Chapter 73 Chapter 73
11. Mathematical implications
One of the most remarkable features of the analysis of Racah (1949) for the Coulomb energies in the f shell is that the theory works much better and shows more simplifications than could possibly have been anticipated in 1949. As was explained in section 4.3.6, Racah's use of the Lie groups SO(7) and G2 provides explanations for the vanishing of many matrix elements and for the proportionalities that some matrices bear to others; but it also goes beyond a straightforward application of the Wigner-Eckart theorem that such examples represent. The principal surprises are as follows:
(a) In the calculation of the matrix elements of the e, of eqs. (64), multiplicity difficulties are much less of an encumbrance than might have been expected. F o r example, in working out
( f u W UzSLIe2
If:'
w' U'.c'SL), (117)Racah needed to know how many times (40) (the G2 label for e2) occurs in the reduction of the Kronecker product U x U', since this determines the number of possible labels ~? to be attached to the isoscalar factors
(UzL](40)0 + U'z'L);, (118)
of eq. (51) when the generalized Wigner-Eckart theorem is used to evaluate the matrix element given by (117). But, although (40) occurs up to five times in the reduction of those products U x U' appearing in the f shell, Racah found that each set of isoscalar factors given by (118), for a given y and various z, r' and L, is proportional to every other set; with, however, the sole exception of the set for which U = U' = (21), where two labels 7~ and 72 are required.
(b) The calculation of the matrix elements of e3 is enormously simplified by introducing an operator 12, given by 2-tL a - 12G(G2), where G(Ga) is Casimir's operator for Gz (briefly mentioned in section 8.3.3). Racah brought g2 into play because its labels WU are the same as those for e3, namely (220)(22), and because its eigenvalues can be immediately written down. It could thus be used to find a set of isoscalar factors (UrL[(22)0 + U'z'L) for a particular U and U', and hence define an initial ~. By what must have seemed like an astonishing piece of good luck, Racah discovered that the matrices of the combination e3 + f2 (for a given U and U') are all proportional to one another. Thus no multiplicity labels are required at all for the operator e3 + f2, although (22) occurs up to three times in the reduction of the relevant U x U'.
(c) All matrices of e3 + £2 vanish for states whose seniorities v (defined in section
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 177
3.6) are 6 or 7. Others (of specified U, U', v and S) are related by coefficients depending only on N and v.
(d) The following sum is valid:
14-v'
(fNvUSL]e3 + Y2[fNvU'SL) = 0 . (119)
N--V
More unexpected simplifications were discovered when the matrix elements of other operators (like Hso or V "~) were evaluated. The casual reader who opens the book of tables compiled by Nielson and Koster (1963) might be surprised to find large numbers of null matrix elements. Most of these correspond to selection rules on W or U that come immediately from an application of the generalized W i g n e r - Eckart theorem, but the others constitute an unnervingly large residue. It was clear by the 1960's that they were the indicators of unrecognized group structure in the f shell.
11.1. Quasi-spin
Of the two 2D terms in d 3, one possesses no 1S character in its parentage (see section 3.6), and corresponds to a seniority v of 3. The other, for which v = 1, can be imagined as being produced by adding a ~S term to 2D of d ~. It can be written (in an unnormalized form) as (a*a*)~°°la*]O). The use of the fermion creation operators guarantees proper antisymmetry. The 2D term of d 5 with seniority 1 can be constructed by a further application of (a*a*)~°°~; in fact, we can continue to d 7 and finally to d 9, thereby obtaining five 2D terms for which v = 1 in all cases. We can return to 2 D of d 1 by means of (aa) ~°°~. The c o m m u t a t o r of (a'a*) ~°°~ and (aa) ~°°~
yields an o p e r a t o r proportional to (a*a)~°°l + (aa*) I°°l, and the collection of three operators closes under commutation. With suitable prefacing coefficients, the com- mutation relations can be made identical to those for the three components Q +, Q and Qz of an a n g u l a r - m o m e n t u m vector Q, which is called the quasi-spin. An analogous operator was first constructed by K e r m a n (1961) for nuclear shell theory and elaborated by Flowers and Szpikowski (1964) and by Lawson and Macfarlane (1965). With the help of that work it was not difficult to extend the method to the atomic case (Judd 1967b).
