Modern trends in the spectroscopy of multicharged ions

PHYSICS REPORTS (Review Section of Physics Letters) 164, No. 6 (1988) 315—375. North-Holland, Amsterdam

MODERN TRENDS IN THE SPECTROSCOPY OF MULTICHARGED IONS L.N. IVANOV, E.P, IVANOVA and E.V. AGLITSKY Institute of Spectroscopy, Academy of Sciences of the USSR, 142092. Troitsk, Moscow Region. USSR Received January 1988

Contents: I. Introduction 2. Experimental investigations of multicharged ion spectra 2.1. High-temperature plasma as a source of multicharged ions 2.1.1. Laser plasma 2.1.2. Low-inductive vacuum spark 2.1.3. Tokamaks 2.2. Beam—foil method and recoil spectroscopy 3. Isoelectronic sequence of lithium as the simplest fewelectron system for testing QED and nuclear effects 3.1. Review of experimental and theoretical results 3.2. One-particle wave functions 3.3. Nuclear potential 3.4. General scheme of the calculation 3.5. Vacuum polarization 3.6. Self-energy part of the Lamb shift

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3.7. Results of the calculation for the energy levels of Li-like ions 3.8. Hyperfine splitting of levels 3.9. Isotope shift 4. Isoelectronic sequence of helium 5. Electron satellites in the spectra of nuclear -y-radiation 6. Isoelectronic sequence of neon 6.1. Review of experimental and theoretical results 6.2. Relativistic perturbation theory with zeroth-order approximation 6.3. Imaginary part of the secular matrix 6.4. Numerical results for the energy structure. Cornparison with experiment 6.5. Highly excited levels References

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Abstract: Modern experimental and theoretical methods to investigate multicharged ion spectra are reviewed, including the study of radiation and cooperative electron—nuclear effects. A theoretical approach to study the spectroscopic characteristics of heavy and superheavy multicharged ions is presented. Some new results for He-, Li- and Ne-like systems are listed.

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MODERN TRENDS IN THE SPECTROSCOPY OF MULTICHARGED IONS

L.N. IVANOV, E.P. IVANOVA and E.V. AGLITSKY Institute of Spectroscopy, Academy of Sciences of the USSR, 142092, Troitsk, Moscow Region, USSR

I NORTH-HOLLAND

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AMSTERDAM

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1. Introduction

Modern experimental spectroscopy is characterized by increasing experimental accuracy, so adequate theoretical investigations have to include a detailed analysis of the interelectron correlation, relativistic and quantum electrodynamic (QED) effects. In recent years a tendency has developed which emphasizes the investigation of systems with heavy and superheavy nuclei, for which relativistic, QED effects and also the corrections for the finite size of the nucleus are of special importance. The intensive investigations of the processes in the L and K electron shells of heavy atoms put qualitatively new requirements on the theoretical apparatus. On the one hand, the spectroscopy of highly charged ions has become one of the main spheres of application of consistent QED theory of fermion systems in a strong external field; on the other hand, the spectroscopy has itself become an effective tool for testing the predictions of QED theory. It was traditionally supposed that the most suitable object for the investigation of fine relativistic, QED and nuclear effects are one-electron hydrogen-like ions. However, as was noted in ref. [1], some peculiarities of the spectra of helium- and lithium-like systems simplify the observation of both effects and, consequently, there is a need for precision calculations of these systems. In this connection, new approaches based on the consistent QED theory of few-particle systems has experienced a considerable development in the last years. Zigelman and Mittleman [2] state that the relativistic theory of heavy atoms “for historical reasons” has been developed “in a configuration space Hamiltonian (single-time) framework,. so a necessary step is the extraction of the configuration space Hamiltonian from the Fock space formulation of QED”. In the case of two-particle systems (H-like ions, for example) the quasipotential method has been suggested, which accounts for the QED vacuum effects by using an energy dependent quasipotential [3]. In ref. [4], the modified Coulomb potential has been accepted as a suitable quasipotential, and the ~corrections of the rigorous QED theory have been obtained. The spectrum of two bound particles with arbitrary spins and masses has been studied in ref. [5]. Further development improved the effective potential method by using a special linear nonunitary transformation that transforms the quasipotential equation to a Schrodinger type of equation, with the potential not depending on the energy [5]. The radiative transition probabilities for one- and two-electron ions have been examined in ref. [5]as illustration. A many-electron Hamiltonian derived from “first principles” includes three-particle terms; their contribution to the binding energy of Li-like heavy ions (Z = 80—137) turned out to be very small (—1.1 x 1O~—9.1x 10_i eV) [2]. Modern calculation procedures allow one to accomplish consistent QED calculations only in lowest order of perturbation theory. The traditional methods are based on an expansion in natural physical parameters: a, a Z, and liZ. The modern tendency is to avoid the aZ-expansion for heavy few-electron ions, and to avoid the 1/ Z-expansion for many-electron ions. The basic function determining all the atomic properties is the electron Green function. A simplification of the procedure to calculate it is one of the most important problems today. This procedure must be relativistic (without using the a Z-expansion), must be valid for an arbitrary central one-electron potential and for arbitrary complex energy parameter. Some attempts to construct such a procedure are made in refs. [6, 7], where calculations are presented for the self-energy shifts for H-like ions. A thorough investigation of few-electron systems is also important for simulation of processes which take place in complex atoms. For example, in order to predict nonstationary plasma kinetic characteristics one has to know at least which of the decay channels are open. Remember that for states with Kand L-vacancies an important role is played by autoionization Koster—Kronig decay with injection of . .

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outer electrons [8]. This channel prevails over all other decay channels if it is energetically open; however, in order to clarify this question precision calculations of the inner shells are often required which include QED and nuclear effects. Thus, already for Z ~ 40 the contribution of QED corrections to the energy of the 2s—2p transition is comparable with the binding energy of the outer electron. In ref. [9] we have calculated the spectra of hydrogen-like ions and special attention was paid to heavy and superheavy ions including ions with nuclear charge close to the critical value (Zcr 170). Here we partially present our results for Li-like ions. In this review we also try to analyse calculational and experimental possibilities to clarify electrodynamic effects. However, we do not consider the problem of testing QED theory, since the associated questions are considered in detail in ref. [1]. The relative contributions of different effects has been analysed in a number of papers (see references in ref. [10]);these results are summarized in the review of Drake [11] and in the monograph of Zapryagaev et al. [12]. Systematization of the energy levels is the most important goal in studies of the spectroscopic characteristics of highly ionized atoms. A detailed description of the spectrum observed experimentally requires numerical values of the rates of various processes; however, the energy levels of atomic systems yield fundamental information that can be used as a basis to reproduce the wave functions of the appropriate ion and (with the use of adequate theoretical models) to calculate the cross sections of such processes as radiative and nonradiative (autoionization) decay, dielectronic recombination, collisional excitation and decay of the atomic state, shifts of levels in external fields, scattering of particles by an ion, radiation transfer, etc. The spectra of states with one and two quasiparticles (electron or vacancy) outside of the core of closed shells, such as 0-, F-, Ne-, and Mg-like ions, resemble those of few-electron systems. However, the theoretical analysis of their properties from “first principles” is complicated by core polarization effects. The situation may be simplified due to the availability of extensive experimental information concerning the level positions of the appropriate one-quasi-particle systems. Such information allows one to construct a formally exact perturbation theory with the zeroth-order approximation generating the optimal one-quasi-particle basis and to introduce the effective interquasi-particle polarization interaction, all the one-quasi-particle relativistic, QED, finite nuclear size effects and one-quasi-particle polarization effects being accounted for automatically in the zeroth-order approximation. This approach allows high-accuracy calculations of more complicated atomic systems with a few quasi-particles [13—15]. The spectroscopic characteristics of highly ionized atoms have a wide range of application. Traditional problems are those where the characteristics of highly ionized atoms are used for the diagnostics of laboratory and astrophysical plasmas, directed towards the prediction of the macroscopic properties of the plasma. The study of spectroscopic data of ions is closely connected with the development of lasers for the ultraviolet and X-ray ranges using ions as the working medium; the highly ionized plasma is interesting here as a medium which enhances emission at the transitions of highly ionized ions. The first point in these investigations is a search for inverted atomic level populations. This phenomenon has been observed recently for some levels of H- and He-like sulphur [16], and Liand Be-like copper. The wavelengths of the transitions under consideration are: A ~ 42A in sulphur and A 24—28 A in copper. A design for producing a soft X-ray laser via 3p—3s transitions in Ne-like selenium (Az~s200 A) is described in ref. [17]. Many problems are reduced to the theoretical modelling of the radiation spectrum of stationary and nonstationary plasmas [18].This complicated spectrum includes quantized radiation from different ions as well as bremsstrahlung. A variety of elementary processes: radiation, collisions, autoionization, and

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their interference must be accounted for in this theory. The quantized radiation can be spontaneous or stimulated by the inner super-high-frequency (SHF) pulses (or by the laser field). At plasma temperatures exceeding 1 keV some new cooperative electron—nuclear reaction channels open in addition. These circumstances drastically complicate the theoretical analysis. It can be simplified by treating all the elementary processes in a unified manner. An adequate apparatus must be based on consistent QED theory [19—24]. One of the possible versions of such an apparatus is the so-called energy approach [23,24]. Well known in nonrelativistic theory is the field form procedure, connected with the diagonalization of the secular matrix for the calculation of the energy shifts t~Eof degenerate atomic states [25]. The procedure can be based upon the Gell-Mann—Low adiabatic formula and the perturbation expansion of the scattering matrix. An analogous scheme with the electrodynamic scattering matrix is a valuable tool in relativistic theory [26]. In contrast to the nonrelativistic case the relativistic secular matrix elements for excited states have an imaginary part already in second order of the electrodynamic perturbation theory (the first nonvanishing approximation for i~E),which is totally connected with radiation decay. The next, fourth-order imaginary contribution includes terms corresponding to one- and two-photon radiation decay and autoionization decay for autoionizing states; the interference terms appear in higher order. In the energy approach the whole procedure of level position and decay rate calculations reduces to the diagonalization of the complex secular matrix. In this scheme two-electron recombination can be regarded as the background process with respect to autoionization. Inclusion of a perturbation by an external field (laser field, for example) allows one to account for stimulated transition effects, such as the field shift of lines and line shape deformation [27]. An example of such a complex investigation was accomplished by Jacobs et al. [21].Using Möller scattering operator techniques they calculated the intensities, width and Fano parameters for the 21’21”—1s21 and 1s21’21”—1s221 dielectronic satellite lines in argon as functions of plasma temperature and density. Such factors as electron collisions, two-electron recombination, spontaneous radiation and autoionization decay, as well as the interaction with the quantized radiation field permitting different process channels were taken into account. Extensive theoretical and experimental information about the atomic level positions and the velocities of the elementary reactions was included in the system of kinetic equations. Much of this information can be obtained experimentally from Tokamak plasma investigations. The Tokamak plasma parameters can be essentially varied by the choice ofthe regime of SHF heating; there are additional possibilities of scanning over the plasma and over the time of observation with respect to switching on of the SHF pulse [64]. It is clear now that an immensely large number of physical channels must be accounted for to reproduce the real plasma kinetic picture. Moreover, as was pointed out above, the interference of channels of a different physical nature can play an important role [22]. Under these circumstances successive enlargement of the set of channels could hardly lead to a complete solution of the problem. A more realistic approach will apparently be connected with the renormalization of the scattering matrix by introducing effective (mixed) reaction channels. Presently we do not know of consistent

theoretical analyses of this kind. A great part of our review deals with He- and Ne-like ions. These ions play an important role in plasma physics because these stages of ionization are dominant for wide ranges of temperature. This is due to an appreciable jump in the ionization potential, considered as a function of the degree of ionization. The lively interest in the ions of neighbouring degrees of ionization is often supported by the fact that in a real plasma, they coexist and their level populations are tightly connected through

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elementary processes. The spectral lines emitted by these systems form a group of adjacent satellites, often unresolvable. Information about relative intensities and shapes of the resonant and satellite lines is now widely employed in the diagnostics of stationary and nonstationary plasmas occurring in fusion energy and astrophysical research.

2. Experimental investigations of multicharged ion spectra In the current experimental spectroscopy of multicharged ions, two main directions can be noticed. The first one serves the purposes of systematic spectroscopy, namely, accumulation of diverse spectroscopic information not oriented at a particular problem, but which can be used in certain diagnostic or fundamental experiments. For this direction a compromise between a large number of transitions of various ions under investigation and the required accuracy of the determination of atomic constants (as a rule, the wavelengths of spectral lines) is characteristic. This approach can be exemplified by extensive investigations of X-ray spectra of the laser plasma in the early 70s [28—30]. Experiments of the second kind aim at a solution of a particular problem of fundamental importance [10]. Here the requirement of high precision is dominating. Recent examples of such work are measurements of the Lamb shift in multicharged hydrogen- and helium-like ions using different methods [31], including classical spectroscopy [32]. Investigations of this kind stimulate systematic spectral measurements. Presently the precision criterion for measurements of multicharged ion spectra is that the accuracy be comparable with the contribution of quantum electrodynamic corrections to the energies of the levels under investigation. The term “precise measurements” will be understood in this sense. The experimental realization of any spectroscopic investigation consists of several stages: (1) obtaining multicharged ions and excitation of the transition under investigation; (2) recording of the spectra; (3) precise measurement of wavelengths and intensities and identification of spectral lines; (4) obtaining physical information on the basis of a theoretical model. The method by which the first step is realized influences all subsequent steps and determines both the maximum ionization multiplicity and the precision attained. Until recently the main source of spectroscopic information on multicharged ions were hightemperature laboratory plasmas. Occasional use of observations of solar flares [33] for this purpose is limited by both the chemical composition of the Solar corona and by the difficulties of precise wavelength measurements for the spectral lines being investigated. Within the last few years some alternative methods to obtain multicharged ions have been developed extensively. Among them beam—foil and recoil spectroscopy methods are worth outlining. The greatest number of results have been obtained by means of the beam—foil method based on the ionization of an ion beam passing through a thin foil at relativistic velocities. We shall consider briefly the possibilities and restrictions which arise at the excitation of multicharged ions both in a hot plasma and by the beam—foil method. 2.1. High-temperature plasma as a source of multicharged ions X-ray spectra of multicharged ions are actively investigated in plasma setups of three types: (1) laser plasmas (LP):

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(a) sharp focusing of laser emission onto a flat target; (b) spherical laser irradiation and compression of the content of the shell target; (2) electric discharge devices leading to the formation of micropinches: (a) exploding wires (one- and multi-wire configurations); (b) plasma focus; (c) low inductive vacuumspark (LIVS) (including the pulsed input of gas); (3) Tokamaks.

However, from the above range of plasma setups precision.spectroscopy uses mainly: (a) a laser of energy 10—100 J with pulse duration 0.1—10 ns and diameter of the focusing spot onto a plane target 50—100 p~m;(b) low-inductive vacuum sparks with maximum discharge current 100—200 kA; and (c) Tokamaks of the types PLT, T-10, TFTR, TFR, ALCATOR, JET a.o. It should be noted here that the latter are mainly used only for precise relative measurements of spectral line wavelengths for multicharged ions which are components of the impurities. Exceptions are refs. [34, 35]. The maximum ionization multiplicity in plasma setups is determined by both the electron temperature (Te) and the value of the parameter PIer. The temperatures attained are 0.5—1.5 keV (LP); 1—4 keV (LIVS) and 1—3 keV (Tokamaks). In a recent paper [36]a temperature of Te ~—20 keV was reported. 3 for practically all types of modern The parameter fleT ranges within the limits plasma setups. Then the ion residence time in the 10U_1012 hot zone siscm 10_li s for LIVS, i0~s for a laser plasma and of the order of fractions of a second in a Tokamak plasma. Accordingly, the electron density in a laser plasma is 1020_1021 cm3 and in a LIVS micropinch it reaches 1022_1023 cm3 (and, perhaps, higher). This predetermines an appreciable distortion of the contours of the lines under investigation by microfields in the plasma. The qualitative difference in the spectra of multicharged ions excited in various plasma setups is also determined by the peculiarities of the directed and turbulent motion of the plasma as well as by its nonstationarity. It is well known that the spectra of multicharged ions with Z 10—20, which correspond to transitions with a change of the main quantum number n, lie in the soft X-ray range (A <20 A). In this region practically the only way of dispersing the radiation under investigation into a spectrum is the use of selective Bragg reflection from crystals. Plasma sources are characterized by considerable variation in the dimensions of the active region (—1 ~mis the “hot point” of the LIVS, 100 iim is the focal region of a laser plasma and 0.1—1 m is the plasma region of a Tokamak within the field of view of a spectrograph). Depending on the properties of the radiating object, precision spectroscopy of hot

plasma multicharged ions also uses various geometric configurations of the crystals (flat and curved) in combination with various recording systems (from photofilms to image intensifiers and position-sensitive counters). Precise wavelength measurements for spectral lines emitted by the plasma are based on the use of characteristic line radiation of the classical X-ray tube. The relative error ofthe reference lines is within 104_10_6 [37].In many cases internal or secondary standards are also used. The precision of the measurements in this case, naturally, decreases. Determination of the precise transition energy of a multicharged ion encounters also a serious difficulty caused by the presence of so-called satellites, which accompany practically any line, and, in some cases, cannot be separated from it. This circumstance is encountered also in the beam—foil method, but it redoubles at an appreciable width of the line emitted by a high-temperature dense plasma. Extraction of information requires a special procedure in each particular case, based on theoretical calculations of the parameters of satellites [38,39] and on an adequate model of the plasma.

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2.1.1. Laser plasma In the investigations of isoelectronic sequences which are of interest to us, the laser plasma results hold a special place by the number of measured transitions. As soon as the density of the heating radiation flux reaches 1013_1014 Wicm2, it becomes possible to obtain X-ray spectra of H-like ions with Z = 10—16, of He-like ions with Z = 10—23, of Li-like ions with Z 22—26 and of Ne-like ions up to Z = 42 [28, 42]. Note that the laser thermonuclear program oriented toward the creation of powerful multibeam systems to irradiate balloon targets has not provided as yet appreciable progress, as a side result, towards large ionization multiplicities (except for ref. [43], where the spectra of Ne-, F- and 0-like Xe were studied). This is, apparently, due to the technological difficulty of injection of the substance under investigation into the microballoon. The laser plasma as an emission source for short-wavelength spectroscopy has the following important experimental advantages: (a) easy transport of the heating emission energy to any point of the vacuum chamber of practically any configuration, which facilitates combinations with a wide range of various spectrographs; (b) stability of the spatial position of the compact emitting region, which allows, in particular, the use of signal accumulation on the photofilm without appreciable deterioration in the quality of the spectrograms; (c) high spectral brightness of the laser flare in the X-ray region. The last two circumstances have made it possible to use relatively simple crystal spectrographs without focusing in the dispersion plane: with a plane crystal [29, 43, 44], the de Broglie scheme with a convex crystal, which allows one to obtain high-quality round-up spectra [28], and a concave crystal, which focuses the emission in the plane perpendicular to the dispersion plane [45]. The spectral resolution in this case is determined by the transverse dimension of the plasma of the order of the focusing spot diameter. The possibility of obtaining qualitative spectra of laser plasmas by means of a fiat crystal has influenced the choice of the technique to measure wavelengths. As a rule, previously measured wavelengths of multicharged ions excited in a laser plasma are used as internal standards. The precision attained in this case is not high. The use of a line of the characteristic X-ray spectrum of the classical X-ray tube as a standard reference is more reliable. However, this presupposes the use of either sharp-focus X-ray tubes in combination with nonfocusing spectrographs or of simpler wide-focus X-ray tubes, but in combination with focusing spectrographs. Neither of these alternatives has been realized as yet. Other limitations on the precision of the wavelengths being measured are determined by physical properties of the laser plasma. Thus, for example, a certain part of the thermal energy turns into kinetic energy of directed motion of multicharged ions with a velocity of 106_107 cm/s, depending on the zone of the laser flare where this ionization multiplicity exists. This leads to an additional broadening and at some observation angles to a Doppler shift. The most favourable opportunity from the viewpoint of the precision of wavelength measurements is offered by the microdot technique being developed within the last few years [46]. It is based on laser irradiation not of a massive target, but of a deposition on a mylar layer of the substance being investigated with a diameter less than the focusing spot. Several results are simultaneously achieved by this: the transverse dimension of the plasma propagating in the given direction is decreased, a small optical density is reached and when the observation is made at a right angle to the propagation direction both the Doppler shift and some broadening connected with symmetric expansion can be avoided. This method was used to obtain the spectra of Mg-, Na-, Ne-, F- and 0-like ions of Br with rather narrow —

