Pauli theory

Introduced in the Pauli theory, the Lande factor G is defined as follows... 

The first term is the non-relativistic result derived from Breit-Pauli theory, and is the Lande -factor, gji, while the second is the first-order relativistic correction, Ajk, which is proportional to the small-component electron density. We compare, in Table 2, the values which we have calculated directly from Eq. 

It is known from Pauli theory that two one-electron terms should emerge in the nonrelativistic limit, as is evident from Eq. (5.143). One of these terms is quadratic (bilinear) with respect to the vector potentials. 

In chapter 5 we mentioned the Lamb shift in passing and the difficulty of calculating it due to the renormalizations required. In fact, there is a well-developed perturbation theory of the Lamb shift in the same framework as Breit-Pauli theory. We do not propose to derive the expressions here, for which the reader is referred to Bethe and Salpeter (1957). Instead, we report the results for the lowest-order terms, which turn out to be expressible as corrections to the Darwin and spin-orbit one-electron operators. The combined operators may be written... 

The calculation of properties using direct perturbation theory follows exactly the same lines as we used for Breit-Pauli theory. As we noted above, stationary direct perturbation theory leads to precisely the same equations we would have obtained by simply expanding the perturbed wave functions in the set of eigenfunctions of the zeroth-order Hamiltonian, and on this basis we proceed with the development of multiple direct perturbation theory for properties. 

The two perturbations will have a different form for direct and Breit-Pauli perturbation theory. The wave function also will be different in form two components in Breit-Pauli theory and four components in direct perturbation theory. In addition, the metric must be expanded in the direet scheme,... 

A similar situation arises for the magnetic perturbations. Here there is a formal difference, in that the perturbation operator has a relativistic correction in the Pauli theory but in DPT there is no relativistic correction the operator is simply... 

The conclusion is that there are real differences between Pauli and direct perturbation theory for the lowest-order relativistic corrections to both electric and magnetic properties, which do not vanish even for exact wave functions. Direct perturbation theory is in general to be preferred because it is convergent and therefore can be used to higher order, in contrast to Breit-Pauli theory. 

It has been suggested by Brooks (1983) that the deviations from theory observed in Np and Pu may in part be explained by the relativistic volume effect (sect. 3.8). If there is a preferential occupation of the j = f band, there is a 7 = contribution to the pressure in eq. (54), which, in the limit where the spin-orbit interaction is much larger than the 5f bandwidth, vanishes for Mf = 6, i.e. between Pu and Am. It turns out that this limit is not realized in any of the metals Np, Pu or Am, but the effect is sufficiently large to cause deviations from Pauli theory. Even with spin-orbit interaction included, the calculated radii of Np and Pu are too low compared with experiments. The remaining deviation may be due to correlation effects not included in our one-electron scheme. Both Np and Pu are known to be nearly magnetic (Brodsky 1978), which indicates strong f correlations. An additional reason for the discrepancies might be found in the crystal structures. 

---