Chapter 73 Chapter 73
6. Racah's 1964 lectures at the Colli~ge de France
6.1.2. Single-electron excitations
It is a r e m a r k a b l e fact that single-electron excitations lead to more complicated effective operators than do two-electron excitations. The analog of (77) is
Y~ Y~ lz~ k'~ (tl). ~k,~tt,t))(~I. k~tlr), z~ ~J (u)).
rCs i # j
(79) For excitations from the closed shell l '~ we must have s = i, and (79) becomes a sum
over true three-electron operators (since r :~ i C j ¢ r). The existence of effective three-electron operators had already been pointed out by Rajnak and Wybourne (1963). Their method was to actually evaluate the matrix elements of the exciting and de-exciting operators, form their product, and ask for the operator whose matrix elements within l N could reproduce the product. A detailed reconciliation of the two approaches for a variety of interacting configurations can be found in the article of Racah and Stein (1967).
In his lectures, Racah showed that the use of the commutator (78) in simplifying (79) leads to z tensors for which the ranks k, k' and k" are all even. He then stated that he expected that the four non-trivial possibilities (222), (224), (244) and (444) for (kk'k") in the d shell would lead to four three-electron operators and hence four new parameters. In that he was unduly pessimistic. In the summer of 1965, Feneuille (1966) showed that one linear combination of the four operators always had null matrix elements, while another possessed matrix elements proportional to a com- bination of those coming from the Coulomb interaction between the d electrons. A similar situation occurs for the lanthanides. At first sight we would expect single- particle excitations to require nine new operators, acting solely within the f shell, to reproduce the effect of the excitations, since, in this case, we have (kk'k") = (222), (224), (244), (246), (266), (444), (446), (466) and (666), respectively. However, it turns out that one linear combination possesses vanishing matrix elements, while those of two others are proportional to those of the operators e2 and 6eo - 7el, respectively, where the ei are defined in eqs. (64) (Judd 1966a). The remaining six operators, which are called ti (with i = 2, 3, 4, 6, 7 and 8), are associated with six parameters T i, just as the e~ are associated with the E ~. Each t~, like each e , corresponds to a distinct pair of irreducible representations W and U of SO(7) and G2. In fact, the analysis leading to the construction of the ti employs the same group-theoretical arguments that Racah made use of in his 1949 article. It is now standard procedure to include the parameters T ~ when fitting the terms of the configurations 4f N (see Goldschmidt 1978). These six T ~, taken with the four E; and the three Trees parameters ~, fl and ";, give 13 electrostatic parameters in all.
6.2. Configuration ordering
After describing the effects of configuration interaction in the d shell, Racah began a discussion of the status of the spectroscopy of free lanthanide ions. His opening statement was that we know very little; but he qualified that remark by pointing out that something was at least known about a few special cases, particularly some second spectra, and a certain amount of information on the fourth spectra could be gleaned from the work on crystals. The basic difficulty for a theorist is the fact that the 4f, 5d and 6s electrons all possess comparable energies. In the third spectra the levels of the configurations 4fN-15d and 4f N compete for the ground level; in the second spectra the competition is between 4fN-15d6s, 4fu-15d 2 and 4fN0s: while for the first spectra (those of the neutral atom) we again have three low configurations to contend with, namely 4fu-~5d6s 2, 4fN-~5d26s and 4fN6s 2. Racah put the con-
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 125 figurations 4f u, 4fU6s and 4fN6s 2 in class B, and the rest in class A. F o r a given N all the configurations in a single class possess the same parity.
Something of the difficulty facing spectroscopists in the mid-1960's can be appreciated when we ask for the ground levels of the cerium and praseodymium atoms and ions from the data available at that time. F r o m the work of Harrison et al. (1941) it was known that low levels o f C e II existed of both types A and B. It was Racah (1955) who found the connection between the two systems and thus confirmed the opinion of Harrison and his co-workers that the lowest level of Ce II is of type A (principally 4f5d 2 4H7/2, in fact). On the other hand, the work of Rosen et al. (1941) indicated that the ground level of Pr II is 4f36s 514, thus belonging to type B. Can we deduce anything about the lowest level of Ce I? We know that the ground level of the adjacent ion to Ce II, namely La II, belongs to 5d 2 (see section 2.1). The addition of a 4f electron converts it into the ground configuration of Ce II.
