Chapter 73 Chapter 73
7. Ligand effects
7.3.1. Anisotropic ligands
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 141 the other hand,
E (II(L 2) ~LVI3)~2)IO ~ E Y2o( OL, tiDE ) y6 0,
L L
where we have used the fact that any tensor of rank two must transform like the spherical harmonics Y2q. Thus a term in (91) of the type
(G[(L 2) *LV(3)~(2)10 V(_+ 2) E-T-
has non-vanishing matrix elements (provided there is a large enough departure from perfect Russell-Saunders coupling to overcome the change in S), and a line should be seen in the c~ spectrum. The D 3 case studied by K u r o d a et al. (1980) is not so clear- cut as the example presented here, since there are terms in the crystal field that mix 15D2, M s = _+2) into [5D2, M s = -T-l) when D 3 symmetry obtains. The effects of the anisotropic ligands are thus blended with those coming from a more mundane source. An extrication of these contributions and a detailed parameterization has been carried out by Dallara et al. (1984) for similar trigonal Eu 3+ systems.
It is natural to wonder whether changes need to be made to the selection rules worked out by Hellwege (1948). It would be strange if the powerful group- theoretical arguments he developed required modification just because some of the atoms in the crystal were not perfectly isotropic. The resolution of the paradox lies in the actual transition that was chosen to be studied. We concealed the fact that the D4 crystal field must mix a small amount of, say, ]5D4,Mj =-T-3) into
]SD2,Mj=
_+1). The operator YL(Y~)VI4))~p 1) c a n connect 7F 0 to the doublet ]SD4, Mj =-T-3) provided 5ZLY54(6)L,fiOL) does not vanish; and, indeed, for an arbitrary set of coordinates (OL, qSL) consistent with D4 symmetry it does not. Even in the electrostatic model, then, the transition from ~Fo to ]SD2,Mj = _+1) is allowed, albeit very feebly. The new operators (for which k = t) can produce large changes in intensity but they do not violate any selection rules.In the process of working out the implications of expression (91), it was mentioned that terms of the type given by (90) could occur when we take isotropic ligands and set n = 0. Since their origin is different, the relative weighting of the two tensors (Y(k) vIk-+lt)") as determined by the electrostatic model no longer obtains.
Reid and Richardson (1983b, 1985a) used this property to disentangle the two mechanisms, and Reid et al. (1983) showed that in many cases the dynamic-coupling mechanism plays the dominant role. However, Poon and Newman (1984) have pointed out that the absence of any consideration of overlap and covalency in that kind of work still leaves something to be desired.
7.4. Vibronic parameters
It is only within the last few years that attempts have been made to set up a detailed theory for the intensities of vibronic transitions. T o understand what is involved we can return to expression (72) and manipulate it in much the same way as we did the corresponding electronic linkage given by (71). This somewhat lengthy procedure can be circumvented by noticing that any operator producing an
electronic transition in virtue of ligands situated at R L can be modified to produce a vibronic transition by the simple expedient of making the substitution
g L --~ g L -+- dE, where d E is the displacement of ligand L from its equilibrium position
R E . Methods for coping with the general problem of expressing the spherical harmonics Ykq ((~L, tit)L) as sums over coupled products in which d E appears have been given elsewhere (Judd 1975, ch. 5). In the present case we only require terms lin- ear in dL in order to produce a single-phonon transition. A further simplification is that Y~L k~ in expression (90) is multiplied by R E k 1, thus providing a product that satisfies Laplace's equation. Under these conditions it is straightforward to show that the substitution RL --~ RE + dE implies
R E k-1 YL k) -~ [(k + l)(2k + 1)] 1:2 R E k 21v(k + l) •(1)) ~ L ~ L (93) so that expression (90) generates terms proportional to
RLk-2((yLk+I)d[1))(R) V"))(1)'E (k odd, t even). (94) Such terms are equivalent to those obtained by Satten et al. (1983) in their study of the excitations of the T~u vibrational mode of the octahedral complex UCl~- at 260 cm-1. They were aware that their approach would work in a similar way for lanthanide complexes; their choice of an actinide was partly determined by their familiarity with the UC16 complex. Another important consideration was the high symmetry of the uranium site, which greatly reduces the number of intensity parameters required in the analysis. To assess the situation we note first that the possible pairs (k, t) in expression (94) are (1, 2), (3, 2), (3, 4), (5, 4), (5, 6) and (7, 6). The number of Tlu representations of Oh, that occur in the representations ~ k of 0(3) are l, 1, 2 and 2 for k = 1, 3, 5 and 7, so we expect 9 parameters in all. Satten et al.
(1983) reduced that number to 6 by carrying out a complete closure over all virtual electronic states, which has the effect of fixing the ratio of the two operators (94) for which t = k + 1. They were able to account quite well for the relative intensities of some 20 or so transitions of the type Aig(3H4)~Tlu( 2s+ JLj), where 2s+ ILj is a level of 5f 2.
