J Dembczy

M Elantkowska

B Furmann

J Ruczkowski

Stefa Ńska

Critical analysis of the methods of interpretation in the hyperfine structure of free atoms and ions. Case of the model space (5d + 6s) 3 of the lanthanum atom

Keywords: numbers: 32, 10, Fn

Critical analysis of the methods of interpretation in the hyperfine structure of free atoms and ions: case of the model space (5d+6s)3 of the lanthanum atom

• T (K) n describes the partition of the observed hyperfine splittings into the contributions of ranks K=1,2 and 3 within the experimental accuracy, while the operator

T (κk)1 e • T (1) n = µ 0 µ B 2π N X i=1 » b l i ˙r-3 ¸01 - √ 10
" b s i b C 2 i " (1) ˙r-3 ¸12 + b s i ˙r-3 ¸10 - • T

Introduction

On the basis of our earlier works, both experimental and theoretical [1][2][3][4][5][6], as well as work of another author [START_REF] Büttgenbach | Hyperfine Structure in 4d-and 5d-Shell Atoms[END_REF], we feel qualified to submit a proposition, that the existing theories (e.g. [START_REF] Wybourne | Spectroscopic properties of Rare Earths[END_REF][START_REF] Woodgate | [END_REF][10][11][12]), which provide a semi-empirical description of the hyperfine interactions in the structure of a complex atom, are not sufficiently precise to yield a correct description and interpretation of the measured hyperfine splittings within the up-to-date experimental accuracy. Also the results of ab initio calculations for the hyperfine structure (hfs) parameters, such as magnetic dipole interaction constantA and electric quadrupole interaction constant B, usually differ, in some cases substantially, from their experimental counterparts [13,14].

The main aim of this work is, among other things, an attempt to answer the following questions:

1. Is it possible to theoretically describe the hyperfine interactions within an atom with an accuracy comparable to the accuracy achieved in experiment?

2. What is the main reason of the observed discrepancies between the experiment and the theory: the applied approximations in the analysis of the atomic structure or incorrect experimental data?

3. Are the commonly applied assumptions concerning the mechanism of interaction between the nucleus and the electron shells (e.g. regarding the nucleus as a point magnetic dipole and the assumption that the quantum number I, which describes the angular momentum of the nucleus, is a "good" quantum number) correct?

It was proved [15][16][17][18][19] that the main source of discrepancies mentioned are difficulties with a precise description of the interactions of configurations system investigated with distant electron configurations originating from excitations of one or more electrons from open to empty shells or from closed to open shells. Moreover, Fenuille and Armstrong [20] showed, that correlation and relativistic effects in atomic hyperfine structure are of additive nature, and that effects of configurations interaction (contributions of the second order perturbation theory) can be treated within effective operator approximation, and thus they proved their SL-dependence. Thus one may expect, that a precise definition and development of a mathematical formalism, which yielded a complete description of these effects, should explain the observed discrepancies.

Another problem may be insufficient precision of determination of eigenvectors amplitudes describing particular electron states. The wavefunctions are determined on the basis of the atomic fine structure ( fs) investigations. In this case availability of verified experimental data concerning energies of electronic levels combined with a correct assignment of J quantum number is a crucial point. Scarcity of experimental data has to be considered a severe shortcoming, if the structure of an atom is regarded in multiconfiguration approximation and all important theoretically predicted interactions are to be included. A serious problem constitute the erroneous data concerning energy values of electronic levels -sometimes energies of nonexisting levels are given in literature (e.g. [21]), or incorrect J values are assigned to correct energy values (a critical analysis of some data of this kind was performed by Windholz [22]).

Another factor to be blamed for existing discrepancies is in some cases the method of determination of the hyperfine structure constants A, B, C and D from the measured hyperfine intervalsa simplified description of "repulsion" effect for the hyperfine structure sublevels with the same values of quantum number F yields a substantial distortion of the information concerning quadrupole and higher order interactions. This was first noticed by Casimir [START_REF] Casimir | On the Interaction between Atomic Nuclei and Electrons (Teyler's Tweede Genootshap[END_REF], who referred to them as "pseudoquadrupole" or "pseudooctupole" interactions. Therefore, in order to guarantee the correctness of determination of the hyperfine structure constants A, B, C and D in our approach, described below, a segment called "diagonalization of the hyperfine structure energy matrix" within the basis of states Ψ(conf iguration, vSLJF ) was introduced. In this way we eliminate one of the possible sources of discrepancies.