The five 2D terms of the d shell for which v = 1 constitute a quasi-spin quintet:
that is, Q = 2. As we run from d to d 9, adding pairs of electrons coupled to iS, the eigenvalue M e of Q~ changes in integral steps from - 2 to + 2. F o r states of seniority v in l u, we have, in general,
Q = ½ ( 2 1 + l - v ) , M Q = - ½ ( 2 1 + l - N ) . (120) All properties of the seniority v can now be interpreted in terms of angular- m o m e n t u m theory. The vector Q is scalar with respect to S and L as well as to the generators of G2 and SO(7). O p e r a t o r s can be assigned quasi-spin ranks K;
obviously K = 1 for Q, while K = 0 for S and L. However, Q does not c o m m u t e with V "~ when t is even (and nonzero), and for these tensors it turns out that K - 1.
More to the point, a detailed analysis reveals that K - - 0 for e 2 and K = 2 for e3 + Q. This knowledge allows us to understand many of the puzzles listed under (a)-(d) in section 11 above. For example, the vanishing of e 3 + ~2 for states for which v = 6 or 7 corresponds to the vanishing of a tensor of rank 2 when set between states for which Q = ½ or 0. Again, we have only to apply the Wigner-Eckart theorem in quasi-spin space to the matrix elements appearing in eq. (119) to convert that equation to
~ ( Q M Q I 2 0 , QM e)--O,
MQ
a result that is merely a statement of the orthogonality of C G coefficients of argument 2 to the corresponding ones of argument 0, all of which are equal to 1.
Once the idea of quasi-spin had been introduced, it did not take long for all kinds of properties to be put in perspective. F o r example, Racah's observation that every W occurs in the f shell with two pairs ($1, v l) and (S 2, v2) turns out to correspond to a spin-quasi-spin pair ($1, Q~) and its companion ($2, Q2) for which $2 = Q1 and Q2 - S~. Other properties of the spin-quasi-spin interchange have recently been developed to explain, among other things, the inversion of the octets and sextets of f7 compared to the terms of corresponding L in f2 (Judd et al. 1986). Similar methods can be used to account for some of the properties of e2 mentioned in section 11, note (a), above. However, not all of the unexpected simplifications listed in section 11 yield to this approach.
11.2. Spin-up and spin-down spaces
The idea of dividing the electrons of the f shell into two classes according to their spin orientation was mentioned in section 3.5 as being the method that Racah (1942b) used to calculate the term energies of f3. We can go further and imagine that f electrons come in two kinds: those with their spins up (fA), and those with their spin down fiB)- The configuration fN is thus a collection of the configurations
fr A'B fN-r for r = 0, 1 . . . N. The Pauli exclusion principle need only be imposed on the fA electrons or on the fs electrons separately; and we can introduce Lie groups such as G2A for the fA electrons or SOB(7) for the fB electrons [Judd 1967b). Several options are available for coupling the spin-up and spin-down spaces. It was found that a coupling at the G2 level was particularly useful in accounting for the reduction in the number of multiplicity labels 7 of section 11, notes (a) and (b), (Judd and Armstrong 1969). Of course, our operators must also be given a classification in the A and B spaces, but this is usually not too difficult to do. F o r example, e2 has non-vanishing matrix elements in f2 only for the singlets, that is, for states for which U -= (20) and M s = 0. These belong t o fA fa, which has a G2A × G2B structure of (10) x (10). For
(((10) × (10))(20)I(U A × Us)(40)[((10 ) x (10))(20)> (121) not to vanish, we must have U A ~ U B ~- (20), as can be confirmed from the table of Kronecker products given by Wybourne (1970). So e 2 ~ ((20)A × (20)B)(40). The
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 179 operator e3 is somewhat more complicated to treat: it needs the superposition of four products (UA X UB)(22).
