Optical Properties of Plasmas of High-Z Elements

Transcription

1 Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer Institute This wor was perfored in the frae of the WTZ cooperation agreeent under WIE 0. Forschungszentru Karlsruhe GbH, Karlsruhe 1999

2 Table of Contents Introduction. 3 Calculation of Optical Properties of Tungsten Plasa in LTE Approxiation... 3 Features and Assuptions for Calculation of Optical Properties of Tungsten Plasa.. 4 Spin-Orbital and Electrostatic Splitting.. 6 Calculation of Ion Concentration and Optical Properties of Tungsten Plasa. 9 Results and Discussion 11 Theory of Electrostatic and Spin-Orbital Splitting. 16 Calculation Methods for f, a, b Characteristics.. 18 Calculation of Spin-Orbital Splitting.. 0 Nuerical Calculation of Splitting. 1 Calculation of Relative Line Intensities with Taing into Account the Splitting.

3 Calculation of Charge States and Populations of Levels. 3 Calculation of Optical Spectra... 5 Optical Spectra of Tungsten... 7 Lebesgue Method of Radiation Transfer in Plasa of high-z Eleents Description of the tables 4 Coparison of different averaging types.. 4 References 5 1. Introduction The optical properties of tungsten were presented in the previous wors [1,]. In this wors, the spectra were investigated with taing into account the electrostatic and spin-orbital splitting of the unfilled shells of equivalent electrons. The interaction of two unfilled shells was also taen into account. In this case, the linear transitions between two configurations are splitted into a large nuber of coponents. The validity range of these results is liited by the teperature up to T 0 ev. Such a teperature restriction is depends on a nuber of reasons. The approach of local therodynaic equilibriu (LTE) was used in these 3

4 calculations. This approach liits the validity range of odel in the teperature. q The ethod taing into account the splitting in spectra is liited by the d and q p configurations, where q is the nuber of electrons in configuration. At q teperatures higher than 0 ev the 4 f configuration is occurred. In the given wor the collisional radiative equilibriu odel (CRE) was developed with the q q account of the electrostatic and spin-orbital splitting for the unfilled p, d and q f shells [3]. Thus, the given odel has no restrictions on the validity in teperature and density. Below the detailed theory to tae into account the splitting is presented. Below in Sec. the probles associated with account of the electrostatic and spin-orbital splitting in an ato are discussed. Various typical radiative transitions are considered. Soe results of calculations in the LTE approxiation and a coparison with data of other authors are presented. The theory for calculation of therodynaic and optical characteristics of the heavy eleents in the CRE approxiation is discussed in ore details and nuerous results are shown in Secs The odel is applicable for the unfilled p, d and f shells. In Sec. 8, the statistical theory of radiation transfer applicable for coplex spectra of the heavy eleents is presented. Radiation fluxes calculated in spectral and statistical Lebesgue approxiations are copared.. Calculation of Optical Properties of Tungsten Plasa in LTE Approxiation In the LTE approxiation, the collisional processes (excitation and ionization) are only taen into account. All the radiative transitions are treated as negligible sall and are not taen into consideration. The validity range of this approxiation is liited by high densities and low teperatures. In our calculations, the upper teperature boundary was chosen about 0 ev. At the first stage, the calculations of optical properties were carried out in the LTE approxiation with the account of the splitting for the p and d shells. In this case, the structure of levels can be calculated with use of the perturbation theory. As the basis, the energy levels and wave functions obtained in the HFS or HF approxiation for the average ter are taen. Further, using these wave functions the agnitude of electrostatic splitting for the total set of L, S quantu nubers and also the spin-orbital splitting for all quantu nubers of the total oent J are calculated. The theory to calculate these splittings is relatively siple for the 4

5 unfilled q q q p, d shells, but is very coplex for the f shells. At the first stage, the p q, d q shells were only considered. The deterined structure of levels is used to calculate the ionization plasa balance and the populations of levels. In the LTE approxiation the proble is reduced to the solution of the Saha equations N i+ 1 N N i e U i = U + 1 i T e πh 3 I i exp T. Here, N i, N i + 1 are the concentrations of i th and i +1 th ions, U i, U i + 1 are the statistical sus of corresponding ions deterined by the expression U i M ax = g M = 1 M exp( E M / T ), where g M is the statistical weight of the level M, E M is the level energy, I i is the ionization potential of i -th ion, T is the plasa teperature..1. Features and Assuptions for Calculation of Optical Properties of Tungsten Plasa To solve soe applied probles the nowledge of therodynaical and optical properties of construction aterial plasas (for instant tungsten, iron, titaniu, etc) is required. To calculate the optical properties of these eleents in the LTE approxiation it is necessary to tae into account soe features, which are listed below. The self-consistent Hartree-Foc (HF) or Hartree-Foc-Slater (HFS) calculation ethods becoe ore coplex for heavy eleents because of the great nuber of electrons. The presence of unfilled d and f shells leads to decreasing of accuracy and increasing of iteration nubers. For the heavy eleents, the spin-orbital and electrostatic splitting can not be neglected in the calculations. In soe cases, the agnitude of the electrostatic splitting can exceed half an ionization potential fro the ground state. In this case, all the atoic levels have to be calculated independently with the procedure of self-consistency being perfored for each level. As the available experiental data on the atoic structure for the heavy eleents are fragentary, the sei-epirical ethods can not be used to calculate the optical properties. It is also ipossible to identify spectra of the heavy eleents. Significant difficulties appear when the plasa odels (for exaple CRE) are used for the heavy eleents because a lot of energy levels should be taen into account. For the heavy eleents the diensionality of the inetic equation syste, which have to be solved, increases significantly. A lot of ultipole and 5

