The present status of our knowledge of the structure of the spectra of the doubly and triply ionized spectra of the rare earths is derived partly from experimental data of the emission spectra of the free ions which provide the energy level scheme in great detail but are difficult and laborious to analyze. For the lower levels knowledge of the structure comes from the crystal absorption and fluoresence spectra. In all cases approximate theoretical calculations of the energies are essential.
© 1963 Optical Society of America
The rare earths, in common with the actinides, have the most complicated spectra of any of the elements. This is because the incomplete 4f-shell produces a very large number of low-lying levels. The transitions between these give a many-line spectrum without any apparent regularities. Even multiplets, which are such prominent features of the spectra of the transition elements, are in general unrecognizable. For these reasons the rare-earth spectra have long been neglected. Because of the growing importance of the rare earths in recent years, their spectra have received renewed attention. In the present article we shall restrict ourselves to the spectra of the doubly and triply ionized elements for several reasons.
Most atomic spectra are obtained by vaporizing the element, exciting the vapor by an electric discharge, and thus obtaining the emission spectrum of the free ions or neutral atoms. The triply ionized rare earths, and to a lesser extent also the doubly ionized ones, can be incorporated into crystal lattices where they retain their sharp energy levels only slightly modified by the internal crystal field. This makes it possible to obtain some of the energy levels, particularly the important low ones, from the absorption and fluorescence spectra of crystals containing these rare earths.
An extensive program of the study of the spectra of doubly and triply ionized rare earths is under way at Johns Hopkins University and both the emission spectra of the free ions and the crystal spectra are investigated. One of the chief purposes of this study is the comparison of the levels of the free rare-earth ion with those of the same ion in a crystal lattice, in order to be able to obtain an exact evaluation of the influence of the crystal field. The present paper is a report of the work carried out or planned for the immediate future at Johns Hopkins University.
II. Experimental Techniques
The free-ion spectra are obtained from a controlled spark in a rare-gas atmosphere at reduced pressure. Figure 1 shows the development of the spectrum of gadolinium as the circuit parameters are changed. The 3-A spectrum is that of a dc arc, and even at this low-excitation lines of the doubly ionized ion appear. The other spectra are of unidirectional pulsed discharges, the duration varying from about 2 msec for 100 A to 100 μsec for 1500 A. Above 100 A there is no trace of the second (singly ionized) spectrum, and as the excitation is increased the third also gradually gives way to the fourth. In no case is the spectrum completely pure for a given ionization stage, but by comparing one which contains only third and fourth (“hot spark”) with one containing only second and third (“mild spark”) the fourth spectrum lines may be identified. The lower excitation of an arc or microwave discharge is also needed to eliminate the first and second spectra, but for our purposes it is not necessary to distinguish between the two. It is important for this separation that the pulsed discharge not be allowed to oscillate as this will superimpose a lower excitation discharge on the high and make the suppression of the second spectrum difficult.
Figure 2 shows a comparison of hot and mild sparks for various wavelength regions of the neodymium spectrum. In the upper section Nd iv lines predominate for the hot spark (A) although some very weak Nd iii may be found. A slightly longer wavelength region (center) is very rich in Nd iii. Even longer wavelengths (bottom) show mainly ii and i in the mild spark (B) and very little in the hot. This separation by wavelength is also very significant for the analysis and is discussed further in Sec. IV.
Accurate wavelength measurements are of the greatest importance as the possibility for finding spurious regularities is proportional to (Δν)2, where Δν is the uncertainty in the frequency measurements. The use of large gratings is therefore most appropriate. In our work a 17.8 cm wide, 6.64 m f.l. concave grating with 1200 lines/mm is used for most of the work, in the first order for the longer wavelengths (above 5000 Å), in the second or third orders for the shorter ones, which makes it possible to obtain wavenumbers accurate to about 0.01 cm−1 for most of the spectrum above 2000 Å. The use of an interferometer would not contribute much because, particularly in the hot spark, the lines are not sharp enough for utilizing the greater inherent accuracy of the interferometer. The fact that thousands of lines must be measured in each spectrum makes the use of modern automatic measuring techniques imperative.
