The atomic properties of heavy, moderately-charged ions are important for a wide variety of applications, including precision tests of fundamental physics and for the study and development of atomic and nuclear clocks. In these systems it is known that relativistic effects, such as the Breit interaction and radiative quantum electrodynamics corrections, are important for an accurate understanding of atomic properties. It is also known that inclusion of correlations alongside the Breit effect is crucial. In this work we include the Breit interaction into all-orders calculations of energy levels and fine structure intervals of ions in the Cs and Fr isoelectronic sequences. This requires modifying the electron Green's function to account for Breit within the all-orders correlation potential method, which sums dominating series of perturbation diagrams exactly using a Feynman diagram technique. We find that Breit corrections to the energies of moderately ionized ions along these sequences are very large, particularly for the f states. We also observe a significant deviation from experiment for these levels. Incorporating Breit into the all-orders correlation potential provides a significant additional contribution beyond including Breit at the second-order level alone. While this does not resolve the disagreement in the energy levels, it does substantially improve the fine-structure intervals beyond what is achieved by including Breit only at second order. Furthermore, we include the frequency-dependent Breit interaction into the Dirac-Fock procedure, and find that this does not significantly modify the energy levels at this order of approximation.
An accurate theoretical understanding of atomic systems has far-reaching applications in many areas of physics. For example, high-precision calculations of atomic parity violating processes inside atoms provides one of the most stringent tests of new physics beyond the Standard Model, and one of the highest precision tests of the electroweak sector of the Standard Model (SM) in general, see, e.g., Refs. [1][2][3]. In this context, it is known that the common starting point for atomic structure calculations, the Dirac-Fock approximation, is not sufficient on its own for high accuracy calculations, and that correlation and relativistic effects must be taken into account. The Breit interaction gives the leading order relativistic correction to the Coulomb interaction and is one of the most important relativistic effects inside a heavy atom such as Cs. For example, combining experiment [4] with calculations [5,6] of atomic atomic parity violation in Cs indicated a 2.5 σ deviation from the Standard Model. Derevianko [7,8] and others [9][10][11] demonstrated that the Breit effect (along with radiative QED corrections [12][13][14][15]) resolved this disagreement.
Current calculations of atomic parity violation indicate reasonable agreement with the SM [16][17][18]. Calculations typically only include the one-body part of the Breit corrections at the level of second order in the Coulomb interaction. While this level of inclusion clearly leads to very satisfactory theoretical predictions in Cs, there is currently ongoing work towards studying parity violation in Fr, Ra + and heavier ions [19][20][21], and there have also recently been several important experimental * b.roberts@uq.edu.au breakthroughs towards the construction of the first nuclear clock using the nucleus of a thorium atom [22][23][24]. This further motivates precision atomic-structure and symmetry-violation studies in heavy and highly charged systems, including proposals to exploit highly charged ions as sensitive probes of fundamental physics [25]. In parallel, complementary efforts on atomic clocks based on heavy ions [26][27][28] highlight the broader potential of these platforms for precision metrology and tests of fundamental symmetries.
In this work we will demonstrate that in such ions the deviation from experiment for the theoretical energies of f states are unusually large. We will show that the Breit corrections to these levels are very large, being the right order of magnitude to be the cause of this issue. This problem is particularly relevant for the nuclear clock project, as one way to investigate the long-term viability of thorium for a nuclear clock is through highly precise studies of its hyperfine structure. One accessible way to do this is through studies of the Th 3+ ion, whose ground state is 5f 5/2 , and so being able to accurately predict the f states in heavy, moderately charged ions would be important in such studies.
Due to the importance of treating Breit and second order correlations concurrently, including Breit and allorders correlations on the same footing may also lead to important corrections in cases where Breit and all-orders corrections are large, such as heavy ions. As such, in this work we include the one-body part of the Breit interaction into the Green’s function in the all-orders correlation method developed in Refs. [29,30], thereby including the Breit interaction into an important series of nonperturbative correlation effects to all orders (we stress that we mean all orders in the residual Coulomb interaction; Breit should be treated to first order only). We also show that inclusion of Breit into the all-orders correlation potential method greatly improves the fine structure intervals, leading to near-perfect agreement with experiment. However, the large disagreement between theory and experiment for the energy levels still remains. We also include higher order relativistic corrections into the Dirac-Fock equation, in the form of the frequencydependent Breit interaction, however, we find that the resulting corrections are very small.
The starting point for modeling single-valence atomic systems is the Dirac-Fock equation,
where h 0 is the single-particle Dirac Hamiltonian including the nuclear potential and V DF is the usual frozencore Dirac-Fock potential due to the N -1 core electrons. Correlations between the core and valence electrons can be taken into account by adding a non-local, energy-dependent operator Σ(ε), known as the correlation potential, to the Dirac-Fock equation for the valence electrons,
whose expectation value is the correlation correction to the energy from treating the residual electron-electron interactions perturbatively. We may calculate Σ to lowest (second) order or to all-orders, as discussed below. The solutions to this equation are known as Brueckner orbitals and the self-consistency of this equation includes an additional correction, known as the chaining of the self-energy
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