Satten et al. (1983) were not the first to attempt to fit vibronic lines, though their method is the easiest to describe. Some five years earlier, Faulkner and Richardson (1978a) had set up a vibronic-coupling model for the octahedral complex EuC13- in Cs2NaEuCI6. In addition to the terms of the type (94) coming from the electrostatic model, they took into account contributions coming from the dynamic-coupling mechanism (or, equivalently, the mechanism based on an inhomogeneous dielectric).
For us, we have only to make the substitution (93) in expression (91), thereby getting operators of the type
RL 5 {'~t")~U'L '--L(V(4) "Ldtlllt3~<k)' , V~2~}<').E. (95) Faulkner and Richardson (1978a) assumed isotropic ligands, for which n = 0. Thus the rank k in expression (95) is limited to 3. After making various estimates of the charges, polarizabilities, and bond lengths, they were able to obtain reasonably good values for the relative intensities of the three vibronic lines
Alg(VFo) -~ Eg (5D2), Atg (TFo) --~ T2g (5D2), Tlg (7F1 -~ TIg(SD1)
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 143
of EuC136-. In a companion study, they examined the magnetic-dipole and vibroni- cally induced electric-dipole intensities of the 5D4 ~ 7 F j transitions of Tb 3 ÷ in CszNaTbC16, with generally satisfactory results. A more complete analysis of the vibronic fluorescence spectrum of Cs2NaEuC16 was later carried out by Flint and Stewart-Darling (1981).
With fewer transitions to analyze, Faulkner and Richardson (1978a,b) did not attempt to fully parameterize their spectra. Their articles are thus different in style from that of Satten et al. (1983). In conjunction with various colleagues, they were nevertheless able to carry out similar analyses for other RC13 systems with considerable success (Faulkner et al. 1979, Morley et al. 1981, 1982a,b, Hasan and Richardson 1982). After the appearance of the article of Satten et al. (1983), Reid and Richardson (1984) decided to try to extend the parameterization of Satten et al.
(1983) in a parallel way to the analysis of Newman and Balasubramanian (1975). A physical realization of this is provided by anisotropic ligands, which permit terms for which n = 2 in the operator given by (95). Ranks of 2 and 4 for k are now possible, and the number of parameters is increased from 9 to 11. However, the fit obtained by Reid and Richardson to Satten's UC16 data was not improved when the two new operators were included, thus indicating that the superposition model (with C/v symmetry for individual U-C1 pairs) is adequate. One of the interesting results of the nine-parameter fit was the discovery that the relative signs of the parameters associated with the operators for which t = k _+ 1 could not be accounted for in the electrostatic model. Other mechanisms are evidently important.
Some impression of the continued interest in the energy levels of lanthanides at cubic sites can be gauged from the exchange of views between Tanner (1986) and Richardson (1986) concerning T m 3 + in the CszNaYC16 system.
7.5. E x t e r n a l maonetic fields
A large part of the theory of the effect of an external magnetic field on lanthanidc ions in crystals was developed in the context of paramagnetic resonance (Bleaney and Stevens 1953). It is that field that bequeathed optical spectroscopists the spin Hamiltonian and its various elaborations. This is not to say that such devices have ever been taken much advantage of. After all, the perturbation Hamiltonian f i l l . (L + 2S), where fl is the Bohr magneton and H the applied magnetic field, is particularly simple to evaluate, since the quantum number S and L are used in defining all lanthanide states.
A basic theorem was worked out by Kramers (1930b), who showed that any atomic system possessing N electrons and subject only to external fields of an electrostatic kind must exhibit at least a double degeneracy in its energy levels when N is odd. A crucial step in Kramers's proof involved taking complex conjugates.
T o d a y this would be done by invoking the time-reversal operator T. It has the advantage of immediately drawing a distinction between electric potentials, which are time-reversal invariant, and the operators S and L in the Zeeman Hamiltonian, for which
T L T -1 = - L , T S T -1 = - S . (96)
These properties lead to the linear splittings of the doublets when the external magnetic field is applied.
It might be wondered how N can be considered an integer at all for an ion in a crystal, where inter-penetrating electronic orbitals are a commonplace. The answer is that the exact superposition of a multitude of doublets, each one being provided by an identical lanthanide ion for which N is odd, becomes smeared into a narrow band. Its width is usually small compared to fill, and so the effect on the optical spectrum is negligible.
The pair of eigenstates corresponding to a given doublet are often called Kramers conjugates. In an extensive tabulation of the number and kind of the various sublevels arising from a level J (~<8) in a crystal field, Prather (1961) listed the Kramers conjugates for all crystal point symmetries. Of course, certain site symmetries permit doublets when N is even, as we have already seen (in section 7.3.1) for the components of 4f 6 5D 2 for which Mj = +1 in D 3 symmetry. The action of T on an eigenstate must produce another eigenstate corresponding to the same energy when H = 0; however, when N is even we have no guarantee that the two eigenstates are ~ distinct. In the special case just mentioned, the two states ]Mj = + 1) are indeed distinct, but TIMj = 0) is equivalent (to within a phase factor) to [Mj = 0 ) itself.
