 2 tallies with the situation for the free ion Ce III (see section 6.2.2). The evolution of the ground state was described by Alig et al. (1969) in the following way. The e state of 5d is coupled to the combination a2 + tl of 4f via the electrostatic interaction, thus neglecting, in the first instance, dt2 and ft2. The first-order spin-orbit interaction is then included, followed by a second-order contribution that mixes 1T2 and aT1. The relative intensities of the transitions 4f5d ~ 4f 2 could be reproduced rather well, and Dq was estimated to be around 1700cm-1.
The europium chalcogenides (EuO, EuS, EuSe and EuTe) exhibit absorption bands corresponding to 4f v ( 8 S ) ~ 4f6(TF)Sd(t2g + eg) of Eu 2 +. These crystals have been the subject of entire chapters in the present H a n d b o o k series (Wachter, 1979) and in the book of Htifner (1978). Two broad peaks are observed, corresponding to an energy separation between t2g and eg running from 1.1 eV for EuO down to 0.6eV for EuTe. Equating these figures to lODq gives a value for Dq of around 1000cm -~ or less. For Eu2+:SrS, Eu2+:KBr and EuF 2, the lower band (cor- responding to 4f 7 ( s S ) ~ 4f6(TF)5dt2g) exhibits a structure that is highly suggestive of the spin-orbit splitting of 7F (see, e.g., Hiifner 1978, fig. 41). The detailed analysis of Weakliem (1972) (already referred to in section 7.5.2) indicated that the Coulomb interaction between the 5d electron and the 4f 6 core, although substantially reduced compared to its strength for the free ion, is far too large to permit the energy levels of 5d(% + t2g) and 4f 6 7 F j tO be simply superposed. The multiplet-like structure on the lower band must be fortuitous. Hernandez et al. (1980) studied the band shapes of the corresponding transitions for Eu 2 + in NaCI, KC1, RbC1, KBr, RbBr and KI.
They found that the experimental shapes of both the lower and upper bands could be accurately reproduced by a Hamiltonian in which the Slater integrals Fk(4f, 5d) and Gk(4f, 5d) are reduced from their free-ion values by factors of roughly 2, thus confirming Weakliem's analysis. However, reduction factors of that magnitude conflict with the earlier results of Piper et al. (1967) on Yb 2+ and Alig et al. (1969) o n C e 2 +, where free-ion Slater integrals were assumed. Presumably the positions (and intensities) of the lines for 4f ~4 ~ 4fm35d and 4f5d ~ 4f 2 are insensitive to the values of F k and G k.