Finite Nucleus Effects on Properties

Thus far in our discussion of relativistic expressions for properties we have assumed that the nuclei are represented by point charges. However, schemes for actual calculation of relativistic wave functions normally use nuclei with finite size in order to avoid problems with the weak singularity of the Dirac equation at the nucleus—and also because the nucleus really does have a finite size. The use of a point nucleus to calculate properties therefore appears somewhat inconsistent. At the very least we should know what errors we incur by using a point nucleus, and we will therefore discuss the low-order effects of finite nuclear size for electric and magnetic fields. 

The source of the electric field can be an externally applied field, or it can originate in the components of the nuclear potential that are not included in the internal component of the field (that is, the nuclear potential V). Such components arise from the nonspherical nature of the nucleus, the lowest-order term of which is the quadrupole moment. The implementation of a finite-nuclear model is quite straightforward we simply expand the nuclear charge distribution in a series  

The collection of all the C q for a given value of k then yields a vector Q with 2fe - -1 components. For the tensor function above we get 

We next consider the effect of finite nuclear size on the nuclear spin Hamiltonian. The electric moments were derived by considering the Coulomb interaction of the nuclear charge density, expanded in a multipole series, with the electrons. By analogy, the magnetic moments are derived by considering the Gaunt interaction of the nucleus with the electrons. It is at this point that we must consider, at least as a formal entity, the nuclear wave function, and from it obtain a nuclear spin density that interacts with the electron spin density. 

The power of r is necessary to give the correct form for reduction to the point nucleus expression, and can be justified from a consideration of the part of the nuclear wave function that gives rise to the nonspherical terms. 

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