Chapter 73 Chapter 73
9. Non-linear spectroscopy
oscillator strengthsf of three lines of Er I 4 f 12 3H6 (6S 2 __~ 6s6p) with quoted errors of 25~, and found them roughly half as large as the corresponding HF values.
Szynarowska and Papaj (1982) measured f f o r a single line of Gd I, thus scaling the 138 relative f values found by Penkin and Komarowsky (1973)for that spectrum.
The latter authors were able to find the absolute values of f for the two lines 4f14(6s 2 -* 6s6p 3P 1, 1P1) of YbI as 0.014 and 1.12, respectively (Komarovskii and Penkin 1969), which they compared to the figures 0.0167 + 0.008 and 1.30 + 0.06 obtained by Baumann and Wandel (1966) from the lifetimes derived by means of the zero-field crossing technique. A later measurement by Gustavsson et al. (1979) using the pulse modulations of a CW dye-laser beam yielded f = 0.0159 + 0.005 for the first of the two transitions, thus indicating that the figure of Komarovskii and Penkin (1969) may be somewhat too low. Similar discrepancies occur for E u I (Gustavsson et al. 1979, Komarovskii et al. 1969). However, the differences between the experimental results are small compared to their deviations from the ab initio calculations. Loginov (1984) evaluated the radial transition integrals for EuI 4fT(aS)6s(5d ~ 6p) using the Hartree-Fock-Dirac method and found them to be two to three times larger than those deduced directly from experiment. It is somewhat disconcerting, to say the least, that better results are obtained by the Coulomb approximation (Loginov 1984). That method is empirical by nature, being based on Heisenberg's form of the correspondence principle for non-relativistic matrix elements (Naccache 1972) and developed with considerable ingenuity by Picart et al. (1978) and Edmonds et al. (1979).
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 171 energy levels below 22000cm -1 (other than the ground state 4f 78S7/2 of E u 2 + and some presumably irrelevant vibronic levels), taken with the dependence of the emitting intensity on E 4 (rather than E2), made it clear that the fluorescence arose from a preliminary two-photon absorption followed by a cascading to the fluoresc- ing level of 4f6(TF)5d, whence the final transition to 4f 7 ~$7j2 was made. Bayer and Schaak (1970) studied the anisotropy of this two-photon process, without, however, being able to provide a unique interpretation of their data. Several two-photon transitions for tripositive lanthanides in C a F 2 were reported by Apanasevich et al.
(1973), who recognized that the two-photon process is similar in structure to that represented by expression (71) provided we set k = 1. Thus the effective transition operator is proportional to (Eli)El1)) (2). V (2). Another way of seeing this result is to note that (EE) (°~. V ~°~ cannot produce a transition since V (°~ is a number, while (EE) tl~. V "~ vanishes because (EE) "~, being the vector product of two identical commuting vectors, is zero. Any closure procedure that brings the two vectors D together to form an equivalent operator of the type V "~ must thus lead us to (EE)(2) • V(2).