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lines; this permitted a sufficiently detailed study of satellite structures closely related to resonance transitions [47]. 2.1.2. Low-inductive vacuum spark In spite of its simplicity, the LIVS is presently the source with the maximum ionization ability among plasma devices. Thus in refs. [48, 49], He-like ions up to MoXLI were recorded. The authors of ref. [50]point to possible recording of He-like Xe in the plasma of a powerful setup with pulse injection of the gas. Thus, it is of interest to use LIVS for precision measurements of the spectra of ions with large nuclear charges. In the LIVS plasma the source of X-ray emission of multicharged ions is the so-called “hot point”, i.e. a dense hot plasma formation which arises in the course of the charge pinching between the electrodes. Presently this process is well described by the radiative collapse model [51]. One of the important peculiarities of the “hot point” is the indefiniteness of its place of origin. Therefore, precise measurements of wavelengths by means of the LIVS requires the use of spectrographs with focusing in the dispersion plane (the schemes of Johann [52], Johansson and Caushois [53], spherically curved crystals [54], etc.). At the same time this allows the use of the characteristic emission of simple wide-focus X-ray tubes as reference lines. Another circumstance that influences the quality of the spectrograms obtained is an appreciable difference in physical conditions at the “hot point” from one shot to another, up to the complete absence of a micropinch [55].When spectrograms are obtained by accumulation of the emission of a large number of bursts (100—1000), distortion of relative intensities and deterioration of the signal-tonoise ratio occurs. A natural way to eliminate this defect is the use of an image intensifier, which allows recording of the LIVS spectra with just one shot [56].This also has allowed one to obtain the spectra of elements which cannot be used to manufacture the electrodes required for multiple repetitions of LIVS discharges (such as Ga, Ge, As, Se, Br, etc.). The recording of the LIVS spectra per burst revealed that the “hot point” is moving as a whole with a velocity of i07 cmis [57]. Thus, e.g., the value of the Doppler shift for the resonance line of He-like iron is 1 mA, and exceeds the value of the Lamb shift for Z = 26. Compensation of this effect is reached by the simultaneous use of two focusing Johann-type spectrographs placed on the same straight line on different sides of the moving “hot point”. It is evident that in this geometry the signs of the projections of the velocity of the “micropinch” on a general direction will be different; accordingly, the signs of the shifts will be different too and the true wavelength is obtained as half the sum of the A values measured in two recording channels. However, the above improvements do not permit elimination of the appreciable widths (i~AIA’—’ 10~)of the spectral lines inherent to the spectra which are caused, apparently, by rapid compression or expansion in the course of the LIVS plasma pinching [52]. This restricts the choice of the object of investigation to relatively simple spectra of hydrogen-, helium- and neon-like ions with lines spaced sufficiently far from one another. Under such conditions, separation of satellites of the type ls2nl lsnl’21”, n 2, near the resonance lines of He-like ions is possible only by means of a special procedure of modelling the experimental spectrum on the basis of theoretical calculations (see ref. [52]). The measurement of the position of the resonance line 1s2—ls2p presupposes also elimination of the shift in the spectral maximum, which is measured due to the use of satellites with n 3. Presently the relative accuracy of wavelength measurement of the spectral lines emitted by a LIVS “hot point” is i0~, and He-like yttrium (ionization potential 20 keY) is the “hottest” ion whose resonance transition wavelength has been measured precisely. —

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2.1.3. Tokamaks The spectra of multicharged ions excited in Tokamak plasmas are free from some disadvantages inherent in a laser plasma and LIVS. They are not distorted by high density. Presently there are no indications of any appreciable influence of microscopic ion motions on the shapes and positions of the spectral lines being measured. Thermal Doppler broadening is the main mechanism which controls their width [58]. A large volume of emitting plasma presupposes the use of focusing Johann-type spectrographs with large curvature radius ~1 m to obtain spectra with high resolution. This provides an appreciable size of the operating area of the crystal (several cm2) required to increase the recording efficiency. On the other hand, the quasistationary character of the Tokamak plasma allows the use of position-sensitive multiwire proportional counters [58—60]in the recording system. Their spatial resolution is, as a rule, not very high; therefore, in order to obtain high spectral resolution it is necessary to increase the dispersion of the instrument and this also requires large values of R. An interesting instrument constructed according to the von Hamos scheme was used in ref. [61].In this experiment high efficiency and spatial resolution are supplied by vertical focusing and spectral resolution is reached by placing the slit on the curvature axis of the crystal. The above techniques for the study of X-ray spectra of Tokamak plasmas are widely used for diagnostics [58—61]and for precise relative measurements [60]. An example is determination of the position of the lines near resonance transitions, whose wavelengths are taken from the theory [62]. In the first experiments, the objects of investigation were restricted to impurities entering the plasma from the materials from which Tokamaks are constructed, and also by accidental impurities. In this case the range of H- and He-like ions with Z = 16—26 (S, Cl, Ti, Cr, Fe [60, 61, 63—65]) was covered, along with Ne-like ions up to Mo [66]. The development of techniques to introduce arbitrary materials into the Tokamak plasma has broadened the range of elements to be investigated. It has become possible to control the amount, the moment and the duration of the injection of the substance being introduced and to obtain the spectra under controlled conditions. In some cases, one can increase their intensity. The gas (Ar) under investigation was introduced into an ALCATOR Tokamak plasma by means of a pulse valve [61]. Laser evaporation was used to introduce Ag into a PLT plasma and Sc, V, Cr into TFR [35, 67]. Thus, it has become possible to investigate the spectra along isoelectronic sequences under comparable conditions. In this case, for the range being investigated, the problem of calibration of the wavelengths is partly solved as the lines of hydrogen-like ions, whose wavelengths can be calculated sufficiently well, can be used as reference lines. In some cases this precision is quite sufficient, as, e.g., for the investigation of Ne-like Ag excited in a PLT plasma. Absolute measurements with the use of an X-ray tube as a source of reference lines were made in only one work [34], where the energies of the resonance doublet of hydrogen-like chlorine were determined with high precision. The spread of techniques of absolute measurement for present-day setups in combination with the introduction of arbitrary substances into a plasma makes Tokamaks a powerful tool for fundamental investigations. 2.2. Beam—foil method and recoil spectroscopy Apparently the beam—foil spectroscopy method of obtaining multicharged ions presently offers the most favourable opportunities for precise measurements of spectral lines, wavelengths, and lifetimes of energy levels. Though ionization requires that the ions be accelerated to high energy (from dozens to

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325

hundreds of MeV/nucl) the available systems (Gânil, France; Bevalac, USA; CSI INULAC, FRG) meet these requirements. The distribution of the ions obtained with respect to ionization multiplicities and also the excitation of the necessary transitions are determined by the material and the thickness of the foil being used [68]. The range of the ions of H- and He-like sequences that are obtained extends up to uranium (Z = 92) [68, 69]. The absence of collisional deexcitation permits the study of electric quadrupole, magnetic dipole and other strongly forbidden transitions. A classical scheme for recording the spectra of ions excited by the beam—foil method includes a plane-crystal spectrograph, a slit arranged behind the foil to isolate the radiation region of the transition under investigation and a position-sensitive detector of the dispersed emission [69]. Wavelength calibration is made by means of an X-ray tube or with the lines of hydrogen-like ions [32,69]. An apparent restriction for the precision of wavelength measurements by this method is caused by the inaccuracy in the determination of the Doppler shift arising from the directional motion of the ions being focused at relativistic rates (v/c = 0.1—0.5). Therefore precise spectroscopic measurements require ion beam energy measurements no less precise, for example, by means of a magnetic spectrometer. Also, one must investigate energy losses after passing through one or two foils. The error in wavelength measurement is also influenced by such factors as the divergence of the ion beam and the dependence of the Doppler shift on the angle between the velocity and the observation direction. A decrease in the angular aperture of the recording system increases the precision of the measurements, but, naturally, deteriorates appreciably the efficiency of the recording. In ref. [70]an unusual method was proposed to eliminate the uncertainty of the angle of ion emission without a decrease in the recorded emission intensity. The method is based on the use of two Johann-type crystal spectrographs with the total dispersion plane perpendicular to the axis of the beam of ions stripped in the foil. When the crystals are symmetrically positioned with respect to the emitting region, any deviation from perpendicularity of observation may be eliminated by averaging the wavelengths measured in two recording channels due to the equality of the long-wavelength and short-wavelength shifts. The average wavelength must be corrected for the Doppler shift caused by the directional motion of the excited ions; this shift is not compensated by the above-mentioned scheme. The spectra obtained by the beam—foil method, in a manner similar to the plasma spectra, are not free from satellites. However, there exist some methods to minimize their influence by choosing the material and the thickness of the foil in which the transitions being investigated are excited [71,72]. Direct observation of allowed transitions of ions with large nuclear charges by the beam—foil method is impeded by strong decay probabilities (up to _~1016s~1in the case of H-like uranium). Even at

relativistic velocities of the emitting ions the distance over which relaxation of excitation occurs is a fraction of a micron. In ref. [10]the Lamb shift for n =2 in He-like uranium was determined by a method which does not require precise measurements of wavelengths. In this case the lifetime of the 23P 0 state was measured, 3P 3S which depends on the energy difference 2 0—2 1, from which, in its turn, the Lamb shift was evaluated on the basis of theoretical calculations [73—76]. The recoil spectroscopy method is free from the problems introduced by the Doppler shift, because in this method the sources of emission are ions arising in a rarefied gas target when a relativistic ion beam goes through it. In this case, the incident ion must be massive to transmit a small portion of its kinetic energy to the excited target atom. Apparent restrictions are determined by the necessity of using gas targets and also by the fact that the maximum charge of the ion being formed cannot exceed the charge of the incident ion. In refs. [77,78], recoil spectroscopy was used for very precise measurements of the wavelengths of

326

L.N. Ivanor et a!., Modern trends in the spectroscopy of mu!ticharged ions

resonance transitions in H- and He-like argon ions. The contribution of satellites to the spectrum under investigation was minimized by carefully choosing the pressure of the gas target.

3. Isoelectronic sequence of lithium as the simplest few-electron system for testing QED and nuclear effects Hydrogen-like ions and lithium-like ions considered here have a similar energy level structure; for lithium-like ions the problem is complicated due to the presence of the polarizable electron core is2. Precision calculations of hydrogen-like ions have been made up to Z = 170 [6, 7, 9], whereas the isoelectronic sequence of lithium is less investigated. As shown in ref. [1], this sequence is one of the most promising from the viewpoint of testing theoretical predictions of corrections in many-electron atomic systems. This refers primarily to the systems with a large nuclear charge, for which the contribution of the above corrections is appreciable. The early calculations of QED corrections were based on expansions in a series in the parameter aZ, whose convergence already in the region Z ~ 40 is not sufficient. Presently techniques are being developed for QED corrections in which expansions in aZ are avoided as far as possible. Thus, in refs. [79—81]a complete calculation of the Lamb shift is made with due regard to the first nonvanishing approximation of QED-theory without aZ-expansion for hydrogen-like ions with Z = 10—100 for the states is, 2s, 2p 1123~2.A point atomic nucleus was assumed. The same problem for a finite nucleus was considered in refs. [6, 82]. One of the urgent problems in modern atomic theory is the generalization of QED calculations for systems with a finite nuclear size, where the number of electrons exceeds one. Such calculations are associated with considerable computational difficulties. They require detailed checks, and an experimental check is especially important. To study QED effects it is desirable, on the one hand, that they make an appreciable contribution to the energy transition searched for (heavy atomic systems); on the other hand, it is important that these corrections are not masked by inaccuracies in the calculation of the “pre-Lamb” contributions, i.e. inaccuracies in correlation and relativistic corrections. In this sense the hydrogen-like sequence is ideal, and the first calculations of QED effects were made for it (see the review in ref. [ii]); a comparison with experiment (Z ~ 26) [6] demonstrated excellent agreement of the theoretical and experimental values of QED corrections. In this connection, we mention one recent work on the formation of a beam of H-like ions of krypton in the excited state with the apparatus “Gânil” (Caen, France) [83]. The absolute value of the Lyman-a1 line was measured with an accuracy of 36 ppm by means of a spectrometer with a plane crystal. The value of the Lamb shift for hydrogen-like krypton, 11.95 ± 0.5eV, agrees with high accuracy with the Mohr theory (11.856eV) [81]. In heavy atomic systems a considerable role is played by the finite nuclear volume and also by many-body QED effects, or the effects of mutual screening of electrons. It will be shown below that for the study of the latter effects the most convenient objects are 2s—2p transitions in Li-like ions. This conclusion is valid from both the theoretical and experimental viewpoints. 3.1. Review of experimental and theoretical results In refs. [84, 85], 2P3I2,l/ 2~If the beam—foil technique was used to measure the energies and the lifetimes of the lower 2~ 2 transitions in Li-like krypton. The accuracy of the energy measurement for the

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

327

lowest 2p112—2s112 transition in Kr33~(A = 174.15 ±0.26 A) allowed us to judge the quality of various theoretical calculations in this region of Z. Thus, it was found that the calculations on a nonrelativistic basis, including relativistic corrections in the Breit operator approximation, yield for the energy of the

above transition a result whose error appreciably exceeds that of the experiment. From all theoretical work discussed in ref. [84]the calculations in the multiconfiguration Hartree—Fock—Dirac method [86] are strongly preferred. Analysis of theoretical calculations and their comparison with experimental data has permitted determination, with an accuracy of several per cent, of the Lamb shift in the one-particle approximation for the 2s state in Li-like krypton: it is 13 940 cm Note that for Z = 36 the contribution of the finite nuclear size effect to the energy of the 2s state is about 500 cm and for 2Pt/2, 3/2Lamb is negligibly 2 in this ion decreases the shift by small; about the effect of screening of the outer electron by the core is the same value. Since these corrections are of the order of the experimental error [82,85], and of the uncertainty in the calculation, they cannot be clearly isolated by a detailed comparison of theoretical and experimental results. In theoretical calculations, the correlation energies of the outer electron of Li-like ions are evaluated with an accuracy of several hundreds of cm~1up to Z 90. In the calculations using Dirac wave functions, the relativistic part of the energy is determined with approximately the same accuracy. In U89~the QED effects and those of the nuclear volume together make up more than 10% of the energy of the 2p 1. The many-body 1 12—2s112 i.e.effects) a valueareof several several per hundreds thousands of cmthe accuracy of the QED effects (and transitions, the screening cent ofofthis value. Thus, calculations for the energies of the lower transitions of Li-like ions in the region of large Z ~ 90 is determined by the choice of the nuclear size, and the method of taking into account many-body effects - ~.

- ~,

-~

of the Lamb shift.

Recently the accelerator of super-heavy nuclei, “Hilac” [1] at the University of California, Berkeley, was used to provide highly charged uranium ions: a beam of bare uranium nuclei passed through a uranium target. By choosing the target thickness and the velocity of the nuclei appropriately, it was possible to achieve an output of predominantly uranium ions with one, two or three electrons. The emerging ions were in the excited state. In this work [1], it is stated that the spectra of uranium ions of the above three types were recorded both in the X-ray and vacuum ultraviolet regions. For testing QED theory, the most suitable are the lines which correspond to the transitions 23P 3S 0—2 1 in He-like uranium, and to 2 P1 /2—2 S112 in Li-like uranium. Since the latter transition is to the ground state, its intensity is about an order of magnitude higher than that of the transition in the He-like ion; moreover, ions of Li-like uranium are formed at lower velocities. Therefore of the is appreciably less. Asan stated in ref.of[1], 2P the2Sbackground radiation 89~may, in target principle, be measured with accuracy 0.2 the eV. energy of the 2 1~2—2 1~2 transition in U The total energy of this transition, according to the calculated data [86], is 281—285 eV. An inaccuracy of several eV is caused by inaccuracies in the inclusion of the finiteness of the nuclear volume, and inaccuracies in the calculation of the Lamb shift for a three-electron atomic system with large Z, for

which two-particle QED effects start playing an important role. The present paper is aimed at the development of a calculation with a model for the nucleus in which the charge distribution is introduced explicitly, followed by a calculation of all energy levels 21j, 3lj, 4lj of the outer electron in the nuclear field, and the core is2 electrons. The main QED effect, vacuum polarization and self-energy correction for the Lamb shift, should be taken into account completely and successively in the calculation. A basic set of spectral characteristics of Li-like ions with Z ~ 29 has been obtained by Aglitsky,

328

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

Boiko et al., in the Lebedev Institute using a 50GW neodymium laser [28], which were compiled later in NBS by Sugar and Corliss [87]. A round-up crystal spectrograph has been used to study the lines in the spectral interval 6—17 A. In a nonstationary plasma, the ion multiplicity distribution lags behind the electron temperature. Under these conditions, an intensive, well-pronounced Li-like spectrum exists for those elements whose He-like spectrum is well pronounced too. For heavier elements, the rich Be-like spectrum appears almost simultaneously with the Li-like spectrum. This complicates the identification of the latter. Copper (Z = 29) is the heaviest element whose Li-like spectrum has been studied in detail using modern plasma installations for T~ 1 keV. The accuracy achieved in ref. [28] is 0.001—0.003 A. It is enough to identify clearly the Li-like spectrum, which usually coexists with other rather amplified L-shell spectra. The real accuracy, however, can deteriorate because of occasional or systematic overlap of lines radiated by ions of different multiplicity. By now there is rather complete information about the 2—3 transitions for ions with Z ~ 29, but the 2—4 and higher transitions are barely classified. For example, the doublets 2s—npj (j = 1/2, 3/2), 2p312—ndj (j = 3/2, 5/2) with n >3 are practically unresolved in -~-

plasma spectra. Thus, additional information about transitions between highly excited states is needed. Experimental data about transition energies in Li-like ions with Z >30 are very scarce. In ref. [88], the wavelengths of the lower transitions were measured for CuXXVII the PLT tokamak in 2P two 2S 2P in 2S Princeton: for the 2 112—2 11, transition, A=224.8A, and for the 2 3,2—2 115 transition, A= 153.6 A. In ref. [89] 4—3 transitions in Li-like ions with Z = 12—22 were investigated; these lines were generated by means of a 4 GW Nd laser irradiating solid targets. The measured wavelengths were extrapolated along the isoelectronic sequence up to FeXXIV; for the extrapolation, ab initio calculations were used [90]. In ref. [91] 2—2 transitions are reported with the Tokamak DITE for Ti, Cr, Fe, Co, and Ni ions of various ionization degrees. It is shown that the results for Li-like ions confirm the semi-empirical calculations made by Edlén [92]. Afterwards, a group working with DITE [93]obtained specified wavelengths for 2—2 transitions in ionized of the transitions 2P Ti. For 2S TiXX the wavelengths 2P 2S were measured with an accuracy of 4—10 mA: 2 3~2—2 112:A=309.072(i0)A; 2 3/2—2 1/2: A= 259.272(4) A. An important criterion for the correctness of the experimental data is their smoothness along the isoelectronic sequence determined by the first, second etc. differences of the energies as a function of Z. In the work by Edlén [92], the experimental data were analysed for 2—2 transitions. They were smoothed, and also extrapolated semi-empirically up to Z = 36. Beginning from CuXXVII (Z = 29) data on 2—3, 2—4 transitions are lacking in the literature. The data on these transitions for ions of lower ionization multiplicity given in the literature do not always satisfy the smoothness conditions and, consequently require an additional analysis. Theoretical methods used to calculate the spectroscopic characteristics of Li-like ions may be divided into three main groups: (a) The multiconfiguration Hartree—Fock method, in which relativistic effects are taken into account in the Pauli approximation [90, 94, 95]. This gives a rather rough approximation, which makes it possible to get only a qualitative idea on the spectra of Li-like ions. (b) The multiconfiguration Hartree—Fock—Dirac approximation (the Desclaux program [96]) has been, in the last few years, the most reliable calculation scheme for many-electron systems with a large nuclear charge; in these calculations one- and two-particle relativistic effects are taken into account in a practical and precise manner. For few-electron systems, equivalent results can be obtained by the simpler method of relativistic perturbation theory starting with the one-electron approximation [97]. In ref. [86], the positions of the lower levels are calculated in Li-like ions for some Z-values over the range

L.N. Ivanov et al., Modern trends in the spectroscopy of multicharged ions

329

3—92. The QED effects are included in the calculation with regard to the effect of the screening of the outer electron by the inner shell is2. The calculation program of Desclaux is compiled with proper account of the finiteness of the nuclear size; however, a detailed description of their method of investigation of the role of the nuclear size is lacking. We feel that the data for Z > 60 presented in ref. [86]require an additional analysis. In the region of small Z, the calculational error in the MCDF approximation is connected mainly with incomplete inclusion of correlation and exchange effects, which are only weakly dependent on Z. (c) In the study of lower states for Z s 40, the expansion in a double perturbation series in the parameters liZ, aZ has turned out to be quite useful [101—103]. This permits evaluation of the relative contributions of different expansion terms: nonrelativistic, relativistic, and quantum-electrodynamic, as a function of Z [109]. 3.2. One-particle wave functions

The procedure outlined below reduces the calculation of all atomic characteristics to solving one system of ordinary differential equations (DE). It includes the equations for all one-electron potentials

and other operators, Dirac equations for one-electron orbitals, equations for radial integrals of the Slater type and for other matrix elements. The integration starts from a finite value r 0 of the radial coordinate (usually r0 = Ri 100, where R is the nuclear radius). The starting values of all functions at r = r0 are defined using corresponding Taylor expansions. Thus, the whole calculation procedure is one dimensional, which unifies the calculation of different atomic characteristics and saves computer time. The one-particle wave function is found from the solution of the relativistic Dirac equation, which can be written in a central field in two-component form as

(1)

r + (1— K)

+

(e

—

m

—

V)F =0.