6 intercobinative transitions, which are neglected for the light eleents, should be taen into account also. Rates of these transitions are not fully investigated and ethods for calculations of the are only estiating with low accuracy. In soe cases such processes as ipact excitation and ionization through auto-ionized levels, double auto-ionization should be taen into account also. There are a lot of splitted spectral lines in the spectra of the heavy eleents, which leads to significant difficulties in radiation transfer calculations. Therefore, the self-consistent ethod can not be directly used to calculate the atoic structure of the heavy eleents as well as the CRE odel to calculate the optical properties. In this case, soe assuptions and siplifications of the odels are done to decrease the aount of calculations. Below the used assuptions are listed. The spin-orbital and electrostatic splitting are fully neglected. The atoic structure is calculated using either the HFS ethod in the configuration assuption or the HF ethod in the average ion assuption. (Both ethods give the sae results). In this case, the specter consists of the isolated strong lines, which are the su of splitting coponents. This specter is siilar to the light eleent one. It should be noted that such a odel is a rough approxiation and can be used for ulti-ionized ions where the splitting is not so large. However, the use of these optical properties in the radiation transfer calculations can lead to rough istaes. The next approxiation is the statistical ethod, which taes into account the splitting [4,5]. In this ethod, the atoic structure is calculated either for the average ter or for the configuration with the splitting width being estiated. Further, it can be assued that line coponents are set out within the splitting width and are distributed in accordance with the noral law. Its relative intensities are also distributed in accordance with the noral law. The integral over the counter of the set of lines is equal to the line strength obtained using the average ion assuption. In effect, the group of line coponents is changed by the single line with the effective width being close to the splitting width. This odel is valid if a nuber of line coponents is large and the intensities of coponents correspond really to the noral law. Additionally, the distances between coponents have to be coparable with the line width, otherwise the averaged coefficients will be calculated with a low accuracy, for exaple, ean Rosseland coefficients will be overestiated. The reference q data [6] confir this conclusion. Using of this ethod is restricted by the f ters with 4 < q < 10. The next approxiation is the developent of previous odel for the p and d shells as well as for the f shells having a low nuber of electrons or for the closed f shells. In this case, the atoic structure is calculated using the average ter approxiation, but the splitting width and intensities of the 6

7 coponents are obtained directly using the quantu nubers of each coponent and the Racah techniques. Further, each coponent is taen, as an isolated line having own width and the specter is the su of all the coponents. This ethod is ore coplex then previous one but the ean Planc and Rosseland coefficients are obtained with a good accuracy. Authors developed the effective procedure to calculate all the above-entioned characteristics and specter construction. This ethod was used to calculate the absorption and eission coefficients of the tungsten plasa... Spin-Orbital and Electrostatic Splitting In this ite in the LTE approxiation the electrostatic and spin-orbital q q q splittings are considered for the p and d shells of the heavy eleents. The f shells are not taen into account. The electrostatic splitting is deterined by the dependence of energy on the total orbital L and total spin S oenta. The HFS ethod in the configuration approxiation is used to calculate the energy levels and wave functions. In this approxiation only one level, which is independent on the L, S quantu nubers of the ter, corresponds to each configuration, for exaple, to the ground state 5d 4 6s of the tungsten ato. Tungsten has the large charge nuber Z = 74, therefore the relativistic correction should be taen into account. In our odel, the perturbation ethod with the non-relativistic HFS wave functions is used to calculate the relativistic corrections. All the ain relativistic effects, ass-dependence on velocity, contact and spin-orbital electron interactions, were taen into account. A coparison with detailed Dirac-Foc calculations shows a satisfactory accuracy of this ethod. Thus, energy levels and wave functions are calculated using the configuration approxiation. The wave functions can be used to calculate the different integrals which does not contain the angle variable, for exaple, atrix eleents required to calculate the oscillator strengths, photo-ionization cross sections, broadening constants, etc. For exaple, the direct and exchange F and G Slater integrals deterining the electrostatic splitting scale as well as the constant ξ nl of the spin-orbital splitting are very iportant. Expressions to calculate these characteristics will be presented in Sec. 3. Beside these characteristics, the splitting width depends significantly on angle characteristics. Both the set of quantu nubers of ters and the genealogical structure of ters deterine the angle characteristics. In this case, the nuerical factor before the integrals is the coplex algebraic expression consisting of 3nj -sybols and genealogical coefficients. 3nj -Sybols allow us to calculate the total oentu as a su of the several oenta. The genealogical coefficient deterines the probability of ter foration when one electron is reoved fro the shell. There are direct and exchange angle coefficients too. They depend significantly on the orbital quantu nuber l and a nuber of equivalent electrons in the shell as well as whether an interaction is considered within the 7