If, then, we add a 4f electron to the ground configuration of L a I, that is, to 5d6s 2, we might expect to obtain 4f5d6s 2 as the ground configuration of Ce I. The validity of that argument was confirmed by Martin (1963), w h o found the lowest level to be 1 6 4 (thus demonstratin.g an exception to Hund's rule). The bare recitation of this discovery does not do justice to the excitement of the time, when active experimental groups working with atomic beams of lanthanide a t o m s were thwarted from making an u n a m b i g u o u s determination of the ground level of cerium by the presence in their beams of other low-lying populated levels.
O u r argument from lanthanum to cerium cannot be repeated for Pr II because the configurations do not match in the same way. However, we can confidently predict the lowest level of Pr I I I to belong to 4f 3 because, if 4f 3 were not lower than 4fZ6s, we would expect the ground configuration of Pr II to be 4f26s 2, which it is not. The work of Sugar (1963) confirmed 4f 3 as the ground configuration of Pr Ill.
At the time of his lectures, Racah was also aware of the Ce III analysis of Sugar (1965a), which established 4f 2 as the ground configuration for Ce 2+. Thus the ground configurations of Pr I I I and Ce III are both of type B, in contrast to the second spectra, where the types are different. Racah asked whether any sense could be made of results such as these. In particular, can we make predictions for other lanthanides?
6.2.1. The third spectra
The basic idea developed by Racah in his lectures is to use our knowledge of what happens at both ends of the lanthanide series to interpolate towards the middle. We can allow for the widely differing energy spreads of the configurations 4f N by choosing the centers of gravity Eav of the various c o m p o u n d configurations as the crucial energies to study across the series. In his lectures Racah began with the third spectra, for which he proposed the following formulas:
Ea v (4fN- 16p) - Eav (4t "N - 16s) = 36524 + 873(N - 8), (80) Eav(4fN-~5d) - Eav(4fN-16S) = --3446 + 1459(N - 8) + 58[(N - 8) 2 - 33], (81) Eav(4f u-~ 5d) - Eav(4f N) = 23686 + 3660(N - 8) - 307[(N - 8) 2 - 33], (82)
where the coefficients are in units of c m - L In addition to the data already mentioned for the early members of the lanthanide series, Racah could use the results of Meggers and Scribner (1930) for L u l I I (for which N = 15), of Bryant (1965) for Yb I I I (for which N = 14), as well as the d a t a of Callahan (1963) for G d I I I 4f 7(aS)5d, 6s and 6p. It is impossible, today, to reconstruct the arguments that led Racah to choose the numerical coefficients in eqs. (80)-(82). Only three d a t a points (Ce III, Pr III and Yb III) were available to establish eq. (82), since it was not until almost a decade later that Johansson and Litz6n (1973) discovered the position of the ground term 7F of 4f a relative to 9D of 4fT5d in G d III. N o t all the levels of 4f 3 and 4f25d were found by Sugar (1963), so Racah had to rely on the theoretical work of Trees (1964) to fix the relative positions of the centers of gravity of these configurations. The recognition today of the importance of configuration mixing gives us an additional problem to weigh. Should we find Eav(C) for a configuration C by averaging over those levels whose principal c o m p o n e n t s belong to C, or should we perform a diagonalization of the mixture C + C' + C" + " ' , and use as Eav(C ) the average of the entries appearing on that portion of the diagonal associated with C?
