Isoelectronic sequences

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

Kossel-Sommerfeld law spect The law that the arc spectra of the atom and ions belonging to an isoelectronic sequence resemble each other, especially in their multiplet structure. kas-al zom-sr.felt, 16 )... 

C. Valdemoro, L. Lain, F. Beitia, A. Ortiz de Zarate, and F. Castano, Direct approximation to the reduced density-matrices calculation of the isoelectronic sequence of beryllium up to argon. Phys. Rev. A 33, 1525 (1986). 

Fio. 12-2.—The melting points and boiling points of isoelectronic sequences of hydride molecules. 

This valence effect shows up strikingly in the comparison of interatomic distances for isoelectronic sequences. In crystals containing covalent bonds the interatomic distance remains almost constant... 

With increase in valence there are corresponding great increases in hardness, melting point, and other properties of ionic crystals of an isoelectronic sequence. 

It is very important for semi-empirical calculations to have experimental data measured accurately and identified correctly. Otherwise, it is impossible to achieve good correspondence between calculated and measured quantities. To overcome these dangers one has to analyse the structure of the eigenfunctions, energy levels and oscillator strengths of electronic transitions along the isoelectronic sequences. As a rule, all these quantities vary fairly smoothly and, therefore, the occurrence of any inconsistency in their behaviour usually indicates that there are uncertainities or errors in the data utilized. 

As was shown in Chapter 4 (formula (4.22)), relativistic corrections of the order a2 to the intercombination 1-transitions in length form for accurate wave functions compensate each other. It follows from formulas (4.18)-(4.20) that for the velocity form of the 1-transition operator the relativistic corrections are of the order a2 and may be presented in length, velocity and acceleration forms. Calculations of the 1-radiation for the Be isoelectronic sequence (Z = 4 92) indicate that these relativistic cor-... 

Fig. 30.1. Dependence of gf of the 1-transition ls22s2p lPi — ls22s2 So on gauge K in the Be isoelectronic sequence.

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

In the previous chapter we have briefly discussed the use of the stationary many-body perturbation theory of the effective Hamiltonian, sketched in Chapter 3, to account for correlation effects. Here we shall continue such studies for electronic transitions on the example of the oxygen isoelectronic sequence. Having in mind (29.32) and the approximation Q(1) iiji the matrix element of the 1-transition operator Oei between the initial state... 

Multiply charged ions represent a peculiar world, essentially differing from the world of neutral atoms or their first ions. In this chapter we shall try to describe these peculiarities. Main attention will be paid to the need for a combination of theoretical and experimental studies of the spectra of highly ionized atoms, and to the role of relativistic and correlation effects for multiply charged ions, and to the regularities and irregularities of the behaviour of their spectral characteristics along isoelectronic sequences. 

Studies of isoelectronic sequences are very useful and fruitful in theoretical, semi-empirical and experimental investigations of atomic spectra. They reveal important regularities and pecularities in the behaviour of various physical characteristics of many-electron atoms and ions, help to identify and classify their energy levels with an optimal coupling scheme, find initial values of semi-empirical parameters, etc. 

Let us mention a few peculiarities of the spectra of some groups of highly ionized atoms. For certain Z values in an isoelectronic sequence very strong interaction between the electrons of some special sorts of configurations, e.g. n0l tnjh (no < m < 2) and no/o,0+1ni/f ... 

In a similar way the pecularities of electronic transitions may be investigated along isoelectronic sequences. Line and oscillator strengths as well as transition probabilities usually change fairly smoothly along them. As... 

Calculations indicate that about 1% of the transitions between complex configurations has values of F < 10-4. Computed line strengths may have large percentage errors for F < 0.1. Because of such interference of separate terms (so-called angular effects) some spectral lines disappear for certain ions. Figure 31.2 illustrates this statement for the example of the transition 2p53d — 2p6 for the neon isoelectronic sequence. Indeed,... 

Fig. 31.2. Dependence of oscillator strengths of transition 2p53d — 2p6 on Z in neon isoelectronic sequence.

Asymmetry of the spectrum, defined by skewness jq in (32.7), is positive, if the majority of levels are in the lower part of the spectrum, and negative, if the opposite is true. The excess K2 of the spectra of the majority of atoms with one open shell for 1 < N < 41 + 1 is also positive hence, the density of the levels, while approaching the average energy, grows more rapidly in comparison to the case of normal distribution of the levels. The results of the Hartree-Fock calculations of a2, q and K2 for isoelectronic sequences with an open ndN shell indicate that a grows almost linearly with Z, whereas jq and K2 decrease with increase of Z, more rapidly for larger n. 

Having explicit formulas for a number of first moments we can approximately restore the envelope line of the radiation spectrum without its detailed calculations. If lines in the spectrum have one symmetric maximum, then its envelope line is approximated by a normal function whose reconstruction requires only the mean energy and variance of the spectrum. Such an approach is useful for the case of complex spectra consisting of many lines, which, due to low resolutions as well as Doppler and collisional broadening or large natural width, form continuous or quasi-continuous bands. Studies of variation of these statistical characteristics along isoelectronic sequences give a wealth of information on intra-atomic interactions. 

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