8 shell or between different shells. There are a lot of expressions for all the cases. Because of an inconvenience of the, we don't list these expressions. In the practical probles all the above entioned situations are realized. Let us tae tungsten as an exaple. Line transitions of tungsten can be classified as follows 1. The transition fro the ground state 5d 4 6s - d 6s6 p There are unfilled inner shell 5d and closed outer shell 6s in the ground state. In this case, the interactions between the groups of d d, s s, s d electrons are calculated. Each of these transitions produces the own splitting structure and the resulting splitting is its superposition. The excited configuration has the following electron interactions d d, d s, d p, s p. The splitting structure is a su of the ground state splitting and the excited configuration with taing into account the selection rule for transitions in the quantu nubers. The total nuber of coponents can exceed several hundreds. It should be noted that 4 the structure of the 5d shell is not changed through transitions. This shell has the largest splitting scale but this splitting does not influence on the splitting of lines. The scale of the line splitting is deterined by the wea s s, s d interactions. All the transition coponents are located within the relatively narrow interval deterined by these two interactions.. The transitions between two excited configurations, for exaple, 5d 4 6s6 p- 5d 4 6s6d. 4 In this case the 5d and 6s shells are not changed through transition. The splitting scale is deterined by the outer electron. All the transition coponents are located within the narrow interval. 3. The transition fro an inner shell, for exaple, 4 5d 6s -5d 6s 6 p 3. 4 This case differs fro above entioned one. The splitting scale of the 5d 3 S and 5d shells are different. Moreover, there are transitions fro an each L ter 4 S1 3 of the 5d shell to several deferent L 1 ters of the 5d shell. Weight of the each transition is proportional to square of the corresponding genealogical coefficient. The specter of the transition consists of several separate groups of lines, each group being siilar to transitions of first or second type. Moreover, each group can be located in the different frequency interval and the intensities can significantly differ. 4. The transitions between two groups of equivalent electrons, for exaple, the transition f 5d - 4 f 13 5d for W 3 ion. This type of the transition is ost coplex. Soe coponents can have the large splitting and appear in the autoionization region in depending on the nuber of electrons in the shell, ion charge, etc. 8

9 In our odel we tae into account only first three cases and oit fourth one because it is not significant for tungsten. Fourth case can be significant for other heavy eleents, for exaple, uraniu. The special algorith was developed for all the possible quantu states of lower and upper configurations, for calculations of the angle factors, and the genealogical coefficients, which is included into the TOPATOM code. For tungsten, the spin-orbital splitting should be taen into account in addition to the electrostatic one. Because of large charge of the tungsten nucleus, they are coparable. It should be noted that the spin-orbital splitting has another eaning then for description of the relativistic corrections. In the latter case the single electron interaction between the electron spin and its own orbit is taen into consideration. In this case, the interaction between the total orbital and total spin shell oenta with the further splitting onto J coponents is taen into account. Here, there are also a lot of different situations. Magnitude and structure of the splitting which depend on whether the interaction within the shell or between different shells are considered. Classification of splitting depends on the splitting scale within each group of electrons. These groups are changed through a transition. Here, we don't list the classification because it siilar to the classification of the electrostatic splitting but soe significant features are pointed out. The splitting structure depends essentially on the coupling type in an ato. Tungsten is the eleent for which the pure type of coupling is not realized. In this case, the interediate coupling should be taen into consideration with diagonalization of an energy atrix for all the type of transitions being allowed. Such the procedure is used for a precision calculation with several levels and can not be used for total calculation because of the large laborious. In our odel ether the L L or J J type of coupling is used to calculate the splitting in dependence on which type of coupling doinates. Such a siplification does not ae worse the optical properties because the total nuber of the coponents and the splitting width reain the sae. However, the structure of line coponents is changed a little. If the distance between the coponents on frequency is less then the plasa teperature, then the ean absorption and eission coefficients are not changed. After a re-noralization, the specter becoes ore realistic. If soe experiental energy levels are used together with the experiental ionization potentials, then the experiental values of Slater integrals can be deterined and used to calculate the other coponents of the splitting. It should be noted that such the re-noralization procedure of the calculated energy levels is essential for neutral atos and low ionized ions when the HFS and HF ethods give the results with a low accuracy. The accuracy of both ethods increases with increase of the ionization degree. In this case, the relative scale of electrostatic splitting decreases and the re-noralization procedure is not required. On the other hand, the experiental data [7] exist only for a sall 9

