Lithium-like ion

In Tables 5 and 6 there are displayed the results for the hyperfine coupling constants in the lowest excited states of Li-like ions. In Table 5 we compare the results of our calculations with those from papers [11, 22] for magnetic dipole coupling constants in the ground state ls 2s of a few lithium-like ions. 

Helium is not the only three body system under study, but it is likely the most complicated one. The electron-electron interaction is comparable with the electron-nucleus interaction and cannot be considered as a perturbation. The situation is different with helium-like (and lithium-like) ions, where the electron-electron interaction is as small as 1 jZ with respect to the interaction of an electron and the nucleus. 

Fig. 1. Hydrogen-like and lithium-like ions with spinless nuclei...

The possibility of the direct experimental study of these higher-order two-loop terms shows the importance of physics of moderate Z. Medium-Z theory provides a possibility to develop both high-Z and low-Z approaches. In case of low-Z technics, one can expand over (Za) and see if our assumptions on higher-order terms is appropriate or not. Using high-Z methods one can perform a 1/Z expansion and treat the electron-electron interaction in few-electron ions as a perturbation. The study of the g factor of lithium-like ions offers an experimental test of present ideas on how large higher-order corrections electron-electron interaction can be. 

A wide variety of plasma diagnostic applications is available from the measurement of the relatively simple X-ray spectra of He-like ions [1] and references therein. The n = 2 and n = 3 X-ray spectra from many mid- and high-Z He-like ions have been studied in tokamak plasmas [2-4] and in solar flares [5,6]. The high n Rydberg series of medium Z helium-like ions have been observed from Z-pinches [7,8], laser-produced plasmas [9], exploding wires [8], the solar corona [10], tokamaks [11-13] and ion traps [14]. Always associated with X-ray emission from these two electron systems are satellite lines from lithium-like ions. Comparison of observed X-ray spectra with calculated transitions can provide tests of atomic kinetics models and structure calculations for helium- and lithium-like ions. From wavelength measurements, a systematic study of the n and Z dependence of atomic potentials may be undertaken. From the satellite line intensities, the dynamics of level population by dielectronic recombination and inner-shell excitation may be addressed. 

Now we turn to lithium and three-electron lithium-like ions. Again we start with the normally-ordered no-pair Hamiltonian given in Eq. (132), and choose the starting potential to be the Hartree-Fock potential of the (Is) helium-like core. We expand the energy of an atomic state in powers of the interaction potential... 

Figure 9. Second-order and third-order energies of 2si/2, 2pi/2, and 2ps/2 states of lithium-like ions.

When we solve these equations iteratively, we recover the first-order Breit energy together with second-, third-, and higher-order RPA contributions. We plot Brpa against Z for n = 2 states of lithium-like ions in the lower panel of Fig. 10. 

Figure 10. Srpa, the residual contribution, and the Brueckner-orbital contribution to Bi are plotted against Z for n = 2 states of lithium-like ions. Notation solid lines represent 2si/2, dotted lines represent 2pi/2, and dashed hnes represent 2py2- Units a.u..

The expectation value of the mass-polarization operator, calculated as described above, is presented for 2s and 2p states of lithium-like ions in Fig. 11. It should be mentioned that for the special case of lithium-like ions, with a (Is) core, the RPA corrections to P identically vanish. Therefore, for lithium-like ions, Prpa = P - Moreover, for ns states of lithium-like ions, both P > and P q vanish. Thus, P(2s) = P (2s) for lithium-like ions. 

Further comparisons of experimental and theoretical values of the 2s ji — 1v li Lamb shift in lithium-like ions with nuclear charges ranging from 10 to 92 are shown in Fig. 12. The experimental values of the 2si/2 — 2pi/2 intervals are taken from the review of Ph. Bosselmann et aJ. [52]. The corresponding MBPT values are calculated using the prescription given in Ref. [53] the difference gives an experimental Lsimb shift. The theoretical Lamb shift shown in the plot is from Ref. [51]. 

In these examples, the perturbation series converged rapidly and one was able to iiffer accurate values of the QED corrections from the difference between theory and experiment. For neutral lithium or light lithium-like ions, the convergence of MBPT is much slower and one must resort to all-order methods to obtain precise theoretical energies. We briefly describe the all-order single-double method for lithium and lithium-like ions in the next subsection. 

Single-Double (SD) Equations for Lithium-like Ions... 

X-rays from a few electron systems such as hydrogenlike, helium-like, lithium-like ions are observed in hot... 

Figure 11 Energy diagram for 2s and 2p satellite transitions in lithium-like ions corresponding to He-a and He-p transitions in helium-like ions.

Predictions of [13] also include 2s 2p levels of some ions isoelectronic with O, N, C, B, Be, and Li. The elements Os, Ir, and Pt are not considered, but relevant data can be derived by interpolation. Isoelectronic trends in the n=2 —n=2 transition probabilities are sketched in [14] for boron-like ions by the relativistic parametric potential method and in [15] for beryllium-like ions by a 1/Z perturbation method. In a survey of the lithium-like sequence, Steiger reports spontaneous emission rates and energies for three forbidden transitions of lr + [16]. A relativistic model potential method was used by Gogava etal. for deriving the lowest 15 energy levels of all lithium-like ions [17]. 

---