Isoscalar factor

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two 0(4) and two 0(3) representations, respectively. They can be evaluated either analytically using Racah s factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2... 

As discussed for the one-dimensional model, we can expand the coupled basis in terms of uncoupled states [see Eq. (3.7)]. In the 0(4) case, we obtain a similar expansion, in which, besides the usual 0(3) D 0(2) Clebsh-Gordan coefficients, we need the 0(4) D 0(3) transformation brackets, often referred to as 0(4) isoscalar factors. This is typically written as... 

Sample calculation of isoscalar factors using Slater determinants... 

The symbols a and y are multiplicity labels that are required if A and B occur more than once in the respective decompositions of A, x A2 and B, x B2. The symbol Pi is a multiplicity label of a different kind it is required when the reduction G yields more than one identical irreducible representation B,- in a given A,. Like the proof of the Wigner-Eckart theorem, the derivation of (51) depends crucially on Schur s lemma. The second factor on the right-hand side of eq. (50), for want of a better expression, has become known over the years as an isoscalar factor, in analogy to the situation for SU(3) (Edmonds 1962). 

The factorization of the isoscalar factors and hence of the cfp, by itself, brings us no nearer to finding their numerical values. It is here that the detailed properties of the representations of G2 and 80(7) play a role. Racah, of course, had to discover many of these properties for himself today we can refer to the tables of Wybourne (1970) for much of what we need. Because of the variety of aids now at hand, it is quite difficult to get a sense of the problems that Racah faced. No tables of 3-j or 6-) symbols were available in 1949, though their algebraic forms were known, of course. It therefore seems worthwhile to show how some non-trivial isoscalar factors can be calculated by elementary techniques. We cannot know in detail how Racah built up his tables, but we can at least glimpse the kinds of methods available to him. 

These numbers can be normalized by multiplying them by (2/3) In this way we find that the isoscalar factors (57) are given by... 

All kinds of isoscalar factors can be calculated with sufficient ingenuity. The proportionality factor —(2/3) of section 4.3.3 is itself an isoscalar factor for... 

U(14) => SOj(3) X U(7). Many more isoscalar factors can be calculated by generalizing that particular symmetry property of an SO(3) CG coefficient that interchanges a component with a resultant 9ji... 

The linear combination of the Pn(cosco,j) corresponding to a particular WU entails the isoscalar factors (lPl/01 (200)(20)/c +(200)(20)k) as well as the renormalization factors for the Only one complication remains, and it is a very minor one. The constant PofcostOy) is also present in the expansion of and this produces a second operator for which WU = (000)(00). The particular mixtures picked by Racah (1949) for and ei simplify the expressions for the energies of the terms of maximum multiplicity. Putting the pieces together, we can relate the parameters and operators of eqs. (45) and (46) by writing... 

Racah needed to know how many times (40) (the G2 label for e ) occurs in the reduction of the Kronecker product V x U, since this determines the number of possible labels y to be attached to the isoscalar factors... 

It should be remembered, of course, that Lie groups have an independent existence 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 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 representations (iiiHi 2) of SO(14) with dimensions 2 (Judd 1970). These are the analogs of the spinor representations of eq. (130), and (21) describes the quasiparticle basis of the configurations (p-t-h)". The spinor representations of SO(3) and (Hiii) of SO(ll) provide the quasi-particle bases for the p and h shells respectively and their SO(3) structures, namely S 1,2 and 512+ 912+ is/2> when coupled, must yield the L structure of (21) of G2, namely D-l-F-l-G-l-H-t-K-t-L. In this context, the F and G terms of (21) are associated with the different irreducible representations Sj,2 and S9/2. It is this property,... 

Since the U(n) G-T basis relies on the canonical chain (1), a C-G coefficient can be expressed as a product of the U(m) D U(m — 1) isoscalar factors with m ranging from 1 to n. The scaled C-G coefficients are then given by a product of scaled isoscalar factors. 

As we have seen above, the well-known Wigner-Eckart theorem represents a powerful tool for the evaluation of MEs. Thus, the MEs of any U(2 ) tensor that may be decomposed into the irreducible tensors of U( ) and SU(2) can be expressed as a product of three factors (1) the RME that depends on the relevant tensors and irreps of U(2n), U( ), and SU(2), (2) the U(n) C-G coefficient, and (3) the SU(2) C-G coefficient. In a multiplicity-free case for U(n) irreps, the U(n) C-G coefficients can be further factorized into simple products of isoscalar factors, yielding the ME segmentation formalism for spin-dependent operators. We shall see that this is exactly the case for one-body operators (36). 

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