10 nuber of ions of the heavy eleents, for exaple, for tungsten there are data only for W 1 and W ions. The above-described procedure is used to correct the spectra of the W 1 and W ions. Six ters for the ground configuration 5d 4 6s are listed in the table [7] for W 1 ion. Those are the following ters (in order of energy increase) 5 D, 3 P, 3 H, 3 G, 3 F, 3 D. Really, there are 16 ters for the ground configuration. A nown nuber of the ters is not sufficient to deterine all the energy characteristics. In our calculations the ionization potentials and the direct Slater integrals ( F, F, F ) are deterined only for the d shell. For this purpose, the values of energy levels of the 5 D, 3 P, 3 D ters are used. The siilar procedure is used for the ters of the ground configuration 5d 4 6s of W ion. For other ions, such the procedure is not applicable because of the absence of experiental data..3. Calculation of Ion Concentration and Optical Properties of Tungsten Plasa The CRE odel was used to calculate the ion plasa concentrations and level populations, i.e. the balance equations taing into account the nuerous collisional and radiative transitions were used. It is evident that the splitting of energy levels has to be taen into account in the inetic atrix. The nuber of inetic equations corresponding to the nuber of energy levels increases significantly due to the splitting. As it was entioned above, there are soe difficulties to calculate the rates of the collisional transitions. There is the following way to siplify this proble. The ratio between the radiation and collision rates depends significantly on the agnitude of the transition energy. The rate of spontaneous transition is proportional to E, where E is the transition 3 energy. The rate of collisional transition is proportional to E and E is proportional to Z, where Z is the ion charge. The resonance transition energy is very large for the light eleents. Its agnitude can exceed half of the ionization potential. For this reason, the equilibriu close to the coronal one is quicly established with the teperature and average plasa charge increase, when the radiative rates are higher then collisional ones. For the heavy eleents the situation is another when the splitting is taen into account. The energy structure is deterined by a lot of levels having sall value of E between the. This structure is still valid for ulti-ionized ions. Thus, the collisional transitions doinate upon the radiative ones in the wide teperature and density ranges, and ion concentrations and level populations can be obtained using the Saha-Boltzan assuption. These arguents are valid if the energy structure consists of the set of closed levels but they are not valid for inner electrons having a large energy scale without the splitting as well as for ions with closed shells. There is no splitting for the closed shells and the specter is siilar to hydrogen-lie one. For tungsten, 6 14 there are the following closed shells 5p, 5s, 4 f, etc. 10

11 For tungsten, the calculations of the ion concentrations were perfored in the teperature range fro 1 up to 0 ev and in a wide density range. Here, we are restricted by the Saha-Boltzan odel. To calculate the absorption and eission coefficient the procedure siilar that for the light eleents is used. The total absorption coefficient is the su of absorption coefficients fro all the levels taen with a weight deterined by the level population. Soe features of calculations that are typical for the heavy eleents are listed bellow. For the light eleents the absorption cross section is deterined by the oscillator strength, line broadening contour, and frequency. For the heavy eleents, it was assued that the su of oscillator strengths for the total coponent is equal to the oscillator strength of the transition between the configurations. Therefore, to deterine the oscillator strength of the coponents it is sufficient to calculate the oscillator strength of transition between the configurations using either the HFS or HF ethods. Relative intensities of the coponents are calculated using the siilar characteristics as for calculations of the splitting, that is the Slater integrals, genealogical coefficients, 3nj -sybols, etc. The expressions are not presented here due to its inconvenience. It is also assued that all the line coponents have the siilar broadening and the contours are the sae. In this case, it is also sufficient to calculate paraeters of broadening and to noralize the contour for the non-splitting transition between the configurations. The relative distance between the coponents is deterined by the splitting scale of the lower and upper configurations. Thus, due to the splitting a single transition between the configurations is splitted into a lot of coponents having the different intensities. All of the above entioned features refer to the continuu. The absorption in continuu (except the bresstrahlung) is deterined by the su of processes fro the levels. The absorption fro each level is deterined by the photo-ionization cross section, which can be calculated using either the HFS or HF ethod. The splitting of the levels leads to the splitting of the photo-ionization thresholds. The relative weight of each threshold is proportional to the probability of its realization. Here, the following assuption is ade. The photo-ionization cross sections of the coponents are equal and correspond to the photo-ionization cross section fro the configuration..4. Results and Discussion The above entioned ethod of calculations was realized as a coputer code which was used to calculate the ion concentrations of a tungsten plasa in the wide teperature and density ranges: teperature range fro 0.5 ev up to 0 ev, density range fro up to c -3 which cover the toaa disruption proble odeling. Validity liits of the used odel are restricted by the sae ranges. It should be noted that an investigation of the heavy eleent spectra is 11

12 very coplex proble. At present, there are not fully developed conceptions to solve this proble. Therefore, this report should be treated as the first step. As it was assued, the structure of absorption coefficients of the tungsten plasa is very coplex. In Figs.1-3, one can see the absorption coefficients of the tungsten plasa for several teperatures and densities. For the low frequencies ( E < T ) the absorption is deterined by the bresstrahlung and is a sooth function of the frequency. The ain part of the line transitions is non-uniforly located within the T < E < 10 T range with line intensities having a non-regular character. Soe parts of spectra include a lot of lines fro a quasi-continuu. For frequencies higher then photo-absorption fro the inner shells and are sooth functions of the frequency. As it has been entioned, above all the calculations were perfored using the Saha-Boltzan assuption. In this case, the eission coefficients β are 10 T the absorption coefficients are deterined by the associated with absorption ones κ by eans of the following expression β = κu p, where U is the Planc function. This expression is valid for frequencies up to p 10 T, i.e. for the ground and excited levels. However, it can not be used for the inner shells. The real eission coefficients have to be less then calculated ones using this expression. However, a axiu of the Planc function is located about 3 T. Therefore, the contribution of frequency range E > 10 T into the total eission is not so large and an error in eission coefficients will not leads to the large error in the total plasa eission. It is interesting to copare our result with others. Coparisons with data [8] are presented in Figs In wor [8] authors use the ethod close to that described in ite, i.e. the statistical ethod of the splitting was used. The resulting specter consists of the very broaden lines (transitions between configurations with the width being equal to the splitting one), bresstrahlung, photo-continuu. One can see that there is a good agreeent in the bresstrahlung, therefore ion concentrations are siilar. There is sufficient agreeent in the lines. The fors and aplitudes of the lines are different due to the different approach for taing into account of the line splitting. In our calculations, all the coponents with real width of the splitting were taen into account. Due to this reason, the lines have large aplitude in the centers and a quic decrease in the wings. Opposite, in the wor [8], the aplitudes of lines are not so large and there is not detailed structure of lines (see Figs. 4-5). To copare both results ore carefully we perfored averaging of our results over the spectral groups having the width coinciding with the splitting width. In result, we have obtained the spectra reinding that fro the wor [8] (see Fig. 6). The coplete agreeent can not be achieved because the averaging procedure of the detailed specter is not adequate to the procedure of statistic description of the splitting. A coparison of the absorption coefficients obtained using the coon CRE odel without the splitting with the results with taing into account the splitting is 1