Here the fine structure constant a 1—(l+1) ~

K—1

=

1. The moment number

forj>l, forj
(2)

For large K, the radial functions F and G vary rapidly at the origin. For a point nucleus, 2 a2Z2)1’2 (3) F(r), G(r) ~ y = (K This leads to difficulties in the numerical integration of the equations in the region r—~0. To prevent the —

-~

.

,

integration step becoming too small, it is convenient to turn to new functions isolating the main power dependence:

f=

Fr~”,

g = Grl~H.

The Dirac equation for the F and G components are transformed as

(4)

330

L.N. !vanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

f’= —(K+fK~)flr—ciZVg—(aZEfl,(+2IaZ)g, (5) g’

=(K

—

K~)gIr aZVf —

+

aZEflKf.

Here Coulomb units (Cu) are used; 1 Cu of length = 1 au~Z; 1 Cu of energy 1 au~Z2. In Coulomb units, the atomic characteristics vary weakly with Z. Enk is the one-electron energy without the rest energy. The system of equations (5) has two fundamental solutions. We are interested in the solution regular as r—~0. The boundary values of the correct solution are found by the first terms of the expansion in a Taylor series: g=[V(0)—Efl~]raZI(2K+1), f=1

for K<0,

f=[V(0)—EflK —2/a2Z2]aZ,

for K>0.

(6) g

1

The condition f, g 0 as r ~ determines the quantized energy of the state EflK. At the correctly determined energy E~,,the asymptotic forms off and g for r—* ~ are —~

—~

f, g~~~exp(_rIn*); =

(7)

(1I2~E~j)112 is the effective principal quantum number.

Equations (5) were solved by the Runge—Kutta method. The initial integration point r 0 the end of the integration interval is determined as rk 30n ~

6, and =

Rib

3.3. Nuclear potential In ref. [104],we calculated some characteristics of hydrogen-like ions with the nucleus in the form of a uniformly charged sphere; analogous calculations by means of an improved model were made in ref. [9], where a smooth Gaussian function for the charge distribution in the nucleus was used. The use of a smooth distribution function instead of a discontinuous one simplifies the computational procedure and permits a flexible simulation of the real distribution of the charge in the nucleus. As in ref. [9], we model the charge distribution in the nucleus p(r) by a Gaussian function. With regard to normalization we have p(r, R)

=

(4y312IV~)exp(—yr2),

j12 dr r p(r, R)

=

1,

j13 dr r p(r, R)

(8)

=

R,

(9)

0

where y = 4 ii~R2,and R is the effective nuclear radius. The following simple dependence of R on Z is assumed: R=i.20xi0~3ZH2cm.

(10)

Such a definition of R is rather conventional: we assume it as a zeroth order approximation. In the work

L. N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

331

the derivatives of various characteristics which describe the interaction of the nucleus with the outer electron are calculated; this permits recalculation of the results when R varies within reasonable limits. The Coulomb potential for the spherically symmetric density p(r, R) is V~~~1(r, R) = —(ilr)

J

dr’ r~p(r’,R) +

J

dr’ r’p(r’, R).

(ii)

Below we call the potential (11) for brevity the “smeared Coulomb interaction”. It is completely determined by the following system of differential equations: 2)

(1 1r

J

dr’ r’2p(r’, R)

(1 ir2)y(r, R),

V~uci(r, R)

=

y’(r, R)

=

r2p(r, R),

p’(r, R)

=

—(8y5’2rI’./~)exp(—’yr2) = —2yrp(r, R) = —(8rilTR2)p(r, R),

(12)

with the boundary conditions 312i1T”2 = 32/ir2R3. (13) 1(0, R) = —4/(ITR), y(O, R) = 0, p(O, R) 4y The system of equations (12) includes the equations for the density distribution. A spherically symmetric function p(r, R) is accepted here. This does not mean that our considera-

V~~~

tions are restricted to a spherically symmetric nuclear model. Below, nonsphericity is in fact accounted for by phenomenological parameters such as the electric quadrupole nuclear moment Q, and the gyroscopic Landé factor g 1. Besides, the “smoothing” of the electric and magnetic moments over the nuclear volume is accounted for in our model. The introduction of different radii for the density distributions for the nuclear charge, and the electric and magnetic moments makes the theoretical model more flexible. The nuclear finiteness cannot be accounted for by perturbation theory, in which the difference between (ii) and the bare Coulomb potential is a perturbation and which starts with the point nucleus approximation. This statement is valid for the lightest atoms too. The reason is that in spite of the possibly small absolute correction to one or another atomic characteristic, nuclear finiteness leads to a radical reconstruction of the wave function just in the region contributing to the matrix elements of the corresponding perturbation theory, which violates the condition of applicability of perturbation theory. The correct approach is represented, for example, by the matching method: the electron wave function must be smoothly matched at the nuclear boundary, the corresponding equations being solved with the use of perturbation theory in the parameter RiRa (Ra is of the order of the atomic radius). This problem can be solved analytically for the simplest nuclear models (see, for example, ref. [105]). Different atomic characteristics are sensitive to the behaviour of the wave function in the corresponding r-regions. Thus, to correctly account for relativistic effects it is important to have an accurate wave function up to r0 = a iZ (in atomic units). For systems with R < r0, relativistic effects as well as nuclear-size effects can be regarded subsequently. For Z> 60 this condition is violated and nuclear finiteness must be introduced from the very beginning. It is very important to introduce nuclear finiteness correctly in the calculational procedure. It is nuclear finiteness, for example, that controls the

332

L.N. Ivanov et a!.. Modern trends in the spectroscopy of mu!ticharged ions

Z-dependence of the whole energy, the hyperfine correction integrals, and provides the stability of superheavy systems with regard to electron—positron pair creation up to Z 170. The following question can be posed: how sensitive are the electron spectra to the model of the nuclear shape? There are two aspects: the dependence on the radius for a given functional form of the density distribution p(r, R), and the dependence on this functional form at a given mean radius R. We calculated the nlj-state energies of hydrogen-like ions with Z = 30—170, n = 1—3 [9]. Two alternative forms of p(r, R) were considered: the function (8) and the model of a uniformly charged sphere. For Z >90 the discrepancy in the total bound state energy for the is state lies between Z2 and 10Z2 cm~, to be compared to a total energy of i05Z2 cm~.For the ground states this discrepancy does not exceed Z2 cm For the nlj—nl‘j and nlj—nlj’ splittings this discrepancy is less by one order of magnitude than that for the total energy. In the case of the hyperfine constants (these will be considered below) the calculation is so accurate that no discrepancy shows up. In any case, the results of tables 10, ii coincide for both models to all figures. The dependence on the nuclear radius in Li-like ions will be discussed below. ‘.

3.4. General scheme of the calculation Consider the Dirac—Fock equations for a three-electron system ls2nlj. Formally they are one-

electron Dirac equations for the is and nlj orbitals with the potential V(r)

2V(r~ls)+ V(r~n1j)+ Vex(r) + V(r, R),

=

(14)

where V(r, R) includes the smeared Coulomb and the polarization potentials of the nucleus; the

components of the Hartree potential are

V(rli)

=

~

J

dr’ p(r’~i)i~r r’~,

(15)

—

where p(rJ i) is the distribution of the electron density in state i), Vex is the interelectron exchange interaction. The main exchange effect will be taken into account if in the equation for the is orbital we assume V(r)

V(rjls) + V(r~nlj),

=

(16)

and in the equation for the nlj orbital V(r)

=

2V(rIls)

(17)

.

The rest of the exchange and correlation effects will be taken into account approximately. The electron density is usually determined iteratively. At the first step of the calculation, a Li-like ion in the ls2nlj state is considered as a one-electron system; the motion of the outer electron is described by the Dirac equation (5) with the central potential V(r, R)

=

V~cr(?~, R) +

Vnuci(r,

R) + U(r, R)

.

(18)

L.N. Ivanov ci a!., Modern trends in the spectroscopy of muiticharged ions

333

v~,UCI~ 1”scr and U denote, respectively, the smeared Coulomb potential, the Hartree—Dirac potential of the electronic K-shell and the nuclear polarization potential. At the second step of the calculation, the exchange and correlation interaction of the nlj electron with the core are taken into account by means of perturbation theory. In fig. i the diagrams describing these interactions in the lowest two orders are presented. In first order, the exchange interaction is calculated as the exchange matrix element of the “operator”

In eq. (18) the potentials

exp(iwr12)(i

—

â1â2)1r12

(19)

.

The exponential factor takes into account the retardation of the Coulomb interaction of the electrons; a2 are the Pauli matrices; w = E15 E~11. The expression for the appropriate contribution of diagram B is: tX~,

—

B~=>.Q0(isnlj;isnlj),

(20)

where QaO~5nlj; =

is nlj) Ra(i5 nlj; is nlj)S0(is nlj; is nlj) + Ra(iS nlj; is i j)Sg(is ni]; is nil) + Ra(i5 nif; is nlj)Sa(ls nlj; is nlj) + Ra(iS nh; is fllj)Sa(iS ,zlj; is nlj) + ~ R,(is nlj; is nlj)S~(isnlj; is nil) + ~ R1(ls nil; is nlj)S~(I~ nh; is nil) —

~ R1(is nil; is nlj)S~(1snh; is nil)

—

~ R,(is nlj; is nlj)S~(isnlj; is nlj).

(21)

In eq. (21) the first four terms are responsible for the Coulomb exchange interaction, and the last four for the magnetic interaction. Note that the contribution of the magnetic term to the potential (nonexchange, diagram A) interaction equals zero since a closed shell has no magnetic moment. The

-1

Coulomb part of diagram A is included in the calculation by the Hartree potential ~

9

P99

~1

E

F

Fig. 1. (A), (B) First-order one-electron diagrams. (C)_(G) Second-order one-electron diagrams.

Let us

334

L . N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

determine the angular function S. The Coulomb and magnetic angular functions are factored: from the four components, two-component fragments Sa(~,j), S~(i,j) are isolated: 5a(12,43)5a(13)~5a(42),

(22)

S~(i2,43)=—S~(13).S~(42),

where /‘ .h J~ Sa(13){hi l3a}~112 —1/2

S~(i3)= {l~13

~(

1/2

~ [213+1+

a~ o)’

—1/2 (2j1

~)(2a +

[2a(a + i)]~2

1)1/2

+ 1)(_1)u1~13~]( a

+ (_l)ts+tl÷~2(~

~

1)

1

(23)

~)(_i)/3+J3~2].

{J1 J2 J3} denotes the triangle condition for three moments and the parity condition for their sum. The two-component fragments in eq. (22) have permutational symmetry: 5a(31)

=

Sa(13)

,

S~(3i)= S~(13)(—1)~’~~

(24)

.

The Coulomb two-component fragment is invariant with respect to replacement of the large components by the small ones: 5a(i3)

=

(25)

Sa(13).

In the magnetic two-component fragment, one index always refers to the large component and the other to the small component of the one-particle state. The 3-j symbols in eq. (23) can be defined by simple analytical formulae. We give the expressions for the combinations required: 1/2

(2a~~i))

(~

i/[(1

+ 1)U2(~

~

=

(21+1)1/2

The symbols

(i~

13

~1/2

—1/2

1)(2/

+ i)]~2,

1I[1(1 + 1)]h/2, 1/[l(21+1)]

=

~

(2a

+

a 0

were calculated by the usual formulae.

_(l+i)H2

{l?/;,

:=

,

a=/+1 ~ 1.

a

=

1

+

1,

a = 1, a=1—1,

(26)

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

335

The necessary Slater integrals Rk(is nlj; is nlj) =

Jf

dr1 dr2 r~r~flS(rl)fflI3(r2)Wk(rlr2)flS(rZ)ffl,J(rl),

are expressed in terms of the Bessel functions of the first and second kind: Wk(rl, r2)

=

Wk(rl,

=

~

(_i)k(2k + i)Jk÷l/2(aZw,rl)J_k_l/2(aZu, r2)

,

r1
\1’7~_T-(_i)k(2k + i)Jk+l/2(aZw, r2)J_k_l/2(aZw, r1)

,

r1 > r2

(27)

____

r2)

-

The calculation of the Slater integrals has been reduced to the solution of a system of ordinary differential equations for the functions 3’2 Y1’(r) =ffllJ(r)Jk÷l/2(aZw,r)f1,(r)r Y~(r)=f 3’2 1,(r)J~÷1,2(aZw, r)f~11(r)r Y~(r)= ~(—i)”1r[Yl(r)flS(r)J_k_l/ 3’2 2(aZw,r)f~,1(r)+ Y2(r)Jk+l/2(aZw, r)f1~(r)]r ,

(28)

All the functions go to zero as r—~’0.This system also includes the equation for the Bessel functions and Dirac wave functions. The required integral is found as Y 3(co).

The calculation of the second-order correlation and exchange diagrams C, D, E for the 2s and 2p states is known in the nonrelativistic approximation (it Z expansion [60]): 2~(2s)E~2~(2slex) + E”~(2sIcorr)= i440 cm~1 E~ (29) E~2~(2p) = E~2~(2p~ex) + E~2~(2pIcorr) = —5690cm’ -

For ns, np states with arbitrary “n” one expects E~2~(nbj) = E~2~(2l)(E~,,iE

3’2 .

(30)

21)

Here E

2, is the nonrelativistic 21 bound state energy, E~11is the one-electron bound state energy found

from the Dirac equation with the one-electron potential (18). Such a simple form for the correlation and exchange contributions can be justified by the weak interference ofthe correlation and higher-order 2~(n1j) exchange nuclear Z. sizeFor andthe relativistic Thei the relative contribution of E’~ decreases effects rapidlywith with the increasing nil-stateseffects. with 1> correlation and higher-order exchange contributions were omitted. 3.5. Vacuum polarization Current calculations of the radiation polarization correction are restricted to the first term of the

a-expansion [8i]. This quantity is represented as a matrix element of the potential VH which in its turn

336

L. N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

is represented as a sum [81], VH=VH1+VH3+---.

(31)

Each term VH~in eq. (31) includes “n” field insertions to the electron loop of the polarization diagram of second order; each VH~is usually calculated with the use of an expansion in an additional parameter a Z. 7These for of H-like ions showed for order, Z ~ 80which all known of this compensate expansion upone to makecalculations contributions approximately thethat same to a terms large extent (aZ) another. Actually, in many cases the inclusion of higher-order terms of the expansion in a Z in eq. (31) leads only to a loss of accuracy. This has been shown, e.g., for the ground state of a H-like ion with

Zse~40[i06]. It is known that the first term in eq. (31) takes into account completely the first three terms of the exact expansion in a Z and with sufficient (by present-day standards) precision the remaining part of the contribution. Thus, for any arbitrary large Z S 170, the error of such an approximation does not exceed 10%. The first term in eq. (31) is known as the Uehling potential and is usually written as follows: U(r)

=

—

~~-~--

f

dtexp(—2rt/aZ)(i + i/2t2)

t

1

2 ns—~——C(g), g=riaZ.

(32)

In ref. [104] a computational procedure is proposed which permits regular specification of the contribution made by the Uehling potential by inclusion of additional terms in an expansion in the parameter na Z. A simpler approximate model of the Uehling potential was proposed in ref. [9]. In the present section, we shall show that the Uehling potential determined as a quadrature, eq. (32), may be

approximated with high precision by a simple analytical function. For the derivation, the same as in ref. [9],we determine the asymptotic form of the function C( g) in two limiting cases: C(g)—+ C

1(g) =ln(gi2) + 1.410 548— 1.037845g,

g—*0,

(33)

3’2, g—~cs (34) —1.8800exp(—g)/g The first term in eq. (33) is well known [2]; the other two were obtained by us [9]. In ref. [6], the two C(g)—+ C,(g)

=

limiting expressions for C( g) were matched as follows: C(g) = C 1(g)C2(g)/[C1(g) + C2(g)].

(35)

For the ground state of a hydrogen-like ion the error caused by the approximation (35) did not exceed 2—5% of the total polarization shift over the whole range Z = iO—i70; the main part of the error resulted from inaccuracies in the determination of C2(r) in eq. (35). Here for a more precise approximation of the Uehling potential, we make numerically a complete calculation of the function C( g) in eq. (32). Its graph is shown in fig. 2, and given in numerical form in table 1. Now, using precise !alues of C( g), we perform the matching (35) with a function C2( g) which we obtain by multiplying C2( g) by a polynomial f( g). The coefficients of this polynomial will be determined in such a way that the result of matching gives the best agreement with the exact function

L.N. Ivanov et a!., Modern trends in the spectroscopy of mullzcharged ions

337

C (~)

2.0

1.5

1.0

as

0.5

1.0

/

Fig. 2. The function C(g) of the Uehling potential: U(r)= —2C(g)/(3irr), g= r/aZ.

C(g). Thus, the corrected form is: C(g) = ~1(g)~2(g)t[~1(g) + ~:2(g)],

(36)

C2(g) = C2( g)f( g),

(37)

f(g) = (i.1022tg

(38)

—

1.3362)/g + 0.8028. Table 1 Numerical values of the Uehling potential as a function of g = rlaZ in three approximations for C(g), see eq. (32) (~(g)

((g)

C(g)

analytical

adjusted

exact

asymptotic

asymptotic

calculation

2.000

0.079

0.0350

0.03496

1.500

0.15

0.0757

0.07574

1.000 0.750 0.500 0.400 0.300 0.200 0.150

0.22 0.28 0.43 0.55 0.73 1.03 1.27

0.100 0.075

1.64

0.177 0.277 0.466 0.597 0.789 1.096 1.334 1.688 1.950 2.330

0.1766 0.2832 0.4831 0.6160 0.8083 1.1126 1.3472 1.6948 1.9453 2.3225

g

=

r/aZ

0.050

1.91 2.30

338

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

Table i presents the numerical results for the two approximations of the function C(g): C( g) from eq. (35) and C(g) from eq. (36). The use of the new approximation for the Uehling potential permits one to decrease the calculation errors for this term down to 0.5—1%. The use of such a simple analytical form of the function approximating the Uehling potential allows its easy inclusion into the system of differential equations which also includes the Dirac equations and the equations for the matrix elements. For specific calculations with a finite nucleus it is necessary to turn from the point Uehling potential to the bulk one, i.e. to smearing of the potential over the bulk of the nucleus; as a result of this procedure the Uehling potential, similarly as the Coulomb potential, becomes finite as r—~0. The mathematical procedure to distribute the polarization potential over the nuclear sphere is expressed as U(r, R)

=

J

U(~r’ r~)p(r’)dr’.

(39)

—

In our calculational scheme, we must only write out the differential equation for the function U(r, R),

U(r, R)

=

U(r)

J

dr’ p(r’, R) +

J

dr’ U(r’)p(r’, R).

(40)

For actual determination of the potential the differential equations U’(r, R)

=

U’(r)Y(r, R)

Y’(r, R)

,

=

r2p(r, R)

(41)

,

were solved with the boundary conditions U(0, R) =

f

dr’ U(r’)p(r’).

(42)

Table 2 gives an idea of the magnitude of the correction for the finite volume of the nucleus when the

polarization shift is calculated in the Uehling approximation. Table 2 Energy shift (cm~)due to vacuum polarization in a point nucleus, U(r), and a finite nucleus, U(r); data obtained in calculation for Li-like ions. For the is state the difference Ufr) — U(r) is given is Z 20 30 41

112 ~E

36 200 780 59 3800 69 8400 79 18200 92 59200

U(r) —158 —826 —3000 —14470 —30070 —59810 —144000

2s,,2 U~(r) —154 —803 —2900 —13940 —28860 —56880 —133600

2p112

U(r)

U~(r)

U(r)

U~(r)

—1 —10 —78 —893 —2726 —7605 —27030

—i —10 —76 —865 —2635 —7309 —25590

—0.2 —2 —14 —122 —309 —690 —1713

—0.2 —2 —14 —120 —304 —677 —1677

L.N. Ivanov ci a!., Modern trends in the spectroscopy of multicharged ions

339

3.6. Self-energy part of the Lamb shift The Lamb shift for the first three excited states 2s1 2’ 2p112, 2p312 of H-like ions for Z = iO—ilO was calculated in refs. [6, 7] using the covariant regulator method. The contribution to the energy of the second-order diagram was found by direct successive numerical calculation of the exchange matrix element with the relativistic Green function of the Dirac equation, i.e., in refs. [6, 7] the aZ expansion was not used. The computational details are given in ref. [i07]. The main part of this task is the calculation of the integrals /

JJ

dr1 dr2 [F0(r1) G0(r1)]G(~,K, T~,n2)[ ~

(43)

]A(r<)B(r>,

where F0, G0 are two radial components of the state under investigation, G is the radial Green function, ~is an energy parameter which runs over a finite segment of the real axis and the whole imaginary axis, and K is the Dirac quantum number of the virtual states, while A, B are known functions. Finally, r< r> are the smaller and larger of r1 and r2, respectively. The Green function is constructed using two fundamental solutions F, G and E, O of the radial Dirac equations with complex energy~.For a finite ,

nucleus, the first solution is regular as n

—~

0, the second is regular as r

solution with K <0 has a weak divergence as r

—~

—~

~

For a point nucleus, the first

0; with both solutions being represented by Whittaker

functions, they are calculated in a preliminary manner in ref. [107]. It is possible, in principle, to find them numerically from the Dirac equations with the simultaneous calculation of the integral (43) in the same system of differential equations. Thus, the whole procedure becomes one dimensional. For the first solution F, G (regular as r 0), the boundary conditions can be easily found from the Taylor expansion. This is not true for the second one (regular as r—s cc). The corresponding recurrence procedure for the Taylor coefficients terminates, so the coefficients determining the asymptotic —~

behaviour as r—* ~ remain undetermined. Thus, additional regulating conditions must be introduced for the functions F, G. Let us consider two relations connecting both fundamental solutions, W= FO

-

EG = (44)

W=FG+EG—*0

asr—,co.

