Transposition of spin and quasispin quantum numbers

The analogy between spin and quasispin (q = s = 1/2) enabled Judd to introduce another unitary transformation - the operation R [12] 

A relation of the type (16.53) between states has been noted by Racah [24] who established that two states avyLSi and aV2LS2 appear to be coupled, if 

For the matrix elements of a qls considering that R is a unitary operator, we obtain 

Applying the Wigner-Eckart theorem in all three spaces, we establish the property of SCFP in relation to interchanges of the spin and quasispin quantum numbers 

For practical purposes, it is only sufficient to find the phase cp in equation (16.54) relating the SCFP. 

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CFP (9.11) also have a simple algebraic form. In the previous paragraph we discussed the behaviour of coefficients of fractional parentage in quasispin space and their symmetry under transposition of spin and quasispin quantum numbers. The use of these properties allows one, from a single CFP, to find pertinent quantities in the interval of occupation numbers for a given shell for which a given state exists [92]. 

Using (16.16) and the above relations, we can work out algebraic expressions for SCFP, and hence for CFP, in the entire interval of the number of electrons in the shell existing for given v. Taking account of the symmetry of CFP under transposition of spin and quasispin quantum numbers further expands the number of such expressions. Formulas of this kind can be established also for larger v, but with v = 5 and above they become so unwieldy and difficult to handle that this limits their practical uses. They may be found in [108]. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

This expression for /-electrons can be derived using the phase relations established for isoscalar parts of factorized CFP with different parities of the seniority number [24]. It turned out [91] that phase (16.55) provides sign relations between the CFP in the tables for d- and /-electrons, but it is unsuitable for the p-electrons. In this connection, in what follows all the relationships derived using the symmetry properties under transposition of the quantum numbers of spin and quasispin are provided up to the sign. 

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