Chapter 73 Chapter 73
5. Crystal spectra
5.1. The impact of Racah's 1949 article
The ideas developed by Racah (1949) were so novel and intricate that it is not surprising that it took some time for them to be appreciated. Adding to the difficulty of assimilation was the terse style of the article itself, which made no concessions to the general reader. There were no examples of how to put to use the many tables or how to calculate the energy matrices for a particular configuration fN. Racah realized, of course, that his methods need not be limited to atomic shell theory, and he suggested to Jahn that the group chain U(5) ~ SO(5) ~ SO(3) could be useful in classifying the states of the nuclear d shell. The resulting article (Jahn 1950) is much easier to understand than Racah's. Reliance is placed on the calculus of the Young tableaux, and all the methods are illustrated by simple examples. Furthermore, Jahn's initial program was more limited than Racah's, and he could appeal to Racah's work for theorems that he needed. 19lore remarkably, the short-range attractive force between d nucleons turned out to be represented by a term in the Hamiltonian that is scalar with respect to the generators of SO(5). Thus the irreducible representations of SO(5) play the role of good quantum numbers, and SO(5) is a true symmetry group of the Hamiltonian. Jahn and his colleagues exploited these properties in a series of articles (Jahn 1951, Jahn and van Wieringen 1951, Flowers 1952a,b, Elliott and Flowers 1955) that led eventually to SU(3) being used by Elliott (1958) to reconcile nuclear shell theory with the collective motions of the nucleons.
Racah took the opportunity to more fully explain his methods in a series of lectures given at the Institute for Advanced Study at Princeton in 1951. The mimeographed notes of Eugen Merzbacher and David Park circulated widely and were reproduced as a C E R N report in 1961 before being included in a Springer Tract in Modern Physics (Racah 1965). Among the standard sources for back- ground material we find the book by Eisenhart (1933), which evidently played an important part in helping Racah develop his ideas. In describing how to use the theory of Lie groups to construct the energy matrix, Racah (1965) chose an arbitrary two-particle interaction rather than a Coulomb potential. He was thus able to set his methods in the framework of nuclear shell theory, thereby supplementing the
work of Jahn (1950). It is sometimes wondered why Racah, with his background in atomic and nuclear physics, did not play a role in the application of Lie groups to particle theory. For one reason or another, the moment when he could have made a major impact on that field passed. A compilation by Gell-Mann and Ne'eman (1964) of the early articles on the symmetries of the strong interactions (the so-called eight- fold way) scarcely mentions Racah's work. The style of their presentation - for example, the method they chose to distinguish the irreducible representations of SU(3) by the dimensions rather than the highest weights - shows that their achievements owe little to Racah. The overwhelming development of group- theoretical methods in particle physics has tended to obscure Racah's work; for example, in the introduction to his book on Lie groups, Gilmore (1974) assigns the date of the realization that non-symmetry (i.e., non-invariance) groups could be useful to around 1960, eleven years after Racah's 1949 article appeared. Another example of an oversight occurs in the chapter in the book by Hamermesh (1962) that describes applications of Lie groups to atomic and nuclear problems. Although reference is made to Racah's article, there is no recognition of the importance of non-invariance groups, and the generalized Wigner-Eckart theorem is not brought into play. States are classified by the irreducible representations of Lie groups and left at that. The reader is led to believe that Lie groups are useful in defining quantum states, but the computational advantages are not pointed out.