For the crystal absorption spectral light from a continuous source, usually a high-pressure mercury or xenon lamp, a tungsten ribbon lamp or a zirconium arc depending on the wavelength region, are passed through a properly oriented crystal immersed in liquid helium. The Zeeman effect is often important, so the dewar vessel must be constructed so that it can fit between the pole pieces of a magnet. At low enough temperature the wavenumbers of the absorption lines directly represent the position of the excited energy levels. The electric crystal field produces a Stark effect which splits all levels into a maximum of 2J + 1 components for an even number of electrons and J + ½ components for an odd number. The number and spacing of the components depends on the symmetry and intensity of the crystal field. In most cases the total splitting of a free-ion level is of the order of a few hundred cm−1, in general small compared to the spacing between adjacent multiplet components.
A sharp-line fluorescence spectrum is obtained, containing in some cases several hundred sharp lines, by illuminating the crystal by an intense light source. When the coupling between the electronic level and the crystal lattice is large the excitation energy is dissipated before fluorescence can take place. This is the case for most hydrated salts except for the central rare earths from Sm to Dy. In other lattices all rare earths fluoresce strongly but not from all levels. The anhydrous chlorides and bromides, the garnets and the oxides are among the crystals that show general fluorescence. Figure 3 illustrates this for a crystal of Y2O3 containing 1% erbium. From theoretical considerations and previous experience we expect to find a group (called the A group) of excited levels at about 10,000 cm−1 above the ground state which would correspond to the single free ion 4I11/2 level (see Sec. III, especially Fig. 7). As the number of electrons for Er3+ is odd, and as the local crystal field symmetry about this ion is low, we expect to find the Stark-split level to consist of J + ½ = 6 components. The ground state 4I15/2 will be split into 8, but if the crystal is cold enough, only the lowest of these will be occupied. Absorption transitions occur between this lowest level and any higher level unless forbidden by selection rules. All six appear in the left-hand spectrum of Fig. 3. Fluorescence from the lowest Stark component of 2P3/2 (P group), shown on the right, confirms all six levels. Very low levels, such as the higher Stark components of the ground state, for which the absorption spectrum is not in an accessible region, must be confirmed by additional fluorescing levels.
In complicated fluorescence spectra, single levels can be excited by monochromatic illumination, and this greatly helps in the analysis of such spectra. The fluorescence spectra are extremely sensitive for very small amounts of impurities, and great care must be taken to identify such impurity lines which occur in even the purest available materials.
For the crystal spectra the task of obtaining the energy level system from the observed wavelengths of the absorption and fluorescence lines is in general direct and simple. For the free-ion emission spectra this is a much more elaborate and difficult problem. It can be solved much more expeditiously when one knows approximately what to expect. In fact, a satisfactory analysis of these complicated spectra would be virtually impossible without a thorough theoretical study of the structure of the energy levels. Such a study is based on the general theory of atomic structure, as for instance set forth in the book by Condon and Shortley or those by Slater, systematized for the complicated rare ion cases by Racah. The specific situation for the rare earths has been clarified by a comparison with the empirical results so far obtained from the crystal and free-ion spectra.
III. Level Structure of the Rare-Earth Ions
It has been known since the early days of the Bohr theory that the 4f, 5d, and 6s electrons have very nearly the same energy for the rare earths. This is true especially for the neutral and singly ionized atoms. This means that configurations like 4fn, 4fn−15d, 4fn−16s, 4fn−26s2, etc., may overlap and create an extremely complicated set of low-lying energy levels which it is difficult to disentangle. Fortunately, the situation is much clearer for the divalent ions and even more so for the trivalent ones. The reason for this is that because of the higher nuclear charge and the consolidation of the inner shells the screening is more perfect and the levels appear more nearly in the order of their principal quantum numbers. For the fourth spectra the 4f orbits always have the lowest energy, then come 5d, 6s, and 6p. This means that without exception the 4fn configuration is lowest, then come 4fn−15d, 4fn−16s, and 4fn−16p, with several other configurations coming in with energies near that of 4fn−6p. Figures 4 and 5 show for ions with three external electrons how this situation changes as we go to the lower stages of ionization.