13 presented in Fig. 7. There is a large difference in absorption coefficients that can leads to a large divergence in results of the radiation transfer calculations. Absorption and eission coefficients are usually used for calculations of radiation transfer in the plasa. The radiation transport in the lines is a coplex proble. For the adequate calculations of the radiation transport within a line the rather large nuber of spectral points have to be used. The detailed description of a line is not possible if there are a lot of lines in a specter. In this case, either the Planc or Rosseland group coefficients have to be used in the radiation transfer calculations. As it has been entioned above, only the detailed description of the splitting allows obtaining the ean Rosseland coefficients with a good accuracy. Either absorption or eission line spectra are soeties used to identify the individual lines or for the plasa diagnostics. However, calculated absorption tungsten spectra can not be used for these purposes due to its coplexity and there is not sufficient accuracy to deterine the individual line coponents. 10 Absorption coefficient (c -1 ) Fig.1 Tungsten plasa T=1 ev, N=10 17 c -3 13

14 10 Absorption coefficient (c -1 ) Fig. Tungsten plasa T=1 ev, N=10 17 c -3 Absorption coefficient (c -1 ) 10 4 Present data [8] Fig.3 Tungsten plasa T=10 ev, N=10 19 c -3 14

15 Absorption coefficient (c -1 ) 10 3 Present data [8] Fig.4 Tungsten plasa T=10 ev, N=10 17 c -3 Absorption coefficient (c -1 ) 10 3 Present data [8] Fig.5 Tungsten plasa T=10 ev, N=10 17 c -3 15

16 10 3 Present data [8] Absorption coefficient (c -1 ) Fig.6 Tungsten plasa T=10 ev, N=10 18 c CRE - odel with splitting CRE - odel without splitting Absorption coefficient (c -1 ) Fig.7 Tungsten plasa T=1 ev, N=10 17 c -3 16

17 3. Theory of Electrostatic and Spin-Orbital Splitting The atheatical technique of atoic physics and its terinology is very coplex. We shall briefly reind soe basic concepts to facilitate the understanding of further exposition. The independent variables in the Schrodinger equation of an ato can be divided. Thus, the radial wave functions can be found fro the HF or HFS equations [9,10]. The angular functions are found separately with use of the atheatical technique of suation of the electron oents. It should be noted that the orbital and spin oent suation in the quantu echanics is not coutative operation, i.e. the result of suation depends on the sequence of perfored operations. In principle, there are ( n )! ways to carry out the suation for n electrons with an orbital and spin oent. However, there are the ost probable ways of a oent suation. First, the oents with the greatest energy are sued, and then step-by-step the oents with saller energy is added in the order of their decrease. Such a ethod of oent suation was naed as the principle of genealogical schee. Depending on the ratio of energies of two electrons the different schees are realized: first, the suation with respect to orbital oents of all the electrons is perfored, second, all the spin oents is separately sued, and finally, the total orbital oent is added to the total spin one. It is the LS -coupling. The suation of the orbital electron oent with the intrinsic spin and then the consecutive addition of these oents are the JJ -coupling. The different interediate cobinations are also possible. In the case of low- Z eleents (up to Z 30 ) the LS -coupling is valid with a good accuracy. In the case Z 90, the JJ -coupling is valid with a greater accuracy. For the group of equivalent electrons in the q nl shell, all the electrons have an identical energy. In this case, the principle of genealogical schee is a poor approxiation. The ter with quantu nubers LS can be obtained in several ways. The LS ter is obtained by addition of one electron ls to the frae with quantu nubers L S : L S + ls = LS If the l value is large, then a nuber of variants of the L S 0 0 nubers is also large. The suation schee of equivalent electrons is deterined by a linear cobination of different variants of suation. The realization probability of that or other anner of suation is deterined by the genealogical coefficients. The suation of two groups of the equivalent electrons is carried out in the approach of genealogical schee. The calculation anners of genealogical coefficients and the classification of groups of the equivalent electrons are naed the Racah technique [11-13]. The transition coefficients fro one type of coupling to other 17