The first (Wronskian) relation (44) is automatically satisfied by any two properly normalized solutions. As calculations show, the value of W is weakly influenced by errors accumulated during the integration of the Dirac equations. The second relation is specific for the pair of solutions we seek. Practically, with F, G known, one can find from (44) the asymptotic forms of the functions F and G. To smooth the transition from the “exact” (satisfying the Dirac equations for intermediate r) F, G to their asymptotic form, it is worthwhile to retain in W.. the first nonvanishing (--~ i/n) term with the appropriate numerical

coefficient. In addition, the nuclear size effect in the relativistic Green function was accounted for in the is states for Z > 40. However, the development of general methods suitable for calculations with arbitrary one-electron potentials taking into account nuclear size effects, and also the screening of the nucleus by

the atomic electrons, remains for the future. The problem of finding an appreciable simplification ofthe calculational procedure remains urgent too. We do not present here a new calculation of the self-energy correction. We only describe a method to use the results of a calculation for H-like ions with point nucleus including nuclear size effects and

340

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions Table4and 3 the quantity F(AIZ) = n3EsE(A~Z)I Relativistic energy, (0.027 148Z4) for H- the and parameter Li-like ions,~ =forE~ the state n = 2, K = —i (2s,, 1 2); units are cm Z —E,~(H~Z) —ER(LiIZ) ~(HIZ) ~(Li~Z) F(HIZ) F(Li~Z) 20 30 40 50 60 70 80 90 100 110

73870+0 379377+0 122399+1 307 137+1 659636+1 127703+2 230 103 + 2 394570+2 655435+2 107337+3

56654+0 318859+0 107568+ 1 277 139+1 604422+1 118084+2 214 485 + 2 369449+2

16.4861 24.8180 33.2616 41.8632 50.6787 59.7792 69.25% 79.2558 89.9771 101.7857

15.4280 23.7629 32.2048 40.8013 49.5833 58.6202 68.0533 77.9630

3.5063 2.8391 2.4550 2.2244 2.0948 2.0435 2.0650 2.1690 2.3870 2.7980

2.7810 2.4459 2.1969 2.0350 1.8528 1.8292 1.8646 1.9683 2.1661

Table 4 Self-energy contribution to the Lamb shift for ls2nlj states of Li-like ions

Nz

nlj ‘NN

20

21

22

23

24

25

26

27

28

29

30

31

1510 —39 35 447 —12 10 189 —5 4

1816 —44 47 538 —13 14 227 —5 6

2161 —49 61 640 —15 18 270 —6 8

2549 —56 77 755 —17 23 318 —7 10

2982 —65 96 883 —19 28 373 —8 12

3463 —75 118 1026 —22 35 433 —9 15

3997 —87 142 1184 —26 42 500 —Ii 18

4586 —100 170 1359 —30 51 573 —12 21

5234 —114 203 1551 —34 60 654 —14 25

5946 —129 240 1762 —38 71 743 —16 30

6723 —145 281 1992 —43 83 840 —18 35

7571 —161 329 2243 —48 97 946 —20 41

32

33

34

35

36

37

38

39

40

41

42

43

8494 —177 383 2517 —52 113 1062 —22 48

9496 —193 443 2814 —57 131 1187 —24 55

10580 —209 511 3135 —62 151 1323 —26 64

11752 —224 587 3482 —66 174 1469 —28 73

13016 —238 671 3857 —70 199 1627 —30 84

14377 —249 765 4260 —74 227 1797 —31 96

15838 —259 870 4693 —77 258 1980 —32 109

17406 —266 985 5157 —79 292 2176 —33 123

19085 —270 1113 5655 —80 330 2386 —34 139

20880 —270 1253 6187 —80 371 2610 —34 157

22797 —266 1407 6755 —79 417 2850 —33 176

24840 —257 1575 7360 —76 467 3105 —32 197

2s 112 2p 3/2

3p 312 4p 4s112 12 4p3,

Nz n!j N\ 2s11,

3s112 4s,12

Nz n!1NN 2s112 2p,,2 3s,12

44

45

46

47

48

49

50

51

52

53

54

55

27016 —242 1759 8005 —72 521

29329 —220 1959 8690 —65 581 3666 —28 245

31785 —192 2177 9418 —57 645 3973 —24 272

34391

37151

40073

43161

46006

49043

—156

—111

—56

8

118

235

2414 10190 —46 715 4299 —20 302

2670 11008 —33 791 4644 —14

2946 11873 —17 873 5009 —7 368

3245 12788 3 961 5395 1 406

3578 13632 35 1060 5751 15 447

3934 14531 70 1165 6130 29 492

52282 360 4312 15491 107 1278 6535 45 539

55733 498 4716 16513 147 1397

59405 650 5147 17602 193 1525

6967

7426

62 590

81 643

3377 —30 220

334

L.N. Ivanov et a!., Modern trends in

the

341

spectroscopy of multicharged ions

K-electron screening, and also for interpolation of the results for arbitrary Z. The corresponding results of refs. [6] and [7] are practically equivalent for this purpose. Our method is based on the idea 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 [6] as follows: Table 4 (cont.)

Nz

nljNN 2s,,

2

2p312 3s,,2

~1/2

56

57

58

59

60

61

62

63

64

65

66

67

63311 820 5607 18759 243 1661 7914 103 701

67461 1012 6097 19988 300 1807 8433 126 762

71865 1228 6620 21293 364 1962 8983 154 828

76536 1473 7177 22677 436 2127 9567 184 897

81486 1750 7777 24144 519 2303 10186 219 971

86725 2064 8404 256% 612 2490 10841 258 1050

92267 2419 9077 27338 717 2690 11533 302 1135

98124 2819 9794 29073 835 2902 12266 352 1224

104309 3269 10556 30906 969 3128 13039 409 1319

110834 3774 11365 32840 1118 3368 13854 472 1421

117713 4338 12226 34878 1285 3622 14714 542 1528

124959 4968 13139 37025 1472 3893 15620 621 1642

68

69

70

71

72

73

74

75

76

77

78

79

132586 5667 14108 39285 1679 4180 16573 708 1763

140608 6443 15134 41662 1909 4484 17576 805 1892

149039 7300 16222 44160 2163 4807 18630 913 2028

157895 8245 17374 46784 2443 5148 19737 1030 2172

167188 9284 18592 49537 2751 5509 20899 1160 2324

176935

187151 11668 21241 55452 3457 6293 23394 1459

197851 13027 22677 58623 3860 6719 24731 1628 2835

209051 14506 24192 61941 4298 7168 26131 1813 3024

220767 16112 25789 65412 4774 7641 275% 2014 3223

233015 17853 27472 69042 5290 8140 29127 2231 3434

245813 19735 29244 72833 5847 8665 30727 2467 3655

Nz ~I/2

4s~

Nz

n1j~”N 2s,,2

19880 52425 3088 5890 22117 1303 2485

2655

80

81

82

83

84

85

86

87

88

89

90

91

259175 21767

288102 26354 35272 85364 7809 10451 36013 3294 4409

303692 28976 37493 89983 8585 11110 37961 3622 4686

320092 31843 39802 94842 9435 11793 40011 3980 4975

337350 34974

76793 6449 9217 32397 2720 3888

273278 21958 33143 80971 7099 9820 34160 2995 4143

99955 10364 12503 42169 3272 5275

355514 38400 44678 105337 11378 13238 44439 4800 5585

374633 42135 47238 111002 12484 13996 46829 5267 5909

394758 46203 49876 116965 13690 14778 49345 5775 6235

415942 50630 52589 123242 15001 15582 51993 6329 6574

438237 55439 55372 129848 16426 16407 54780 6930 6922

461699 60656 58223 136800 17972 17251 57712 7582 7278

92 486384 66307 61137 144114 19647 18115 60798 8288 7642

93 512350 72418 64110 151807 21457 18996 64044 9052 8014

94 539654 79016 67138 159898 23412 19893 67457 9877 8392

95 568358 86130 70217 168402 25520 20805 71045 10766 8777

96 598523 93787 73341 177340 27789 21731 74815 11723 9168

97 630211 102016 76506 186729 30227 22669 78776 12752 9563

98 663487 110849 97707 196589 32844 23617 82936 13856 9963

99 698417 120315 82938 206938 35649 24574 87302 15039 10367

100 735066 130446 86195 217797 38650 25539 91883 16305 10774

31108 3s,,2 3p 112 3p,,2 ~ 4P 112

10423

42198

Nz n1j~’N 2s,,2 2p 112 2p,,5 3s,,2

L.N. Ivanov et a!.. Modern trends in the spectroscopy of mu!ticharged ions

342 ESE(H~Z, nlj)

=

0.027 148(Z4In3)F(H~Z,nlj).

(45)

The values of F are given for Z = 10—110, nlj = is, 2s, 2p hese resultsinare modified 2nlj of Li-like ions. It is supposed that for any112, ion 2p3~2. with aTnlj electron addition to ahere corefor of the states ls closed shells, the quantity sought may be presented in the form ESE(Z,

nlj)

=

0.027 i48(Z4/n3)f(~,nlj) cm’

(46)

.

The parameter ~ = ~ ER is the relativistic part of the binding energy of the outer electron; the universal function f( nlj) does not depend on the composition of the closed shells and the actual potential of the nucleus. The procedure to generalize the results of ref. [6] to the case of Li-like ions with a finite nucleus consists of the following steps: (1) calculation of the values ER and ~ for the states nlj of H-like ions with a point nucleus (in accordance with the Sommerfeld formula); (2) construction of an approximating function f( nlj) using the known value of the reference ~ (discussed below) and the appropriate F(H~Z,nh); (3) calculation of ER and ~ for the states nlj of Li-like ions with a finite nucleus; (4) calculation of ESE for the desired states by formula (46). The approximating function is determined as ~,

~,

f(~,nlj)=X

2+X 3. (47) 0+X1I~+X2I~ 3I~ The parameters X 0, X1, X2, X3 are determined by four specially chosen reference points. The energies of the states of Li-like ions were calculated twice: with the conventional fine structure constant a = 1/137 and with ~ = a/1000. The results of the latter calculations were considered as nonrelativistic. This permitted isolation of ER and The best interpolation is, apparently, achieved if for various regions of Z their own reference points are used. In table 3, with the state 2s112 (K = —1) taken as an example, the behaviour of the quantities under investigation is shown as a function of Z. An analogous investigation was carried out for the states 2p112 (K = 1)3.and = —2). For the appropriate high The2p312 final (K results of extrapolation are given in levels with n = 3, 4 we used the approximation ESE —~1/n table 4. A detailed evaluation of their accuracy may be made only after a complete calculation of E~E(Li~Z,nlj). It may be stated that the above extrapolation method is more justified than the use of expansions in the parameter a Z. ~.

3.7. Results of the calculation for the energy levels of Li-like ions In table 5 the theoretical results for the relative positions of the n experimental ones. The deviations E(25i/ 2Pi/ 25ti 2Pii 2~

2)theor

—

E(

2~

=

2 levels are compared with the

2)expare represented in fig. 3. One

can see a considerable deviation of our results from the experimental data [851for Z = 22, 24, 26. A smoothness analysis of the Z-dependence reveals inexplicable irregularities in the experimental transition energies just for this region. Nevertheless, the remaining theoretical, and compiled results of Edlén are in best agreement with the experimental data just here. For Z > 40 there are three sets of theoretical results: those of the multiconfigurational Hartree— Fock—Dirac method [86], the expansion in lIZ and aZ [108], and the present calculation. The discrepancies between these results rapidly increase with increasing Z. Thus, for Z = 70 the present results and those of ref. [86] for the ~ 2—2p112 transition differ by 8000 cm’, and by 16 000 cm for Z = 92. For the 2p112—2p312 splitting the difference exceeds 70 000 cm~(Z = 92). -

L.N. Ivanov et al., Modern trends in the spectroscopy of multicharged ions

343

Table 5 Comparison of different calculations with experiment [85]for the energies of the 2s

1 12—2p112 transition and of the

2p,,2—2p,,2 splitting (cm’)

Z

Present work

[86]

[108]

[103]

[92]

[85](exp.)

289970 323392 357286 391699 426680 462277 573251

290057 323530 357505 392034 427179 462994 575137

290023 323468 357359 392003 426985

E(2p112—2p212) 41034 62343 91158 129027 177757 239317 523685

40844 62145 90915 128768 177495 239078 523712

40884 62275 91071 128722 177537

E(2s,,2—2p112) 20 22 24 26 28 30 36 41 59 69 79 92

291610 324470 358650 392960 427890 463450 574550 672100 1094670 1389200 1740100 2263600

291226 324759 358782 393349 428432 464248 576276 675717 1099427 1396756 1751903 2279524

289786 323209 357124 391582 426635

20 22 24 26 28 30 36 41 59 69 79 92

40860 62140 90900 128740 177400 238920 523400 914650 4414000 8825000 16323100 33621800

40706 61965 90686 128482 177143 238648 522969 914259 4416072 8832958 16345446 33692315

40797 62093 90863 128722 177462 239062 523827 915728 4423638 8849877 16382500 33800872

574380

523315

We have tried to analyse the source of these discrepancies. For low Z, the main difference is apparently connected with the way of including the electron correlations. The correlation contribution depends weakly on Z, and does not exceed 6000 cm’ for all Z, i.e., the way of accounting for radiative and nuclear size effects becomes a decisive factor for Z > 40. The contribution of the radiative shifts is represented in table 6. The subtraction of these corrections from the final results increases the differences between the present results and those of ref. [86] still more. Unfortunately, in ref. [861the method of taking into account the form of the nuclear charge distribution is not described. Inclusion of finite nuclear size in the calculation makes a negligibly small contribution for Z 20, while for Z = 70 it considerably compensates the contribution of the vacuum polarization, and for Z = 92 it approximately equals the total radiative shift (see table 7). It is known that the finite size of the nucleus provides the stability of the atom up to Z = i73. Special attention should be paid to the role of the nuclear size for Z = 92: variation of the nuclear byofseveral per cent leadscalculations to a changeare in the transition energies by 1 theradius results the corresponding presented in table 8. The dozens of thousands of cm15 levels are shown in table 9. Some experimental results for intermediate Z final results for the lower can be found in refs. [28, 87, 110]. A smoothness analysis of the experimental 2s—31 and 2s—4l transition energies shows that their inconsistencies amount to a few thousands of cm This can be seen from fig. 4, where the deviations of the experimental values from our theoretical results are shown. The deviation of the theoretical values from the exact ones is supposed to have a smooth Z-dependence. - ~.

344

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

8 theor

8

exp, cm 1

1500

x •

X

.~(

1000 .

.

a 500

0

P 4

9 4

A

26

28

30

32

.

~

34

36

z

0

—500

0

—1000

Fig. 3. The deviations of the theoretical transition energies from the experimental ones. I present calculation; x multiconfigurational Hartree—Fock—Dirac [86];0 expansion in liZ and aZ [108], A l/Z-expansion [103],0 compiled data of Edlén [92]. All experimental results are from ref. [851.

Table 6 2n!j energy states of Li-like ions counted from the is2 core. Contributions of QED effects (in cm’) to the ls Contributions of QED effects to the 2s—2p energy transitions are shown separately. VP: vacuum polarization shift, SE: self-energy shift, LS: Lamb shift (LS =VP + SE) 2s

2p

2p1,2

1,2

Z

—VP

SE

—VP

20 30 41 59 69 79 92

113 607 2278 11626 25020 51306 126647

1510 6720 20880 76540 140600 245800 486400

0.5 —39 7 —145 53 —270 635 1470 1989 6440 5708 19700 21128 66300

SE

3,2

—VP

SE

0.1 1 8 64 159 365 921

280 1250 7180 15100 29200 60900

35

—LS(2s12—2p12)

—LS(2s,2—2p312)

present work [86]

present work [86]

1436 6265 18925 64079 111129 180502 314581

1558 6572 19553 69130 119490 192929 334468

1362 5834 17360 57798 100639 165659 299774

1517 6386 19112 69386 122850 204666 374691

L.N. lvanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

345

Table 7 Contributions of the nuclear size effect to the lowenergy transitions, and effective nuclear radius (in ~i)

Z

2s

112—2p,,2

2s,,2—2p3,2

R

20 30 41 59 69

—14.9 —114.6 —653 —6490 —20340

—15.0 —115.1 —663 —6730 —21420 —65900 —284200

3.26 3.73 4.14 4.68 4.93 5.15 5.42

79

—61300

92

—253500

10i3

cm

89~energy levels (cm’) calculated with Tabledifferent 8 nuclear radii (in i0~cm). All excited Li-like levels U are counted from the 2s-level. The self-energy part of the Lamb shift is not included NR nlj ‘

4.88

5.15

5.42

5.69

5.97

2s 1,2 2p112 2p312 3s,,2 3p 1,2 3p,,2 3d3,2 3d,,2 ~~lI2 4P 1,2 4p312 ~3,2 ~5,2

—265257 + 3 272098+1 363528+2 151828+3 152550+3 162605+3 162721+3 165275+3 203 184+3 226422+3 207665+3 207716+3 208803+3 208806+3 209323+3

—265236 + 3 270191+1 363315+2 151813+3 152529+3 162584+3 162700+3 165254+3 203 166+3 226401+3 207644+3 207695 +3 208781+3 208785+3 209302+3

—265 214 + 3 268310+1 363104+2 151798+3 152509+3 162562+3 162679+3 165232+3 203 148+3 226380+3 207623+3 207674+ 3 208760+3 208763+3 209280+3

—265 192 + 3 266316+1 362881+2 151782+3 152488+3 162540+3 162656+3 165211+3 203 128+3 226358+3 207601+3 207651+3 208738+3 208741+3 209258+3

—265 170+ 3 264312+1 362656+2 151767+3 152466+3 162518+3 162634+3 165188+3 203 109+3 226336+3 207578+3 207629+3 208716+3 208719+3 209236+3

3.8. Hyperfine splitting of levels The hyperfine interaction in atoms is sensitive to the nuclear form factors and this circumstance is widely used for their experimental determination by optical and radiospectroscopic methods. The hyperfine structure may be investigated both to test fundamental physical principles and to solve specific

spectroscopic problems of an applied character. For example, Sunyaev and Churazov [111]proposed observing radiation from a number of supernova remnants and the central parts of clusters in the millimeter wavelength range. Here one is concerned with a hot (up to 10 keV), low-density astrophysical plasma. Transitions between sublevels57Fe of the of hydrogen-like nitrogen (A = (A =hyperfine 3.06 mm) structure were pointed out as promising candidates 5.64 mm) and the lithium-like iron isotope [111]. Here we present the results of calculations of the electric quadrupole constant B, and the magnetic

L.N. lvanov et a!., Modern trends in the spectroscopy of multicharged ions

346

Table 9 2n!j states of Li-like ions Energy levels of the is

Nz ~

n!j

2s,,, 2p,,. 2p,,. 3s,,, 3p,,. 3p 3~. 3d,,, 3d,~, 4p, ~~I2 1, 4p,., 4d,,, 4d,. 4f,, 4f,,,

20

~

3p,, 3p, 2 3d,,1 3d,,2 4s,,,

—114983+2 324737+0 386877+0 646803+1 655760+1

535221+1 538368+1 538745+1 706401+1 709704+1 710213+1 711526+1 711685+1 711751+1 711831+1

594768+1 598091+1 598560+1 785546+1 789039+1 789670+1 791055+1 791252+1 791 323+1 791422+1

657597+1 661103+1 661679+1 869031+1 872716+1 873490+1 874952+1 875194+1 875269+1 875391+1

723716+1 727409+1 728110+1 956860+1 960741+1 961682+1 963222+1 963516+1 963596+1 963744+1

29

30

31

32

—208673+2 —224402+2 445589+0 652064+0 463452+0 702435+0 117158+2 125972+2 08391+2 127254+2 119002+2 119477+2 119669+2 157548+2 158055+2 158312+2 158511+2 158592+2 158602+2 158643+2