As an example of the method, consider the two sets of matrix elements
CL = (f5 4(211)(30)Lie 2 if5 4(211 ) (30)L) (122)
and
DL = (f5 2(221)(30)Lle 2 if5 2(221)(30)L), (123)
where L runs over the possibilities P, F, G, H, I, K and M. In spite of the fact that (30) × (30) contains (40) twice (Racah 1949, table V), thus implying the potential need for two symbols 7 in the isoscalar factor (118), it turns out that CL = 4DL. Can we understand how an equation of this simplicity c a n come about? We first note that the possible representations UA in fA ~' (or, equally, UB in fa u) are i10) for N = 1, [10) + (11) for N = 2, and (00) + (10) + (20) for N = 3 or 4 (Racah 1949, table i).
Picking M s = 3 (corresponding to f4fB), eq. (122) becomes
CL = ( ( ( 2 0 ) × (10))(30)L1((20)× (20))(40)1((20) × (lO))(30)L). (124) We can equally well pick M s = ½, but first we have to make the expansion
If 5 4(211) (30)(M s = ½)L) = all(20) × tll))(30)L) + b[(120) × (10))(30)L), (125) getting an equation of the form
CL = a2EL + 2 a b F t + b2GL, (126)
where EL, FL and GL are defined in a similar way to CL in eq. (124). In fact, GL has exactly the same structure as CLexcept that (20) in the bra and ket refers to f3 rather than f~, while (10) refers to f2 rather than fa. The first corresponds to a hole- particle interchange in spin-up space, the second to passing from 2F of fl tO 3F of f2.
F o r the o p e r a t o r e2, which possesses the structure (20) × (20) corresponding to even-rank tensors in the A and B spaces, these two changes produce the factors - 1 and - ] (Nielson and Koster 1963, p. 70). So GL = ]CL. The orthogonal expansion to (125) gives if5 2(221)(30)(Ms = ½)L), so we have
D L = b 2 E L - 2abF L + a2GL. (127)
Since e 2 is diagonal with respect to S, the cross term must vanish: that is,
ab(EL - GL) -- (a 2 -- b2)FL = 0. (128)
Equations (126)-(128), together with GL = ± C 3 L, are enough to show that CL, DL, EL and GL are all proportional to one another. In particular, CL is proportional to DL, the result we want. F o r readers interested in following the argument in detail, the additional information a = _+½x/8 and b = _+1 (from Donlan 1970, p. 54) may be useful.
The line of reasoning given above is a good example of how group theory can be used to establish connections across the entire f shell. The spin-up and spin-down spaces lead to a better understanding of the properties of the C o u l o m b interaction - an interaction which, in itself, makes no reference to the orientation of the spins of
the electrons. Because exchange and direct interactions are of comparable magni- tude for 4f electrons, one would not expect the A and B spaces to have any direct application to experiment. Most remarkably, however, it turns o u t - for reasons that are still obscure today - that the three 5D terms of 4f 6 for which Ms = 2 are almost pure states of the type IrA 5 (L g)fB, D), where L g - H (for the lowest 5D term), L g --- F (for the middle 5D term), and LA =-- P (for the highest 5D term). These results are important for understanding the multiplet splittings of the 50 terms, since it turns out that (f and 2 of eq. (32) are related by
,~ = (f[-LA(L A q- 1) - La(L B + 1)]/2MsL(L + 1). (129) So 2 is large for the lowest 5D term, very small for the middle one, and negative for the highest one. The large observed separations between 5D0, 5D 1 and 5D2 of E u 3 +
4f 6 are thus a direct consequence of the purity of the lowest 5D term in Ll coupling.
Equation (129) also provides an explanation for ( f 3 2 ( l l ) H [ H s o [ f 3 2 ( l l ) H ) being zero, a result presented as a curiosity in eq. (36) and mentioned again in section 5.2.
F o r f] fa, the representation (11)can only be formed from the product (10) A x (10)a of the representations of G2A × G2B (Wybourne 1970, table E-4). So L A = L B = 3, and 2 = 0 from eq. (129). A similar argument can be used to explain 2(f 3 zp) = 0 and 2(f6 sp) = 0, both of which would be tedious to account for otherwise.
