Nucleon Pairing and Seniority

The calculation of fractional parentage coefficients described in Section 15 starts from a certain principal parent. It was explained that antisymmetric states arising from different principal parents need not be independent. For example, in the ( 5 2 ) 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1569.tif"/> configuration of identical nucleons there is only one antisymmetric state with j = 5 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1570.tif"/> (and given M). The same state can be obtained from ( 5 2 ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1571.tif"/> principal parents with either J ₀ = 0, J ₀ = 2 or J ₀ = 4. In other cases, however, there may be several independent states with the same value of J (and M) in the jⁿ configuration. For example, in the ( 9 2 ) 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1572.tif"/> configuration of identical nucleons there are two antisymmetric states with j = 9 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1573.tif"/> (and M). In such a case it is necessary to find an orthogonal basis for these two j = 9 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1574.tif"/> states. Another example is the ( 7 2 ) 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq1575.tif"/> configuration of identical nucleons which has two states with J = 2 and two states with J = 4. It is convenient to find basis states in such cases which can be characterized (or labelled) by the eigenvalues of a simple operator. Such quantum numbers then uniquely characterize the various states. The Hamiltonian matrix within the jⁿ configuration of identical nucleons has then its rows and columns labelled by these quantum numbers in addition to J (and M). The submatrix for a given J (and M) which should be diagonalized, has rows and columns labelled by these additional quantum numbers. We shall now describe a scheme that can distinguish 350in simple cases of practical importance, between states with the same value of J. It will also facilitate the calculation of matrix elements of various operators. It will turn out that this scheme and its generalizations give a very good description of certain nuclear states and energies.