It took time, of course, for atomic spectroscopists to come to grips with Racah's approach. One of the most active theoretical groups in the early 1950's was that of Ufford at the University of Pennsylvania. The single volume 91 of the Physical Review for 1953 contains articles by Innes (1953), Reilly (1953) and Meshkov (1953) that apply Racah's methods to the electronic spin-spin interaction, to the matrix elements of the Coulomb interaction for f4, and to fractional parentage coefficients for mixed configurations such as (p + d) N, where (p + d) denotes the 16 states provided by a p electron and a d electron. All acknowledge the guiding hand of Ufford. The reprintings in 1951, 1953, 1957 and 1959 of the book by Condon and Shortley (1935) scarcely helped the dissemination of Racah's methods. Nor did the book by Fano and Racah (1959). Admittedly, it contains one fascinating new idea:
in appendices H and I a connection is made between projective geometry and angular-momentum theory, which is shown to be Desarguesian. However, the book was not accepted as a standard text by the spectroscopic community. The language and notation throughout are almost perversely at variance with the classic articles of Racah (1942b, 1943, 1949). Phase differences abound; in some instances pure imaginary factors appear where everything had been real before. The use of gothic characters, of W, V and X coefficients, and the very title itself, which introduced the word - uncommon to spectroscopists - 'sets', together with the formidable quali- fications 'irreducible' and 'tensorial', all conspired to make the book less attractive.
The monograph of Edmonds (1957) had a greater impact, partly because of the wide range of examples that were provided. The only eccentricity was the use of boldface characters in the scalar proportionality factors (i.e., the reduced matrix elements) appearing in the Wigner-Eckart theorem.
The mid-1950's also saw a detailed description by Brinkman (1956) of the
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 113
symbolic method of Kramers (see section 2.2.2). The final chapter of the book describes the calculation of the Coulomb energies of the terms 2S+~L of the atomic configuration l112, and this makes it possible to assess the value of Kramers's approach. Instead of the Racah coefficient W(ll 12 Ii 12; Lk) (mentioned in section 3.4), a generalized hypergeometric function appears. Apart from additional factors, the two forms must be equivalent, of course. What has happened is that the Racah coefficient has been reconstructed ab ovo in Brinkman's analysis; and, in calcu- lations involving more than one Racah coefficient, they would all be similarly reconstructed. The fatal difficulty with Kramers's method is that the building blocks - the spinor invariants - are too small and too many of them are required. The availability, later in the decade, of the tables of 6-j symbols prepared by Rotenberg et al. (1959) made it easy to apply Racah's methods to actual physical situations.
The decade ended with the publication of the two-volume work of Slater (1960) on atomic structure. A cautious treatment is given of tensor operators and fractional parentage. Expansions in determinantal product states are still resorted to, and there is no discussion of Racah's use of groups. A little later, the writer attempted to remedy that deficiency (Judd 1963). His prefatory assurance that the needs of the experimentalist were borne in mind was sourly commented on, in a private conversation, by Edl6n. The recollection that the book of Condon and Shortley (1935) gave similar problems to Russell (its dedicatee), provided some solace.
5.2. Interplay between theory and experiment
In spite of various problems of identification of individual lines or bands, the general features of the spectra of the tripositive lanthanide ions in crystals were becoming clear by the early 1950's. The sharp absorption lines in the visible region were recognized as being produced by transitions within particular 4f N con- figurations. The more diffuse lines associated with them were seen for what they are:
transitions in which an odd vibrational mode of the lattice is excited at the same time as the parity-forbidden electronic transition within the 4f shell takes place. The occasional overlapping of these two kinds of lines could conceivably blur their distinctive features and lead to the extra levels mentioned in section 2.1. The onset of more intense absorption in the blue or beyond could be safely assigned to transitions of the type 4f -~ 5d, which are allowed by the Laporte parity-change rule.
