Relativistic effects in the search for new intra-atomic force with isotope shifts

Isotope shift of atomic spectra is considered as a probe of new interaction between electrons and neutrons in atoms. We employ the method of seeking a breakdown of King’s linearity in the isotope shifts of two atomic transitions. In the present work, we evaluate the magnitudes of the nonlinearity using relativistic wave functions and the result is compared with that of nonrelativistic wave functions from our previous work. It turns out that the nonrelativistic calculation underestimates the nonlinearity owing to the new interaction in the mass range of the mediator greater than 1 MeV. Further, we find that the nonlinearity within the standard model of particle physics is significantly magnified by the relativistic effect in the $\text{p}_{1/2}$ state. To get rid of this obstacle in the new physics search, we suggest avoiding $\text{p}_{1/2}$ and that e.g. $\text{p}_{3/2}$ should be used instead.

Regular Approximations

Perturbation theory is a useful tool for evaluating small corrections to a system, and as we noted in the preceding chapter, relativity is a small correction for much of the periodic table. If we can use perturbation theory based on an expansion in 1/c we assign much of the work associated with a more complete relativistic treatment to the end of an otherwise nonrelativistic calculation. The problems with the Pauli Hamiltonian—the singular operators and the questionable validity of an expansion in powers of p/mc—are essentially circumvented by the use of direct perturbation theory. For systems containing heavier atoms it is necessary to go to higher order in 1/c perturbation theory, and possibly even abandon perturbation theory altogether. If we could perform an expansion that yielded a zeroth-order Hamiltonian incorporating relativistic effects to some degree and that was manifestly convergent, it might be possible to use perturbation theory to low order for heavy elements. If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r: that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters? The expansion we consider in this chapter has roots in the work by Chang, Pélissier, and Durand (1986) and Heully et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term “regular approximation” because of the properties of the expansion. The Pauli expansion results from taking 2mc2 out of the denominator of the equation for the elimination of the small component (ESC). The problem with this is that both E and V can potentially be larger in magnitude than 2mc2 and so the expansion is not valid in some region of space. In particular, there is always a region close to the nucleus where |V − E|/2mc2 > 1. An alternative operator to extract from the denominator is the operator 2mc2 − V , which is always positive definite for the nuclear potential and is always greater than 2mc2.

Core Approximations

When we think of chemical bonding, we usually think only of the valence orbitals. It is these orbitals that form bonds, and the core orbitals are not involved in the chemistry. To be sure, this is only a qualitative picture, but it raises the question of whether we really need to consider the core orbitals in our calculations. For heavy elements, the number of core orbitals is not small. An element such as platinum, from the third transition series, has 30 orbitals in the first four shells that could be classed as core orbitals. If these remain essentially atomic over some region of the molecular potential energy surface, they might as well be fixed in their atomic form. That is, we would make a frozen-core approximation, and all that the core orbitals are doing is supplying a nonlocal static potential that could be evaluated once and used for the remainder of the calculation. As relativistic effects are to a large extent localized in the core region, they could be included in the frozen-core potential. We could then treat the valence orbitals and the orbitals on the light atoms nonrelativistically, as we did in the previous chapter. This would save all the work of calculating the relativistic integrals, and the calculation would be as cheap as a nonrelativistic calculation. There is one main difficulty with this idea, and that is the orthogonality of the rest of the orbitals to the frozen core. The basis sets we use in molecular calculations are not automatically orthogonal to the core of any one atom: we must make them so by some procedure, such as Schmidt orthogonalization. But this involves taking linear combinations of the core and valence orbitals, and then we not only have to calculate all the integrals involving the core, we also have to transform them to the orthogonal basis. The reintroduction of the core integrals means that we have to calculate all the relativistic contributions that we had previously put into the frozen-core potential. Obviously, this is not a satisfactory state of affairs. Two solutions to this problem are in common use.