Chapter 73 Chapter 73
7. Ligand effects
7.1.1. Crystal-field invariants
The numerical values of the crystal-field parameters
Akq
depend on the choice of coordinate axes. F o r Cah symmetry, for example, the two terms A66Y66 and A6 6Y6-6 appear in the crystal-field Hamiltonian for a single 4f electron, but it is found that, in calculations of the splittings of the levels, the two sixth-rank parameters always appear in the combination A66 A 6 -6- It is not difficult to see why that should be so. Under a rotation of the x and y axes by E about ¢, the three-fold axis of symmetry, we have 4, -~ q~ + ~, and so Y6+6 --' e-+6~ Y6+6. This is equivalent to making the substitutions /[6_+6 --~ e+6~A6+6 in the original Hamiltonian, and so the p r o d u c t A66 A 6_ 6 is an SO(2) invariant, since we have complete freedom in picking E. The splittings of a level are physical quantities and cannot depend on E: hence the sixth-rank parameters can only influence the splittings through the single quantity A 6 6 A 6 - 6. Had we started with the more familiar formA~(Y66 +
Y6-6), we would have concluded that the sign of the (necessarily) real parameter A66 cannot be determined from the energies of the sublevels. In their analysis of the effect ofconfiguration interaction on the crystal splittings of Pr 3+ 4f 2 in LaC13: Pr 3 +, Morrison et al. (1970a)considered an additional term of the type iB6(Y66 -- Y 6 - 6 ) , but excluded it on the grounds that the local symmetry at a Pr 3 ÷ site approximates closely to D3h. It was not necessary to use that argument, since the effect of including B 6 is equivalent to making the substitutibn A 6--, [(A6)2 + (B66)2] 1/2 everywhere, the replacement being the new SO(2) invariant.
When rotations of the coordinate frame in the full three-dimensional space are considered, other invariants appear. If A ~k~ denotes the collection of parameters A~ ( - k ~< q ~< k) the crystal field Hamiltonian for a single 4f electron can be writ- ten as Z, krk(A ~k)" ytk)). Although it may seem strange to construct a tensor out of nu- merical coefficients, there is nothing to prevent our doing so; after all, the point- charge model gives the Akq directly in terms of the ligand coordinates (R L, OL, @L), which necessarily change too when the coordinate frame is rotated. Viewed in this light, our approach makes it clear that other kinds of invariants can be constructed:
all we have to d o is ensure that the final rank in a multiply-coupled product of the tensors A ~k~ be zero. Kustov et al. (1980) pointed out that, for a given level 2s+ ~L, the energies Ei of the sublevels relative to the center of gravity of the level are related to the invariants AtkJ.A ~k~ by an expression of the form
~ g i E 2 = ~ hk(Atk~'A(k~),
i k=2,4,6
where gl is the degeneracy of sublevel i. The coefficients h k depend on the level under consideration. Kustov et al. (1980) determined the three invariants Atk~" A tk~ from the observed splittings of Nd 3 + 4f3(4F3/2,419/2, 4Ill/2), the process being repeated for 43 different crystals containing Nd 3÷ . Qualitative differences between garnets, scheelites, perovskites and fluorides were noted, and the method was extended to Eu 3+ in N a B a Z n silicate glass subjected to varying excitation wavelengths.
A more elaborate form of analysis was presented by Leavitt (1982), who found that corrections due to the J mixing of neighboring levels, if sufficiently small, could be taken into account by including the invariants of the type (A~k'~A~k"~) ~k~" A ~k~ in the sum above. Yeung and Newman (1985) called attention to the difficulty of finding the crystal-field parameters Ak~ for low-symmetry crystals, giving Er 3÷ :LaF3 as a striking example of three widely varying sets of parameters extant in the literature.
They showed that values of the A t k ) ' A tk~ found from experiment, taken with a detailed knowledge of the crystal structure, could be used to derive the intrinsic parameters associated with the superposition model. In later work, they classified the quartic invariants
(A(kl)A(k2))(k).
(A(k3) A(k4))(k),
pointing out that the most general f-electron lanthanide crystal field is described by 27 parameters, so some of the quartic invariants are not independent (Yeung and Newman 1986a). The quartic invariants have been used by Yeung and Newman (1986b) to test the consistency of the fits to 17 levels of Er 3+ 4f 1~ in YAIO3 under different constraints on the spin-correlated crystal-field parameters Ck (defined in section 7.6.1).