All of this was anticipated by Van Vleck (1937). It was Hellwege and Hellwege (1951) who finally established that the absorption band of Pr 3÷ in the yellow corresponds to the transition 3 H 4 ~ 1 U 2 of 4f 2, thus resolving a long-standing problem (see section 4.1). This work was carried out with crystals of the so-called double nitrates, P r z M a ( N O 3 ) 1 2 " 2 4 H 2 0 , where M = Mg or Zn. Almost as signi- ficant was the discovery by Hellwege and Kahle (1951a,b)that the absorption bands at 16900, 19020 and 21480cm -1 in salts of E u 3+ correspond to, respectively, 7F 1 ~ 5D o, 7F o -~ 5 D 1 and 7 F 0 --* 5D 2 of 4f 6. The importance of these results lay in the information that they provided on the Slater integrals F k ( o r , equivalently, on the Racah parameters Ek). The erroneous term scheme for La II 4 f 2, a s reported by Condon and Shortley (1935), was now seen to be qualitatively different from that for
Pr 3 + 4f 2, indicating that some adjustment had to be made to the ratios F4/F 2 and F6/F 2. The latter had been thought to be extremely small (see section 2.2.3), but, through a misinterpretation of a decimal point for a multiplication symbol in the number 7361.64 of eqs. (22), which was printed, in the British convention, as 7361.64 in C o n d o n and Shortley's book (but otherwise in their 1931 Physical Review article), the small value of F6/F 2 appeared to be supported by calculations based on a physically realistic 4f wavefunction, such as that corresponding to a hydrogenic 4f orbital, for which the result F6/F 2 ~- 2.5 × 10 -4 was given (Trefftz 1951). The eccentricities of British t y p o g r a p h y had also confused Laporte and Platt (1942) some years earlier. In their article on the degeneracies a m o n g terms deriving from configurations of equivalent electrons, they gave D~/2 of eq. (22) correctly as 429, but then stated that the figure of 7361.64 of C o n d o n and Shortley was an error. Once D 6 is taken for what it actually is, the hydrogenic ratios become
F4/F 2 = 41/297, F6/F 2 = 175/11583 = 0.015. (67)
These ratios (or values close to them) account for the position of 1D 2 in Pr 3+ 4f 2 rather well (Jargensen 1955a, Judd 1955).*
Another significant step forward was the analysis of the absorption spectrum of N d ( B r O 3 ) 3 . 9 H 2 0 by Satten (1953). The ion Nd 3+ possesses the ground con- figuration 4f 3, for which the terms of m a x i m u m multiplicity are separated by multiples of E 3 (see sections 3.1 and 3.5). The same parameter determines the separation of 3p and 3H in 4f 2, so only a small adjustment for the effect of the increased nuclear charge is required to predict the positions of all the quartets of 4f 3.
This enabled Satten (1953) to identify the bands that had been earlier reported by Ewald (1939). Satten's discussion of the positions of the doublets was impeded by an error in converting the inequality F6/F 2 < 1 (which follows from the definition of the Slater integrals) to F6/F 2 < 0.00306 rather than the correct form F6/F2 < 0.0306. This mistake was not noticed because a small value of F6/F 2 was consistent with current thinking in the early 1950's. An exchange of letters between Satten (1955) and Jorgensen (1955b) cleared the matter up.
By 1955, Jorgensen (1955c) had penetrated at least some of the mysteries of the article of Racah (1949) and had begun calculations on the low-lying terms of m a n y 4f N configurations. His analysis represents the first application of Racah's group- theoretical methods to ions in crystals. It is flawed only by the assumption that terms of low seniority could be ignored, which led to his finding it difficult to understand how 5D of 4f 6 could be as low in the energy scale as it appeared to be.
An analysis based on determinantal product states circumvented that difficulty (Judd 1955). It also provided independent checks on Racah's approach, so that the piecing together of matrix elements from his tables could be carried out with a
*Condon and Shortley (1935) inserted the denominators D k at only a few critical points in their tables 16 and 26. Their implicit presence elsewhere was not appreciated by Rao (1950), who was thereby misled so far as to claim that the calculation of Racah (1942b) of the term energies of fa was in error. See Racah (1952b).
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 115 greater degree of confidence. With such aids, and on the assumption of hydrogenic F k ratios [as given in eq. (67)], Elliott et al. (1957) compiled a list of the energies of the terms of maximum and next-to-maximum multiplicity for all configurations 4f N.
The s p i n - o r b i t splitting factors 2 [defined in eq. (32)] for such terms were also calculated as multiples of ~ in the limit of Russell-Saunders coupling. The occur- rence of unexpected simplifications [as exemplified by eqs. (36)] provoked McLellan (1960) to give a group-theoretical treatment of the s p i n - o r b i t interaction, Hso. The group labels for SO(7) and G2 turn out to be (110) and (11), respectively, and the vanishing of the second matrix element in eqs. (36) is due to the absence of the identity representation (00) of G2 in the decomposition of the triple Kronecker product (11)× (11)x (21). Slightly more intricate arguments are needed to explain the vanishing of the first one (see section 11.2). G r o u p theory helped considerably in understanding the unexpected simplifications; but it also exposed new ones.