7.5.1. Transverse Zeeman effect jbr uniaxial crystals
F o r many point symmetries at a lanthanide site, the effective Zeeman Hamiltonian for a doublet can be written as
gllflHzSz + g±fl(HxSx + HySy), (97)
where the components Sx, S r and S~ of the fictitious spin S act in a space for which S = ½, corresponding to the doublet in question. It is easy to confirm that for H ± c (where the crystal axis c is parallel to the z-axis), the splitting of the doublet is independent of the azimuthal angle q~ defining the direction of H in the basal plane.
But although the splittings of both the upper and lower doublets involved in an optical transition may be invariant with respect to rotations of the crystal about c, the spectrum is not necessarily invariant too. Spedding et al. (1965) examined the variation of the absorption spectrum of erbium ethylsulphate when c, H and k (the direction of the incident light) are mutually perpendicular. They found 60 ° per- iodicities in both the energies of the levels and the intensities. A theory, based on the mixing by the Zeeman Hamiltonian of neighboring sublevels, was developed by M u r a o et al. (1965). A subsequent article by M u r a o et al. (1967) introduced an antiunitary operator whose associated eigenvalues could be used to classify the C3h doublets (appropriate for the ethylsulphates) according to two types, A and B. Syme et al. (1968) developed the theory further without, however, making it more accessible to the casual reader. The field of study was extended to systems possessing an even number of electrons by Spedding et al. (1968), who reported results for holmium ethylsulphate. Additional theoretical developments, still cast in terms of the properties of the antiunitary operator mentioned above, were described by K a m b a r a et al. (1972, 1973). They showed that for sites of D E symmetry large variations in the intensities of transitions of the types A ~ A and B ~ B occur as the
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 145
crystal is rotated, while transitions of the types A --, B and B ~ A usually show only small variations. A key element in their analysis is the classification of the states of the (split) doublet by the groups C~h and C ~ whose symmetry axes are fixed in the crystal, make 9 0 with respect to each other, and coincide with H from time to time as the crystal is rotated. When that happens they become symmetry groups of the total Hamiltonian and their irreducible representations can be used to label the states. If the labels assigned to the upper and lower components of a particular doublet are reversed in passing from C~h to C~h the doublet is of type B: no reversal corresponds to type A.
The reversal or constancy of the C2h labels of a doublet as the crystal is rotated has to be used in calculating the relevant transition probabilities. It is here that an added feature occurs: for the odd-parity part of the crystal-field Hamiltonian responsible for mixing states of opposite parity into the f shell itself possesses certain symmetry properties with respect to rotations about c, and these must be taken into account when the effective operator producing the transition is considered. It turns out that the transitions A ---, A and B ---, B exhibit the wide fluctuations in intensity as the crystal is rotated. The details of this analysis for D2d symmetry have been described elsewhere (Judd and Runciman 1976). Similar results obtain for the other two tetragonal groups D4 and Cav, except that the striking alternations in intensity now occur for A ~ B and B ~ A. For hexagonal crystals, the variations in the intensities of the Zeeman components can only occur if the magnetic fields are strong enough to mix neighboring sublevels.
7.5.2. Magnetic circular dichroism
The differential absorption (or emission) of left- and right-circularly polarized light of a lanthanide crystal subjected to a magnetic field has not received the attention one might have expected. The reference list in the recent book by Piepho and Schatz (1983, p. 616) contains not a single article that refers specifically to a lanthanide spectrum. Part of the reason for this is the ease with which Zeeman data for the f ~ f transitions can be obtained from the pure electronic spectra. However, magnetic circular dichroism (MCD) possesses several advantages when the lines are broad, as is often the case for those with a vibronic origin.
The theory of Faraday rotation for lanthanide ions was developed by Shen (1964) and applied to several ions in CaF2 by Shen and Bloembergen (1964). At the Zeeman Centennial meeting in Amsterdam in 1965, Margerie (1967) described his work on the ions Sm 2 ÷ in CaF2, where the breadth of several lines originating from 4 f 6 7 F 0 ~ 4 f 5 5 d was too great to permit their g values to be determined by conventional means. The host lattice CaF2 was again used by Weakliem et al. (1970) for a more extensive survey of the M C D spectra of the divalent lanthanides. Entire bands were found to exhibit a c o m m o n circular dichroism, which was attributed to a single dominant electronic state and a sequence of vibrational components possessing the same symmetry. The energy levels of Eu2+4f6(7F)5d(eg) were worked out as a function of the Coulomb exchange integrals Gk(4f, 5d) by Weakliem (1972), who found that values given by roughly one half of the corresponding free- ion values were required to fit the data. Schwartz and Shatz (1973) chose the
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).