For the third spectra the order of the configurations is still essentially the same as for the fourth, but the beginning of the rare-earth series 4fn and 4fn−15d are nearly coincident and 4fn−15d and 4fn−16s cross over near Dy (see Fig. 6). For the second and first spectra, as Fig. 5 shows, the situation is very confused, and this makes the analysis of these spectra much more complicated than that of the higher ionization stages.
It is, of course, not sufficient to know merely the position of the centers of the configurations as indicated in Fig. 6. We must know their total width and the arrangement of the individual energy levels. The method for doing this can be briefly indicated as follows.
The outer electrons are regarded first to be in a central field provided by the nucleus screened by the 54 electrons of the completed xenon-like shell. For the rare earths the assumption that this central field is a Coulomb field determined by one parameter, the effective nuclear charge Z*, leads to practically as good results as the more general assumption of a nonrestricted central field and simplifies the calculations materially. After first considering the outer electrons as completely independent, their electrostatic repulsion and their spin-orbit interactions are introduced by the potentials
expressing the electrostatic repulsion between the outer electrons and the spin-orbit interaction, and the influence of these perturbing potentials on the energy and wave functions calculated. In performing these calculations interactions between states of only one configuration are taken into consideration, but no further restrictions are made. For the higher stages of ionization, other configurations of the same parity are far away for the lowest configurations and we make no fatal mistake by leaving out interconfiguration interactions, although they are by no means negligible.
The systematics of handling the very large number of individual states have been developed by Racah. It is evident that only states with the same L and S can interact through H1, only those with the same J through H1. This and other symmetry considerations greatly limit the number of matrix elements of H1 and H2 that have to be calculated and restrict the order of the interaction matrix that has to be diagonalized to the number of different states with the same J occurring in the configuration. The situation is simplest for the 4fn configurations. The calculations, which are somewhat tedious, but not unduly so for a modern computer, are carried out numerically as a function of the spin-orbit parameter ζ and contain a scaling parameter F2 which is determined by the effective charge Z*. The two numerical parameters F2 and ζ are obtained by comparing the calculated set of levels with some well-identified empirical ones.
This has been carried out for all the crystal spectra of the trivalent rare earths, and Fig. 7 gives an example of the agreement that is reached. The 29 lowest of the 41 possible levels of the 4f11 configuration of Er3+ are shown. It is clear that for the first few multiplets there cannot be any possible doubt about the identification. For the higher ones crystal splittings also have been used for identifying the levels. It is seen that in the crowded regions the mere position of a level is not enough for identification. Here the levels are very sensitive for the particular choice of the parameters, and the fact that the calculations are only approximate ones puts a limit to the accuracy one can expect.
The situation is similar for the other trivalent rare earths.– The references give in each case first the papers which give the most recent empirical data and then the ones containing the calculations. In all cases the first few multiplets have been identified beyond possibility of doubt. For the higher energies the situation is often so complicated that positive identifications cannot be made without a great deal of further study. Figure 8 gives a summary of the results obtained so far. Table I lists the parameters F2 and ζ derived from these data.
The energy levels obtained in this way are, of course, those of the ion in a particular crystal lattice. Using the center of the Stark components eliminates the influence of the crystal field in first order. This, however, is not sufficient to obtain the free-ion levels with any high degree of accuracy. Comparison of the same levels in different lattices shows that there may be shifts of several hundred cm−1 and we can trust the levels obtained from the crystal spectra to agree with the free-ion levels at most to this extent.
The configurations where not all outer electrons are in 4f orbits, such as 4fn−15d or 4fn−16p, can be dealt with in similar fashion although the calculations are somewhat more involved as more parameters enter into the calculations. Figure 9 shows the situation for the 4f135d configuration of Yb2+, and again it is seen that the comparison of the empirical levels with the calculated set gives a positive identification.
Knowledge of the position and extent of the two lowest configurations 4fn and 4fn−15d is particularly important as these two configurations may be expected to furnish the strongest lines.