In our group a procedure (schematically outlined below, Figure 1) of analysis of experimental data concerning the structure of complex atoms was developed. In the updated procedure the segment "parametrization of one-and two-body hyperfine interactions" was improved with respect to the earlier applied method [2,[START_REF] Dembczyński | [END_REF]25]. Now we include all one-and two-body contributions to hfs constants A and B which exhibit a dependence on the quantum numbers SL as well as on N (the number of equivalent electrons in an open shell) and which originate from the excitations "open shell -empty shell" or "closed shell -open shell". Our earlier method [START_REF] Dembczyński | [END_REF] was developed for the system (3d+4s) N , and thus excitations of the kind "closed n 0 d shell -open nd shell" are not occur.

In the lanthanum atom the configurations system (5d+6s) N is well separated from the excited configurations. The closest excited configuration 4f6s6p begins at ca. 29000 cm -1 . Configurations originating as a result of "closed shell -open shell" excitations (core-excitations) certainly lie much above 100000 cm -1 and have not been observed in lanthanum yet. Thus in the case of the lanthanum atom it can be assumed that the configurations system considered (3d+4s) N is well isolated from any disturbing configurations and the conditions for application of the perturbation theory are fulfilled. It yields an excellent possibility of an alternative analysis of the contributions mentioned within the second order perturbation theory according to the excitation model either "open shell -empty shell" or "closed shell -open shell". A simultaneous application of both models is not possible because of the fact that in both models implicit linear dependence between angular coefficients corresponding to the certain radial parameters have to occur, which make the solution of a redundant set of linear equations impossible and thus hinder the determination of the respective radial parameters. It provides an excellent test confirming the correctness of the complex formulae derived, which in the case of consideration of e.g. configurations with three open shells require re-coupling of five or more angular momenta and strict observance of the electron permutation rules, in particular for interconfiguration matrix elements. When both the fine and the hyperfine structure are considered independently within the frame of both excitation models, two independent sets of radial parameters describing the atomic structure are obtained. On the basis of the theoretically predicted relations between the radial parameters originating from both models one can prove the correctness of the obtained description of the atomic structure, as well as precisely define the information provided by the radial parameters determined from the experimental data. In Section 5 the relations are given, which allow to recalculate the parameters obtained for the model space (5d+6s) 3 within both excitation models: "closed shell -open shell" (c-o) and "open shell -empty shell" (o-e) into the radial parameters characteristic of the individual configurations 5d 3 , 5d 2 6s and 5d6s 2 .

In this analysis we apply experimental data available in the literature, in particular those obtained with Rabi method [26] and the method of double laser-rf resonance [13], as well as the results obtained within this work. In our experimental investigations it was not possible either to confirm the existence of the electronic level with the energy 25414.63 cm -1 and the value of quantum number J = 5/2 or to measure its hyperfine splitting; we also did not manage to find the level with the energy ca. 25752 cm -1 and the value of quantum number J = 3/2. For the model space (5d+6s) 3 the information concerning the hyperfine structure of the levels 5d 3 2 D with J = 3/2 and J = 5/2 is still missing; however, in spite of the lack of these data we managed to describe quantitatively the one-and the two-body interactions in the fine and the hyperfine structure of the electronic levels which form this system.

Experimental details

Measurement of the hyperfine structure constants for electron levels belonging to the configuration 5d 3 of the lanthanum atom is a difficult task, first of all because of the lack of strong spectral lines involving those levels. In the commonly available tables of spectral lines [27] in the visible and near infrared regions no such lines can be found. Transitions to the levels belonging to configuration 5d 3 , which could be conveniently excited with commonly available tunable lasers, involve odd-parity levels with energy values within the range 32000-40000 cm -1 . However, most such levels belong to configurations differing from the configuration 5d 3 with states of two electrons. Because of selection rules electric dipole transitions in such a system of configurations can only occur as a result of "mixing" of the configurations in question with other configurations, for which electric dipole transitions to configuration 5d 3 are allowed. Since the admixtures of such configurations are usually minor, the resulting spectral lines are, as a consequence, very weak. In this situation the only chance of successful application of the experimental method based on laser induced fluorescence or optogalvanic spectroscopy consists in recording of the hyperfine structure for possibly great number of the strongest among the weak spectral lines and the reduction of experimental errors through statistical evaluation of the results.

