Chapter 73 Chapter 73
3. Consolidation and compression
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 93
5d6p and 6s6p on 4f5d into account, she obtained
FZ(4f, 5d) = 15855 _+ 105, F4(4f, 5d) = 12266 + 208, Gl(4f, 5d) =11025 _+ 35, G3(4f, 5d) = 10080 _ 157,
GS(4f, 5d) = 6098 + 305, I18)
for L a l I 4f5d + 4f6s + 5d6p + 6s6p. A similar analysis for LaI1 5d 2 + 5d6s + 6s 2 + 4f6p + 4f 2 + 6p 2 + 6d6s yielded
F2(4f,4f) = 24043, Fg(4f,4f) = 21964, F6(4f,4f) = 14668, (19) with mean errors of around 2};. It is evident that the numbers obtained by Condon and Shortley (1931, 1935) have stood the test of time pretty well, with one exception:
F6(4f,4f). This parameter is almost an order of magnitude too small in eqs. (17), a result which Goldschmidt (1978) attributes to 6p 2 ID2 being mistaken for 4f 2 ID 2.
A clue that something is amiss is the failure of the F k of eqs. (17) to satisfy the requirement F z > F 4 > F 6, which follows from the very definition of the Slater integrals (see, e.g., Condon and Shortley 1935, p. 177). Goldschmidt's analysis confirms that the lowest term of 4f5d has 1G as its principal component rather than 3H. The latter would be expected from a naive application of Hund's rule. Since the lowest configuration of La II is 5d z, whose Hund term, 3F, is indeed the lowest for that configuration, not much attention has been paid to the apparently anomalous situation for 4f5d in La II. The appearance of 4f5d as the ground configuration of CeI and C e I I I is much more striking (see sections 6.2 and 6.2.1).
The hyperfine structures of E u I and L u I I reported by Schiller and Schmidt (1935a,b) were analyzed by Casimir in his 1936 prize essay for the Teyler's Foundation in Haarlem. The later reprint (Casimir 1963) has made this work more accessible. Racah (1931a,b) had worked out a method for using the electronic spin- orbit coupling constants (which could be easily deduced from experiment) together with some relativistic correction factors to find the hyperfine interaction strengths.
The procedure was further elaborated by Breit and Wills (1933). Casimir (1963) was able to use this method to deduce actual numerical values for the magnetic-dipole and electric-quadrupole moments of the europium and lutetium isotopes. His results for the quadrupole moments provided the first quantitative figures for the non- spherical distribution of electric charge within the nucleus.
Nothing as striking as this occurred at that time in the interpretation of the spectra of rare-earth ions in crystals. The deceptively low value of F 6 given in eqs.
(17) was to impede progress for several decades. N o detailed analyses of crystal spectra had been attempted by 1935, though Bethe (1930) had examined in a qualitative way some Zeeman data of Becquerel (1929). Bethe (1929) had also set up a formal theory for the splittings of atomic levels produced by the crystal field, but no applications to rare-earth ions were possible.
operator H involves expanding both bra and ket according to the symbolic scheme ]TJ mj ) -o Zl~,Sms L m c ) ---, Z {K1 K2""KN }. (20) For ground levels of some lanthanide ions the sums in (20) can be reduced to a single term. Thus for Ho 3+ 4f 1° 518 we have
[ 4 f 1 ° 5 I s , m s = 8 ) = 14f1° 5I, Ms = 2, m c = 6 )
= { 3 + 2 + 1 + 0 + - 1 + - 2 + - 3 + 3 2 - 1-},
where the numbers in the curly bracket give ms values and the signs indicate ms = +½ or - ½ . In this case the matrix element of H a m o u n t s to an integral over a product of two Slater determinants and H itself. On the other hand, the state ] 4 f 6 x p 1 , M j = l ) of Eu 3+ requires a sum over no fewer than 114 Slater de- terminants. If H were a two-electron operator like the C o u l o m b interaction it would be expressible as a sum over 15 operators of the type h~, where 1 ~<i < j ~< 6 for the six-electron configuration f6. T o find the C o u l o m b energy of 1P of f6 we should have 114 × 114 × 15 ( = 194940) integrals to evaluate, each one of which is a linear combination of four terms involving F o, F 2, F 4 and F 6. Hermiticity constraints and various selection rules would reduce the work, of course, but it is clear that much would still have to be done.