4p,~,

4p,,, 4d,~. ~5/2

4f,,2 4f7,, n!j ~‘~N 2s1, 2p,,, 2p,,, 3s1, 3p, 3p,, 3d 3~, 3d,,2

4s,, 4~112

4p,~, ~3/2

4d,,, 4f,,, 4f71

23

—103909+2 308010+0 358666+0 584780+1 593271+1

~‘\\

2s1,, 2p,, 2p,,,

22

—934110+1 291611+0 332472+0 525982+1 534013+1

Nz n!j

21

38

127961+2 128454+2 128676+2 169412+2 169940+2 170238+2 170443+2 170537+2 170548+2 170595+2 39

—240727+2 481489+0 756730+0 135121+2 136453+2 137268+2 137778+2 138034+2 181727+2 182276+2 182619+2 182832+2 182939+2 182951+2 183005+2 40

—372012+2 613361+0 127348+1 208769+2 210465+2 212422+2

—393244+2 632957+0 137090+1 220692+2 222442+2 224631+2

—415113+2 652898+0 147555+1 232978+2 234783+2 237223+2

213048+2 213659+2 280844+2 281542+2 282366+2 282627+2 282885+2 282900+2 283028+2

225272+2 225954+2 296887+2 297607+2 298528+2 298796+2 299084+2 299099+2 299243+2

237880+2 238639+2 313417+2 314159+2 315186+2 315460+2 315780+2 315796+2 315956+2

24

—126632+2 —138852+2 341725+0 358647+0 417226+0 449542+0 712052+1 780486+1 721483+1 790389+1

26

27

28

—151650+2 375768+0 484348+0 852165+1 862545+1

—165028+2 392959+0 521700+0 927091+1 937952+1

—178989+2 410361+0 561975+0 100529+2 101663+2

—193536+2 427892+0 605334+0 108678+2 109861+2

793078+1 796950+1 797794+1 104898+2 105306+2 105419+2 105580+2 105616+2 105625+2 105643+2

865757+1 869808+1 870816+1 114547+2 114975+2 115110+2 115279+2 115321+2 115 330+2 115352+2

941761+1 945989+1 947186+1 124634+2 125081+2 125242+2 125418+2 125468+2 125478+ 2 125503+2

102112+2 102553+2 102694+2 135161+2 135629+2 135818+2 136001+2 136061+2 136070+2 136100+2

110387+2 110845+2 111010+2 146132+2 146619+2 146840+2 147031+2 147101+2 147 111+2 147146+2

33

34

35

36

37

—257651+2 —275179+2 499710+0 815246+0 518118+0 878295+0 144608+2 154436+2 145990+2 155868+2 146925+2 147451+2 147745+2 194497+2 195066+2 195460+2 195679+2 195803+2 195815+2 195877+2

156935+2 157479+2 157813+2 207724+2 208314+2 208764+2 208990+2 209131+2 209144+2 209215+2

41

42

—437622+2 —460776+2 673096+0 693558+0 158775+1 170794+1 245628+2 258646+2 247489+2 260562+2 250202+2 263572+2 250874+2 251717+2 330434+2 331198+2 332340+2 332621+2 332977+2 332993+2 333170+2

25

264258+2 265192+2 347943+2 348731+2 349997+2 350284+2 350679+2 350695+2 350891+2

—293314+2 536724+0 946198+0 164604+2 166089+2

—312060+2 555531+0 101929+1 175120+2 176657+2

167303+2 167863+2 168243+2 221412+2 222023+2 222534+2 222768+2 222928+2 222941+2 223021+2

178031+2 178608+2 179038+2 235564+2 236197+2 236775+2 237016+2 237197+2 237211+2 237302+2

—331421+2 —351403+2 574551+0 109794+1 593788+0 118250+1 185983+2 197198+2 187573+2 198840+2 189124+2 189717+2 190202+2 250184+2 250838+2 251491+2 251739+2 251943+2 251957+2 252059+2

200586+2 201195+2 201740+2 265275+2 265951+2 266686+2 266940+2 267170+2 267184+2 267299+2

43 —484579+2 714282+0 183657+1 272034+2 274007+2 277337+2

—509039+2 735296+0 197416+! 285796+2 287827+2 291504+2

278039+2 279071+2 365950+2 366760+2 368161+2 368455+2 368891+2 368908+2 369124+2

292220+2 293357+2 384458+2 385292+2 386839+2 387139+2 387619+2 387636+2 387875+2

—534160+2 —559950+2 756593+0 778194+0 212118+1 227818+! 299938+2 314461+2 302026+2 316609+2 306077+2 321062+2 306807+2 308058+2 403473+2 404331+2 406035+2 406341+2 406869+2 406887+2 407149+2

321807+2 323179+2 423001+2 423882+2 425756+2 426067+2 426647+2 426665+2 426953+2

347

L.N. Ivanov et al, Modern trends in the spectroscopy ofmulticharged ions

Table 9 (cont.)

Nz n!j “N~ 2s,,

2 2p,,2 2p312 3s112 3Pi,z ~

47

48

—586414+2 —613560+2 800102+0 822332+0 244569+1 262428+1 329 372 + 2 344 674 + 2 331579+2 346942+2 336 465 + 2 352 292 + 2

49

50

—641394+2 —669923+2 844893+0 867800+0 281 453+1 301 706+1 360371 + 2 376469 + 2 362700+2 378860+2 368550 + 2 385244 + 2

51

52

53

—699159+2 —729106+2 —759770+2 891516+0 915576+0 939993+0 323294+ 1 346236+ 1 370603+1 392976 + 2 409 892 + 2 427 224 + 2 395430+2 412412+2 429810+2 402387 + 2 419980 + 2 438 030 + 2

54

55

—791162+2 —823289+2 964786+0 989952+0 396464+1 423893+1 4-44 977 + 2 463157 + 2 447629+2 465876+2 456546 + 2 475 535 + 2

3d,, 2 3d,,2 4s,,2 4p1,2 4p312 4d,,2 4d,,2 4f512 4f7,2

Nz nlj ~ 2s112 2Pi, 2 2p312 3s,,2 ~ ~

3d,,2 3d,,2 4s112 4p 1,2 ~ ~3,2 ~5~2

4f,,2 4f712

337 224 + 2 338727+2 443046+2 443951+ 2 446007+2 446325+2 446959+2 446978+2 447293 + 2

353065 + 2 354707+2 463614 +2 464544+2 466795 +2 467119+2 467813+2 467832+2 468 176 + 2

369336 + 2 371 128+2 484712 +2 485666+2 488 127+2 488456+2 489213+2 489233+2 489608 + 2

386044 + 2 387995+2 506344 +2 507324 +2 510009+2 510344+2 511169+2 511 189+2 511 597 + 2

403 199 + 2 405320+2 528522+2 529528+ 2 532453+2 532794+2 533690+2 533711+2 534 154 + 2

420 805 + 2 423 106+2 551 248+2 552280 +2 555462+2 555808+2 556781+2 556802+2 557 282 + 2

438868 + 2 441 361+2 574529+2 575587+2 579043 +2 579395+2 580449+2 580470+2 580991 + 2

457396 + 2 460093+2 598371+ 2 599456 +2 603204 +2 603561+2 604702+2 604723+2 605286 + 2

476396 + 2 479311+2 622782+2 623894+2 627954+2 628316+2 629548+2 629570+2 630 178 + 2

56

57

58

59

60

61

62

63

64

—856159+2 101552+1 452967+1 481 769+2 484557 + 2 495006+2 495878+2 499022+2 647770+2 648909+2 653301+2 653668+ 2 654997+2 655020+2 655675+2

NZ iilj ‘~N~ 65 2s,,2 2p,,2 2p,,2 3s112 3p 1,2 ~

3d,,2 3d,,2 4s,,2 4p1,2

~i3,2 4d,,2 4d,,2 4f512 4f7,2

—889783+2 —924170+2 104148+1 106787+1 483765+ 1 516370+ 1 500819+2 520315 +2 503667 + 2 523 242 + 2 514966+2 535425+2 515850+2 536319+2 519236 +2 539963 +2 673342+2 699506+2 674509 +2 700701+2 679253+2 705820+2 679625 +2 706 197+2 681057 +2 707738+2 681080+2 707761+2 681785+2 708520+2 66

—118711+3 —122799+3 126525+1 129542+1 802837+1 853286+1 669817+2 693129+2 673265+2 696655+2 693404+2 718210+2 694364+2 719178+2 700249+2 725453+2 899992+2 931227+2 901394+2 909847+2 910253+2 912745+2 912771+2 913990+2

932660+2 941706+2 942116+2 944773 +2 944799+2 946099+2

67

—959329+2 —995271+2 109467+1 112193+1 550866+1 587345+1 540261+2 560666+ 2 543 260 + 2 563 737 + 2 556391+2 577874+2 557296+2 578789+2 561 212+2 582992+2 726271+2 753646+2 727495 +2 754898+2 733011+2 760837+2 733393 +2 761 223+2 735049 +2 763001+ 2 735073+2 763025+2 735887+2 763898+2 68

—126974+3 —131236+3

69

—103201 + 1 114963+1 625 897+1 581 536+1 584680 + 2 599885 +2 600809+2 605 315+2 781 639+2 782921+2 789307+2 789697 +2 791 603+2 791628+2 792563+2 70

—106955+3 —110790+3 117780+1 120645+1 666620+1 709614+1 602878+2 624701+2 606 097 + 2 627995 + 2 622432+2 645 527+2 623366+2 646470+2 628 190+2 651 630+2 810260+2 839520+2 811 572+2 840861+2 818431+2 848222+2 818826+2 848620+2 820867+2 850804+2 820892+2 850830+2 821894+2 851900+2 71

—135588+3 —140031+3 —144566+3

132613+1

135738+1

138919+1

142158+1

145453+1

906451+1 716953+2 720560+2 743610+2 744585+2 751271+2 963 143+2

962451+1 741301+2 744988+2

102142+2 766180+2

108348+2 791602+2

114878+2 817577+2

769950+2

795455+2

821514+2

769618+2 770600+2 777715+2 995751+2

796246+2 797234+2 804801+2 102906+3

823508+2 824503+2 832542+2

964607+2 974279+2 974692+2 977524+2 977550+2 978933+2

997247+2 100758+3 100800+3 101 101+3 101104+3 101251+3

103059+3 104162+3 104204+3 104525 +3 104528+3 104684+3

72

—114709+3 123560+1 754983+ 1

647011+ 2 650 381 + 2 669 181+2 670132+2 675646+2 869427+2 870798 +2 878690+2 879092+2 881 426+2 881452+2 882594+2 73

—149196+3 —153921+3 152224+1 128966+2 871228+2 875337+2 909251+2 910261+2 919855+2

106309+3

851420+2 852420+2 860954+2 109786+3

148809+1 121745+2 844115+2 848138+2 879996+2 881001+2 890053+2 113336+3

106465+3 107642+3 107684+3 108025 +3 108028+3 108194+3

109946+3 111199+3 111241+3 iii 603+3 111606+3 111782+3

113499+3 114835+3 114877+3 115261+ 3 115264+3 115450+3

117129+3 118550+3 118593+3 119000+3 119003+3 119200+3

116963+3

348

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions Table 9 (coOt.)

nlj

‘N.~

2s,, 2p 2 1,2 2p,,, 3s,,~

3p,,2 3p,,2 3d,,2

3d,,2 4s,,2 4p 12 4p,,2 4d,,2 4d,,, 4f5,2 4f7,2

74

75

—158744+3 —163666+3 155701+1 159238+1 136556+2 144530+2 898927+2 927224+2 903124+2 931511+2

“\.~

2s1,2 2Pi,,

77

78

—168690+3 162838+1 152907+2 956133+2 960510+2

—173817+3 166499+1 161703+2 985666+2 990135+2

—179049+3 170222+1 170939+2 101584+2 102040+3

79

80

81

82

—184388+3 —189837+3 174006+1 177851+1 180632+2 190804+2 104666+2 107816+3 105132+3 108290+3

—195398+3 181741+1 201474+2 01033+3 111517+3

—201074+3 185681+1 212666+2 114320+3 114814+3

939203+2 940217+2 950378+2 120667+3 120836+3 122348+3 122391+3 122822+3 122825+3 123034+3

969869+2 970886+2 981638+2 124450+3 124623+3 126230+3 126273+3 126729+3 126732+3 126953+3

100127+3 100229+3 101366+3 128314+3 128490+3 130197+3 130240+3 130723+3 130726+3 130959+3

103341+3 103443+3 104645+3 132259+3 132439+3 134252+3 134295+3 134805+3 134808+3 135055+3

106633+3 106735+3 108004+3 136289+3 136473+3 138396+3 138439+3 138978+3 138981+3 139241+3

110003+3 110106+3 111445+3 140405+3 140592+3 142631+3 142675+3 143243+3 143246+3 143521+3

113455+3 113557+3 114969+3 144609+3 144799+3 146961+3 147005+3 147604+3 147607+3 147896+3

116989+3 117092+3 118580+3 148902+3 149096+3 151386+3 151430+3 152062+3 152065+3 152369+3

120610+3 120712+3 122280+3 153288+3 153485+3 155909+3 155953+3 156619+3 156622+3 156942+3

83

84

85

86

87

88

89

90

91

NZ nh

76

—206866+3 189667+1 —212778+3 193692+1

—218811+3 197752+1

—224970+3 201838+1

—231257+3 205945+1

—237676+3 210061+1 —244228+3 214175+1

—250919+3 218273+1

—257751+3 222340+1

2p,, 2

3s1,2 3p,,2 3P,,2

224401+2 07679+3 118183+3 124318+3

236703+2 121112+3 121624+3 128116+3

249599+2 124620+3 125142+3 132007+3

263114+2 128204+3 128737+3 135994+3

277276+2 131869+3 132410+3 140079+3

292114+2 135614+3 136165+3 144266+3

307658+2 139443+3 140003+3 148558+3

323940+2 143358+3 143927+3 152958+3

340995+2 147362+3 147940+3 157469+3

3d,, 2

124420+3

128218+3

132108+3

136095+3

140181+3

3d,,,

126070+3

129954+3

133934+3

138013+3

~ 4P1,2

157767+3 157968+3

162343+3 162547+3

167017+3 167225+3

171792+3 172004+3

142195+3 176671+3 176886+3

144367+3 146481+3 181657+3 181875+3

148658+3 150876+3 186752+3 186973+3

153057+3 155383+3 191959+3 192184+3

157567+3 160006+3 197281+3 197509+3

160533+3 160577+3 161278+3 161281+3 161618+3

165261+3 165305+3 166042+3 166045+3 166399+3

170094+3 170138+3 170913+3 170917+3 171289+3

175036+3 175080+3 175895 +3 175898+3 176289+3

180090+3 180133+3 180990+3 180993+3 181402+3

185258+3 185301+3 186200+3 186203+3 186633+3

190544+3 190587+3 191 531+3 191534+3 191984+3

195952+3 195995+3 196984+3 196986+3 974459+3

201484+3 201527+3 202564+3 202568+3 203062+3

93

94

98

99

4p,, 2 ~4/2

4d,,2 4f,,2 4f,,2

NZ nlj ~ 2s,,2 2p1,2 2p,,2 3s1,2 3p1,2

3p,,2 3d,,, 3d,,, ~ 4P1,2

4P,,2 ~3I2

92 —264729+3 226357+1 358856+2 151456+3 152043+3 162095+3 162193+3 164747+3 202723+3 202953+3 207145+3 207188+3

—271856+3 —279136+3 230304+1 234136+1 377562+2 397149+2 155645+3 159929+3 156239+3 160531+3 166840+3 171707+3 166937+3 171804+3 169612+3 174603+3 208286+3 208520+3 213975+3 214211+3 212938+3 218868+3 212981+3 218910+3

95

96

97

—286574+3 237863+1 417663+2 164314+3 164923+3 176702+3 176798+3 179726+3 219794+3 220032+3 224938+3 224980+3

—294175+3 241437+1 439143+2 168801+3 169417+3 181828+3 181923+3 184985+3 225746+3 225986+3 231154+3 231195+3

—301943+3 244809+1 461637+2 173394+3 174015+3 187091+3 187185+3 190386+3 231837+3 232078+3 237520+3 237561+3

226227+3 226230+3 226821+3

232500+3 232503+3 233120+3

238925+3 238928+3 239572+3

—309884+3 —318002+3 247950+1 250801+2 485192+2 509859+2 178097+3 182913+3 178723+3 183543+3 192495+3 198045+3 192587+3 198136+3 195932+3 201629+3 238069+3 238312+3 244449+3 244692+3 244041+3 250722+3 244082+3 250762+3

4d,, 2 4f5,2 4f7,2

208274+3 208278+3 208795+3

214119+3 214122+3 214663+3

220102+3 220105+3 220670+3

245507+3 245510+3 246183+3

252251+3 252254+3 252956+3

349

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions 1. nhj.3p1/2.

S Etheor_EexpIcm~

6000

S

4000 •

2000 I

I 24

I

20

21

22

23

I 25

•

6000

I

I

26

27

28

29

Z

n1j~3p3/2.

• 4000 •

2000.

•

S

I 21

20

I

22

23

24

I

I

25

26

I

27

I

29

28

8000.1

6000 n].j—4d5/2

6000.1

nljs3d5/2

4000

•

•

21

22

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40001

•

2OOO-~

2000

•

I

I

20

21

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22

23

I

I

24

I•

25

I

26

I

27

I

20

23

24

25

~6

a

Z

27

I

28

8

6000

n].j~4d3/2

S

6000

•

4000

nuj~3d3/2

.

a

•

4000

2000 •

20

2000

21

22

23

24

25

26

27

Z

S

6000

5

20

I

I

22

23

21

i

n].j~4p1/2

i

24

25

26

27

28

Z 4000

•

•

• I

2000 •

4000.

•

20

n1j~3e1/2

2000.

6000

0.

4000

—2000.

•

21

22

23

24

25

•

20

-4000

Z

•

21

22

23

24

25

2000.

—6000.

27

nhj.4p3/2

2000

S

26

•

•

26

27

Z

nlj—4a1/2

S

-I

•

_120001 20

21

22

23

24

25

26

27

28

0~ Z

20

21

22

23

24

~5

2~6 2,7

Z

Fig. 4. Deviations E(2s 1,2—n!j),heo, calculation.

—

E(2s~,2—n!j)~~~ for different experimental data [28, 87, 110]. All theoretical data are

from

the

present

350

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

dipole constant A with inclusion of nuclear finiteness and the Uehling potential. Analogous calculations of the constant A for ns states of hydrogen-, lithium- and sodium-like ions with nuclear charge Z < 53 were made in ref. [112]. In ref. [112] another model of the charge distribution in the nucleus was accepted and another method of numerical calculation for the Uehling potential was used. Experimental values of the nuclear radius were used. In the phenomenological theory the influence of the inner nuclear structure on the electron spectrum is described by two constants: the nuclear quadrupole moment Q and the gyromagnetic ratio g1. They are related to the electric quadrupole constant B and the magnetic dipole constant A. The appropriate contributions to the energy are W0=[zl+C(C+l)]B,

W,,~=0.5AC,

(48)

where I(J’1~I1)’

~=_~(4K2_1)

C=F(F+1)—J(J+1)—I(i+1).

(49)

In eqs. (48), (49) I is the nuclear spin, F is the total moment of the system, J is the total moment of the electron. The constants of the hyperfine splitting are expressed in terms of radial integrals. The relation between the constants A, B with the nuclear parameters Q, g is well known for the case of a point nucleus [113]. A straightforward refinement of the model is smearing of the electric and magnetic moments over the bulk of the nucleus. In this case 2Kg

A=

3.489Z 2 4K

1

1

(RA)2,

(50)

3Q B = (4K2 1)1(1 1) (RA)_ 0.00588Z —

3

,

(51)

where the radial integrals

(RA)2

=

J

2F(r)G(r)U(1 1r2, R), dr r (52)

(RA)

3=

J

2[F2(r)

dr r

+

G2(r)]U(1 1r3, R),

are calculated in Coulomb units (3.57 X 1020Z2 m2 and 6.174 x 1030Z3 m3 for the integrals of the appropriate dimension); F, G are the radial parts of the two components of the Dirac function for the electron moving in the potential (18). If Q is given in barns, the result obtained will be in cm~.For the calculation of the smeared potentials of the hyperfine interaction the following differential equations

351

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

were solved: U’(1Ir~,R) = —nY(r, R)/r’~’.

(53)

Note that the distribution functions of the nuclear quadrupole and magnetic dipole moments may not coincide with the charge distribution function (8). At least, the distribution radius may differ. This allows one to introduce an additional independent parameter to describe the electron spectrum. In ref. [112] models of the charge distribution and of the surface distribution of the magnetic moment in the nucleus are used. As the contribution of the finite size effect of the nucleus to the hyperfine splitting increases rapidly with Z, for large Z the model of a point nucleus becomes, in general, inapplicable. In fact, for R = 0, U(11r2,0)

=

hr2

,

U(11r3, 0) = hr3 .