In spite of these difficulties, it is interesting to see how Racah's most vulnerable formula, eq. (82), which determines the lowest levels of the doubly ionized lan- thanides, has stood the test of time. F o r N = 1 we get, from that equation,
Eav (La I I I 5d) - Eav(La I I I 4f) = - 6 8 4 6 c m - L
This is in excellent accord with the value of - 7 0 9 0 found by Sugar and K a u f m a n (1965). C o m p a r i s o n s with other lanthanides are not so easy to m a k e because the levels are only partially known. The best way to proceed appears to be to first pick an observed low-lying level (or entire multiplet) whose designation is as pure as possible. A theoretical expression is then worked out for its energy relative to the center of gravity of the configuration. The various Slater integrals Fk(4f, 5d), Gk(4f, 5d) or Racah parameters E k (for the C o u l o m b interactions between the 4f electrons) that are needed can be found by linear interpolation between their values for Pr III 4f 3, Er | I I 4f 12, Pr I I I 4f25d and T m I I I 4fIZ5d, as given by Goldschmidt (1978, tables 1.27, 1.31 and 1.41), which yields
El(4f N) = 5410 + 232(N - 8), EZ(4f N) = 25.8 + 1.15(N - 8),
E3(4f N) = 520 + 22(N - 8); (83)
El(4fN-15d) = 6078 + 229(N - 8), E2(4f N 15d) = 28.6 + 1.17(N - 8),
E3(4fN-15d) = 580 + 20(N - 8); (84)
Fz(4f, 5d) = 188 + (N - 8), F4(4f, 5d) = 20.4 - 0.2(N - 8), Gl(4f, 5d) = 244 - 8(N - 8), G3(4f, 5d) = 29 - 0.6(N - 8),
Gs(4f, 5d) = 3.9 - 0.1(N - 8). (85)
O u r occasional need for a p p r o x i m a t e s p i n - o r b i t coupling constants can be satisfied by using the data for Ce I I I 4f5d and Yb III 4fa35d (Goldschmidt 1978, table 1.36) 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 1 2 7
E .g
~4 e"
.g
I
r---
g
I II
~D
+
I
I
I
~ . ~ ~ +
~ o . < . < t I
I I I 1 f I I I I I f I I
II tl II II tl II II It II II II II II
~l ~ . I
+ +
t ~
-- N I I I m ~
+ + ~7 ~ ~ + +
N~ I I I --
I I
I I I . I I
I I I I 1 I I I I I I I ~ I
o a
, , " =~ a: n a
- - = = - = = = _ = = = _=_= - - = = E E
--- g,
" ~ ~ Ca >. " "
eO - -
. ~ 2 w £ s
g N ~ = ~, s : >,
the forms
~f(4fN-15d) = 1779 + 190(N - 8), ~d ( 4fN- 15d) = 918 + 39(N -- 8), (86) which neglect a small quadratic term. By means of eqs. (83)-(86) we can deduce the positions of the centers of gravity of the configurations 4fu-15d and 4f N from the experimentally known energies of a few (or even just one) low-lying level. The tabulation of energy levels given by Martin et al. (1978) proves extremely helpful here, since these compilers had access to unpublished material from many labora- tories. In fact, the sheer size of table 2, where our results are collected, gives an indication of the considerable activity in rare-earth spectroscopy in the twelve or so years after 1964, the date when Racah only had C e l I I , P r l I I and Y b l I I to work with. Some of the spectroscopists at the National Bureau of Standards at Gaithersburg (Kaufman, Sugar and Martin) have already been mentioned. Other important groups that contributed to the experimental work on the third spectra were sited at The Johns Hopkins University (Becher, Bryant, Callahan and Dupont), the Zeeman Laboratorium, Amsterdam (Van Kleef, Meinders), the Laboratoire Aim6 Cotton, Orsay (Blaise, Camus, Wyart), the Argonne National L a b o r a t o r y (Crosswhite), and Lund (Johansson, Litz6n). The reader is referred to the tables of Martin et al. (1978) for a detailed list of references.