Figures 10 and 11 give a survey of these configurations for the third and fourth spectra, respectively, obtained from preliminary empirical data and very approximate calculations. Because of the particular couplings involved we must in general expect the strongest lines to be transitions from the bottom of one configuration to the bottom of the other one, from the middle to the middle, etc. We see that for the fourth spectra all 5d → 4f transitions must lie in the vacuum ultraviolet and shift from 2000 Å for Ce iv to 1000 Å for Yb iv. The 6p → 5d transitions remain at about 1300 Å for all fourth spectra. The 6p → 6s transitions, however, lie in the more easily accessible part of the ultraviolet and shift from 2500–2800 Å for Ce iv to 1700–2200 Å for Yb iv.
The situation for the third spectra is considerably different. For La iii the 4f and 5d levels must practically coincide and the transitions between them lie in the far infrared. For this reason they have not yet been found. They move apart with increasing Z, and the strong 5d → 4f lines lie in the visible for Pr iii and near 2000 Å for Yb iii. Figure 10 shows that because of the varying width of the configurations the situation for the intermediate elements may be irregular. For Gd iii for instance the strong lines must again be expected in the infrared, and a search for them (Callahan) has revealed that they certainly cannot lie in the visible or ultraviolet. For the third spectra also the transitions between the other low configurations lie mostly in the visible or accessible ultraviolet where precise wavelength measurements can be made more easily.
We may now come back to the crystal spectra of the divalent ions. The sharp-lined crystal spectra in the visible and adjacent regions of the trivalent and divalent ions are due to normally forbidden transitions between levels of the 4fn configuration. These are made possible, except for rare cases of magnetic dipole radiation, by the admixture through the crystal field of parts of opposite parity in the wave functions. These transitions have f numbers of the order of 10−6 and would be completely blotted out if they occurred in the region of the allowed 5d → 4f transitions. We have seen that for the trivalent ions the latter lie usually in the vacuum ultraviolet which leaves the visible and near ultraviolet free for the crystal spectra. Figure 10 shows that for the divalent ions the 5d → 4f transitions usually reach into the visible which leaves much less room for the crystal spectra, and we must expect these chiefly in the infrared. Sm and Eu have long been known to form divalent ions. The fluorescence spectrum of Sm2+ is known in the extreme red and infrared and resembles that of Eu3+ with all the energies roughly reduced by 20%. Something is also known of the spectrum of Eu2+. Recently it has been found by Kiss and others that in crystals that contain trivalent rare-earth ions these can be converted into divalent ions by irradiation with gamma rays. This makes it possible to obtain. the lowest levels of all the divalent rare-earth ions from crystal spectra data.
IV. The Interpretation of the Free-Ion Spectra
The energy levels derived from the crystal spectra are obtained without ambiguity and their interpretation is now possible without difficulties. The data so obtained are of necessity very limited, as they cover only the lower portion of the 4fn configurations. While further studies in the ultraviolet may extend the known crystal lines somewhat, Fig. 11 shows that we cannot expect to go much further without coming in conflict with the strong continuous absorption to the 4fn−15d levels which may reach farther to the visible than the figure indicates because of the crystal field broadening of the 5d levels. Also other constituents of the crystal lattice may absorb in the far ultraviolet. In order to obtain a more complete picture of the energy levels, it is necessary to go to the free-ion emission spectra.
These as we have seen earlier are very complex We have set ourselves the task of obtaining the four lowest configurations 4fn, 4fn−15d, 4fn−16s, and 4f5−16p of the third and fourth spectra as completely as possible. Experience with the spectra which have so far been analyzed,, has shown that the electron in a 6s or 6p orbit (and even 5d for the heavier elements) is only loosely coupled to the core, which makes the so-called J1j coupling scheme proposed by Racah most appropriate. Strong transitions occur only when the core does not change and the orbital quantum number of the excited electron changes by ±1. This has a marked influence on the appearance of the spectrum. As the Δl transitions are not much affected by the particular core value, there will be one such transition at approximately the same wavelength for each choice of core. For instance for the one-electron La iii spectrum the 6p3/2 → 6s and 6p1/2 → 6s transitions occur at 3172 and 3517 Å, respectively. In Ce iii the corresponding 4f6p → 4f6s transitions occur as groups centered near 3100 and 3400 Å; for Pr iii at 3000 and 3350 Å. For Nd iii we can, therefore, expect that the strong group of lines between 2900 and 3050 Å will belong to 4f36p3/2 → 4f36s and those between 3200 and 3350 Å to 4f36p1/2 → 4f36s. In a similar way the strong group between 2050 and 2400 Å may be associated with 6p → 5dtransitions.