18 one are deterined by the values, which are naed 3nj -sybols. The 3 j, 6 j, 9 j and 1 j -sybols are used ore often. The sybol nuber denotes the nuber of oents and interediate suands. Below we use the approxiation of the LS -coupling and genealogical schee. The deviation fro this approxiation will be especially discussed. In the calculations of the spectroscopic characteristics, the HFS ethod is q used. The HFS ethod gives a possibility to calculate the energy levels of the nl configurations, where n is the principle quantu nuber, l is the orbital quantu nuber, q is the nuber of equivalent electrons in the shell. In the case of filled shells, the total orbital oent L and total spin oent S are equal to zero. In the case of unfilled shells, the variety of the L, S nubers is possible. The energy E ( n, l, q, L, S) of each ter depends on these nubers. The additional quantu nuber v is used to describe the d -shells. This nuber is necessary to differ soe ters with the sae LS. The nuber v defines the iniu value q for which the ter with given value L, S is occurred for the first tie. The additional quantu nubers U and W are introduced to describe the f -shells [13]. The sense of these nubers has not the siple explanation as v. The exact value of the energy level is deterined by all the quantu nubers n, l, q, L, S, v, U, W. It is necessary to use the ulti-configuration Hartree-Foc (MHF) ethod for each set of quantu nubers to calculate the characteristics of energy levels. In the case of the high- Z eleents, the spin-orbital splitting becoes coparable with the electrostatic one. In this case, it is necessary additionally to use the total oent J. For the d 5 -shell a nuber of possible ters is N = For the 7 f -shell, this nuber reaches N = 119 (without taing into account the splitting in J ). The interaction of two unfilled shells produces a nuber of ters N * N 1. Thus, the use of the MHF ethod is associated with a very large aount of wor. The opportunity to use the MHF ethod with q q reference to the d and f -shells is still poorly investigated. In the present wor, the ethod of the perturbation theory was used. The q following assuption was ade. All the ters of the nl -shell have the sae radial wave functions. The radial wave functions are calculated using the HFS-ethod. The angular wave functions are different that for different ters. In this case, the interaction energy inside this shell is deterined by the forula [10,13] 18

19 E( n, l, q, L, S, α) = F ( nl, nl) f ( l, q, L, S, α). Here, α is the set of quantu nubers deterined by the forula v, U, W. F is the Slater integral F ( n1 l1, nl) = Pn 1l1( r) Pn l ( r1 ) P 1 n1l1( r) Pn l ( r1 ) drdr + 1. r r Here, P n1l1 and P nl are the HFS radial wave functions for the n 1 l 1 and n l quantu states. The r and r values are the saller r and greater r 1 radiuses of electrons, respectively. Index deterines the ultiplicity 0 l. The value f ( l, q, L, S, α) is the angular ultiplier, which is deterined by the ethods of the Racah technique. Below the ethod to calculate these values is described in ore details. The interaction between the shells is described by [13] E ( n1l 1q1L1 S1α 1; nlqlsα ) = F ( n1l 1, nl) a ( n1l 1q1L1 S1α 1; nlqlsα ) + G ( n1l 1, nl) b( n1l 1q1L1 S1α 1; nlqlsα ). in Except the Slater integral F there is the exchange integral G G ( n1 l1; nl ) = P ( r) Pn l ( r1 ) P 1 n1l1( r1 ) Pn l ( r) drdr1, n1l1 r + r l +, 1 l l1 l where a and b are the angular functions of interaction between the shells. Many physical values are calculated using the HFS radial wave functions. It is the F and G integrals, oscillator strengths for transitions between the n1l 1 nl configurations, the photoionization cross-section. These characteristics are calculated in the first part of a proble. Then, these characteristics are stored in the special ban of spectroscopic data for all the type ions of the plasa. The f, a and b values depend on all the set of quantu nubers. The calculation of these values is the ost difficult proble. These values are calculated in the second part of proble together with of the absorption and eission spectra Calculation Methods for f, a, b Characteristics 19

20 The f a, b, functions are expressed by coplex algebraic forulas. These forulas contain soe standard functions of the theory of coplex spectra. Those are genealogical coefficients, reduced atrix eleents of the spherical function, subatrix eleents of the U and V tensor operators, 3nj -sybols. Let us describe these functions LS α the genealogical coefficient G L1S1α 1 deterines the realization probability of the ter with the L S quantu nubers when the electron is reoved fro the shell with the L S 1 1 quantu nubers. The genealogical coefficients are q found as a result of solution of the syste of algebraic equations. For the p, q d, f f and f f -shells the genealogical coefficients are contained in the tables and are used to calculate the ter splittings. The atrix eleents deterined by eans of integrals fro different wave functions should also depend on the agnetic quantu nubers. The nuber deterines a value of oent projection on the given axis. For atrix eleents of an ato the division of variables with the l and quantu nubers (the theore by Eccart-Wigner) is possible. The atrix eleents which don't contain the dependence on is referred below as the reduced atrix eleents [13]. the reduced atrix eleents of the spherical l C l ) function were calculated ( 1 for a large nuber of orbital nubers l and the ultiplicity, and are contained in the tables. the 3 nj sybols are calculated by eans of algebraic ethods. The 3 j, 6 j and 9 j sybols are usually used. In the given wor, the calculation ethod of all these values was realized. q q q 1 q the subatrix eleents of the l LSα U l L S ) and l LSα V l L S α ) ( 1 1α 1 ( tensor operators deterine the interaction between the different ters of one configuration. These values are expressed by coplex algebraic forulas. These forulas include the genealogical coefficients, the 3nj -sybols, and the reduced atrix eleents of the spherical function. In the given wor, the calculation of these values was also realized. Taing into account the above discussion the expression for the interaction energy inside the shell can be presented by the following forula [13] f q ( l C l) 1 l + 1 = ( l (l + 1) q L + 1 L1S1 q q LSα U l L1S 1α 1) 1. 0