The reaction of the m o d e r n research worker to this kind of problem is to enlist the aid of the computer without further ado. It is often surprising to see to what extent the tedium of writing a computer p r o g r a m is preferred to bringing one's imagination into play. The situation is particularly piquant for atomic spectroscopy because all fundamental interactions between particles are known with high pre- cision. If one merely wanted to calculate the energy levels of an a t o m one might well choose to use as an analog computer a second a t o m of precisely the same kind as the first. An important c o m p o n e n t of theoretical spectroscopy is the nature of the methods themselves. The opportunity for invention appeals to anyone with a taste for applied mathematics. In the 1930's, of course, there was no option but to try to simplify the theory. We now turn to some of these approaches to see how far problems specific to the rare earths could be reduced to manageable proportions.
3.1. Diagonal sums
Since the C o u l o m b interaction e2/rij between electrons i and j c o m m u t e s with S and L, the energies of the terms 2s + 1L of an atomic configuration are independent of Ms and ML. This was put to good use by Slater (1929), who recognized that a progressive calculation of the matrix elements of the total electronic C o u l o m b interaction, taken between a determinantal product state and its complex conjugate, could be m a d e to yield the energies of the terms themselves. Consider, for example, the terms of f3 with Ms = 2 z. The one with m a x i m u m L is 4I, for which the c o m p o n e n t with ML = 6 is represented bv the single Slater determinant {3 + 2 + 1 ÷ . J "
The C o u l o m b energy is the sum of the C o u l o m b energies for {3 + 2 + }, {3 + 1 + } and {2 + 1 + }, which, from tables 16 and 2 6 of C o n d o n and Shortley (1935, pp. 179, 180),
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 95 turns out to be given by
E(4I) = 3F0 - 65F2 - 141F4 - 221F 6, (21)
where Fk = Fk/Dk, with
D O = 1, D 2 = 225, D4 = 1089, D6 = 7361.64 = 184041/25. (22) We n o w hold M s at 2 ~ and step M L d o w n to 5. O n l y one d e t e r m i n a n t can be con- structed, n a m e l y {3 + 2 + 0 + }. There is thus n o 4H term in f3, and the sum of the C o u l o m b energies for { 3 + 2 + } , { 3 + 0 + } and { 2 + 0 + } gives eq. (21) again.
Continuing, we step M L d o w n to 4. There are n o w t w o determinants, {3 + 2 + - 1 + } and {3 + 1 + 0 + } . O n e linear c o m b i n a t i o n of these c o r r e s p o n d s t o 4I, the other, o r t h o g o n a l to the first, c o r r e s p o n d s to 4G. T h e strength of the d i a g o n a l - s u m m e t h o d n o w appears. We d o n o t need to k n o w these linear c o m b i n a t i o n s in detail. Instead, we simply use, for the energies E,
E ( 4 G ) + E ( 4 I ) = E { 3 +2 + - 1 +} + E { 3 +1 + 0 +}
= 6F0 - 75F2 - 216F4 - 1443F6.
F r o m (21) we c a n d e d u c e that
E(gG) = 3Fo - 10Fz - 75F4 - 1222F6. (23)
Continuing, we get
E(4S) = E(4F) = 3F0 - 30F2 - 99F4 - 858F6, (24)
E(gD) = 3Fo + 25F2 - 33F4 - 1859F6. (25)
This p r o c e d u r e can be repeated for M s = ½. However, the terms 2D, 2F, 2G and 2H all occur twice in f3, a n d the m e t h o d can only give the sums of the energies of like terms. In addition to this limitation, the m e t h o d is quite long for terms of low S and L. Thus, for the 1P term of f6 m e n t i o n e d in section 3, we would have to calculate the d i a g o n a l energies of n o t only the 114 determinantal p r o d u c t states a p p r o p r i a t e t o 1p but also the several h u n d r e d determinants for which ( M s , ML) = (0, 1), (0,2), (1, 1) and (1,2) in order to be able to separate out E( 1 P) from the various d i a g o n a l sums. Such applications of the d i a g o n a l - s u m m e t h o d as appear in the literature are for m u c h simpler configurations such as d3s (see B o w m a n 1941, Catalan and Antunes 1936).
3.2. D i r a c - V a n Vleck method
The configuration Is comprises two terms, 1L a n d 3L. The energy separating them is p r o p o r t i o n a l to the exchange integral G~(/,s). We can r e p r o d u c e this energy separation by introducing an effective H a m i l t o n i a n of the type xG~(l,s)sl "s2, where x is a suitably chosen constant. It turns out that x = - 2 / ( 2 / + 1). The general a p p r o a c h , in which scalar p r o d u c t s involving the spin and orbital angular m o m e n t a of the electrons are i n t r o d u c e d to provide effective H a m i l t o n i a n s , was developed by Van Vleck (1934) f r o m a statement of Dirac (1929) relating Sl" s2 to the p e r m u t a t i o n
operator for two electrons. It has been extended and described in great detail by Slater (1960).
