He-like ions

He- like ions Z (w)2(J - ) Correlated wave function of type III. 125 Exact... 

The wave function III. 144 has a very simple form, and applications to the He-like ions show that one can still expect a surprisingly high accuracy. Let us consider a singlet wave function of the form ... 

TABLE IX. Energies of the He-like ions for a Wave Function of the Form (l-far12)(w, v) (in at.u.)... 

The data show that, in the case of the He-like ions, a combination of the method using correlation factor and the method with different orbitals for different spins leads to excellent results. 

At the end of the paper, condensed tables of the higher approximations have been carried out with respect to atomic and molecular systems. For atoms, the tables are arranged after the number of electrons involved, which means that, e.g., N = 2 refers to the series of He-like ions H", He, Li+, Be2+, etc. For molecules, there is a table for H2 a table for other simple molecules (LiH, BeH+, H20, NH3, etc.) with all or almost all electrons treated, and finally a special table for the n electron systems in the two latter cases, the references to the best SCF data available are also contained for comparison. 

Hart, J. F., and Herzberg, G., Phys. Rev. 106, 79, Twenty parameter eigenfunction and energy values for the ground state of He and He-like ions. ... 

Lowdin, P.-O., and Redei, L., Combined use of the methods of superposition of configurations and correlation factor on the ground states of He-like ions."... 

The effect of the Debye screening on the spectra of a few He-like ions was studied by Saha et al. [194] using time dependent perturbation theory. Detailed analysis has been made for the behavior of the orbital energies, polarizabilities transition energies, oscillator strengths and transition... 

Table 2. Summary of high precision measurements for helium and He-like ions...

Further Hylleraas-type calculations with basis sets of increasing size and sophistication, culminating with the work of Pekeris and coworkers in the 1960 s (see Accad, Pekeris, and Schiff [26]) showed that nonrelativistic energies accurate to a few parts in 109 could be obtained by this method, at least for the low-lying states of helium and He-like ions. However, these calculations also revealed two serious numerical problems. First, it is difficult to improve upon this accuracy of a few parts in 109 without using extremely large basis sets where roundoff error and numerical linear dependence become a problem. Second, as... 

For numerical evaluation (to sum over the entire spectrum of Dirac equation) B-splines are used [28], in particular the version developed by I.A. Goidenko [29]. Earlier the full QED calculations were carried out only for the ground (lsi/2)2 state He-like ions for the various nuclear charges Z. At that ones used either B-splines or the technique of discretization of radial Dirac equations [27]. As well as in [27] we used the Coulomb gauge. For control we reproduced the results of the calculation of (lsi/2)2 state and compared them with ones of [27]. Coulomb-Coulomb interaction is reproduced for every Z with the accuracy, on average, 0.01 %, Coulomb-Breit is with the accuracy 0.05 % and Breit-Breit (with disregarding retardation) is with the accuracy 0.1%. The small discrepancy is explained by the difference in the numerical procedures applied in [27] and in this work. 

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. 

Most of the experiments have been performed on He-like ions. This is due to the large range of plasma temperatures, where the fractional abundance is dominated by He-like ions, and in addition they are ideal for the diagnostics of hot plasmas. On the one hand, the spectra of He-like ions are rather simple to be calculated accurately, on the other hand, they are complicated... 

In this contribution, we will show that the spectra of He-like ions can be modeled with high accuracy, using physically relevant parameters only, such as ion and electron temperatures, plasma motion and relative ion abundances. It is organized as follows in Sect. 8.2 there is a brief discussion on X-ray spectrometers used on fusion experiments, Sect. 8.3 contains the detailed theoretical description of the He-like system, in Sect. 8.4 some results are shown as obtained by a self-consistent procedure based on the detailed modeling of the spectra. 

The spectra are dominated by triplet and singlet transitions in He-like ions and associated satellite transitions in the Li- and Be-like system. The spectrum of He-like argon and iron ions is shown in Fig. 8.2. 

The k and j satellites are the strongest dielectronic satellites to the He-like lines, the q, r, s and t satellites have strong contributions due to inner-shell excitation from the Li-like ground state. Besides the dominant collisional excitation of He-like ions in the ground state, recombination processes (radiative, dielectronic and charge exchange) of H- and He-like ions, inner-shell excitation of the Li-like ions and, in the case of the z line, also inner-shell ionization process contribute to the intensity of the He-like lines. We will discuss these processes in detail. 

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 ratio between the w-line, which is predominantly excited by electron collisions (8.1), and the k-satellite, which is populated by dielectronic recombination (8.2), depends on the electron temperature only. The ratio between the w line and the intensity of the collisional excited Li-like satellites (8.5), depends on the density ratio between the Li-like and He-like ions, as the collisional excitation rates for the allowed transitions in the He-like system and in the doubly excited Li-like system are similar. 

Within the last 25 years of X-ray spectroscopy on fusion devices, the theory of He-like ions has been developed to an impressive precision. The spectra can be modeled with deviations not more than 10% on all lines. For the modeling, only parameters with physical meaning and no additional approximation factors are required. Even the small effects due to recombination of H-like atoms, which contribute only a few percent to the line intensity, can be used to explain consistently the recombination processes and hence the charge state distribution in a hot plasma. The measurements on fusion devices such as tokamaks or stellarators allow the comparison to the standard diagnostics for the same parameters. As these diagnostics are based on different physical processes, they provide sensitive tests for the atomic physics used for the synthetic spectra. They also allow distinguishing between different theoretical approaches to predict the spectra of other elements within the iso-electronic series. The modeling of the X-ray spectra of astronomical objects or solar flares, which are now frequently explored by X-ray satellite missions, is now more reliable. In these experiments, the statistical quality of the spectra is limited due to the finite observation time or the lifetime of... 

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