21 The a and b functions have the siilar structure but they are very coplex. All these functions are evaluated during the calculation process of the splitting value. The interaction energy f depends on all the set of quantu nubers and the ultiplicity. Thus, the nuber of the f, a, b characteristics is deterined by a nuber of sets of the quantu nubers. The relative value of splitting is deterined by the difference of these values. The for of f, a, b functions depends on the coupling type between the electrons. All the presented forulas are written for the LS -coupling. The siilar forulas can be written in the extree case of JJ -coupling. These forulas are ore coplex for an interediate type of coupling. In the given wor, the splitting for the interediate type of coupling was studied approxiately. We have used a linear cobination of two extree types of LS - and JJ -coupling. 3.. Calculation of Spin-Orbital Splitting The value of spin-orbital splitting for the high-z eleents is coparable to the electrostatic one. For the ato with one extra electron outside the filled shell, the value of spin-orbital splitting is deterined by the following forula 1 du ξ nl = Pnl ( r) dr r dr, where U is the potential inside the ato, P nl are the radial wave functions. In the given wor, the HFS wave functions are used. The ξ nl values are calculated for all the states and are contained in the ban of spectroscopic data. In the case of LS -coupling for the unfilled shells with the equivalent electrons the splitting is deterined by the Lande rule [10] 1 E J = A( L, S, α ){ J ( J + 1) L( L + 1) S( S + 1) }. The Lande constant A ( L, S, α) for one unfilled shell is deterined as follows A( l q, L, S, α) = ξ nl l( l + 1)(l + 1) ( l S( S + 1)(S + 1) L( L + 1)(L + 1) q LSα V 11 l q LSα) The interaction of two shells is deterined by the coplex algebraic forula containing the A values for different shells. In the case of JJ -coupling the value of splitting is deterined by the forula 1 3 EJ = ξ nl j( j + 1) l( l + 1), 4 1

22 where j = l ± 1. In the case of interediate coupling, it is also possible to use a linear cobination of two above forulas Nuerical Calculation of Splitting In the previous ites, the ethods for calculation of the electrostatic and spin-orbital splitting were presented. The splitting of spectral line is deterined by the splitting of the lower and upper levels. The described above technique was realized for the p q q , d -shells, f f -shells and f f -shells. The 5 9 realization of this technique for the f f -shells is ore difficult. It deterined by the several reasons. The ain difficulty is the calculation of the genealogical 5 9 coefficients for these shells. The spectra of the f f -shells are uch ore coplex than the spectra of the f and f shells. The spectra of the f and 10 f shells contain a very large nuber of the spectral coponents. The intensity distribution of the coponents is close to the noral law. In this case, it is 5 9 possible to assue that the spectra of the f f -hells will be also distributed in accordance with the noral law. In this case, it is necessary to deterine the interval between the inial and axial coponents of the splitting, the nuber of coponents, and to distribute these coponents in accordance with the 5 9 noral law. This procedure was realized for the f f -shells. The splitting calculation is perfored siultaneously with the calculation of the absorption and eission spectra. The calculation of the a and b characteristics is the ost difficult. It should be noted that the existence probability of two unfilled shells in a ulti-charge ion is very low if there is ore than one electron in both shells. For the neutral tungsten ato the 4 5d 6s, 5d 4 6 p and 5d 4 6s6 p states are only realized. For the tungsten ions, the nl q n l 1 1 states are usually realized. In the second unfilled shell, there is only one electron. In this case, the expressions for the a and b coefficients are significantly siplified. In the given wor for all the coplex shells the preliinary calculation of the a and b characteristics is perfored and they are stored in a databan. In a case of one extra electron outside the shell, these calculations are perfored siultaneously with the calculation of the absorption spectra. The splitting of the energy levels are taen into account at calculation of the wavelength of a spectral line and it relative intensity. The splitting is taen into account at deterination of the photoabsorption threshold and level population. 4. Calculation of Relative Line Intensities with Taing