In our two experimental works, which have recently appeared in J.Phys.B [28,29], we presented results of the measurements of the hyperfine structure constants for several hundreds electronic levels of both parities in the lanthanum atom. In those works also results of the measurements of constantsA for 13 levels belonging to the configurations analyzed in the present work were included; these results were obtained for the first time. Since in the work [28] the values of constants A for several hundreds odd parity levels with energies above 33000 cm -1 were also presented, it was possible to extend the investigations for the 13 levels mentioned, belonging to the model space, through the measurements of a greater number of spectral lines including them. Such an extension was performed within experimental investigations in the present work.

With the use of laser induced fluoresecence or optogalvanic method the hyperfine structure of 71 spectral lines involving the levels investigated was recorded. In the case of 27 of the recorded lines both the signal to noise ratio and the observed hyperfine splitting enabled determination of the constantsA with experimental uncertainty below 7 MHz. A compilation of those lines is presented in Table 1. In the case of the odd parity level with the energy E=33799.23 cm -1 , which iis the upper level of the investigated spectral line with the wavelength λ = 594.797 nm, it proved necessary to change the assigned quantum number J from the value of 5/2 to 3/2, since both the calculations of the hyperfine structure (including also the observed number of components) and the presence of such a line (the lower level hasJ=1/2) indicate, that the value J=5/2 encountered in the tables [START_REF] Martin | Atomic Energy Levels -The Rare Earth Elements[END_REF] cannot be correct. This question was already discussed in the work [28].

An additional verification of the results obtained was recording of the remaining 44 spectral lines, which were too weak for precise determination of the constantsA, but their profiles allowed us to confirm the earlier determined values and in particular to eliminate the doubts concerned with the fact that for equal J values for both the lower and the upper level the change of the signs of both constantsA along with the interchange of their values yields the identical pattern of the hyperfine structure of the line.

In the measurements an experimental setup described in the previous works [31][32][START_REF] Furmann | Investigation of electron levels of free atoms and ions of lanthanum, praseodymium and europium with laser spectroscopy methods[END_REF][START_REF] Furmann | [END_REF] was used. This setup was earlier successfully applied in our lab in investigations of praseodymium [35][36][37] and lanthanum [38]. The source of the exciting radiation was a ring dye laser -a modified version of Coherent model CR 699-21. Because of the spectral regions applied the laser was operated in turn with several sets of optical elements with the use of the following dyes: Rhodamine 6G (565-615 nm) and DCM (615-650 nm). According to the width of the spectral line investigated, the dye laser frequency was precisely tuned over the range covered with the line's hyperfine structure, 5-40 GHz. Several elements were used for dye laser frequency control: a wavemeter (Burleigh, model WA-1500), an iodine cell and a mode analyzer. Along with LIF signal a transmission of a frequency marker (a Fabry-Perot interferometer with FSR=1497 MHz) was recorded.

The source of lanthanum ions was a discharge in a hollow cathode lamp. Fluorescence light from the discharge was collected and focused with the mirrors and lenses systems onto the entrance slit of a monochromator (SPM-2). Behind the exit slit a photomultiplier with a preamplifier was placed. Electric signal from the photomultiplier was fed to a lock-in amplifier. As a reference signal for phase-sensitive detection a signal from a mechanical chopper, placed in the laser beam, was used. Exit signal from the lock-in amplifier was A/D converted and recorded by the computer. Determination of the frequency scale on the basis of frequency marker signal and calculation of the values ofA and B constants for the levels involved in the transition investigated were performed with the use of a program Fitter, which was earlier used for such purposes both in our laboratory and elsewhere.

As a result of the investigations the values of the constants A presented in the works [28,29] were confirmed for all 13 levels belonging to the model space. in the case of two levels 5d 3 4 P 1/2 with the energy 16617.30 cm -1 and 5d 3 2 D 3/2 with the energy 18037.64 cm -1 the experimental uncertainties were reduced.