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 137 7.2. Hypersensitive transitions
The existence of many plausible sources for the contributions to the observed crystal-field parameters has an interesting parallel in the extraordinary sensitivity to the environment of certain lines in the absorption spectra of the lanthanides. These lines, the so-called hypersensitive transitions, satisfy the same selection rules as electric-quadrupole radiation: that is, AJ ~< 2. This condition was first noticed when the absorption spectra obtained by Hoogschagen and Gorter (1948) for different kinds of aqueous solutions were compared. In going from solutions of the chlorides to those of the nitrates, the lines 4115/2 - ' ~ 2 H11/2 of Er 3 + and 419/2 __. 4 G5/2 of Nd 3 + show noticeable changes in their intensities relative to almost all other lines. Adding alcohol produces a marked enhancement of these two lines, as was demonstrated to the writer during a visit to Dr. C.K. Jorgensen at his laboratory in Cologny, Geneva, in 1963. All that was required was a test tube containing a solution of neodymium ions and a hand-held spectrograph. Other hypersensitive lines in the spectra of various lanthanides could be picked out without much difficulty. It was immediately recognized that this phenomenon, when cast in terms of the parameters if2 k of section 5.4.2, exhibits itself by the variability of f22. Why should that parameter be more sensitive to the environment of a lanthanide ion than the others?
A number of possible explanations were explored in an early article (Jorgensen and Judd 1964). The extension of the tail of the 4f wavefunction, as represented by the substitution given by (89), was quickly seen to be ineffective in enlarging (r 2 ) to the point where pure quadrupole transitions could take place. Vibronic transitions were at first considered to be prime sources for the intensities in view of the identical selection rule AJ ~< 2 [-see eq. (74)]. An order-of-magnitude calculation fell short by a factor of 10 3, however. Resort was finally made to a mechanism based on the inhomogeneity of the dielectric. The electromagnetic radiation field induces dipoles in the ligands surrounding a lanthanide ion, and the re-radiation that takes place must deviate markedly from a plane wave. This idea led to a value of £22 that was too small by a factor of only 30. The discussion of these various sources was marred by an uncritical acceptance of the fallacious argument of Broer et al. (1945) (see section 5.4), which, when corrected, provides another possible source for (22 in complexes whose symmetry permits harmonics of the type Ylq in the crystal-field potential (Judd 1966b). Objections to this mechanism were later raised by Mason et al. (1975) on the grounds that it could not account for the hypersensitivity of Eu 3 ÷ doped into sites of D2a or D3h symmetry in metal oxide lattices (Blasse et al. 1966), nor for the more spectacular hypersensitivity of the transitions reported by Gruen et al. (1967) for the supposedly planar molecules of the lanthanide trihalides. However, there is some evidence that the structure of the latter are pyramidal (see Giricheva et al. 1967, Charkin and Dyatkina 1964); in which case the mechanism in question might well play a major role.
An apparently more promising mechanism, called dynamic coupling, was in- troduced by Mason et al. (1974, 1975). The idea is that the 4f electrons polarize the ligands, thus giving the entire complex of lanthanide ion and neighbors an extended dipole moment (should the site symmetry allow it) that has an enhanced interaction
with the radiation field. The polarizability of the ligands plays a central role in the dynamic-coupling mechanism, just as it did for the mechanism based on the inhomogeneity of the dielectric. With hindsight, it is perhaps not so surprising that the mathematics is identical for the two mechanisms, and indeed Newman (private communication) suspected as nmch all along. Thus one mechanism is just an alternative verbalization of the other (Judd 1979). Mason et al. (1975) did not encounter the discrepancy of the factor of 30 mentioned above because they used better structural information and polarization data. However, neither approach allowed for the screening by the outer shells of the lanthanide ions, which are able to follow the oscillatory variations of the vector E of the radiation field. The approp- riate reduction factor for ~¢~2 is (1 - 0-2)2; its omission vitiates the formula of Mason [1980, eq. (9)] as well as the reported good agreement between experiment and theory, such as that for the 419/2 ---~4G5/2 transition of tris(1,3-diphenyl-l,3- propanedionato) a q u o n e o d y m i u m ( I I I ) described by Kirby and Palmer (1981).
Another example is the detailed comparison of the dynamic-coupling and the electrostatic mechanisms that Richardson et al. (1981) made for Pr 3 +, E u 3 +, T b 3 + a n d H o 3 + complexes of trigonal symmetry.
Mason (1985) has argued that a compensatory antiscreening exists as a result of an expanded 4f wavefunction, and this seems the most plausible way out of the difficulty. It is always possible, of course, that some crucial component in the theory has not been identified. Faced with a striking phenomenon, the theorist is always tempted to believe that a single key will unlock the mystery. In the case of the hypersensitivity it is more likely that various mechanisms combine or interfere to varying degrees under different experimental conditions. Such was the conclusion reached by Peacock (1975) after an examination of the status of the subject in the mid-1970's. It may still be valid today.