(54)

At the same time the Dirac wave functions F(r),G(r)—r~1, asr—~0,

(55)

Table 10 Constants of the hyperfine electron—nuclear interaction A=Z’g,Acm~, B=J(~~ 1)Bcm~

elf \\ 2s 3s 4s 2p1,2 3p1,2 4p 1,2 2p,,2

A A

A A A A A

3p,,2

B A B

‘4p3,2

A

3d,,2

A

4d3,2

A

3d,,2

A

~,,2

A

4f5,5

A

4f,,2

A

B B B B

B B

20

30

41

59

69

79

92

95 —03 28 —03 16 —03 28 —03 83 —04 35 —04 54 —04 12 —04 16 —04 36 —05 68 —05 14 —05 92 —05 56 —06 39 —05 15 —06 39 —05 24 —06 17 —05 62 —07 12 —05 21 —07 64 -06

106 —02 31 —03 13 —03 32 —03 96 —04 41 —04 60 —04 14 —04 18 —04 41 —05 76 —05 18 —05 10 —04 80 —06 44 —05 33 —06 44 —05 34 -06 19 —05 14 —06 13 —05 46 —07 72 -06

119 —02 35 —03 15 —03 37 —03 ii —03 47 —04 63 —04 15 —04 19 —04 45 —05 81 —05 19 —05 ii —04 88 —06 47 —05 37 —06 46 —05 36 —06 20 —05 15 —06 14 —05 50 —07 77 -06

151 —02 45 —03 19 —03 48 —03 14 —03 61 —04 68 —04 16 —04 21 —04 51 —05 88 —05 22 —05 12 —04 10 —05 52 —05 44 -06 49 —05 39 -06 21 —05 17 —06 15 —05 59 —07 81 -06

180 —02 53 —03 22 —03 58 —03 18 —03 74 —04 70 —04 17 —04 22 —04 54 —05 92 —05 23 —05 13 —04 11 —05 54 —05 47 —06 50 —05 40 —06 22 —05 17 —06 15 —05 61 —07 83 -06

220 —02 66 —03 28 —03 73 —03 22 —03 94 —04 73 —04 18 —04 23 —04 58 —05 96 —05 25 —05 13 —04 12 —05 57 —05 52 —06 52 —05 41 —06 22 —05 18 —06 16 —05 63 -07 84 -06

318 —02 94 —03 39 —03 108 —02 34 —03 14 —03 76 —04 20 —04 24 —04 65 —05 10 —04 28 —05 14 —04 13 —05 61 —05 59 —06 53 —05 43 -06 23 —05 19 —06 16 —06 66 —07 86 -06

352

L. N. Ivanov et a!., Modern trends in the spectroscopy of snu!ticharged ions Table 11 Theoretical results for the magnetic dipole constant. Bulk distribution of the nuclear magnetic dipole (present work), and surface distribution [112]

Z

A

20

9476 1002 1034 1099 1135 1214 1381

25 28 34 37 43 53

(present) —03 —02 —02 —02 —02 —02 —02

A [112] 9290 9872 1020 1087 1123 1203 1370

—03 —03 —02 —02 —02 —02 —02

and from which it follows that the integrals (RA), and (RA)_3 diverge for (aZ)2> (az)2> K2 1, respectively. For example, for the states j = 1/2 the expression for the constant A diverges for Z> 119, for the states j = 3/2 the expression for B becomes senseless for Z > 237. This 1(2

—

~,

—

may be important for the study of superheavy systems. In tables 10 and 11 the computational results for the constants of the hyperfine splitting for the lowest excited states of Li-like ions are summarized. The results of our calculation are compared in table 11 with the results of ref. [112] for the constant which describes the magnetic dipole splitting of the ground state level 1s22s of some lithium-like ions. A systematic 1% difference is observed, which cannot be explained by the uncertainty of the nuclear radius. In the next section calculations of the derivatives dA IdR (table 12) are given, from which it is seen that variations of the nuclear size within reasonable limits lead to variations of the value of A over about three orders of magnitude. Remember

that the main difference between the two works lies in the modelling of the magnetic moment distribution in the nucleus, namely, a bulk distribution is assumed in the present paper, and a surface distribution in ref. [112]. 3.9. Isotope shift The total isotope shift is the sum of the bulk and the mass shift. The latter in its turn is subdivided into the reduced mass correction,

-

m-M (1- (~2)E,

(56)

and the relativistic recoil part. (E is the bound state energy, m and M are the electron and nuclear masses). According to the evaluation of ref. [81] the recoil part does not exceed 103ELs (ELS is the total Lamb shift). Accounting for the recoil energy is not crucial in spectroscopic calculations, but is

important in electrodynamic problems. As a consequence, methods to calculate it are now extensively developed (see, for example, ref. [105]and references therein). For analysis of experimental spectroscopic data the derivatives with respect to the nuclear radius of the various physical characteristics are more convenient quantities than the total bulk corrections. The

method described below allows one to avoid a loss of accuracy when subtracting from one another two relatively large total bulk corrections for different R-values. The calculation of the derivatives is facilitated by using a smooth analytic density distribution function p~(r,R). For any specific electron—

353

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

nucleus interaction (Coulomb, Uehling, hyperfine and so on) the corresponding “smeared” potential is represented by W(r, R) = W(r)Y(r, R) +

J

dr’ r’2W(r’)p(r’, R),

(57)

where W(r) is a known function describing the same interaction with a point nucleus, and Y(r, R) was defined above, eq. (12). The dependence of the corresponding atomic characteristics is described by the matrix elements dW(r, R)/dR. We consider the latter as a function of r, which at the given R satisfies the following system of two ordinary differential equations: [~W(r, R)IdR]~= W’(r)r[ÔY(r, R)19R], (58)

[9Y(r, R) /9R]

=

(2yr2

—

3)p~(r,R) IR.

These equations must be included into the complete system of differential equations described above. The derivatives of the “smeared” Coulomb and Uehling polarization interaction and of the hyperfine interaction constants A, B have been calculated at the specific value of R defined by eq. (10). The results for all nlj states of Li-like ions with the minimal possible values of j are given in tables 12 and 13.

Table 12 Derivatives of some one-electron characteristics with respect to the nuclear radius: d( lvi) IdR = Z’DV (cm ‘1cm); d(~U[)IdR = Z’DU (cm~Icm),dAIdR = Z4g,DA (cm’Icm). V: electric electron—nucleus interaction, U: Uehling potential, A: magnetic dipole constant; cm’ is the energy unit, cm the length unit

\z 20

30

41

59

69

79

92

2s 1,2

DV DU DA

10 +11 16 +06 16 +06

21 +11 15 +06 20 +06

43 +11 15 +06 26 +06

124 +12 21 +06 46 +06

225 +12 27 +06 67 +06

420 +12 39 +06 105 +07

971 +12 67 +06 202 +07

3s,,5

DV DU DA

29+10 46 +05 45 +05

62+10 43 +05 57 +05

13+11 45 +05 76 +05

37+11 62 +05 14 +06

67+11 82 +05 20 +06

125+12 12 +06 31 +06

298 +12 20 +06 60 +06

4s1,2

DV DU DA

12 +10 19 +05 19 +05

26 +10 18 +05 24 +05

53 +10 32 +05

15 +11 26 +05 57 +05

28 +11 34 +05 84 +05

52. +11 48 +05 13 +05

124 +12 83 +05 25 +05

2p,,2

DV DU DA

32 +08 52 +03 57 +03

17 +09 12 +04 17 +04

68 +09 23 +04 44 +04

45 +10 75 +04 17 +05

12 +11 14 +05 37 +05

31 +11 28 +05 80 +05

111 +12 75 +05 23 +06

3p1,2

DV

11

+08 19 +03 20 +03

59 +08 41 +03 59 +03

24 +09 86 +03 15 +04

16 +10 27 +04 61 +04

42 +10 51 +04 13 +05

11 +11 10 +05 28 +05

39 +11 26 +05 82 +05

50 +07 88 +02 87 +02

26 +08 18 +03 26 +03

11 +09 38 +03 69 +03

70 +09 12 +04 27 +04

18 +10 22 +04 59 +04

48 +10 44 +04 12 +05

17 +11 11 +05 36 +05

DU DA 4p1,2

DV

DU DA

19 +05

354

L. N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions Table 13 Derivatives of the magnetic dipole constant B with respect to the nuclear radius, in cm’Icm; 4QDBI[I(21— 1)] dB/dR= —Z n!J\ 2p,~.

DB DB DB

59

69

79

92

17 +02 57 +01 21 +01

27 +02 95 +01 38 +01

40 +02 15 +02 60 +01

71 +02 27 +02 12 +02

These j values are 3/2 for oB/oR and 1/2 for the remaining operators. As for the states with larger j, their characteristics are appreciably less sensitive to the nuclear size; anyway the corresponding effects are not observable in modern experiments. The main power law Z-dependence is pointed out for all derivatives given in tables 12 and 13. The remaining Z-dependence is connected with the relativistic and nuclear size effects in the one-electron wave functions. As mentioned above, these effects cannot be accounted for by perturbation theory, regarding the total difference between the Coulomb and smeared Coulomb potentials as a perturbation. It is worth noticing the strange Z-dependence of the derivatives of the characteristics of the r

1 / 2 states;

for the transuranium elements the volume isotope shifts of the ~1/2 and p1~2levels are of the same order. For Z >60 the level shifts corresponding to 10_I fm do exceed the total hyperfine correction for all states.

4. Isoelectronic sequence of helium The He-like ion spectra are much more abundant than those of the H-like one-electron systems. The

transition energies and the lifetimes vary by large amounts, as new autoionization decay channels arise. These features open new possibilities for the experimental observation of some specific QED effects [1], such as nuclear shape effects and cooperative electron—nuclear processes (see section 5). However, a correct account of the interelectron interaction is needed for the correct interpretation of such an experiment. The calculation of the relativistic part of the correlation energy is the first nontrivial problem appearing in connection herewith. For few-electron systems, the multiconfiguration Dirac— Fock method obviously meets the modern demands in this sense [96]. The state of the art in taking QED and nuclear effects into account is the same as in the case of lithium-like ions. The interdependency of progress in theoretical and experimental methods for the investigation of the much heavier He-like ions, with the increasing role of relativistic and radiative (including two-electron relativistic and radiative) effects, is treated in ref. [10]. The specific role of He-like ions for the formation of spectra of plasmas of different origin was outlined in the introduction. In this respect, systematic investigations aiming at a classification of lines and a determination of cross sections of radiative, collision and autoionization processes are of great interest. These investigations comprise an important trend of plasma physics [19, 91]. The techniques of experimentally determining the kinetic constants of elementary processes with the use of theoretical models have been described in these references. This part of our review deals mainly with the problem of precision measurements of the wavelengths in He-like ion spectra. For Z 10—23 these spectra have been investigated in ref. [116], using a plasma created by a Nd laser. A laser beam with energy 50 J was focused on a flat target, the spot diameter

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

355

1.8 ns, the resulting flux density on the spot —5 x 1014 W/cm2. The observed spectrogram in the range 0.5—19 A was obtained using a spectrograph with a convex mica being —~80~m,the pulse duration

crystal. The high brightness of the laser flare made it possible to obtain the normal blackness of the X-ray film after 5—10 laser flashes even with a defocused spectrograph. The 1s2—lsnp transition lines emitted by He-like aluminium and magnesium have been used as reference lines. They were observed in two orders of the mica crystal reflection spectrum. The reference wavelengths were measured by Flemberg [114],who used a spark spectrum for this purpose. The lines under investigation are in the range of 2.3—10 A. They were registered in high-order spectra. This allowed a measurement accuracy as high as 5 x i0~A in the best cases. Here we refer only to the accuracy of the determination of the wavelengths by the dispersion curve. This accuracy has been evaluated by the reproducibility of reference lines, not included in the least-squares fitting procedure. Some of the 1s231—1s21’31” satellites are distant from the resonance line, which decreases to some extent the influence of the adjacent satellites on the visible position of the line one is searching for. The contribution of the remaining satellites to the line shape has not been evaluated. Another source of

inaccuracy, the Doppler shift, can play an appreciable role in the experimental scheme under discussion, as the angle between the laser beams and the observation directions is 60°(not 90°).It is worth noting that the radiation zone for the aluminium and magnesium ions spreads over a larger region than that for the heavier ions. This can lead to a different Doppler shift for the various ions, as

the expansion velocity of the plasma depends on the distance from the target. One more correction connected with the now accepted relation between kX units and A must be introduced, if the wavelengths are obtained with the help of Flemberg’s data. Some experimental groups have been studying He-like ions for a wide interval of Z which appear in

a LIVS micropinch plasma [52,116]; but, as noted above, the only measurements which can provide adequate accuracy are those that account properly for the motion of the “hot point” perpendicular to the discharge axis [57]. The installation used in ref. [1151for the study of the He-like ion spectra with Z = 16—39 has rather typical parameters: condenser battery capacity 16 p~F,inductance 30 nH, voltage 10—15 kV, maximal discharge current 150 kA. The basic elements of the system for X-ray spectra registration is a focusing crystal spectrograph of Johann type (quartz, R 500—800 mm, 2d = 2.36 A or 6.68 A) optimized for

precision measurements, and an image intensifier PMU-2V module with a luminiferous layer on the input glass fiber washer.

Spectrograms suitable for processing could be obtained with only one LIVS discharge, but He-like ions with heavy nuclear charge do not appear in every discharge. For example, while sulfur ions develop in 100% of the cases, the He-like ions of iron do in only 50%, and yttrium ions in 5% of all cases. A two-channel registration scheme was used to compensate for the Doppler shift, the true A being defined as the mean value of both channels. The characteristic K and L lines emitted by an X-ray tube were employed to achieve exact wavelength calibration [118]. The coordinates of the reference and the desired lines were determined by

extrapolation for the centre chords with the help of an automatic AMD 1 microdensitometer driven by a SM4 computer. The reference lines were chosen so that their distance from the desired line did not exceed a few mA. This provided a sufficient accuracy of the linear extrapolation. The final values ofthe

wavelengths of He-like ions were determined by averaging over a large number of measurements. The value of the interval one can trust is defined by the formula = taN{~

[(A

+

A

2 12)

—

Amean] /N(N + 1)}

356

L. N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

where A11, A12 are the wavelengths found in the two registration channels, N is the number of measurements, A the mean experimental value, and ~ Student’s coefficient. In fact the observed line contours represent the totality of the resonant line and its satellites. To evaluate the corresponding line shifts ~A one must define the dominant contributions of the most intense satellites with n = 3, at least. All satellites to the resonance line and lines to thein lc—ls3p line have 2—ls3p the experimental the same upper state. These two lines are well separated from 1s spectrum. Thus, the information about the relative intensities of satellites to the hs2—ls3p line can be used for the calculation of the resonant line shift SA. This shift has been found in two limiting cases: coronal and thermodynamic, for which overall collisional mixing is assumed. The thermodynamic limit is assumed to be the most adequate to model the real situation in LIVS; the corresponding approximation was used in our computational results presented in fig. 5. The relative accuracy of these results is —h0~. Accurate measurements of resonant lines in He-like ion spectra were made by the beam—foil and recoil methods for sulfur [71], argon [78],iron [72], and krypton [120]. The “München Tandem” was used to investigate ions of sulfur. The beam energy was 85, 105, 127, and 147 MeV per nucleus, the stabilizing accuracy being 0.1°.Aluminium and carbon foils with matter density 50—110 mg/cm2 were used in this experiment to produce the required ions. A flat ADP crystal, supplied with a Soller diaphragm, a system of exact scanning over the Bragg angle and a proportional counter, were used as a spectrograph. The numerous lines of the characteristic X-ray tube radiation were used .for spectrograph calibration. The He-like spectrum was registered practically simultaneously with the Lya doublet in the H-like spectrum, for which high-precision theoretical calculations are available. This allows one to

exp ~thoor’ 2

~

3:1

A

0

j:

-

Ga

C

14].

-

~

K!

~

2~*~~*3+

Sr 351$

A

Fig. 5. Deviation of the theoretical wavelength from the experimental ones (resonant transition in helium-like ions). Theoretical results were obtained by the l/Z-expansion. Experimental results are from: • present work, • beam—foil, L 4 laser plasma. The satellite shift evaluated in the thermodynamical approximation is subtracted from the visible experimental wavelengths.

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

357

estimate the Doppler shift in both spectra. An appreciable decrease in the influence of adjacent satellites has been observed with increasing ion beam energies up to a value exceeding 100 MeV per nucleus. In this experiment, the inaccuracies in the reference line positions (—0.1 eV) are considered as the main source of inaccuracy in the final data. The stated relative accuracy of the final results is

4x105. The recoil method was used in ref. [78]for the study of the spectrum of He-like argon ions. A beam

of U66~ions, after emerging from the UNILAC accelerator, passed through a cell filled with argon gas at a pressure of 700—1000 Pa. This method is free from Doppler shift difficulties. A focusing spectrograph (Johann type) with In—Sb crystal (R = 1280 mm) and position-sensitive detector was used for the registration of the argon ion X-ray radiation. The resolution of this setup is E/~E= 5000

including the detector’s spatial resolution. A fluoresence X-ray tube was taken as the standard radiation source, the wavelengths of the basic lines being taken from the Bearden tables [37]and corrected according to the data of ref. [1211. A decrease of the gas pressure from 8000 to 700 Pa radically suppressed the intensities of the adjacent satellites. This allowed one to make high-accuracy measurements. The final accuracy is 12 ppm, the main part being due to the inaccuracy in the reference positions (10 ppm).

The Super HILAC accelerator (Berkeley) was used to produce He-like ions of iron [68].A beam of Fe18~ions with energy 480 MeV per nucleus passed through two carbon foils. The first foil (300 mg/

cm2) forms a stationary distribution over ionization degrees, the maximum being in the region of Hand He-like ions. The second, thin foil (50mg/cm2), is responsible for the excitation of the resonance transitions in Fe24~,which relax at a distance of 1 p~m.

The spectrum was dispersed by a flat Si 220 crystal placed on a goniometer (3 x i0~deg) located at a distance of 2.6 m from the slit, the latter being placed near the second foil. A position-sensitive detector rotating around the goniometer axis was employed for the registration of the spectrum. The

method of wavelength measurements is based on an accurate definition of the differences between the Bragg angles for the line under investigation, and the standard Lya-doublet of the He-like ion. This technique removes some problems: relativistic aberration, imperfection of the crystal, and nonlinearity

of the detector. All these advantages are due to the fact that all conditions for the radiation and registration of both lines were identical. It was shown that the contribution of adjacent satellites to the interval between the visible centres of two lines depends on the foil thickness. This factor has been accounted for during the optimization of the setup parameters. The absolute error does not exceed 0.25eV. The UNILAC accelerator (CSI, Darmstadt) has been used for the study of the Kr34~and Kr35~ions. The scheme of spectra registration and measurements is similar to that employed in the case of iron, but there is one essential distinction: the doublet Ka Rb generated in the classic X-ray tube has been used as a standard reference. To account for the Doppler shift appearing in this case it was necessary to

measure the ion’s velocity. The time of flight method has been used for this purpose. The total declared inaccuracy in the energy transition is 1.2 eV, the main contribution being connected with the tuning of the crystal spectrograph.

5. Electron satellites in the spectra of nuclear ?-radiation In this part, we discuss the possibility of cooperative electron—nuclear processes in heavy fewelectron ions. These can take place in a hot plasma, for example.