As for table 2 itself, the first impression is that Racah's formula [our eq. (82)-I works extremely well. We can see that the vagaries of the energies of the lowest levels of the configurations 4f N and 4f N-15d conceal the smooth trend of the centers of gravity of the configurations. We can understand why 4fV5d 9D of Gd III should lie lower than 4f 8 7F, thus presenting us with a ground level of type A (in contrast to all other entries in table 2 with the exception of La III). The enormous spread of 4f 7 compared to other 4f N configurations is at work here. Only the two spectra on either side of Gd III, namely Sm III and Tb III, lead to discrepancies larger than a few hundred cm-1. It is likely that a failure in the linearity of eqs. (83) and (84) near the half-filled shell is at least partially responsible for this. Even so, the discrepancies are small compared to the spreads of the configurations 4f 6, 4f v and 4f 8.
6.2.2. The second spectra
Racah began his discussion of the second spectra by regretting that the absence of relevant data precluded the construction of formulas of the type represented by eqs.
(80)-(82). Instead, he suggested that it might be useful (and much more convenient) to compare differences in energy between the lowest levels of opposite parity (the so- called system differences) in the second and third spectra. He noted, for example, that we can write
E(4f 2 3H4) - E(4f5d 1G4) = 3277 E(4f26s, ~) - E(4f5d6s, ~) = - 3854 E(4f 14 xS0) - E(4f135d, 2) = 33386 E(4f146s, 1) - E(4fa35d6s, 5) = 26759
(Ce III), (Ce II), (Yb III),
(Yb II), (87)
so that the differences of the system differences, which can be schematically written
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 129 as
A(III, II) = (A - B)m - (A - B)u, (88)
e v a l u a t e s to 7131 for c e r i u m a n d to the very similar figure of 6627 for y t t e r b i u m . R a c a h suggested t h a t A ( I I I , II) might b e a p p r o x i m a t e l y c o n s t a n t a c r o s s the l a n t h a n i d e series.
T h e r a t i o n a l e for the h y p o t h e s i s of c o n s t a n t A (III, II) w a s n o t discussed by R a c a h , b u t it is e v i d e n t l y b a s e d on the i d e a t h a t all i n t e r a c t i o n s between the 4f e l e c t r o n s a n d b e t w e e n the 5d e l e c t r o n a n d the 4f e l e c t r o n s a r e cancelled b y the differencing p r o c e d u r e . A p a r t f r o m s o m e s m o o t h l y v a r y i n g terms r e p r e s e n t i n g c o n f i g u r a t i o n averages, the o n l y r e m a i n i n g c o n t r i b u t i o n s to t h e energies c o r r e s p o n d to i n t e r a c t i o n s between the 6s e l e c t r o n on the one h a n d and, o n the other, the 4f and 5d electrons. S h o u l d these t o o be s m o o t h l y varying, so s h o u l d be A (III, II). T h i s e x p e c t a t i o n was tested by M a r t i n (1971, 1 9 7 2 ) s o m e six years after R a c a h ' s lectures, when m o r e e x p e r i m e n t a l d a t a h a d b e c o m e available. W e c a n t a k e a d v a n t a g e of the further a d v a n c e in o u r k n o w l e d g e , as r e p r e s e n t e d by the t a b l e s of M a r t i n et al.
(1978), to m a k e a d e t a i l e d e x a m i n a t i o n of R a c a h ' s h y p o t h e s i s . T h i s is d o n e in t a b l e 3. T h e c o l u m n h e a d e d A(III, II) e x h i b i t s n o t m e r e l y a s m o o t h v a r i a t i o n but a r e m a r k a b l e c o n s t a n c y , p a r t i c u l a r l y in the light of the large v a r i a t i o n s in the n u m b e r s listed in the f o u r p r e c e d i n g c o l u m n s . H o w e v e r , it is n o t p o s s i b l e to m a k e an u n a m b i g u o u s a s s e s s m e n t of R a c a h ' s h y p o t h e s i s . T h i s is b e c a u s e the 6s and 5d e l e c t r o n s possess the s a m e p a r i t y a n d , since their energies are c o m p a r a b l e , a p p r e c i - able m i x i n g t a k e s place. T h e J = ~ level of 4f5d6s in eqs. (87), which was used b y R a c a h to j u s t i f y the relative c o n s t a n c y of A from C e t o Yb, is assigned to 4f5d 2 b y
TABLE 3
Differences of differences (in cm ~) between the energies of low-lying levels of types A and B in the second and third spectra of the lanthanides.
