Hydrogen-like ions

For the hydrogen atom, and for the hydrogen-like ions such as He, Li, ..., with a single electron in the field of a nucleus with charge +Ze, the hamiltonian (the quantum mechanical form of the energy) is given by... 

Beier, T., Mohr, P.J., Persson, H., Plunien, G., Greiner, M. and Soff, G. (1997) Current status of Lamb shif predictions for heavy hydrogen-like ions. Physics Letters A, 236, 329-338. 

The exact wavefunction corresponds to a = b = s and a = (3 = Z. The variational ground state energy of Z = 90 hydrogen-like ion in the Dirac-Pauli... 

Our method of calculation is based on an idea by Ivanov-Ivanova [11]. In an atomic system, the radiative shift and the relativistic part of the energy are, in principle, determined by one and the same physical field. It may be assumed that there exists some universal function that connects the self-energy correction and the relativistic energy. The self-energy correction for the states of a hydrogen-like ion was presented by Mohr [1] as ... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

Theoretical investigations on spatial confinement, as well as the screening effect produced by plasma environments of different coupling strengths T on the spectroscopic properties of hydrogen-like ions, have been performed... 

Figure 9 Scaled excitation energies (au) for different hydrogen-like ions against Debye shielding (au) showing explicitly relativistic effects. Reprinted with permission from [173] 2004, American Physical Society...

Figure 16 Plot of the scaled fine structure correction for the ground state of hydrogen-like ions for different plasma densities using Ion Sphere model. Reprinted with permission from [180]...

Precision measurement of energy intervals in hydrogen and helium has been fundamental to the development of atomic theory. Relativistic and quantum-electrodynamic contributions scale with various powers of Z. Hence more information is gained by extending precise measurements to one- and two-electron ions. Laser spectroscopy is restricted to certain special transitions which fall in the infrared, visible or near-ultraviolet, and from which a useful signal can be obtained. However, where applicable, it provides precision tests of theory. The focus of this review is laser spectroscopy of the n = 2 levels of moderate-Z helium-like and hydrogen-like ions. Previous reviews may be found in [1,2,3],... 

Table 1. Lamb shift in mid-Z hydrogen-like ions...

Ground-state Hyperfine Structure of High-Z Hydrogen-like Ions... 

Obtaining experimental tests of the theory of hydrogen-like ions at moderate Z is challenging. Except for hydrogen there appear to be no measurements more precise than current theory. In the next decade, small but significant improvements in precision can be expected for Z = 2, 7 and 14. For moderate-.Z helium-like ions, laser techniques probe relativistic QED effects at higher precision than... 

Precise measurements on g factors of electrons bound in atomic Hydrogen and the Helium ion 4He+ were carried out by Robinson and coworkers. The accuracies of 3 x 10-8 for the Hydrogen atom [5] and of 6 x 10-7 for the Helium ion [6] were sensitive to relativistic effects. Other measurements of the magnetic moment of the electron in Hydrogen-like ions were performed at GSI by Seelig et al. for Lead (207Pb81+) [7] and by Winter et al. for Bismuth (209Bi82+) [8] with precisions of about 10-3 via lifetime measurements of hyperfine transitions. These measurements were also only sensitive to the relativistic contributions. 

The knowledge of nuclear moments from Hydrogen-like ions would allow to determine experimentally the shielding of the outer magnetic field by the elec-... 

Second-Order Self-Energy Calculations for Tightly Bound Electrons in Hydrogen-Like Ions... 

The theory (with unknown higher-order two-loop effects excluded) is found to be accurate enough and we hope the study of helium and nitrogen hydrogen-like ions is a promising way to study in detail the two-loop contributions experimentally. 

In contrast to the study of the Lamb shift and hyperfme structure, it is possible to perform experiments on the g factor of the bound electron in different hydrogen-like ions with about the same accuracy. The experiment [1] is now in progress and some other hydrogen-like ions can be measured soon. This provides a possibility to learn about the bound g factor as a function of the nuclear charge Z and the nuclear mass number A. The ions under study [1] must have spinless nuclei and so they have the most simple level scheme. 

We follow the notation of our paper [4] and present the g factor of a bound electron in a hydrogen-like ion in the form... 

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