358

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

In modern plasma setups, temperatures of several keV are attained. Under these conditions, elementary processes with an energy threshold of 10 keY and more are intense. In particular, the Kand L-shells of such heavy elements as lead (Z = 82) and uranium (Z = 92) may be appreciably destroyed. An energy of the order of 10 keV is also characteristic for many intranuclear processes. Moreover, already for iron ions (Z = 26) the binding energy of K-electrons is comparable with the energy of low-energy nuclear processes. Presently it is difficult to foresee the importance of nuclear excitations for hot plasma physics (for macroscopic plasma phenomena). However, under these conditions cooperative electron—nuclear processes must undoubtedly manifest themselves as well as new discharge channels for metastable nuclear and electron states; modern spectroscopic methods allow one to fix these new phenomena. In principle, for physical investigation purposes a sample containing the necessary number of excited metastable nuclei may be prepared beforehand. When such a sample is transferred into the plasma, processes with the participation of tightly bound K- and L-electrons, which can be sensitive to the nuclear state, gain in importance. This eliminates the necessity to correlate the nuclear excitation energy with the actual plasma temperature and allows one to consider situations that are simpler from the point of view of the requirements on plasma parameters. In some cases, an appreciable population of excited nuclear levels can be achieved in beam—foil experiments owing to the collisions between incident nuclei with the atomic electrons in the foil. Indeed, let the nuclear kinetic energy be 50 meV/A and the foil density 10 mg/cm2. Such parameters are realistic for a beam—foil setup [10, 68]. Under these conditions, the electron—nuclear collision energy is about 25 keY, and the atomic electron density exceeds 1021 cm2. Let the nucleus have a monopole excited state in the range ~20 keV. Its excitation cross section is (10~4/E2[eV]) cm2. Thus, this state can be effectively excited and the beam—foil setup can serve as a source of a beam of heavy few-electron ions with a low excited nucleus. Atomic processes accompanying the discharge of an excited nucleus have long been used to study the properties of inner and outer shells of neutral and weakly charged atoms. Here inner conversion,

Koster—Kronig decay, XK and XL fluorescence of the atomic shells, and modulation of -y-emission by electron shell oscillations (the shell “shaking” effect [117]) are examples. The same processes are, in principle, possible also in highly charged K and L ions. We shall consider one of the simplest manifestations of cooperative electron—nuclear processes, namely the appearance of a system of electron satellites in the -y-spectrum of the nucleus. A system of red (blue) satellites to the central line corresponds to transitions with excitation (deexcitation) of the electron shells. In ref. [40] the positions and intensities of electron satellites in a neutral or weakly charged atom were determined. Also the influence of the electron shell on the decay rate of the metastable nucleus was evaluated and it was shown that for a neutral atom the effect is negligibly small. When going to a highly charged ion, an appreciable change in the characteristics of the electron— nuclear interactions was noted. It is shown in ref. [41]that the intensity of electron—nuclear satellites of dipole nuclear transitions in highly charged ions may reach that of the central line, of the order of ~ 1/2Z is the relative proton mass). Here, in contrast to a neutral atom, there is no small parameter determining the hierarchy of satellite intensities by the multiplicity or the energy of an electron transition. Moreover, the selection rules valid for a purely electron problem are not applicable here. This permits one to include into the considerations, on equal footing with the rest, the transitions between the components of a fine and a hyperfine structure and also the ns—n’s and j = 0—j = 0 transitions for systems with two and more electrons. Thus, the spectrum of electron satellites of a dipole nuclear transition in highly charged ions is incomparably richer than in a neutral atom.

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

359

In a neutral atom under standard experimental conditions, relatively intense satellites are hidden by

the Doppler contour of the central line. In ref. [40]the method of “intra-Doppler” spectroscopy was proposed for their observation. More favourable is the situation in a plasma that contains multicharged ions. Let us evaluate the ratio of the shift of the electron satellite and the Doppler width of the -y-emission line in a thermalized plasma. Let the shell with the principal quantum number n be destroyed appreciably. The kinetic energy of ions in such a plasma is Ek.~s13.6Z2/4n2eV; the Doppler shift is ho~(2Ek/Mc2)”2 0.60 X 1041iw\1~/n,

where M is the nuclear mass. Now we compare this quantity with the transition energies in a one-electron ion. For Z ~ 50: E(ls—2p) 10.5 Z2 eV; E(2s 4 eV; the Lamb shift E(2s 112—2p312) 112— 4 eV. Keep in mind E(2p112—2p312) that the intensity5 xofi0~ theZ above transitions has no small 7 x connected i0~Z parameter, with the transition energy, and does not obey a multipolar hierarchy. For ions with Z = 20—50, the magnitude of the Doppler broadening exceeds the energy of the fine splitting for ~ 60 keV—1.5 MeV, i.e., the fine structure of the atomic levels can be well resolved in the satellite

spectrum of many nuclear transitions. For H-like ions the energy of n—n transitions (in particular 2s~ 2i 2i 2P312) is determined only by fine relativistic interactions. For two- and more-electron 12— 3the /2’ energies 1 12— of n—n transitions increase due to the interelectron interaction. For Z = 20—50 the systems value of the Doppler broadening exceeds the Lamb shift for h&.~3 1—20 keY, i.e., direct observation of 2—2 transitions in these spectra is, apparently, impossible. In section 3.9, a proposal is mentioned about the diagnostics of an astrophysical plasma, through examination of the transitions between the components of the hyperfine structure of the levels in hydrogen- and lithium-like ions [1111.The corresponding transitions are, at the minimum, electrical

quadrupole or magnetic dipole transitions and their probability is extremely low (
three bound particles: a rigid nuclear core, a proton outside of the core and one atomic electron. The masses of the three particles equal, respectively ~ p.~M,and PeM M is the mass of the whole ion, and + + = 1. Radiative transitions between the excited states of this three-particle system are considered. The one-proton model is surely too rough for quantitative calculations of the pure nuclear characteristics, but it is satisfactory for the evaluation of the relative intensities of electron—nuclear ~

processes that are not sensitive to the nuclear model.

The calculation is based on quantum electrodynamic perturbation theory, in which the adiabatic Gell-Mann—Low formula connects the probability of radiative decay with the imaginary part of the level energy. In second (first nonvanishing) order of electrodynamic perturbation theory, the value sought is given by the sum: ImE=ImEc+ImEp+ImEe.

(60)

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L.N. ivanov et a!., Modern trends in the speciroscopy of mu!ticharged ions

Each of the contributions Ea describes the individual interaction of particle “a” with the -y-quantum. The change of the state of the remaining quasiparticles in second order of electrodynamic perturbation theory manifests itself as a shake-up effect of the system following the recoil. Mathematically, this effect is described by the combined electron—proton and electron—core terms of the radiative transition operator. Such terms appear in the operator upon a transition to the centre of mass frame. The model of a neutral atom consists of a strongly bound nuclear subsystem (the core and a proton), and a weakly bound electron. The energy of the fly-quantum approximately equals the energy of a nuclear transition, the cross section of its interaction with the almost free electron is small, and we neglect the corresponding contribution Tm Ee~It is shown in ref. [40] that the intensity of the satellites in the neutral atom connected with the terms Tm E~+ Tm E~may reach i0~ of the central line intensity. The pattern is different when we turn to a highly charged ion, in which the binding energy of the optically active electron increases as Z*2 (Z* is the effective nuclear charge for this electron). Two new features appear: firstly, the tightly bound electron shell of a few-electron ion becomes less sensitive to shake-up during -y-quantum emission (as a consequence Tm E~+ Tm E~decreases as 1/Z*2); secondly, this shell loses its transparency to the -y-radiation and accordingly the contribution of the direct interaction of the y-quantum with the electrons, Tm Ee~increases. This contribution reaches its maximum for WRa 1 (Ra is the atomic radius in relativistic units). It corresponds to wavelengths comparable with Ra, and a -y-quantum energy of =103Z* eV, far in excess of a typical electron transition. In this energy region, optimal for the term Tm Ee~its contribution to the intensities of the satellites of the 25-pole proton transition line is defined by the small parameter w(wp~~R)2” (R is the nuclear radius), i.e., there is an additional small parameter ~ as compared with the pure proton transitions. This causes a rapid decrease of the satellite intensity with increasing nuclear transition multiplicity. We emphasize once more that there is no parameter determining a hierarchy of the Im Ee contributions by the multiplicity of an electron transition. These contributions may be interpreted as an effect of nuclear shake-up in the course of the interaction of the -y-quantum with the electron shell. Naturally, the influence of this effect on the total probability of nuclear decay is small; its contribution equals the total probability for all satellites. Figure 6 shows the results of a calculation estimating relative intensities of some monopole, dipole and quadrupole satellites accompanying electric dipole nuclear radiation in a H-like ion. An energy parameter (&w) is plotted on the abscissa. This parameter, convenient in practical calculations, is related to the -y-quantum energy as follows: h

=

271 Z(aw) eV 4Z(&w) keV.

(61)

Nonrelativistic electron wave functions have been used in the calculation; for each electronic transition the contribution of only the component with the lowest possible multipolarity is taken into account, though for WRa ~ 1, as noted above, the contribution of higher multipolarities may become important. As (&w) the transparency of the electron shell increases and all satellites disappear; as (&w)—~0 the well-known dependence P~-—(hw)2~’ is restored, where A is the multipolarity of the electron transition. The intensity of the monopole transition ls—2s disappears as (~iw)—+0 due to the orthogonality of the orbitals of the initial and final states. This energy dependence of the satellite intensities is qualitatively preserved also for one-electron transitions in two- and more-electron systems. The main difference is determined by the angular coefficient, and does not exceed an order of magnitude. —~

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!rzcharged ions

361

I

~ P(pe)

0.10

vII

0.09 0.08 0.07 0.06

-

i lii

0.05 0.011 0.03

‘1

0.02

______

0.01

—

i.o

i.o

~Iv

i.o

(&~,o)2~

Fig. 6. Contributions of Im Ee to the relative satellite intensities. P(pe) is the satellite intensity, P(p) the pure nuclear transition intensity, js.,, the reduced mass of the proton (p..,, 1/2Z). I: ls—2p 112 electron transition, II: 2s—2p1,2 electron transition, III: 2p112—2p312 electron transition, IV: ls—2s electron transition. The quantum energy 6w, is related to the parameter ~w: 11w 4Z(&w) keV.

In ref. [99], attention was drawn to the fact that electron satellites appear in the spectrum of -y-emission of the nucleus accompanying electron capture from the inner atomic shell (K-capture). These satellites correspond to alternative finite states of the atomic ion. Two groups of satellites can

coexist in this case: those corresponding to shake-up of the external electron shell, and those corresponding to different states of the vacancy in the inner shell. These two groups are due to different physical mechanisms and are characterized by essentially different values of the splittings. As an example of K-capture the transition 163 + I3~ 63 —+

64Gd

63Eu’

was considered. In 30% of all cases the Eu nucleus is in the excited state 3/2k, with an excitation energy of 103.7 keY [100].A vacancy is formed in the atomic shell. The main reaction channel is the cascade process: f3 capture relaxation of the electron shell (radiative and Auger decays with vacancy elimination) with nuclear relaxation. The nucleus makes a transition into the state 5/2k with emission of a ~y-quantum(E2 transition). The lifetime of the nucleus in the excited state is 4 x i0~s, and the —+

relaxation rate of the electron shell is several orders of magnitude higher. In this chain of reactions, only the satellites associated with a low-energy transition in the outer shell may arise. However, there exists an alternative possibility: ~3-capturewith a simultaneous electron—nuclear

radiative transition. This process is described by coupled perturbation theory in the interaction with the

362

L.N. Ivanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

electromagnetic and neutrino fields. Its amplitude is of first order in the interaction with both fields, and it takes into account the correction to the total probability of the process associated with transitions in the inner shells of the atom. This correction has the same physical nature as in the case of a simple -y-transition in the nucleus of a heavy multicharged few-electron ion; its contribution to the relative intensities of the satellites is of the same order.

6. Isoelectronic sequence of neon Ne-like ions are the heaviest few-electron atomic systems for which a number of high-accuracy experimental investigations were performed during the last few years. Here we review the latest experimental and theoretical investigations. The states 2s2p6n1, 2s22p5n1 with ni = 3s, 3p, 3d, and some 2—4 transitions are considered. The original, formally exact method of relativistic perturbation theory, with a model zeroth-order approximation, is presented. This method has been developed especially for relativistic many-electron systems and employed for the calculation of the characteristics of atomic ions (and atoms) with total number of electrons N~ = and systems nuclear of charge Z up to 80 and higher. 22s22p5n1, 1s22s2p6n1 are 0~ the10—70 simplest two quasiparticles outside of excited states theThe many-electron core1sof closed shells: one 2s’ or 2p’ vacancy and one ni electron. As mentioned in the introduction, the theoretical investigation of such systems is analogous to that for two-electron systems, and is facilitated by the availability of reliable information about the spectra of the corresponding one-quasiparticle systems, namely F- and Na-like ions. However, the specific energy level structure leaves its mark in the characteristics of the single ions and of plasmas containing such ions. 6.1. Review of experimental and theoretical results The isoelectronic sequence of neon has been thoroughly investigated, but nevertheless there remains ongoing interest because their spectra contain most important information for the solution of a variety of problems: (1) Development is needed of hot, dense plasma sources that radiate in various wavelength ranges; at a plasma temperature of Te = 1—3 keV atoms with Z ~ 40 are dominantly ionized up to the Ne-like configuration. A review of this problem may be found in Fawcett [118]. Significant progress has been achieved by using laser—plasma techniques [30, 43, 119—122], which permit a systematic investigation of multicharged ions. For heavy Ne-like ions, qualitative measurements of X-ray spectra were made using LIYS [55, 123]. (2) Development is needed of devices that would provide high-precision measurements of wavelengths of ions radiating in plasmas. This problem is of particular importance for measurements in the X-ray region. During the last years, experimental results have been obtained with an accuracy of ±0.002—0.003A. Nevertheless, the discrepancies between the results obtained with various setups often exceed these limits appreciably. High recording efficiency has been attained with the introduction of an image intensifier into the measuring apparatus [123]. (3) Ne-like ions are efficient for diagnostics of thermonuclear plasmas [124, 125]. (4) These spectra are also present in the solar corona radiation [126—129]. (5) Study of the low-lying level structure of Ne-like ions is important in connection with the development of a laser for the far-UY region [17, 130—140]. It has been shown that for an equilibrium

L. N. Ivanov ef a!., Modern trends in the spectroscopy of inulticharged ions

363

plasma there exists a stationary inversion for some 3—3 transitions of Ne-like ions, and the enhancement coefficient increases with Z and reaches a value of 20—40 cm1 for Z = 26 (at optimum plasma parameters: electron density ne = 5 x 10~~ cm3, electron temperature Te = 310 eV) [132].Recently an enhancement coefficient of lOcm’ has been observed for Ne-like selenium (Z=34). At present,

different mechanisms of population of the working levels are under discussion [139,140]. It is stated that under the plasma conditions defined in the work, the population of n = 3 levels occurs through the higher levels, which are populated during dielectron recombination in F-like ions. Investigation of the higher (n > 3), denser spectral region meets with considerable difficulties, both in experimental [121, 141—143] and theoretical studies [144—146]. Information about these states is especially important in the elucidation of the principal limitations of different theoretical approaches. In 1938, Tyren had started precision wavelength measurements of Ne-like ions in the X-ray spectral region (FeXYII, CoXYIII, A = 12—17 A) [147].Thirty years later 2—3 transitions in NiXIX, CuXX, and ZnXXI (A 3 10 A) were identified in a LIVS discharge plasma [148]. The considerable advance up to MoXXXIII was achieved by using a laser plasma setup [28, 30, 122], and Ne-like ions of silver were

excited in an exploding-wire experiment [149].Then, ref. [150]reported identification and accurate wavelength measurements of AgXXXVIII, CdXXXIX, InXL, and SnXLI, registered in a LIVS plasma.

Ne-like ions of xenon were obtained in a laser plasma created by spherical irradiation of a spherical target [43].Finally, using LIVS the spectra of Ne-like Ba, La, Ce, Pr, and Nd were investigated [123, 49]. Recent work of Dietrich et al. permits one to consider the problems connected with the Ne-like ions from a new viewpoint: thus, in ref. [151], using the SuperHILAC installation at Berkeley, for the first time the quadrupole 2p—3p transitions in Ne- and F-like Xe44~were studied. The forbidden lines can be also observed in laser plasma [152] and LIVS [143] spectra. The wavelengths of some 2—3 dipole El transitions were also found in ref. [1511,and this allowed a conclusion to be formed about the energy intervals An 0 for the states 2p53s, 3p, 3d. In the same work, the rates of quadrupole 2p—3p transitions were measured, the values of which can be compared with some dipole 3s—3p transitions. In ref. [153],high-precision data were presented on the spectrum of the BiLXXIV ion, the heaviest ion in

the Ne-like sequence. To solve the above-mentioned problems, the characteristics of satellite spectra for resonance 2—3 transitions are of great importance. The satellite structure of spectra of Ne-like ions can be very useful for diagnostics in the area of high (up to 10 keY) plasma temperatures. Most closely located are the satellite lines radiated by Na-like ions. These lines are due to transitions between the states

2s22p5n

6n 111n212 and 2s2p 111n212 of the three-quasiparticle system. Their theoretical investigation is

surely much more complicated than that of the two-quasiparticle systems. However, in some cases detailed information about the separate satellite transitions is not needed, i.e., the averaged spectral characteristics of satellite formation are of interest. Therefore, a simplified theoretical analysis can be applied, as the averaged values are less sensitive to the approximation used. A number of experimental works have been devoted to the study of spectra of Na-like and other satellites. In ref. [154], a soft X-ray spectrum was obtained in two ways: the plasma was excited by focusing of a laser beam, and by a vacuum spark. These authors studied Na-like ions of iron and

titanium, and found lines which are satellites to the 2p—3s and 2p—3d transition lines in Ne-like ions. In ref. [155], a beam—foil experiment was carried out along with the identification of some strong satellite lines of the C1VIII and ArYIII ions. In ref. [156] observations were made of X-ray spectra of a dense

plasma arising between self-discharging electrodes (Z-pinch). Also, satellite lines radiated by the KrXXVI ion were identified. In ref. [157], the X-ray spectrum of molybdenum was probed. This was

364

L.N. Ivanov et a!., Modern trends in the spectroscopy of inulticharged ions

obtained by wire explosion; satellite lines were noticed, but due to the insufficient accuracy they were not identified. In ref. [158]lines of Ne-like and Na-like ions of Ti, V, Cr, Mn, Fe, Co, Ni, and Cu were studied by means of LIVS. In ref. [159]Ne- and Na-like ions of Ge, Se, and Ze were studied using a laser plasma source. A compilation of all iron ion spectra is given in ref. [160]. One can see the great interest in the study of two-electron satellites of Ne-like ion spectra through ref. [35], in which high-resolution spectra for different silver ions were obtained from the Tokamak installation in Princeton; wavelengths and transition intensities were defined in Ne-like silver, and, apart from dipole transitions, E2 and Ml lines were probed. The characteristics of satellites to 2—3 transitions were determined in Na-, Mg-, and Al-like ions of silver. A recent work [47] was devoted to the experimental identification of two-electron satellites of Ne-like spectra of Br. Research of Br ions in various ionization stages was undertaken, namely, 0-, F-, Na-, and Mg-like ions obtained under laser irradiation of a point target, the intensity of the laser radiation being varied from 3 x iO’3 to 4 X i0’~Wicm2. Due to the spatial resolution of the spectra, an evident sequence of spectral lines dependent on the distance from the target and the radiation intensity was traced, and this made it possible to realize a more careful separation of spectral lines related to the ions of various ionization degree. Identification was made on the basis of calculations with the Dirac—Fock method with configuration superposition (Skofield program). In the papers indicated above, programs by Cowan, Cohen, Crance, Desclaux and Chen were also used; as a rule, one had to scale the theoretical results to reproduce the experimental spectra with better accuracy. Here we deal with systems having quasiparticles of opposite charge, a vacancy and an electron. Their mutual attraction deteriorates the convergence of any perturbation series compared with the case of quasiparticles of like charge, which repel. For example, for Z ~ 30 the contribution of the relativistic and radiative effects to the energy of the resonant transition (Ne-like ion) reaches only several thousands of cm~1considering this energy in terms of the well-known liZ expansion, one may see that this contribution is due to strong compensation of the successive terms of order (aZ)4 and (aZ)3, so that the terms of order (aZ)2, (az)1, and (aZ)° become quite appreciable. This leads to a complex Z-dependence of the transition energies and hampers the use of traditional semi-empirical methods for the interpretation of experimental spectra. The situation does not become simpler for larger Z, where the relativistic and radiative contributions reach tens or hundreds of thousands of cm’. An additional complication is connected with the increasing contribution of the interference of relativistic and correlation terms. The last effect is especially pronounced for states with more than one quasiparticle outside of the core of closed shells. The choice of an adequate zeroth-order approximation generating an optimal one-quasiparticle representation and effective summation of the essential subseries is a straightforward way to diminish the difficulties just outlined. It is impossible to give a complete review of the papers devoted to the calculation of the Ne-like spectra. We only mention some examples representing different traditional methods: liZ expansion [161], Hartree—Fock—Pauli [121, 122, 162], multiconfigurational Dirac—Fock [153, 163—165], relativistic random-phase approximation [166], and the semi-empirical method using the parametric core potential [144]. Below we present a short description of the relativistic perturbation method with zeroth-order approximation, and with effective summation of some higher-order contributions. This method has been used in the last decade to calculate the energy structure of the isoelectronic sequences of Na [13], F, Ne [167], Mg [168], Co, Fe [169], Cu [170], Zn [171], and Ni [15], for a large range of Z values. For rare earth atoms with one, two, and three quasiparticles, the rates of the radiative and autoionization processes have been investigated also. See ref. [172], and references therein.

L.N. Ivanov et al., Modern trends in the spectroscopy of multicharged ions

365

For some sequences this method is the only theoretical approach yielding reasonable results. For other sequences, it proved to be the most economical one. We illustrate the method by the special case of Ne-like ions with two quasiparticles of opposite charge. 6.2. Relativistic perturbation theory with zeroth-order approximation We consider a system with a dense spectrum of degenerate and nearly degenerate states. The principles of perturbation theory for such systems are presented in refs. [25](nonrelativistic theory), and [26](relativistic modification). Following these principles, we calculate and diagonalize the secular matrix M determined by the electrodynamic scattering matrix. As a result, we obtain the state vectors and self-energies E. The adiabatic Gell-Mann—Low formula is used for the construction of the secular matrix. The perturbation theory for the scattering matrix generates an analogous expansion for the energy shift AE, the electron—nuclear interaction being included in the zeroth-order Hamiltonian. Only even-order electrodynamic corrections contribute to AE. The secular matrix elements for excited states have an imaginary part in second order already. This is connected with the radiative decay of the excited state. In the next nonvanishing order, the fourth order, there appear terms describing autoionization decay, and the interference between radiative and autoionization effects are described by the highest-order terms. Two-electron recombination can be regarded as the inverse process of autoionization. It is the essence of the energy approach to the complex atomic problem.