7.3. Transitions between sublevels
The intensity parameters 12k are appropriate for examining the coalesced tran- sitions from level to level of the type J ~ J'. The appearance of any fine structure (or, to use a more appropriate term, crystal fine structure) corresponding to transi- tions from sublevel to sublevel of the type # -~/a' makes it possible to extract more information about the nature of the transitions. All work in that area stems from the initial study of Axe (1963), which has been briefly mentioned in section 5.4.2. In order to appreciate the later developments, we need to recast the theory of section 5.4.2 in terms of tensor operators. We regard the even-rank tensors V ~'~ as arising from the inner part of expression (71) when closure is used to remove the intermed- iate states Zj,, and when Yk (for odd k) and D are coupled to rank t. In the electro- static model the YR part is implicitly multiplied by R - k - ~ Y*q(O, ~ ) of section 5.3 for a ligand at R; so the effective operator driving the transition is proportional to a sum over k and t of various terms of the type
R E k - I ( Y ~ V(t~)tl J" F. (k odd, t even), (90)
where the spherical harmonics Ykq(O, ~ ) have been subsumed into the tensor y~k).
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 139 The subscript L indicates the coordinates of a particular ligand L. The fact that (90) is an SO(3) scalar is predicated on its necessary invariance with respect to the orientation of our coordinate frame. It must also be invariant with respect to inversions and reflections.
F o r an actual crystal, expression (90) must also be summed over the ligands L after including their charges qL" This has the effect of converting (90) to (T(k)V(t))~I).E, where T (k) is an invariant with respect to the point group of the lanthanide site. In the electrostatic model, the relative weighting of (T (k)V ~k+1))(t) and (T (k~ V (k-1~)~1) is determined by the relative importance of the virtual excitations f - , d, d - , f and f - , g. Even without that constraint, it is clear that (T (k~ V(') (1). E is not the most general scalar operator linear in V "~ and E. If we accept that t is even (a result of limiting perturbation theory to second order), there might very well exist tensors T (k) that are scalar with respect to G but for which k = t. This possibility is excluded by the electrostatic model, of course. But other options are open to us. In a remarkable analysis of great generality, in which overlap and covalency were considered, N e w m a n and Balasubramanian (1975) showed that the relative weight- ing of the two t e n s o r s ( T (k) V (k +-
1))(I)could
be relaxed, thereby clarifying the results of Becker (1971) for H o 3+ in YPO4. More remarkably, they also showed that tensors T (k) for which k -- t could occur. In the special case for which G = C~v, however, they are excluded. This is the situation for a single ligand that exhibits cylindrical s y m m e t r y with respect to the axis linking it to the lanthanide ion. T o see why this geometry gives no new terms, take the cylindrical symmetry axis to be the z-axis and xz the vertical reflection plane. Only the c o m p o n e n t T~o k) appears, and this is necessarily invariant with respect to G. The effect of the reflection operator R on E is given by Ex - , Ex, Ey ~ - E y , and E: -~ E~; that is, E~ ) -~ - E ~ ~ and E~ 1) ~ E~ ~.As it stands, there is no way of determining RV~)R-1 unambiguously, since eq. (61) only specifies how V (' behaves with respect to rotations. However, we know that a c o m p o n e n t tensor v ") must ultimately be set between two f-electron wavefunc- tions, and the reflection properties of the latter can be found from the spherical harmonics Y3~. The condition that (fmlv~)lfm ' ) not always vanish yields RV~)R - 1 = ( _ 1)p V([)p. The relation
(k 0, t p l l - p) = ( - 1) k+'+l (kO, t - p l l p)
for the C G coefficients of SO(3) can be used to show
R ( T (k) V " ) ) ( 1 ) . E R - 1 = ( _ 1)k +~ + 1 ( T ( k ) V . ) ) ( 1 ) . E
for C~v symmetry. The invariance of the operator sandwiched between R and R " 1 requires that k :/: t for it to be nonzero. Although no actual crystal possesses C ~ symmetry, the possibility of regarding a crystal as the superposition of isolated ligands, each one interacting solely with the central lanthanide ion in a cylindrically symmetrical way, is an extremely appealing approximation (Newman 1971). The fact that the new operators (those for which k = t) can only describe deviations from that model lessened the impact that the article of N e w m a n and Balasubramanian (1975) had on crystal spectroscopy. An added reason is that they described their method in terms of local and non-local operators without making clear what the distinction was.