The number of levels in these configurations may be quite considerable, as Table II shows. The last column (for some elements) gives the number of lines that one would obtain if the parity and J-selection rules are obeyed, as well as Δl = ±1 rule for the electron that changes orbits. No other selection rules are used for this calculation, but many of the allowed lines may be so weak as to be unobservable. Nevertheless the table shows that the number of lines from these four configurations alone may be very large, and there are of course many other configurations that contribute to the spectrum. Under these circumstances one must proceed with the analysis with a careful plan in order not to be involved in a task that is too time-consuming for the results to be worth the effort.
In the first place it is essential to have as complete a wavelength table as possible with wavenumbers of the highest possible accuracy. The second step is to establish an empirical set of energy levels. Most of the labor of this step can be delegated to a computer. If nothing is known about the spectrum, the computer can be instructed to compute all wavenumber differences between the lines and list those that occur repeatedly within certain limits of accuracy. Many of these will be accidental coincidences, and in this step the accuracy of the wavenumbers is the decisive factor. If the tolerances are narrow, the chances for accidental coincidences become less. By examining all the significant differences, an empirical energy level diagram can be built up, the reliability and completeness of which depends on the extent and accuracy of the wavelength table.
As stated earlier, this process can be carried out in principle without any previous knowledge whatsoever of the structure of the spectrum. The procedure will, however, be greatly simplified if one knows what to expect. Since the energy differences between the lowest 4fn levels of the fourth spectra and some of the third are known approximately from the crystal spectra, the computer needs to look in its first attempt only for differences within a few hundred cm−1 of these values. The 4fn−16 levels occur in pairs, the splitting of which can also be computed with considerable precision. The calculated energy levels will then limit further the range of differences that have to be examined with a great saving in effort and computer time. This search is further aided if one can limit it to a restricted wavelength region as discussed above. Careful studies of excitation variation such as shown in Fig. 1 are also helpful in reducing the number of lines to be handled. Not only is it possible to specify the ionization stage, but also the approximate excitation level for a given ion. This allows the upper electron configuration from which the particular transition occurs to be identified. If enough care is taken, information may be obtained even about the position of the level within the configuration. If necessary, electronic time-decay techniques may be used. After successive efforts of this kind a fairly extensive set of empirical levels can be constructed.
The next step is the interpretation of the levels, which means assigning quantum numbers to them and correlating them with the levels to be expected theoretically. The most important of the quantum numbers is the angular momentum J. This often can be established from the selection rule ΔJ = 0, ±1 which must be satisfied by all lines of the free-ion spectrum. Under favorable circumstances this establishes relative J values of all the levels, which can be turned into absolute values if the J for one level is known. This is always the case for at least the lowest level which is known by Hund’s rule. (Highest L of highest multiplicity, lowest J of this multiplet before Gd iv, highest J after Gd iv.)
For the crystal spectra, the Zeeman effect is useful in providing additional important information, but with these higher ionization stages of the free ion the situation is not so favorable. The spectra are very crowded and the lines somewhat broadened by the violent discharge conditions needed to produce them. Furthermore, the strong J1j coupling makes all 6p → 6s transitions for instance look very similar. Obtaining reliable Zeeman effects under such conditions would add greatly to the burden of wavelength measurements and analysis. We have found that the other methods for the analysis described here give such definite and positive results at least for the low configurations that there has been no need for Zeeman effect data.