Two-photon studies of Pr 3+ in LaF3 were made by Yen et al. (1981), who were able to populate the ' D 2 level and detect the two-photon transition 1D 2 ~ 1S o of 4f z. However, the subject entered a new phase with the quantitative measurements of Dagenais et al. (1981) on G d 3 + 4 f 7 in LaF 3 at Harvard. With E parallel to the crystal axis c, they found that the integrated intensities for the two-photon transitions 887/2 --~6p7/2, 6p5/2, °P3/2 are in the proportions 320:5.4:1. Using the intermed- iate coupling wavefunctions of Carnall et al. (1977), they evaluated the reduced ma- trix elements of V (2' and obtained the theoretical proportions 69:29:1. The dif- ference between experiment and theory led them to conclude that ( 4 f l r t 5 d ) must differ markedly for different J multiplets. It was soon found that a more plaus- ible explanation existed (Judd and Pooler 1982): the vanishing of the diagonal matrix elements of V (2) for the half-filled f shell depresses the contribution to the two-photon transition probability coming from the usual second-order mechanism to such an extent that third-order linkages of the type
(4f 7 8S7/2 [E'DIZ> <zlH,o IZ') ( z ' I E ' D I 4f7 6pj ) (113) must also be considered, where Z and Z' are both states of 4f65d. The most efficient way of evaluating the sums over Z and Z' is to use the method of second quantization. In the notation of sections 6.1.1, the operator product (E. D)H,o (E. D), which results when closures over the angular parts of Z and Z' are performed, is expressed as
E. (a+fad)(o,, {~5~ x / 1 5 (a~ aa)(11 )o + ~4f ,,/42 (af+af) ('' ,o )~ ~d -rl. )~. ,.. + . , (114) Since there is no 5d electron in the ground state •, we can transfer ad to the right by means of the anticommutation relations for fermions until ultimately the equation ad [ ~ ) = 0 can be employed. It turns out that expression (114) is proportional to
20(9¢f + Cd)E 2 ( a / a 0 (11)0 _ x/3 (9¢f - 4~d) (E (1)E (I))(2). (a[af)(11)2,
so that the characteristic second-rank tensor is preceded by a large term, scalar in E, that can directly connect sSv/2 to 6P7/2 and thus enhance the two-photon transition probability. This term disappears for circularly polarized light, since there is no way that E]I~E] 1~ or E~_~] E~_~] can produce a component (E"~E"~)~_~ for which t = 0.
In order to account satisfactorily for two-photon transitions from sS7/2 to the 6D a n d 61 multiplets, a number of extensions to the theory must be made. A detailed account has been given by Downer and Bivas (1983) and by Downer et al. (1983).
The crystal field for a 5d electron contains terms of the type A~°4~.Ia~ad)~°4~, where the various components
Aaq
of A 1041 are proportional to the crystal-field para- meters of sections 5.3 and 7.1. The fourth-order productE. (af* ad )~ol, A,O,,. (ag ad)'°4)(a~ ad)'l~ ,0 E" (a~ad)c01 ) yields terms of the type
((/~E)l°2~A(°4~) ~06~" (af t af) ~16~6, (115)
which can directly link 8S7/2 with 6 b. With the aid of such theoretical devices, Downer and his Harvard colleagues were able to give a remarkably complete description of the intensities and anisotropies of the two-photon transitions for Eu 2 + in CaF2 and SrF 2 as well as G d 3 ÷ in LaF3 crystals and in aqueous solution.
F r o m a theoretical point of view, the success of the multiple closure procedure was one of the most noteworthy results. The presence of varying energy denominators has often deterred the theorist from using the rigorous equation
~:¢1~)<~l
= 1 (which holds for a complete set of states 4) to evaluate a sum such as E¢[~)E~-~(¢I.However, once it was recognized that the angular completeness holds for every configuration C, and that the energies appearing in the perturbation expansion are of the type Ec, the problem evaporated.
Two-photon spectroscopy should continue to develop in the years to come. Since the process conserves parity, the masking of the one-photon transitions 4f N ~ 4f N by the much stronger transitions 4 f N ~ 4fN-~5d can be avoided, thereby allowing the higher levels of the configurations 4f N to be detected. Examples of this have been given for Eu 2÷ 4f 7 by Downer et al. (1983). The IS 0 level of Pr 3÷ 4f 2, which lies very close to the configuration 4f5d, has been found for Pr 3 ÷ in LaC13 by Rana et al.
(1984), and the corresponding transition (namely 3H 4 ~ 1So) has been reported for Pr 3 + in L a F 3 by Cordero-Montalvo and Bloembergen (1984, 1985). The mediating operator is similar to that given by expression (115) except that the superscripted ranks for the coupled creation and annihilation operators are now (15)4 rather than (16)6. Further information on this system has been given by Bloembergen (1984).
9.2. Raman scattering
The theory of Raman scattering is virtually identical to that for two-photon absorption: the principal difference is that El" D for the incident wave need not be identical to E2" D for the scattered wave. Thus (El E2)(1~ is not necessarily zero and terms involving V tl~ can appear in the effective Hamiltonian that drives the transition. Richman et al. (1963) observed the Raman spectrum of LaC13 and
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 173 observed three strong lines at 108, 186 and 211 cm -1, the last exhibiting a shoulder at 217cm -1. These four active lines were confirmed by Hougen and Singh (1963, 1964), who assigned them the irreducible representations E2g, Ag + Ejg, E2g and E2g of the group C6h, corresponding to even-parity normal modes of the LaC13 crystal.