R N AIII(fN- ld) BIll (f N ) All(f N- ldsl Bll(f Ns) A(III, 11)
La 1 0 7195 1895 14148 5058
Ce 2 3277 0 0a[2382] b 3854 7131 [4749]
Pr 3 12 847 0 5855a[7832] b'c 0 6992 [5015]
Nd 4 15262 0 9229a[11 310] b'c 0 6033 [3952]
Sm 6 26 284 0 21508 c 0 4776
Eu 7 33 856 0 30189 d 0 3667
Gd 8 0 2381 0 7992 5611
Tb 9 8972 0 3235 0 5737
Er 12 16976 0 10667 0 6309
Tm 13 22 897 0 16567 0 6330
Yb 14 33 386 0 26 759 0 6627
a Principal component listed by Martin et al. (1978) as 4f N 15d2 b Principal component listed by Martin et al. (1978)as 4fN-15d6s c Listed as tentative by Martin et al. (1978)
dThis level is probably undershot by one for which J = 1 [so far unobserved). Martin (1971) uses the tentative number 29 000 here, thus yielding A(III, II) ~- 4900.
Martin et al. (1978). The adjustments that a correction entails are indicated by numbers in brackets in table 3. At the time of Racah's lectures, the position of the lowest level of type A of P r I I was unknown. F r o m A - B for P r I I I and the expectation that A should be about 7000 cm 1, Racah predicted a low level of type A of P r I I about 5500cm-1 above the ground level. This is in good agreement with the observed figure of 5855, which was not established until 1973. That this level belongs principally to 4f 2 5d 2 (rather than 4f25d6s) does not diminish Racah's prediction. He was fully aware of the fact that both 4f25d 2 and 4fZ5d6s must start quite close to the ground level.
In addition to studies of trends and interpolations, Racah outlined detailed work under way in Jerusalem for Gd II and Ce II. The extensive analysis of Russell (1950) for G d I I had established almost all the levels of 4 f T ( 8 S ) ( 5 d + 6 s ) 2 and 4fT(sS)6p(5d + 6s), where 5d + 6s denotes the twelve states of a 5d and a 6s electron. All 71 levels of 4fv(sS)6p(5d + 6s) had been fit by Zeldes (1953) with 17 parameters, thereby obtaining a mean error of 338 c m - 1 . By treating the 5d and 6s electrons together it was possible to allow for the interaction between t h e , P terms coming from 6s6p and 5d6p, which, if ignored, would lead to an anomalous value for Gl(6p,6s). Racah noted that Russell (1950) had reported the existence of 11 additional levels in the region of 4fv(8S)6p(5d + 6s), to which Racah added six more. In his lectures, he suggested that these 17 levels might belong to 4f 8 (TF)(5d + 6s), supporting his speculation by a preliminary calculation. We now know that these two configurations d o indeed overlap.