Mathematical puzzles of these kinds did not prevent the article of Elliott et al. (1957) from proving to be very useful in interpreting the absorption bands of many lanthanide ions. The situation in late 1964 can be assessed by referring to the monograph of Wybourne (1965a).
5.3. Crystal-field splittings
Paramagnetic resonance experiments were, started in the mid-1940's and were well under way by 1950, particularly at Oxford. They provided an important stimulus to analyze the splittings in the levels 2S+lLj produced by the crystal lattice surrounding a lanthanide ion. In order to understand the magnetic properties of the low components (i.e., the sublevels) of a particular level, it was necessary to treat the crystal field as a perturbation. The most d4rect approach is to imagine the 4f electrons moving in an electric potential V(r). The electron i contributes a term -eV(ri) to the Hamiltonian, which can be expanded in a series of harmonic functions of the type rkykq(Oi, ~b~), each one of which is a solution to Laplace's equation VzV = 0. This was the argument presented by Bleaney and Stevens (1953).
It can be carried further: the coefficients Akq of the harmonic functions depend on the distribution of electric charge in the crystal lattice, and a particular charge at R contributes a term to Akq proportional to R - k - 1 y~'q(O, tp). Everything is thus reduced to electrostatics. However, we can evade the problem of calculating the Akq by regarding them as parameters. This is feasible because only a few of them contribute when matrix elements of the type (4f]V(r)14f) are considered. Parity and angular-momentum constraints limit k to 0, 2, 4 and 6, and the number of possible components q can be minimized by a judicious choice of coordinate axes.
The harmonic functions are examples of the tensor operators T~ k~ of section 3.4.
Their matrix elements, for a given J level, are thus proportional to the C G coefficients (31) with J ' = J. Bethe (1929) had already found some algebraic ex- pressions for some special cases for which k = 2 and 4. Stevens (1952) extended the analysis to k = 6 and showed how to determine the proportionality constants. Since the matrix elements of other tensors characterized by the same k and q must, by the W i g n e r - E c k a r t theorem, be proportional to the same set of C G coefficients, Stevens
found it possible to set up equivalences of the type
~ ( 3 z 2 - r 2) = ~(3J~ - J(J + 1)), (68)
i
where ct is an operator equivalent factor. The importance of Stevens's paper was two-fold: it provided tables of values for such functions as 3M 2 - J(J + 1), which occur when the equivalent operators are put t o use; and it also showed how determinantal product states could be used to find the operator equivalent factors.
Today, these factors would be called reduced matrix elements (and would be differently normalized). Although Stevens's article had considerable practical value, its main effect was psychological. It was no longer necessary to digest the details of the classical works on atomic theory: pencil and paper, and a knowledge of the action of shift operators [exemplified in eqs. (4-6)], were all that were needed to attack a problem. The goal might be far off and the route long and tedious but at least the direction to go in was clear. Trying to understand Racah's methods could be postponed with a moderately clear conscience.
In the mid-1950's crystals of the anhydrous trichlorides RCl 3 became available.
The absorption spectra showed lines of unusual sharpness, particularly when the lanthanide ions were substituted in small quantities for La a ÷ in LaCI3. The point symmetry at a lanthanide site is C3h ; that is, there exists a horizontal reflection plane and a three-fold axis of rotation perpendicular to it. The crystal field can be specified by just four parameters, provided our interest lies solely in raatrix elements within the 4f shell. It was soon found that quite good agreement with the experimental work of Sayre et al. (1955) on praseodymium crystals could be obtained by adjusting the parameters and taking into account deviations from perfect Russell-Saunders coupling in 4f 2 (Judd 1957a). Comparable results were found for other trichlorides and for several double nitrate crystals, where an approximate icosahedral symmetry (established later as coming from the packing of twelve O 2 - ions around each lanthanide ion) was found (Judd 1957b). Two review articles of this period, that of Runciman 0958) and that of McClure (1959), convey the atmosphere of the time.
