H-Like Ions

Bound-free transitions, due to photo-ionization of H-like ions of oxygen etc. ... 

Fig. 1. Feynman diagrams corresponding to the second-order radiative corrections in H-like ions. The double solid line denotes the bound electron and the wavy line indicates the photon...

Table 1. Convergence of PWE approach with the multicommutator expansion (Eq. 16) for the lowest-order SE correction for the ground-state of H-like ion with Z = 10.

The excited levels of the He-like ions are also populated via recombination processes of H-like ions. The line emission due to radiative recombination is expressed as ... 

The process of charge exchange occurs by capture of a bound electron from neutral hydrogen by H-like ions followed by radiative decay. As in the case of radiative recombination or cascades in He-like ions, the contribution of charge transfer is stronger to the triplet lines than to the resonance line w. The emission via charge-transfer is determined by ... 

The Rydberg constant in equation (2.1) carries a subscript M for the mass of the positively charged particle. The reasons are that the nucleus is not infinitely massive, and that the formula also applies to any system containing two particles of opposite charge, i.e. it is valid for H, deuterium, exotic two particle atoms and H-like ions. 

Not only the eigenvalues, but also the eigenfunctions are analytic in c . We demonstrate this here only for the ground state of H-like ions. The normalized relativistic eigenfunction rp is... 

Of course, the Dirac operator for H-like ions has continuum states as well, including ultrarelativistic ones. One can therefore neither expect that all eigenstates are analytic in c , nor that the entire Dirac operator allows an expansion in powers of c. This can at best be the case for the projection of D to positive-energy non-ultrarelativistic states. The paradigm, on which the Foldy-Wouthuysen transformation is based, to construct a Hamiltonian, related to the Dirac operator by a unitary transformation, in an expansion in c ... 

If one evaluates Boi from the original expression (132), and avoids to apply the turn-over rule, there is no need for differentiation in the distribution sense. We illustrate this for the ground state of H-like ions. 

We see that for small r, (f>2 differs from 2 by be. for H-like ions by —Zip jr. Similarly one finds that upper component (p of the exact Dirac eigenfunction of H-like ions behaves for small r as... 

For H-like ions we get the zeroth order upper and lower radial components g(j r) and fo r) from the radial Levy-Leblond equation. We have given them for the Is, 2ps, and Is, 2pi states of H-like ions in section (2.4). From these we obtain immediately 2 as... 

For the ground state of H-like ions the numerical values are... 

The behavior of relativistic wave functions at the Coulomb singularities of the Hamiltonian have been studied [84]. The nuclear attraction potentials don t cause any problem. There are weak singularities of the type r with p slightly smaller than 0, as they are familiar for the H-like ions. The limits r —> 0 and oo commute, and the Kato cusp conditions [85] arise in the nrl. For the coalescence of two electrons the two limits do not commute. An expansion in powers of c is possible to the lowest orders and leads to results consistent with those reported above. 

FIGURE 5.4 Left side X-ray spectra for capture into bare and H-like uranium (forming H- and He-like uranium, respectively) as measured at ESR electron cooler. Right side Schematic presentation of the RR process of free electrons into the initially bare and H-like ions. The energy difference AE = — ha>ne gives exactly the two-electron contribution to the... 

In relativistic Dirac theory, the g-factor of an electron bound to a H-like ion is given by [6,7]... 

Satellite lines These are weaker lines arising from doubly excited ions of higher ionization states. The satellite lines of H-like ions are due to He-like ions, those of He-like ions are due to Li-like ions and so on. For example, transitions in Li-like ions of the type Is nlnT Is nT will appear as a satellite to Is nl-ls resonance transitions in He-like ions (Figure 11). The nT electron is a spectator electron. Due to the presence of this electron. Coulomb shielding decreases, which results in the transition occurring at a slightly lower energy than that of the resonance transition. Since two electrons are involved in such transitions (one active, one spectator), these lines are also referred to as dielectronic satellites. 

The largest separation from a parent line occurs when the spectator electron is in the lowest n level (i.e. n = l) when the Coulomb shielding effect is maximum. These satellites are referred to as 2 s or 2p satellites. For example, the Is 2s 2p - Is 2s transition in Li-like ions will be a 2s satellite for the Is 2p - Is (He-a) transition in He-like ions (Figure 11) Satellites of H-like ions in the Gabriel notation are... 

---