11.3. Group extensions
The development of the theory of the spin-up and spin-down spaces did not end the search for new group structure in the f shell. It has been a theme of recurring interest to the writer for the last twenty years. A summary of the situation in 1969 was provided by Wybourne (1970), though he could not evade the difficulty anyone faces who writes about Lie groups for physicists: how to strike the right balance between giving rigorous mathematical proofs, which is tedious in the extreme, and describing the grand themes of the subject, which can only be achieved through examples if the pace is not to lag. The theorist is continually distracted by results that have aesthetic appeal but which, in all probability, very little utility. F o r example, the collection of 2 ~4 states that make up the f shell can be regarded as the basis for a single irreducible representation of the unitary group U(16 384); but this striking piece of information does not get us very far.
The group SO(28), formed by taking for its generators all pairs of operators a¢a~ a~av , and * * a~a~ that act in the f shell, has fared better. All the states of fN with even N form the basis for the most elementary spin representation ( ! ! . . . ½ ) of 2 2 SO(28); these with odd N correspond to the complementary representation ( 2 ~ ' " ½ - ½ ) . By forming the products of such representations we can deduce that 1_1
all operators must contain parts belonging to irreducible representations of the type ( l l .-. 1 0 " . 0 ) of SO(28) if their matrix elements are not to vanish, thereby restricting the simultaneous assignments of quasi-spin rank K and symplectic symmetry ( a ) when the reduction SO(28) ~ SO 0 (3) × Sp(14) is considered (Judd 1966a). This proved to be of considerable help when the three-electron operators associated with the parameters T i of section 6.1.2. were being studied.
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 181
11.3.1. Q u a s i - p a r t i c l e s
It was discovered in 1969 that the spin-up and spin-down spaces, for either even or odd N, can both be factored into two similar parts (Armstrong and Judd 1970a,b). If, for example, we take N even, the states of fs belong to the irreducible representations Wof SOA(7) given by (000), (ll0), (1 l l ) and (100) for N = 0, 2, 4 and 6, respectively. Their L g content is S for (000), P + F + H for (ll0) (the triplets of f2), S + D + F + G + I for ( l l l ) (the quintets of f4), and F for (100) (the septet of f6). Had we taken odd N, the same Wwould have occurred but in the reverse order.
Now, it can be shown that
( ! ! ! ~ 1111~
2 2 2 ! X ~ ~ ! = ( 0 0 0 ) ~ - (100)-+- (110)+ (111). (130) By taking suitable linear combinations of a,.,m~* and ares ,.~, and coupling pairs of these combinations to odd rank in orbital space, the generators of four new SO(7) groups can be constructed for which SOA(7) = SO~(7) x SO,(7) and SOB(7) = S O d 7 ) x SO¢(7). The two spinor representations t~-~111~ in eq. (130) belong to SOx(7) and SO,(7). These groups conserve neither S nor N; and the basic operators, being linear combinations of both creation and annihilation operators, are said to refer to quasi-particles.
The significance of eq. (130) can be more readily appreciated if it is first noticed (or simply accepted, if the reader is not familiar with Lie algebras) that 1 comprises the eight points (_+½ _+½ _+21) in the three-dimensional weight space of SO(7). Under the reduction SO(7) ~ SO(3), we have ~!!!~ ~ s + f. Thus the terms ~ 2 2 2 !
of maximum multiplicity in f~ (for either N = 0, 2, 4, 6 or N = 1, 3, 5, 7) possess L values that are identical to those provided by (s + f)2. F r o m this point of view, the 4I term of f3 (or, more precisely, its components for which M s = ~) can be regarded as being produced by coupling two f quasi-particles to a rank of 6. The M s =
component of the 4S term of fa, on the other hand, is a superposition of the S states produced by the quasi-particle configurations s 2 and f2. These quasi-particles have no physical existence, of course, but they may be brought into play to simplify calculations. Several mysteries receive ready explanations: in particular, the puzzle of the invariance of the Racah coefficients W(3333; 3k) with respect to even (nonzero) k can be understood (Armstrong and Judd 1970b, p. 42). However, electrons for which l = 3 are too simple to allow the power of the quasi-particle approach to show to its best advantage. Its chief strength is that it allows us to eliminate the undefined classificatory symbols of the type z included in the state (44) for all electrons for which l ~ < 8. The method has been extended to configurations of inequivalent electrons by Cunningham and Wybourne (1969), and to nuclear configurations of the type jN by Elliott and Evans (1970). A detailed description of the quasi-particle method for atomic shells has been given by Condon and Odabasi (1980) in their updating of the classic text of Condon and Shortley (1935).
