Something more about the nuclear spin

Nuclear spin vectors are localized on the nucleus, at least for the purposes discussed here. Therefore they can be treated as point dipoles. We have already shown that they are described by 21 + 1 wavefunctions, each characterized by the value I and a value of A/ . 

In quantum mechanical terms the energy is given by the Hamiltonian operator, which in this case is called the nuclear Zeeman Hamiltonian 

In the absence of an external magnetic field the Zeeman Hamiltonian provides zero energy and all the 11, Mi) levels (termed as / manifold) have the same energy. However, this may not be true for nuclei with / V2. In this case, the non-spherical distribution of the charge causes the presence of a quadrupole moment. Whereas a dipole can be described by a vector with two polarities, a quadrupole can be visualized by two dipoles as in Fig. 1.11. 

The presence of a quadrupole moment can make the /, Af/) levels inequivalent even in the absence of an external magnetic field, provided there is an electric field gradient. Only the wavefunctions with the same absolute value of Af/ are pairwise degenerate in axial symmetry, i.e. Af/ = 1, 2, etc. An example is repotted in Fig. 1.12 for / = 3h. 

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