Only one mode (of the type Ag) escaped detection. Hougen and Singh (1964) repeated their experiments for PrC13; in addition to a virtually unchanged vib- rational pattern of lines, they observed some ten e l e c t r o n i c transitions, which matched the known sublevel structure of 3 H 4 , 3 H 5 and 3 F 2 of 4f 2 rather well. They obtained the selection rules for the electronic Raman effect by performing a complete closure over all intermediate states: that is, by directly coalescing the two vectors D " t in (El" D)(E2" D). Since (D ~itD~l~) "~ is ineffective for t = 0 and vanishes for t = 1, we are left with a second-rank tensor proportional to V ~2~. Thus Hougen and Singh (1964) found that, within the closure approximation, the selection rules should be identical to those for electric-quadrupole radiation. But although rough agreement was obtained for the polarizations of the lines, substantial discrepancies for the intensities were noted.
Axe (1964) showed that, by performing separate (partial) closures over the excitations f , , d and f--.g, contributions to the effective operator V ~1~ emerge.
Electronic Raman spectra of the type 7F 0 ~ 7Fj were observed by Koningstein (1966) for europium yttrium gallium garnet, and by Koningstein and Mortensen (1967) for E u : Y V O , . A continuation of these experiments established the existence of an antisymmetric Raman scattering tensor (Koningstein and Mortensen 1968), which must owe its existence to V ~ . The polarization data necessary to obtain this result were taken further by Finkman et al. (1973) in their work on NdA103. The measurements of G u h a t1981) on T m P O 4 were re-evaluated by Becker et al. (1985) and extended to ErPO4. By that time rather good crystal-field analyses had been made for these phosphates (Hayhurst et al. 1981, Becker et al. 1984), so that the eigenfunctions of the initial and final states of 4f N were known. Finkman et al.
(1973) had similar information at their disposal for their analysis of Nd 3+ 4f 3 4Ij.
Since the proximity of the configurations 4f N- 15d was well established, while the position of any configuration of the type 4f u - ~ng remained unknown, only those intermediate states Z belonging to 4f u - 1 5 d were considered. Both Finkman et al.
(1973) and Becket et al. (1985) found that the calculated strength of the antisymmetric tensor turned out to be very much larger than their experiments suggested. Evidently the strength of V ~ had been over-estimated.
To reduce it, the effects of the g electrons need to be considered. After all, we know from the total closure argument that a suitable weighting of the excitations f , , d and f--, g can be found that eliminates V ~ ~ altogether. As was mentioned in section 5.4.2, Axe (1963) had found it necessary to include g electrons as in- termediaries in his analysis of one-photon transitions within the configurations 4f ~.
Over the years, other were led to a similar conclusion (Krupke 1966, Hasunuma et al. 1984). Of course, the extended ng orbital of a free ion R 3+ cannot exist untouched in a crystal, but, even with a strong perturbation, it is difficult to see how the crucial matrix element
(4firing)
can be large enough. States in the continuum could increase the overlap between 4f and eg, but the energies e labelling these gstates would have to be enormous. At first sight, it seems far from easy to account for the apparent importance of g electrons. As was discussed by Becker et al. (1985), the difficulty arises because the intermediate states X are characterized by I. This quantum number is appropriate for SO(3) symmetries but not for a point symmetry such as D2d (which obtains in the phosphates). The states X that play an important role extend over the central lanthanide ion and the neighboring ligands, and there is no a priori reason why Z should possess more d character than g character. It does not seem appropriate to elaborate this point further because, even with a reduced strength of V "~, the intensities of the individual Raman lines do not agree well with theory. Several theoretically strong lines in T m P O 4 are completely missing. It can only be hoped that the situation will become clearer when data for other lanthanide ions in the same phosphate lattice become available. The experimental results of Williams et al. (1987) for Ce 3 + :LuPO4 should be of considerable help in that regard, since only one-electron configurations are involved.