Angular momentum properties

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

All this is summarized in Fig. 3-12. The energy ordering of the free-ion terms is not determined by consideration of angular momentum properties alone and in general yields only to detailed numerical computation. The ground term - and only the ground term - may be deduced, however, from some simple rules due to Hund. 

A term label like for example, is thus no longer strictly meaningful for it implies constant spin- and orbital angular momentum properties (5 = 1, L = 3). One consequence of spin-orbit coupling is a scrambling of the two kinds of angular momentum. So a nominal term may really more properly be described as a mixture of terms of different spin-multiplicity as, for example, in Eq. (4.10). 

Martin, J.A., W. Greiner Potential Energy Surface Model of Collective States, in High-Angular Momentum Property of Nuclei (N.R Johnson, Ed.) Harwood Academic Publishers, New York, NJ, 1983. 

BEB, 2-BEB etc. We find that such states have much improved angular momentum properties and are not much more difficult to calculate in the HB approximation. In the region above the backbend, where AI for l 0> is increasing rapidly, AI for the state with an excited boson decreases to a minimum. At a still higher spin, the 2-BEB state has the lowest AI value, as seen in Figure 1. 

LEA83] G.A. Leander, S. Frauendorf and F.R. May, Proc. Conf. on High Angular Momentum Properties of Nuclei, Vol. 4, Nuclear Science Research Conf. Series, ed. by N.R. Johnson (Harwood Academic Publishers, New York, 1983) p. 281. 

LEA83] G. A. Leander, et al. in "High Angular Momentum Properties of... 

Since many of the operators that appear in the exact Hamiltonian or in the effective Hamiltonian involve products of angular momenta, some elementary angular momentum properties are summarized in the next section. Matrix elements of angular momentum products are frequently difficult to calculate. A tremendous simplification is obtained by working with spherical tensor operator components and, in this way, making use of the Wigner-Eckart Theorem (Section 3.4.5). A more elementary but cumbersome treatment, based on Cartesian operator components, is presented in Section 2.3. 

There are six electrons to occupy the delocalized space orbitals. By the Aufbau principle, in the ground-state each of the three lowest-energy space orbitals is occupied by two electrons with opposite spins, as depicted by arrows in Figure 21.8. Although these orbitals do not have simple angular momentum properties, we refer to them as pi orbitals. 

The proton has the same spin angular momentum properties as the electron, with a spin quantum number I equal to 1/2. The magnitude of the spin angular momentum of a proton is... 

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