Another important aid for the analysis is the presence of hyperfine structure. This is simplest when there is only a single odd mass number isotope, as is the case for Pr, Tb, Ho, and Tm, particularly with Pr and Ho because of the large value of the nuclear magnetic moment. The magnitude of the hyperfine structure indicates whether or not an s-electron is present and can give other information about the nature of the configuration. Figure 12 shows a few examples of resolved hyperfine structure in the spectrum of holmium. The number of the components gives the value of J if it is below the nuclear spin. The intensities and relative spacing of the components indicate the J values in general, and other important information on the nature of the levels can be obtained from the hyperfine structure.
Obtaining similar information for other elements would require separated isotopes. For these, isotope shifts also could be used for the identification of configurations. So far it has not been necessary to use separated isotopes, but later they may be needed to give enough information to decide doubtful cases.
The most valuable and important aid for the analysis of these complicated spectra has, however, been the comparison of the empirical levels with the calculated ones and this, in general, leads to a positive and unambiguous identification of the levels. While matrix elements for the fn configurations are now available, in general the fn−1l, ones are not. The computation of these is a straightforward, but very tedious process. Here again, we may make use of high-speed computers, and these matrices are being compiled as needed.
V. Results So Far Obtained
While the initial incentive for the investigation was the desire to know the 4fn configurations of the free-ion levels of the trivalent rare earths in order to have a basis from which to evaluate the influence of the crystal field, most results have been obtained so far with the spectra of the divalent ions.
The trivalent 4fn levels can only be obtained through the allowed 5d → 4f transitions which lie in the vacuum ultraviolet, and so do the 6p → 5d transitions which can be used to confirm the 5d configuration. At the present time a new large vacuum spectrograph for our laboratory is under construction which is to be ready for use during the summer of 1963. In view of this it is undesirable to use the much less accurate vacuum ultraviolet wavelengths now available to us. For this reason so far only parts of the Pr iv and Yb iv spectra have been analyzed.
The situation is much more favorable for the third spectra, as the most important lines lie in the more easily accessible region of the spectrum. Moreover, the crystal spectra of the divalent ions also have become of increasing interest recently.
At the present time the goal of obtaining the four lowest configurations has been substantially reached by Sugar for Pr iii and by Bryant for Yb iii. Less complete results have been obtained so far for Gd iii by Callahan. For this ion the configurations are particularly unwieldy, and it cannot be expected that the four ground configurations which contain together nearly 6000 levels can be obtained with anything even remotely approaching completeness. While the parts of the 4f75d, 4f76s, and 4f76p configurations based on the ground term 8S of 4f7 are known, nothing is known, yet about the 4f8 configuration. As Fig. 10 indicates for Gd iii, the 4f8 and 4f75d configurations must be expected very close together, so that the principal lines connecting them must be expected in the infrared. This is apparently true, for none of these lines has been found in the visible or the easily accessible parts of the infrared.
Work on the third spectra of the following elements is in various stages of progress in our laboratory. Nd iii (K. V. Narasimham), Sm iii (F. L. Varsanyi), Ho iii (J. H. McElaney), where the large hyperfine structure is very valuable, Er ii (J. Becher), and Tm iii (G. Gompertz). Work on the remaining elements will be taken up as soon as an opportunity arises. Because of the very large number of lines involved progress is necessarily slow even when all modern aids are available.
It should be emphasized that the preceding article is not a general account of the spectra of the doubly and triply ionized rare earths which would have been impossible in the allotted space, but essentially an outline of the work being carried out in this field at Johns Hopkins University.
The work described in this paper has been carried out with partial support of the U.S. Atomic Energy Commission and of the Office of Aerospace Research of the U.S. Air-Force.
Figures and Tables
1. G. H. Dieke, H. M. Crosswhite, and B. Dunn, J. Opt. Soc. Am. 51, 820 (1961) [CrossRef] .
2. G. H. Dieke, D. Dimock, and H. M. Crosswhite, J. Opt. Soc. Am. 46, 456 (1956) [CrossRef] .
3. G. H. Dieke and L. Heroux, Phys. Rev. 103, 1227 (1956), and subsequent papers on crystal spectra [CrossRef] .
4. See for instance E. H. Carlson and G. H. Dieke, J. Chem. Phys. 29, 229 (1958); [CrossRef] J. Chem. Phys. 34, 1602 (1961); H. M. Crosswhite and G. H. Dieke, ibid.35, 1535 (1961); G. H. Dieke and S. Singh, ibid.35, 555 (1961).