We need not spend much time with the Dirac-Van Vleck method here. Its successes occur for very simple configurations such as p3 and for configurations involving s electrons. F o r us the prime example of the latter is fNs, which can be readily handled by alternative techniques. Although the method excited some attention in the 1930's, it is now little more than a curiosity. It had little impact on the development of atomic theory.
3.3. Tensor analysis
The application of angular-momentum theory to atomic spectroscopy is not limited to bringing eqs. (3)-(7) into play. In their book, Condon and Shortley (1935, ch. 3) developed the theory with particular attention to operators T that are vectors.
They did this by specifying the commutation relations of Twith respect to J rather than by stating the transformation properties of the components of T under rotations. They considered angular momenta J built from two parts (S and L) and obtained formulas for the matrix elements of operators that behave as a vector with respect to one part (say L) and a scalar with respect to the other (S, in this case).
These formulas involve proportionality constants that would be called reduced matrix elements today. Condon and Shortley systematized the methods that had come into current use but which were often only hinted at, if that, by many theorists.
F o r example, Van Vleck (1932) gave the formula
<SLJ MILzlS L J + l M)
= - [ ( J + L + S + 2 ) ( S + L - J ) ( J + S + 1 - L ) ( J + L + 1 - S )
x (J + M + 1)(J - M + l)/4(J + 1)a(2j + 1)(2J + 3)] 1/2 (26) without proof, merely remarking that it 'can be shown' to be the above. It is by no means trivial to derive eq. (26), but we can easily retrace the steps leading to its final form by referring to the relevant parts of the analysis of Condon and Shortley [1935, section 93, eq. (9) and section 103, eq. (2b)].
Work of this kind was given in the language of group theory by Wigner (1931).
For example, we can equally well obtain eq. (26) by using the techniques of his ch.
XXIII. Why, then, did the analysis of Condon and Shortley make more of an impact than that of Wigner? An answer to this question can be sensed by using Wigner's book to actually derive eq. (26). The dependence of the matrix element on M is given by eq. (18) of ch. XXIII. The remaining part is set out in the first of the pair of equations labelled (19a); however, the factor VNSL;N'SL appearing there is not defined.
Nor could it be, since the analysis is for a general vector operator T that commutes with S, not just for L. Two pages later, in deriving the Land~ g factor, VNSL:NSL is given as [ L ( L + 1)]1/2h for T = L, but the reader who is not totally secure in following the arguments leading to that result (in particular, the specialization N' = N) might wonder whether it is applicable to matrix elements off-diagonal with respect to 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 97
Although it provides more detail, the analysis of Condon and Shortley (1935, ch.
3) is not as far-reaching as that of Wigner (1931). In his chapter XXI, Wigner defined an irreducible tensor operator T o f degree (o in terms of its transformation properties under rotations. In m o d e r n notation, a particular c o m p o n e n t p would be written as T~ ~'), where - c o --< p ~< e). Condon and Shortley's analysis is limited to the special cases for which c0-K< 1. The word irreducible m a k e s it clear that, of the 2 u ) + 1 components of T t°'~, no subset of n components, where n < 2oJ + 1, can be formed from the T~ '~J (perhaps by taking linear combinations of them) that transform a m o n g themselves under arbitrary rotations. Because of the formal nature of Wigner's definition, the full force of his results were not immediately recognized. The absence of examples was a contributing factor to the lack of appreciation. The casual reader would have been helped by a statement that the spherical harmonics Yo~, are examples of the tensors T~ "). It was left to Racah (1942b) to give an algebraic treatment for tensors similar to that of Condon and Shortley (1935, ch. 3) for vectors. Instead of their rotational properties, the operators T~ k) (following Racah's use of roman letters rather than the greek of Wigner) were defined by their c o m m u t a t i o n relations with respect to J:
[ Jz, T(q k,] = qT~ k', (27)
[ J + . T~ k)] = . [k(k + . .1) - q(q + 1)] 1/2
T(k)q+_l.
(28)The similarity of eqs. (27) and (28) to eqs. (5) and (6), respectively, is to be noticed. It indicates that the kets l J, Mj ) transform under rotations in an identical way to that of the tensors T~ k) for which k = J and q = M s.