Racah's description of the current work of his student, Goldschmidt, on Ce II gave many of his audience their first glimpse of what can be achieved by properly taking into account the interactions between several neighboring and overlapping configurations. Five odd configurations and seven even configurations were being investigated. Among the former, the three configurations that constitute 4f(5d + 6s) 2 were treated as a group. Of the 122 possible levels, 80 were known and a good fit could be obtained. The two remaining odd configurations, 4f s and 4f26p, were being considered together, but although 57 out of a total of 110 levels were known, those originating principally from 4fZ6p were too sparse to allow a satisfactory analysis to be carried out. More spectacular was Racah's account of the simul- taneous treatment of the seven configurations of even parity, namely,
4fZ6s + 4fZ5d + 4f5d6p + 4f6s6p + 5d 3 + 5dZ6s + 5d6s 2,
where no fewer than 305 levels occur. Goldschmidt (1978), in the first volume of this H a n d b o o k series, has given a first-hand account of how successive fits can be used to find new levels and improve the parameters. Her article gives many details of the modern approach to fitting atomic energy levels, as well as providing a compre- hensive survey of theoretical lanthanide spectroscopy to 1978. Of particular interest are the many diagrams showing the various kinds of coupling that can occur when the Coulomb and spin-orbit interactions of several inequivalent electrons compete in a single configuration. Reference has already been made in section 2.2.3 to her analyses of the configurations LaII ( 4 f + 6p)(5d + 6s) and L a I I (5d + 6s) 2
+ ( 4 f + 6 p ) 2 + 6s6d. In these cases the numbers of parameters are com-
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 131 parable to the numbers of levels, so comparisons with other lanthanide spectra are essential to constrain the least-squares fitting procedure. The direction that this kind of work has taken was foreseen by Racah, and, indeed, it can be said to have grown naturally out of his own analysis of the two-electron spectrum of Th III (Racah 1950).
6.2.3. The first spectra
Racah left until last a discussion of the first spectra of the lanthanides. Before recounting the substance of Racah's lecture, it seems useful to recall the general situation in 1964. Very little had been firmly established. In his listing of the ground levels of the neutral lanthanides (running from lanthanum to ytterbium), Kuhn (1962, p. 320) set six of the ground levels and seven of the configurations in parentheses to indicate that they were uncertain. This corresponded closely to the tabulation of Klinkenberg (1947) and showed only a marginal improvement over the listing of Meggers (1942), where nine ground levels were not specified. In a carping review of Kuhn's book, Meggers (1962) stated that all of the parenthetical entries had been 'experimentally confirmed or corrected'. He did not allow for the crucial period that had elapsed after Kuhn had sent his manuscript to the printers. He also did not specify what he believed the ground levels to be. The ground level of T b I was still uncertain in the present writer's mind at the time of the Zeeman Centennial Conference in Amsterdam in 1965, and it was not until a few years later that Klinkenberg and van Kleef (1970) determined that 4f 8 (7F)5d6s2 8G13/z, the lowest level of 4f85d6s 2, lies a mere 286cm 1 above the lowest level of 4f96s 2, namely 6H15/2. Meggers's confidence in knowing the lanthanide ground levels seemed odd in 1962. Perhaps, as the doyen of the experimental spectroscopists at the National Bureau of Standards, he felt a responsibility for taking a position in the matter, particularly since competing claims were being made by physicists working with atomic beams.
In his lectures, Racah proposed working again with the differences of the system differences, using the second spectra for reference points. The relevant quantity is A(II, I), defined in analogy to eq. (88). But Racah had much less to work with than before. Although (A - B) H for La II was known to be - 12 253 c m - 7, the term 2F of La I 4f was still unidentified. However, from the distribution of spectral lines Racah estimated ](A - B)II to be greater than 17 500, and so, were it actually at that limit, we should get A(II, I) - 5250. The only other suitable data came from Eu. Racah took (A - B)II = 33 780 and (A - B)~ = 27 853 to yield A(II, I) = 5927, which was encouragingly close to 5250. He suggested that we might hope for some guidance in other spectra by using values of A(II, I) of around 5 0 0 0 c m -
We can see today that Racah was pushing the data beyond their limits. His figure of - 1 7 5 0 0 for (A - B)~ in L a I should be - 1 5 197 (Martin et al. 1978), so A(II, I)is actually 2944 for La. As for Eu II, the level at 33780 is not the lowest level of 4f65d6s, the level 9D 2 lying 30 189 above ground (and used in the construction of table 3). The level of E u I at 27853 is aDs/2 and, to judge from E u l I I , lies above at least some of the levels of SH, so far unobserved in Eu I. The problem of mixed configurations in table 3 would recur if any tabulation of A(II, I) were attempted.