11.3.2. O t h e r g r o u p s
The group structures that have been discussed so far do not exhaust all the possibilities. It has already been pointed out (in section 8.2.1) that U(14) of the chain (52) can have Sp(14) as a subgroup. This symplectic group is identical to the one
mentioned in section 11.3 as occurring in SOQ(3) × Sp(14). If, however, the con- figurations 4f N had approximated more closely to jj rather than to LS coupling, we would have needed to classify the states of the configurations s N-M 7 M (3) (3) , for which the direct product Sp(6) × Sp(8) of two symplectic groups is required (Flowers 1952b). This possibility was included by Gruber and Thomas ('1975, 1980) as a. link in one of the three main chains running from U(14) to SOj(3), the other two being the sequence given by (52) and the seniority chain that includes Sp(14). The analysis of Gruber and Thomas (1980) established that no other chains exist (for l ~< 6, at least). No statement of comparable generality can be made for the number and kind of chains leading from U(16 384) to SOj(3), however. It might be thought that, at this late date, there could be little expectation of finding any new group structure. In spite of several techniques - which would take us too far afield to describe here, there remain, nevertheless, some inexplicable zeros in the tables of matrix elements compiled by Nielson and Koster (1963). Similar puzzles occur in the tables of the two-electron double-vector operators needed to evaluate the spin-other-orbit and E L - S O interactions in f4 and f5 (Crosswhite and Judd 1970). The possibility that such zeros are truly accidental cannot be discounted, of course, but to say as much has more the character of an excuse than an explanation. One's uneasiness is compounded by the existence of a suggestive feature of the triple products of creation or annihilation operators (in any combination): if the orbital rank is zero then the G 2 label is also a scalar, namely (00). An example of this general rule is the 4S term of f3, which belongs to (00). Commutators of the triple products must also be G2 scalars; and, although successive commutation produces multiple products of the a~ and a,, the fact that these operators correspond to fermions tells us that the commutation process must ultimately close and hence provide us with the gene- rators of a Lie algebra. If these generators simply put all the terms of a given U and L in a single irreducible representation of the corresponding Lie group, nothing much is gained; but the difficult problem of finding possible subgroups has not been tackled.
It should be remembered, of course, that Lie groups have an independent exis- tence apart from their role in the theory of f electrons in the lanthanides. Some of the bizarre properties that turn up in the f shell might well derive from isoscalar factors that receive a ready explanation in another context. An example of this is provided by the vanishing of the spin--orbit interaction H~o when it is set between F and G states belonging to the irreducible representation (21) of G2. In the f shell, (21) is merely a 64-dimensional representation of no special interest. However, for mixed configurations of p and h electrons, it fits exactly into the spinor repre- sentations CZ__I 1 ! _ i 1 ~ 2 2 2 2 2 2 ..~½) of SO(14) with dimensions 26 (Judd 1970), These are the analogs of the spinor representations of eq. (130), and (21) describes the quasi- particle basis of the configurations (p + h) ~. The spinor representations ~ 1/2 of SO(3) and ~22222p~!±±±±~ of SO(ll) provide the quasi-particle bases for the p and h shells respectively; and their SO(3) structures, namely ~1/2 and ~5/2 + ~9/2 -~- ('~15/2, when coupled, must yield the L structure of (21) of G2, namely D + F + G + H + K + L. In this context, the F and G terms of (21) are associated with the different irreducible representations ~5/2 and ~9/2" It is this property,