The total energy of a state is usually represented as E=ReE+iImE,

ImE=F/2,

(62)

where F is the level width. The whole computational procedure for the level positions and life times is reduced to the diagonalization of the same matrix M with complex elements. The formulae for relativistic (electrodynamic) perturbation theory can be obtained from the nonrelativistic version by replacing the Coulomb interaction of bare electrons by D(1 2)(l

—

(63) D(l 2)

=

2~ ir

r12Z

f

dw exp(iwt12 + iko~r12).

—~

All notation is from ref. [175].Here the photon propagator D( 1 2) is given in the Lorentz gauge. There exists a well-known general method of derivation of subsequent electrodynamic perturbation corrections on the basis of the Gell-Mann—Low formula (see, for example, ref. [26]and references therein). Every nth-order correction contains an n-fold integration over frequencies w~.A part of these integrals, corresponding to the radiative shift (self-energy and vacuum polarization) is divergent. We do not consider them here. All the remaining integrals are convergent, and can be calculated in principle. The one-fold integration can be done analytically in the second, lowest order due to the energy conserving s-function. In this approximation, M is expressed through a linear combination of Coulomb and magnetic radial integrals with angular coefficients similar to eq. (27). Coulomb and magnetic contributions are due to the D(1 2)/r12 and D(l 2)&1&2/r12 terms of the interaction (63). One of the most realistic ways to account for the fourth- and higher-order corrections to Re M effectively is to treat them using the approximation of the semirelativistic equation, i.e. the many-

366

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

electron equation with the Hamiltonian N~ 0,

N~01

H= ~ h(r1)+ ~ 1/r11Z,

(64)

where h(r) is the Dirac one-electron Hamiltonian. The summation in eq. (64) runs over all atomic electrons, those in inner shells included. One can solve the semirelativistic equation using atomic perturbation theory, regarding the interelectron interaction 1/r11Z as the perturbation. The nth order correction from atomic perturbation theory can be regarded as an approximation to the 2nth order of the consistent electrodynamic perturbation theory. The many-electron retardation and many-electron radiation effects and the magnetic interelectron interaction are omitted in this approximation. Modern apparatus allows a direct calculation of these corrections only in second order of electrodynamic perturbation theory. At present there is no detailed analysis of the role of these effects in the fourth and higher orders of consistent perturbation theory. To improve the convergence of the perturbation series, one can account effectively for a part of the electron interactions already in the zeroth-order approximation. This is usually done by introducing the one-electron potential into the zeroth-order Hamiltonian together with the electron—nuclear interaction. It may be the Hartree—Fock—Dirac one-electron potential, that of the density functional method [173], or any other model potential. We use the standard Feynman technique to generate diagrams for the perturbation series, for both the electrodynamic version (time dependent diagrams) and for the semirelativistic equation. This technique was adopted by Tolmachev [25] for the special case of the many-electron degenerate system. Specific details connected with the other peculiarities of the systems under consideration are described in refs. [167, 174]. The zeroth-order approximation is defined by the Hamiltonian N~0~

N~0~

H0 = ~ h(r,) + ~ 1/jr1~/3).

(65)

The central one-electron model potential imitating the independent interaction of separate quasiparticles with the K and L shells of the ten-electron core has the form VK(r) +

V,(r~/3)=

VL(r~/3),

2 VL(r) = 8[1 VK(r) = 2[1

— —

+

0.032r3)j/rZ,

(66)

exp(—213r) (1 + 0.6r + 0.16r exp(—2r) (1 + r)]/rZ.

This form is motivated elsewhere [13]. The only fitting parameter /3 is a characteristic of the core and must depend neither on the number of quasiparticles of any kind, nor on their states. As mentioned above we use the semirelativistic many-electron equation to approximate higher-order corrections. This equation is treated within the framework of the formally exact perturbation theory, with the perturbation containing the Coulomb interelectron interaction and compensation term: N~ 0~

=

~

1/Zr11

N40,

—

~

V1(r~I/3).

(67)

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

367

For an arbitrary many-electron system with p quasiparticles outside the core, the complex secular matrix M may be represented as ~

(68)

M~°~ being the contribution of vacuum diagrams of all orders of perturbation theory, and M”~,~ those of one-, two-,. . . quasiparticle diagrams. The expansion (68) resembles the well-known cluster expansion, but contains only p terms instead of N101. In contrast to the cluster expansion, for all the

terms in eq. (68) the sequence of inequalities M~’~ ~ M~+ 1) is valid for any state of any system, so that different approximations are applicable for calculation of the ~ Moreover, for the system with i + 1 quasiparticles, the contribution of the i-quasiparticle diagrams M~’~ is simply related to the contributions of the same diagrams for systems with only i quasiparticles outside the same core. Thus, information, theoretical or experimental, about i-quasiparticle systems can be used when studying the more complex (i + 1)-quasiparticle system. For example, the real matrix M~°~ is proportional to unity with elements equal to the core electron energy. This quantity is independent of the number of quasiparticles or their states. We assume M~°~ = 0. The diagonal matrix MW is complex, and the real part is equal to the sum of the independent one-quasiparticle energies. The latter can be found from the simplest spectra of one-quasiparticle systems. Using these “exact” one-quasiparticle energy values in the case of two- or three-quasiparticle systems automatically provides the the sum two-quasiparticle of a very important class of diagrams. 2~ describes interactions through the The nondiagonal complex matrix M~ electrodynamic vacuum and through the polarizable core. The latter can be accounted for effectively by modification of the photon propagator (or of the quasiparticle Coulomb interaction in the case of the semirelativistic equation). The corresponding approximation is analysed in ref. [174],where a new form of interquasiparticle polarization interaction has been suggested. After this modification the manyelectron problem becomes effectively reduced to the few-quasiparticle one, which can be solved by one of the methods developed in the theory of few-electron systems. Such an approach accounts to some

extent for the three- and more-quasiparticle interactions, too. The problem to what extent has not been considered in detail yet.

To solve the reduced two-quasiparticle problem we used in refs. [167—172]the simplest method analogous to the Hartree—Fock method [98],which leads in our case simply to a redefinition of the zeroth-order potential. The calculation is carried out in the j—j coupling scheme. The transition to the intermediate coupling scheme is realised by the diagonalization of the secular matrix. Only Re M is diagonalized, the imaginary part is converted by means of the matrix of eigenvectors Cmk obtained by diagonalization of Re M: Im Mmk

=

~ C~Im ~ CIk.

(69)

This procedure is correct with accuracy up to terms of the order of Im M/Re M. 6.3. Imaginary part of the secular matrix

Here we do not give the working formulae for the calculation of the decay probabilities. They can be found elsewhere (see, for example, ref. [172]and references therein). We shall outline only the principal calculational scheme. First of all the whole analytic structure of Im ~ Im ~ is the same as that of the real parts: the lowest order is represented by a finite combination of radial integrals

L.N. Jvanov et a!., Modern trends in the spectroscopy of mu!ticharged ions

368

with angular coefficients, and the next orders include infinite sums (and integrals over the positive and negative continuum). The angular coefficients are the same as in the real part, the only difference being in the definition of the radial integrals. The latter represents a product of two one-fold integrals, each being computed by the same system of differential equations as the real integrals are. Within the framework of the electrodynamic perturbation theory, the radiative processes are determined by the imaginary part of the interaction (63) between active quasiparticle and the vacuum of the electromagnetic field. There are only one- and two-quasiparticle diagrams contributing to Tm M in the lowest (second) order of the electrodynamic perturbation theory. The one-quasiparticle contribution

Im M~’~ = 6,~~ Im M”~(a)= ~ F(a)/2.

(70)

The sum in eq. (70) runs over all quasiparticles, F(a) is the level width of the state with one quasiparticle a outside the core. This width represents the independent contributions of quasiparticles “a” to the total width of the levels of the two- or three-quasiparticle systems. For the Ne-like ions under consideration, eq. (70) accounts for the transitions 2s22p5nl—2s2p6n1, and nl—n’l’. Note that the summation over all possible intermediate and total moments of the final states is accounted for in eq. (70). As one can see, this sum does not depend on the two- or three-quasiparticle quantum numbers of the initial state too. Such a dependence appears in the partial contributions describing the transitions to

concrete final states with right angular symmetry. This is a common property of any representation with a pure coupling scheme, such as f—f, or LS. For an intermediate representation the dependence of Tm Mw on the state is due to the transformation coefficients C’mi in eq. (69), which naturally depend on

all quantum numbers of the whole system. The only two-quasiparticle process occurs in a system of two quasiparticles of a different kind: their

annihilation. 6.4. Numerical results for the energy structure. Comparison with experiment

Extensive experimental information on 2—3 and 2—4 transitions in Ne-like ions and adjacent satellites with Z = 39—60 is presented in ref. [143]. Line identification in this work can be regarded as reliable, in that it is in good agreement with laser plasma measurements [121, 122] and early results on LIYS experiments [56, 150]. Our detailed theoretical investigation for Z = 16—80 was completed in refs. [167,

174], where the results were also compared with available experimental data. The most systematic comparison of theoretical with experimental results may be made for relatively

small Z, Z = 16—28, for which high-precision information on 2—3, 3—3 and some 3—4 transitions is available (see, for example, the work of Litzén, Jupén et al. [142]and references therein). The accuracy of our calculations was evaluated for Z ~ 26 by the absolute deviations of theoretical levels from the experimental ones; these deviations, as a function of Z, depend weakly on Z [174].The comparison for varying Z allows one to draw some conclusions about the computational accuracy for the whole isoelectronic sequence.

In ref. [167],the calculation of the energy structure ofNe-like ions was based on previously obtained empirical information about one-quasiparticle systems: the energy structure of Na-like ions [13]and the total energy of the 2p 312 state of F-like ions, which is the ionization potential of Ne-like ions [167].The large scatter in the one-quasiparticle experimental data used did not allow one to achieve a highaccuracy extrapolation into the region of Z = 70—80. Nevertheless, in the region of moderate Z < 60, the theoretical results of ref. [167]are quite reliable.

L.N. Ivanov et al., Modern trends in the spectroscopy of multicharged ions

369

The high-precision data of Dietrich et al. [153] for Z = 83 (BiLXXIV) allowed us to improve the ionization potentials used in ref. [167]. Recently we completed a new, more refined calculation of Ne-like ion energy levels with improved ionization potentials for Z > 40; besides, the magnetic part of

the interquasiparticle interaction (63) was taken into account. Note that inclusion ofthe magnetic terms of eq. (63) generally lowers the whole energy spectrum of Ne-like ions, by a few thousand cm1 for Z~40and 5 x 1O4—105cm~for Z~80. The discrepancies between the recent LIVS [143]experimental transition energies and refined theoretical values are shown in figs. 7a—7g. In these figures, we also show errors in the corresponding

10

~iscrepancies,

100 cm1.

2s22p53/23d3/2

(3D 1).

_~:

22p53/2 3d3/2 15254

56

60

I.

—20—

—60-

58

a

22p51/23d3/2

z

2s

(3p 1)

120

:1I

(1~~1). 30. 40

2s

~I~

~

—20—

-40

c

—30 —40

L

50b Fig. 7. Energies of excited states 2s22p’j,3n!f 2, 2s2p’3n1f (I = 1). The dots represent the discrepancy between the experimental results of ref. [143] and the model potential method, the solid lines the discrepancy between the MCDF calculations of Cogordan and Lunell [165]and the model potential method. All values are in 100 cm ‘.

370

L.N. Jvanov et a!., Modern trends in the spectroscopy of mu!ticharged ions 63p1/2

20. 10

0

4~46

~

~

J

~—

52— 54—56~8

10.

60— z

~,

—20

—30

-30

d

20.

2s22p51/23s

10

J~44_

40

_____

________________

42

44

4

48

50

54

52

f

5

63p1/2 2s

58

Z

(1~)

2p

(~P 1)

0

ii I

10.

—20

—40

(~p~).

2s22p53/23s(3P 1)

10

2s 2p

46—4 —50—52—54—56—50—60—

10

z

Q.

Jj I

~10e~

44

46

I

48

50

52

54

56

50

Z

—30

—40.

-50.

g

Fig. 7 (cent.)

line measurements. Note that in real spectra, the resonance lines are blended by the transitions of other ions. In the same figures, the discrepancies between the corresponding MCDF data of Cogordan and Lunell [165] and our refined energies are shown. Both curves are in good agreement. However, that cannot be said about the fine splitting calculations. It is worthwhile to note that accounting for the magnetic terms (63) becomes essential in investigations of the level fine structure for Z > 40. The investigation of fine structure effects in highly charged Ne-like ions was started by the beam—foil experiments of Dietrich et al.of [70]. In table 14, we 4” [70]; the results different theoretical give the energies of some 3—3 transitions measured for Xe calculations are presented for comparison. The disagreement between different theoretical results are

L.N. Ivanov et a!., Modern trends in the spectroscopy of muhicharged ions

Comparison of selected An

Intervals

=

371

Table 14 44~,obtained in the beam—foil experiment of ref. [1511,with theory. 0Levels intervals in Xe are designated by (hole state, excited electron) Exp. (±0.5eV)

Theory (eV)

a)

b)

c)

d)

1s52s22p’3p—1s22s22p’3s (89.29) (p 3,2’ p,,2)2—(p3,2, s,,5)1

89.1

89.6

89.8

89.78

(p315, p3,2)2—(p312, s,,2),

182.9

183.2

183.5

183.41

(p.,,2, ~

186.5 ±1

184.2

184.5

181.56

196.5

198.1

198.1

198.21

(p3,2~d,,2)1—(p312, p,,2)2

157.2

160.2

160.3

157.19

(p~,2,d3,2),—(p,,~,p,,2)2

131.3 ±1

130.0

130.0

129.60

~1,2)1 22s22p’3d—1s22s22p’3p 1s (p 312, ~ p,,2)~

89.63 (183.00) 183.45 (184.07) 184.2 (198.24) 198.72 (160.36) 160.51 (129.45) 129.32

‘>Calculation of Scofield in ref. [151]. b) Calculation of Chen in ref. [151]. ~ Calculation of Cogordan and Lunell [1651. d) Calculation of Ivanova and Glushkov [1671;in parentheses the values obtained by a refined calculation accounting for the magnetic terms (63).

rather significant, so that their accuracies obviously are not sufficient enough to confirm the accuracy of the measurements [70]and to elucidate the true values of fine structure atomic effects. In column “d” of table 14, we outline the role of magnetic two-particle terms (63). The numbers in parentheses are obtained after accounting for the magnetic terms (63). The latest more intriguing experiment of Dietrich and coworkers dealing with Ne-like bismuth [153] again challenged theorists from the viewpoint of testing the fine structure atomic effects. In particular, the precise solution of the many-electron QED problem and the development of accurate manyelectron relativistic wave functions are outstanding problems, the effect of nuclear finite size being of great importance. The way ofconsidering these problems was outlined in section 3 while treating Li-like ions as the objects for testing the above described effects. At present the development of this method in order to study other more complicated very heavy atomic systems is rather urgent. 6.5. Highly excited levels Recently very interesting experiments were carried out demonstrating laser enhancement of the 53s 3s—3p transitions selenium the theoretical effect being observed twocollisional transitions: 2p (J = 1)—2p53p (J = in2) Ne-like (A = 209.8 A, 206.4[17], A). The model based for on the excitation population mechanism predicts an enhancement effect for the 2p53s (J = 1)—2p53p (J =0) transition with A = 182.4 A. However, just for this transition the effect has not been observed. We do not know a satisfactory explanation of this fact in the available literature, but cogent arguments for the recombination mechanism of population ofthe excited levels are given in refs. [139,140]. It was suggested in these articles that the recombination of fluorine-like ions with an accidental electron leads to the formation of highly excited neon-like ions, whose cascade relaxation populates the working levels. This mechanism

L.N. Ivanov et a!., Modern trends in the spectroscopy of multicharged ions

372

does not explain, for example, the enhancement of the J—J’ = 1—0 transitions [17], but, nevertheless, the ideas of refs. [139, 140] stimulate the search for new schemes with inclusion of high-lying levels. The configurations 2p541, 2s2p641 have hardly been studied until now. Kastner and coworkers observed several bright 2p—4d lines in the solar spectra. A great deal of information on 2p6—2p54s, 4d (J = 1) transitions can be found in ref. [121].For several 2—4 transitions (Z = 39—42) the wavelengths have been measured using a LIVS installation [143].Reliable data about 3p—4s, 3s—4p, 3p—4d and 3d—4f transitions are given by Jupén et al. [142]. A laser spark was used there as the source of multicharged ions. The largest part of the theoretical data stems from semi-empirical calculations by Crance [144] for the configurations 2p541 (I = 0—2), Z = 14—28. We calculated the level positions for the states 2p541, 2s2p641 (I = 0—2) for Z = 20—42 [176]. One of the purposes of this calculation was to clarify the Z-dependences of the mutual positions of different levels, in particular, the phenomenon of transitions from some bound states to the continuum. The calculation shows that some 1s22s2p641 states are autoionizing for Z ~ 19. The same states lie under the

ionization limit for Z>20, i.e., they become stationary states. The highly excited Rydberg states 2p~ 12nlcan autoionize due to the Beutler—Fano channel: 2p~12nl—+2p~12 + el.

(71)

It is a low-energy (compared with those mentioned above) channel, open only for n > na with ~a depending on Z. Thus, there are a lot of radiative and autoionization decay channels for highly excited

states. The relative values of the corresponding partial rates differ along the Rydberg series, which must determine a rather complicated mechanism of population of the low-lying states. Accurate calculations must be made to determine the set of energetically open channels in every special case. References [139, 140] started this study. References [1]H.

Gould, in: Proc. Atomic Theory Workshop on Relativistic and QED Effects in Heavy Atoms (NBS, New York, 1985) p. 66. [2] B. Zigelman and H.M. Mittleman, in: Proc. Atomic Theory Workshop on Relativistic and QED Effects in Heavy Atoms (NBS, New York. 1985) p. 28. [3] A.A. Logunov and AN. Tavkhelidze, Nuovo Cimento 29 (1963) 380. [4] A.P. Martinenko and RN. Faustov, Teor. Mat. Fiz. 66 (1986) 399. [5] MA. Brown, Teor. Mat. Fiz. 39 (1984) 388. [6] P.J. Mohr, in: The Physics of Highly Ionized Atoms, Proc. Intern. Conf. (Oxford, England, 1985), p. 459. [7] W.R. Johnson and G. Soff, At. Data NucI. Data Tables 33(1985) 405. [8] SI. Salem, S.L. Panossian and R.A. Krause, At. Data Nuci. Data Tables 14 (1974) 91; 0. Keski-Rakhonen and MO. Krause, At. Data Nucl. Data Tables 14 (1974) 139. [9] L.N. lvanov, E.P. Ivanova, AN. Kalinkin, Yad. Fiz. 42 (1985) 355. [10] Ch. Munger and H. Gould, Phys. Rev. Lett. 57 (1986) 2927. [11] G.W.F. Drake, Adv. At. Mol. Phys. 18 (1982) 399. [12] S.A. Zapryagaev, N.L. Manakov and V.G. Pal’chikov, Theory of Multi-charged Ions with One and Two Electrons (Energoatomizdat, Moscow, 1985). [13] L.N. Ivanov and E.P. Ivanova, At. Data Nucl. Data Tables 24 (1979) 95; 35 (1986) 419. [14]E.P. Ivanova and AL. Glushkov, Opt. Spektrosk. 58 (1985) 961. [15] E.P. Ivanova and AL. Gogava, Opt. Spektrosk. 59 (1985) 1310. [16] J.O. Eckberg, J.F. Seely, CM. Brown, U. Feldman, MC. Richardson and ME. Behring, J. Opt. Soc. Am. 4B (1987) 420. [17] M.D. Rosen, P.L. Hagelstein, DL. Matthews, EM. Campbell, AU. Hasi, B.U. Whitten, E. MacGowan, RE. Turner, R.W. Lee, G. Charatis, CE. Busch, CL. Shepard and PD. Rockett, Phys. Rev. Lett. 54 (1985) 106; DL. Matthews, P.L. Hagelstein, M.D. Rosen, M.J. Eckart, N.M. Ceglio, AU. Hazi, H. Medecki, B.J. MacGowan, i.E. Trebes, B.L. Whitten, EM. Campbell, C.W. Hatcher, AM. Hawryluk, R.L. Kauffman, L.D. Pleasance, G. Rambach, J.H. Scofield, G. Stone and TA. Weaver, Phys. Rev. Lett. 54 (1985) 110.

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