Racah's motivation for extending the vector analysis to tensors was his need to cope with matrix elements of the C o u l o m b interaction in a more systematic way than that provided by Slater (1929). The definitit)n of a Legendre polynomial Pk yields
r ~21 = ~, (rk</rk> + 1 ) Pk (cos c012), (29l
k
where r< and r> are the lesser and greater of rl and r2, the lengths of the vectors rl and r2 that define the positions of the two electrons and that are separated by an angle o12. According to the addition theorem (see, e.g., Whittaker and Watson 1927, p. 326), Pk(COSCO12) can be expressed as a sum over the products Ya*(O1, O1)Ykq(O2, ~b 2), SO the problem of finding matrix elements of spherical har- monics arises in a direct way.
3.4. The Wigner-Eckart theorem
Although C o n d o n and Shortley (1935, ch. 3) gave the dependence o n M j , q and M) of the matrix elements
J Ms L M;
(30)for the special cases for which k = 1, they did not point out that it was the same as
that of the CG coefficients
(J M j I k q , J ' M'~). (31)
An explicit formula to that effect was given for arbitrary rank k by Wigner [1931, ch.
XXI, eq. (19)]. He also recognized that his result had a deeper significance, merely remarking, however, that credit f~r a more general form should go to Eckart (1930).
The proportionality of (30) to (31) would today be regarded as a statement of the Wigner-Eckart theorem for the group SO(3).
It took a little while for the full power of the WE theorem to be appreciated. In its most obvious applications, it allows any vector operator to be replaced by another provided a proportionality constant is included. For example, we can write L + 2S -- gJ since L S and J all satisfy similar commutation relations with respect to J. The spin-orbit equivalence,
Hso = ~ ¢(ri) si" li = 2S. L, (32)
i
can be justified by examining the commutation relations of the two expressions for Hso with respect to S, L and J separately. In other words, both forms of Hso given by (32) transform identically with respect to the generators of the three groups SOs(3), SOL(3) and SO j(3).
Racah (1942a) used the WE theorem to evaluate the matrix elements of Pk(COS~O12) for a term L of the configuration 1112. Since there is no preferred direction in space, the result had to be independent of whatever component ML of L was chosen. With some intricate manipulation, Racah (1942a) was able to obtain a symmetrical expression which he later related to the so-called Racah (1942b) coefficient W(l 1121112 ; Lk). Similar work appears to have been done by Wigner, who constructed the 6-j symbol (a slightly more symmetrical quantity than the W function) in an unpublished manuscript of 1940 (see Wigner 1965). The introduction of these SO(3) invariants constituted an enormous psychological breakthrough.
Although, at first, it was tedious to calculate them, advantage could be taken of their orthonormality relations to provide frequent checks. Nowadays we can use modern computing techniques to evaluate whatever 6-j symbols we need, as well as such higher extensions as 9-j or 12-j symbols. The stage was set in 1942 for the development of the modern quantum theory of angular momentum.
3.5. LI coupling
One of the first uses that Racah (1942b) put his W function to was the calculation of the term energies of the lanthanide configuration f3. The limitations of Slater's diagonal-sum method were mentioned in section 3.1. Racah neatly circumvented the problem of satisfying the Pauli exclusion principle by introducing 'spin-up' and 'spin-down' spaces. Two electrons are put in the first space (space A, say) and the remaining electron is put in the second (space B). Thus
Msa = 1, MsB = --~, 1 M s = ½. (33)
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 99 In space A the allowed values of L (say LA) are 1, 3 and 5. Given one of these numbers, the state [L A MLA) is automatically antisymmetric with respect to an interchange of the two electrons in space A. There is no need to impose the antisymmetrization for the interchange of either of these electrons with the one in space B becattse the m~
values are different. A state of fa can thus be written as ](LAf)LML) with condi- tion (33) understood. All calculations can be carried out in what we may term Ll coupling.
Consider, for example, the D terms of f3. Our basis consists of [(Pf)D), I(FfjD}
and [(Hf)D). The roots of the 3 × 3 matrix of the Coulomb interaction yield the energies of 4 D and the two 2D terms. We have already given E(4D) in eq. (25). Its extraction leaves us with a quadratic equation for E(2D), from which explicit expressions in terms of the Slater integrals can be obtained. Formulas for all the terms of f3 were given by Racah [1942b, eqs. (98)].
Although this analysis broke new ground, its generalizations are limited. In the case of ~P of f6 mentioned in sections 3 and 3.1, we would be forced to put three electrons in space A and three in space B (since necessarily M s = 0). Both L A and L B can run over the values 0, 2, 3, 4 and 6, these being the L values for MSA --- 23 and
MsB = --23 (see section 3.1 for the terms of maximum multiplicity of f3). There are thus eight possible coupled states of the type I(LALB)P). The eight roots of the matrix of the Coulomb interaction yield the energies of 5p, 1p and the six 3p terms.