23 into Account the Splitting In the previous sections, the values of the splitting energy were calculated. The total intensity of transition between the configurations is also distributed between the spectral coponents. The value of line intensity is deterined by the oscillator strength of the transition between nl and n 1l1 states [10] E l f ( nl, n1l1 ) R 3h l + 1 R = P r) rp ( r dr. ax =, nl ( n1l1 ) Here, E is the transition energy, is the electron ass, h is the Planc constant, l ax is the axiu of two nubers l,l1. The radial atrix eleent R of the transition is a function of the integral fro the P nl and P n1l 1 radial wave functions. In this case, the HFS wave functions are used. The given value of the oscillator strength is a su of all the coponents. To calculate the specified coponent it is necessary to ultiply the oscillator strength f by the factor [10] L0 + 1 ( G LS L0S 0) Q( SLJ; SL1 J1) Q( L0lL; L0l1L1 ) L + 1 LS Here, GL 0 S 0 is the genealogical coefficient of the shell with the l, L, S, J quantu nubers fro which the transition is occurred. After the electron transition, the residual shell has the L,S 0 0 quantu nubers. In the final state, the syste "residual shell+exited electron" has the l 1, L1, S, J1 quantu nubers. The Q functions are expressed by eans of the 6 j -sybols [10]. (J + 1)(J1 + 1) LJS Q ( SLJ; SL1 J1) =. S + 1 J1L11 Here, LJS J 1 L 1 1 is the 6 j -sybol. The first Q -factor describes the spin-orbital splitting of the intensity coponents. The second Q -factor describes the electrostatic splitting of the intensity coponents. For the intensities of transitions, the selection rules of dipole radiation are used: l = ± 1, L = 0, ± 1, = 0, ± 1 S = 0. J, 3

24 5. Calculation of Charge States and Populations of Levels To calculate the charge states and the population of levels the collision-radiative approxiation is used. In this case, it is necessary to solve the syste of linear algebraic equations with the nuber of equations equal to the nuber of levels. With taing into account the electrostatic and spin-orbital splitting, each nl state can have several tens or hundreds of levels. The total nuber of levels in an ion can reach several thousands. The solution of such the syste of equations is a coplex proble. The radiative dipole transitions between the coponents of one nl configuration are forbidden by the selection rules ( l = 0) but the collision transitions are allowed. Thus, if to neglect by the ultipole radiation, then the Boltzan equilibriu is reached between the splitted coponents of one nl configuration. In this approxiation, it is possible to calculate the charge state and the relative populations of levels between the nl configurations. The Boltzan equilibriu is reached between the specified configurations. In this case, the coon collisional-radiative odel is used. However, there is the essential difference. The transition rates between the nl configurations are the integrated values consisting of a large nuber of the transition coponents. If two configurations have the N 1 and N coponents of levels, then the total nuber of coponents of the collisional transitions between the configurations will be about N * N. Each ion has about nl n l 1 1 transitions. The proble is solved by eans of the ethod of iterations and such a ethod requires an enorous aount of coputer tie. It is necessary to enter further siplifications. Let us consider the ost probable case. There is one extra electron outside the shell. Let it is the 4 f 4 4 5p configuration. The 4 f -shell has 47 ters with the different values of the LS α quantu nubers. At addition of the 5 p -electron, each of these ters fors the new syste of splittings (for the ajority of ters - 6 levels). Hence, the total nuber of levels in this configuration will be ore than 50. Siilarly, it is possible to deterine the nuber of levels for the 4 f 4 5d configuration. It is about 450. The total nuber of coponents of the f 5 p 4 f 5d collisional transitions is about For the accurate account of these transitions, it is necessary to tae into account all 10 5 coponents. It should be noted that the splitting of the 4 f 4 5p configuration has two 4 different scales. The splitting scale of the 4 f shell can be large and coparable with the value of ionization potential. The scale in any respects depends on the 4

25 effective ion charge. The splitting associated with the interaction of the LS α ter with the 5 p electron has the considerably saller scale. At the 4 transition, the LS α quantu nubers of the 4 f -shell are not f 5p 4 f 5d changed. The lengths of waves of all 10 5 transitions will differ by an insignificant value. In this case, it is sufficiently to find the average energy of transition and to choose the characteristic transition rate for all the coponents. 3 4 f For the 4 f 5d or 4 f 5p 4 f 5p5d transitions the situation is uch ore difficult. In this case, the scale of the line splitting is directly deterined by 4 the splitting scale of the 4 f -shell. In this case, it is necessary in details to tae into account all the coponents. In the given wor, the following ethod to solve above a proble is suggested. Let us consider the typical transition rate of the collisional excitation, which is described by the Van Regeorter forula [14] 3. 7 Ry E vσ 1 = f1 exp( E ) p( E ). E T T T 1 The v σ 1 transition rate between the states 1- is proportional to the corresponding oscillator strength f 1, and depends also on the ratio between the transition energy E and the teperature T. Especially the strong dependence on the ratio E T is contained in the exponential function ( P is a slowly varied function). It is obvious that at the large scale of the splitting the σ value is significantly differed fro the v ( E) v ) ( 1 E i i σ 1 value. The first value is the average rate of the collisional transition. The second one is the rate as a function of the average energy of transition. We shall denote δe as the interval between the inial and axial coponents of the splitting. For each set of the E and δ E values it is possible to create the table of the corrections between the v ( E ) σ values for a wide range of E T ratios. In this case, it σ 1 i and v 1 ( E i ) is possible to use the integrated ( ) 3 4 f 4 f 5d v 1 E i σ ratios of transitions, for exaple, the 4 configurations, with the corresponding correction. In the given wor, the calculations were carried out with use of the correction function described above. The exaple of the correction function for the collisional discrete transitions was presented above. The siilar functions were created for the other ratios: collisional ionization, photorecobination, and spontaneous transitions. 5