5. G. H. Dieke and L. A. Hall, J. Chem. Phys. 27, 465 (1957) [CrossRef] .
6. G. H. Dieke, Proc. First International Conference on Paramagnetic Resonance, Jerusalem, Israel (1962).
7. G. H. Dieke and R. Sarup, J. Chem. Phys. 36, 371 (1962) [CrossRef] .
8. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1951).
9. J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960), two vols.
10. G. Racah, Phys. Rev. 61, 186 (1942); [CrossRef] Phys. Rev. 62, 438 (1942); Phys. Rev. 36, 367 (1942); Phys. Rev. 76, 1352 (1949).
11. Ce3+: R. Lang, Can. J. Res. A13, 1 (1935); [CrossRef] Can. J. Res. A14, 127 (1936).
12. Er3+: F. L. Varsanyi and G. H. Dieke, J. Chem. Phys. 36, 2951 (1962); [CrossRef] G. H. Dieke and S. Singh, ibid.35, 555 (1961); E. H. Carlson and H. M. Crosswhite, ref. ;B. G. Wybourne, ref. .
13. Nd3+: E. H. Carlson and G. H. Dieke, J. Chem. Phys. 34, 1602 (1961); [CrossRef] E. H. Carlson and H. M. Crosswhite, Johns Hopkins University Spectroscopy Report 19 (1960); B. G. Wybourne, J. Chem. Phys. 32, 639 (1960); [CrossRef] J. Chem. Phys. 34, 279 (1961).
14. Pm3+: J. G. Conway and J. B. Gruber, J. Chem. Phys. 32, 1586 (1960); [CrossRef] M. H. Crozier and W. A. Runciman, ibid.35, 1392 (1961).
15. Sm3+: M. S. Magno and G. H. Dieke, J. Chem. Phys. 37, 2354 (1962); [CrossRef] J. D. Axe and G. H. Dieke, ibid.37, 2364 (1962); B. G. Wybourne, ibid.36, 2295 (1962).
16. Eu3+: L. G. DeShazer and G. H. Dieke, J. Chem. Phys. (to be published); G. S. Ofelt, ibid.
17. Gd3+: A. H. Piksis and G. H. Dieke, J. Chem. Phys. (to be published); W. A. Runciman, J. Chem. Phys. 30, 1632 (1959) [CrossRef] .
18. Tb3+: K. S. Thomas, S. Singh, and G. H. Dieke, J. Chem. Phys. (to be published); G. S. Ofelt, ref. .
20. Ho3+: G. H. Dieke and B. C. Pandey (to be published); M. H. Crozier and W. A. Runciman, ref. .
21. Pr3+: G. H. Dieke and R. Sarup, J. Chem. Phys. 29, 741 (1958); [CrossRef] J. S. Margolis, ibid.35, 1367 (1961).
22. Tm3+: J. B. Gruber and J. G. Conway, J. Chem. Phys. 32, 1178 (1960); [CrossRef] J. Chem. Phys. 32, 1531(1960).
23. Yb3+: G. H. Dieke and H. M. Crosswhite, J. Opt. Soc. Am. 46, 885 (1956) [CrossRef] .
24. B. W. Bryant (to be published).
25. W. R. Callahan, J. Opt. Soc. Am. (to be published).
27. Z. J. Kiss, Phys. Rev. 127, 718 (1962) [CrossRef] .
28. J. Sugar (to be published).
29. G. Racah, J. Opt. Soc. Am. 50, 408 (1960).
30. H. N. Russell and W. F. Meggers, J. Res. Natl. Bur. Std. 9, 625 (1932).
31. W. A. Hovis Jr., J. Opt. Soc. Am. 52, 649 (1962) [CrossRef] .
32. A. J. Freeman and R. E. Watson, Phys. Rev. 127, 2058 (1962) [CrossRef] .