They came too soon, however, to catalog the vast output of Dieke and his students at The Johns Hopkins University on the anhydrous trichlorides. Most of the sublevels of the lowest multiplets 2S+IL of the lanthanides were found, as well as such excited multiplets as 6p and 6I of Gd 3÷ 4f 7, 5D of Eu3+4f 6, and almost all the sublevels of Tm 3 ÷ 4f 12. A great number of partial splittings were also established. In addition to work on the anhydrous trichlorides, considerable attention was also paid to the ethylsulphates, hydrated chlorides, double nitrates, bromates, sulphates and nitrates. A detailed summary is provided by the book of Dieke (1968), which was assembled by Crosswhite and Crosswhite from published and unpublished data of the Hopkins group and from an almost complete text written by Dieke himself shortly before his death in 1965.
5.4. Selection rules
The proper identification of the crystal-field components (the sublevels) of a J
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 117
level calls for a detailed knowledge of selection rules. Van Vleck (1937) argued that electric-dipole, magnetic-dipole and electric-qu~adrupole radiation might all be signi- ficant. In an experiment involving aqueous solutions of Eu a+ ions, Freed and Weissman (1941)used the different interference properties of electric-dipole and magnetic-dipole radiation to show that the fluorescence in the region of 6100 A is electric-dipole while that near 5880 A is magnetic-dipole. It was suspected (and is now known) that these two bands correspond to 5Do-~ 7 F 2 and 5Do-+ 7F 1, respectively. A little later, Van Vleck's analysis was examined by Broer et al. (1945), who had begun a series of experiments on the absorption spectra of solutions of lanthanide ions. They concluded that Van Vleck had overestimated the importance of quadrupole radiation, while his calculation of the magnetic-dipole intensity, although giving a result in line with their own expectations, contained two omis- sions whose effects roughly cancelled. No major modifications to Van Vleck's calculation of the electric-dipole intensities were suggested by Broer et al. (1945).
Their analysis contains an interesting error, however. They took their origin of coordinates at the equilibrium position of a lanthanide ion, which allowed them to omit terms of the type
riYlq(Oi, c~i)
in the expansion of-eV(ri).
Such terms correspond to a constant electric field; were they present, the lanthanide would move. This would be inconsistent with its being at an equilibrium position. Thus harmonic terms for which k = 3 become the leading odd terms in the expansion of-eV(ri);
as such, they would be primarily responsible for mixing small amounts of excited configurations of the types 4f N- ~nd, 4f u+ ~n'd -1, or 4f N-lng into 4f N, thereby permitting electric-dipole transitions within the almost pure 4f shell to take place.That argument appears innocuous enough. But, for a lanthanide ion to be in equilibrium, there must be no electric field at its nucleus. In order to achieve that desirable state of affairs, each 4f wavefunction might have to possess admixtures of a d or a g character to build up the electric charge on one side of the atom and reduce it at the other. We should clearly retain any terms for which k = 1, should they be permitted by the site symmetry of the crystal, if we wish to take pure 4f eigenfunctions as our zeroth-order basis. However, we can begin with the terms for which k = 3 if our zeroth-order eigenfunctions are taken to be those superpositions of 4f with 5d, 5g, etc., that represent the effect of the constant electric field. A choice of this kind is a familiar feature of perturbation theory, since we have always to decide what is the zeroth-order Hamiltonian and what is the perturbation. This point seems to have escaped Broer et al. (1945), though it scarcely affected the semi- quantitative nature of their calculation. They recognized the important point that the presence of terms with high k in the expansion of - e V ( r i ) leads to permissible electric-dipole transitions for which L and J change by large amounts. We can see that a transition J ~ J ' is permitted (as far as the group SOs(3) is concerned) if the product
d~j, X C..~ k X (~'~1 X ~.~j (69)
contains 90. So AJ (and similarly AL) can be as large as 8, since the condition k ~< 7 holds for excitations of the type f--+ g.