To find E(1p) we would have to perform auxiliary analyses for M s = 2 (putting five electrons in A and one in B) and for Ms = 1 (putting four electrons in A and two in B) in order to extract the roots for the quintet and the triplets. Although this is a great improvement over the method of Slater determinants sketched in section 3.1, it leaves much to be desired.
Part of our sense of dissatisfaction comes from some unexpected simplifications in the f3 analysis, which indicate - or at least strongly suggest - that some unknown structure in the analysis is being overlooked. F o r example, the energy spacings between terms of maximum multiplicity, which can be found very easily from eqs.
(21)-(25), are all simple multiples of the single linear combination
5F2 + 6F4 - 91F6. (34)
Nothing in the analysis so far has prepared us for this amazing result. Again, the three energy separations between the two 2D terms, between the two 2G terms, and between the two 2H terms [as given by Racah (1942b)in his eqs. (98)] are expressible through just two linear combinations of the Fk [one of which is (34)] rather than the three we might have expected. The solution of one problem has brought another to the fore.
3.6. Coefficients of fractional parentage
Shortly after his two 1942 papers, Racah (1943) presented a general method for coping with the Pauli exclusion principle when dealing with equivalent electrons.
We imagine a term SL of l N as being produced by coupling the Nth electron, IN, to a state 7'S'L' of the remaining N - 1 electrons. We write it as ]?,'S'L', Ix, SL), where
the couplings (S's)S and (L'l)Lare implied. It cannot be expected that such a state would be antisymmetric with respect to the interchange of the Nth electron with any of the other N - 1; however, we can superpose a variety of such states with c o m m o n SL to achieve a complete antisymmetry by writing
It N 5 = Y IIIN- ' ',tN,SL ), (35t
where ~'-~ ?'S'L' and ~ - 7SL. The weighting factors (~'1}~,) are the celebrated coefficients of fractional parentage (cfp). The internal symbol [} is meant to stress that the tables of cfp are not necessarily square. It is a corruption of a special symbol chosen by Racah (1943) from an extended set of symbols once made available to authors of Physical Review articles (but long since discontinued).
The idea of fractional parentage stems from work of Bacher and Goudsmit {1934).
These authors were interested in representing the many-particle effects of con- figuration interaction, in particular, the displacements of energy levels. F o r those purposes, only the squares of the cfp are required. The
advantage
of the formal expansion, given by (35), is that states of l N are uniquely defined in terms of the states of l N- l the cfp, and, of course, the C G coefficients that are implicit in any coupled state. Terms like 2S+~L can now be separated and defined by setting up tables of cfp.In his 1943 paper, Racah limited his attention to the configurations pN and d N. No attempt was made to treat the lanthanide configurations fN. Racah (1943) devised a method for separating like terms in d N by means of the seniority number v, but he knew that it was inadequate for as simple a configuration as f3. The two 2D terms of d 3 can be separated by insisting that one of them contains no 1S character in its parentage. Such a state, occurring first in a three-electron configuration, is said to have seniority 3 and we write 2D. A similar device can be used to separate the two 2F terms of fa, but how do we cope with the pairs 2D, 2G and 2H.9 We could, of course, insist that such a cfp as (f21D[}f372G) be zero to give meaning to 7. There are more options here than we need, so the next step would be to experiment with the choices available to see if some work better than others. Should we proceed along these lines, we would be immediately struck by all kinds of unexpected results.
F o r example, if we pick one 2H state by insisting that (f2 aH[}f372H) be zero, then the orthogonal state ]f37' 2H) possesses no component from the parent 3F of f2.
Again, the exclusion of 1S from the parentage of 2F, which defines the state 2F, also leads to the unexpected vanishing of (f23F[}f32F) '
As soon as the cfp are used to calculate matrix elements, more and more surprises occur. F o r example, the two 2H states defined above satisfy the remarkable relations (f3 72HlHso[ f 3 7 2 H ) = (f3 72nlHso[ f37' 2 H ) = 0, (36) where the spin-orbit interaction Hso is defined as in eq. (32). In addition to selection rules like (36), proportionalities abound. F o r example, the diagonal Coulomb matrix elements satisfy the equation
3[E(f3 72H) _ E ( f 3 2 p ) ] = E(f2 3p) _ E(f23H) '