Diagonalization in SLJF basis and determination of the constants A, B and C

The analysis of the hfs of the even configurations of La atom was performed in the basis of 3 configurations, taking into account all possible interactions predicted by many-body fine structure theory. In order to include the J-off-diagonal effects in the hyperfine structure, direct diagonalization of the matrix containing J-diagonal as well as J-off-diagonal elements has to be performed (in the basis of states Ψ(conf iguration, vSLJF )). The effect of "repulsion" of the hyperfine structure sublevels with the same values of quantum number F , first noticed by Casismir, yields a distortion of the obtained values of the hyperfine structure constants of quadrupole and higher order interactions. Diagonalization of the hyperfine structure energy matrix within the basis of states Ψ(conf iguration, vSLJF ) takes the effect referred by Casimir to as "pseudoqudrupole" interaction into account. It requires precision up to 16 significant digits. The diagonal part of this matrix consists of coefficients corresponding to the particular components of the energy of a hyperfine structure sublevel E F : the center of gravity of the hfs energy W J and the experimental hfs constants A, B, C and D. These parameters can be treated as free in the iterative fitting procedure of the experimental and the calculated hfs energies. The differences between E F and E F ±1 values are equal to the experimentally determined hyperfine structure intervals. Values ofJ-off-diagonal hfs matrix elements are fixed.

As a result, we obtain improved values of the hyperfine structure constants, which in turn can be used to determine the radial hfs parameters. The final values are usually obtained after several iterations.

The accuracy of the experimental hyperfine structure intervals [13,26] allows to determine onlyA, B and C constants. If we treat these constants as free parameters, we obtain the agreement between the observed and the calculated interval values within the experimental error (see Table 2). The above result confirms, that the operator

H hfs = 3 K=1 T (κk)K e • T (K) n , (1)
where T (κk)K e and T (K) n denotes the electron and the nuclear operator, respectively, yields a complete description of the observed hyperfine structure splittings, and the constants A, B and C obtained in this procedure are the "true" values, describing the interactions with ranksK =1, 2 and 3.

In Table 3 a considerable "repulsion" effect can be observed for the levels 2 G 7/2 and 2 G 9/2 , for which the constants B were changed as a result of diagonalization by44.05 MHz and 74.40 MHz, respectively.

Parametrization of the configurations interaction effects

The fine structure energy matrix was constructed in the way analogous to the one described e.g. by Cowan [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF]. Moreover, our energy matrix is extended by the elements taking into account electrostatic coupling and electrostatically correlated spin-orbit coupling between the configurations of the system considered and the distant configurations. Generally, for the configurations containing up to three open electronic shells, these matrix elements origin from the second order perturbation theory and can be schematically expressed as follows:

- ψ1 =ψ,ψ [ ψ|G|ψ 1 × ψ 1 |G|ψ ] /∆E = -(angular part) × (radial part) , (2)
where:

ψ = (n 0 l 0 ) 4l0+2 1 S, (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL ψ 1 = (n 0 l 0 ) 4l0+1 2 L 0 , (n 1 l 1 ) N1+1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL ψ = (n 0 l 0 ) 4l0+2 1 S, (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL (3)
for "closed shell -open shell" excitations, and

ψ = (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL ψ 1 = ((n 1 l 1 ) N1-1 S 0 L 0 , n l ), S 1 L 1 (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL ψ = (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SL (4)
for "open shell -empty shell" excitations.

The angular coefficients result from the coupling of angular momenta of the operator G. Radial parameters have denotations which code the interacting configurations and specify the interactions. Denotations of particular radial parameters take the form:

P t (n i l i n j l j , n 0 l 0 n i l i ) P t (n i l i n j l j , n 0 l 0 n i l i ).
In order to specify particular interactions more precisely the symbol P t (where t stands for the order) is replaced respectively by D t in the case of direct interactions, E t for exchange interactions, or R t for interactions involving two equivalent electrons. In our procedure excitations of one or two electrons from a closed shell to all open shells are considered, under the following conditions:

|l i -l 0 | = 0, 2
and

N 1 + N 2 + N 3 = N 1 + N 2 + N 2 (5)
Contributions from the second order perturbation theory for electrostatically correlated spin-orbit interactions (CSO) as well as for electrostatically correlated hyperfine interactions (CHFS), are defined as follows:

- ψ1 =ψ,ψ ψ|G|ψ 1 × ψ 1 |T (κk)K |ψ + ψ|T (κk)K |ψ 1 × ψ 1 |G|ψ /∆E = = -(angular part) × (radial part) , (6)
where:

ψ = (n 0 l 0 ) 4l0+2 1 S, (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SLJ ψ 1 = (n 0 l 0 ) 4l0+1 2 L 0 , (n 1 l 1 ) N1+1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; S L J ψ = (n 0 l 0 ) 4l0+2 1 S, (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; S L J (7)
for "closed shell -open shell" excitations, and

ψ = (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; SLJ ψ 1 = ((n 1 l 1 ) N1-1 S 0 L 0 , n l ), S 1 L 1 (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; S L J ψ = (n 1 l 1 ) N1 S 1 L 1 , (n 2 l 2 ) N2 S 2 L 2 , (n 3 l 3 ) N3 S 3 L 3 S 4 L 4 ; S L J (8)
for "open shell -empty shell" excitations. In these cases also the angular coefficients result from the angular parts of the operators: a two-body electrostatic interaction operator G and a one-body operatorT (κk)K ; the latter may represent either spinorbit (H so ) interaction or a hyperfine (H hfs ) interaction.

The radial parameters P t (n i l i n 0 l 0 , n i l i n i l i ) P κk (n 0 l 0 , n i l i ) specify the coupling between configurations. The radial integral P t (n i l i n 0 l 0 , n i l i n i l i ) describes electrostatic coupling of the configurations, i.e. it specifies the electrons involved and the type of their interaction. In the actual description the symbol P t is replaced, as above, with D t or E t , respectively. P κk is the radial part of a one-body operator T κk , which couples the electrons n 0 l 0 and n i l i . In our procedure excitations of one electron from the closed shell (n 0 l 0 ) 4l0+2 to each of three open shells (n 1 l 1 ) N1 , (n 2 l 2 ) N2 and (n 3 l 3 ) N3 are considered. In the case of the spin-orbit interaction the symbol P κk is replaced by ζ.

The above description of the parameters follows is a simplified notation. The parameters are actually defined as sums over all closed (or open) shells. For example, in the case of the lanthanum atom:

E 2 (n 0 s5d, 5d6s) P 10 (n 0 s, 6s) = 4π 5 n0=1 Ψ 6s (0)Ψ n0s (0) E 2 (n 0 s5d, 5d6s)/∆E, (9)
where ∆E is the energy difference between the relevant closed-and open-shell orbitals.

In the case of CSO the following condition has to be fulfilled:

|l 0 -l 1 | = 0 and κk = 11, K = 0. (10)
The matrix elements determined from relations (2) and ( 6) under conditions ( 5) and (10) were included in the fine structure energy matrix (Figure 1).

In the case of CHFS the following relations hold: for magnetic dipole interactionsK = 1:

|l 0 -l 1 | = 0 if κk = 01, 10 and |l 0 -l 1 | = 0, 2 if κk = 12 (11)
and for electric quadrupole interactions K = 2:

|l 0 -l 1 | = 0, 2 if κk = 02 and |l 0 -l 1 | = 0 if κk = 11, 13. (12)
The matrix elements resulting from relation (6) and conditions ( 11) and ( 12) were calculated by our computer code and included in the hyperfine structure energy matrix.

Determination of one-and two-body fine and hyperfine structure radial parameters

Interpretation of calculated A and B constants

Following our procedure presented in Figure 1, each hfs A and B constant can be expressed as a linear combination of one-and two-body contributions as:

A(ψ) = κk,nl α κk nl (ψ) a κk nl + κk,i α κk i (ψ) a κk i , (13)
B(ψ) = κk,nl β κk nl (ψ) b κk nl + κk,i β κk i (ψ) b κk i , ( 14
)
where α κk nl , β κk nl and α κk i , β κk i represent the angular coefficients at one-and two-body hfs operators respectively, while a κk nl , b κk nl are the radial one-body parameters and a κk i , b κk i are the traditionally used radial two-body parameters taken into account in the considered excitations. In our procedure the above parameters receive a full description introduced in Section 4, e.g.: D 2 (5d5d,5dn d) P 02 (5d,n d) denotes the contribution from excitation from the open shell 5d to the empty shellsn d, or D 2 (n 0 d5d,5d5d) P 02 (n 0 d,5d) denotes the contribution from excitation from the closed shell n 0 d to the open shell 5d.

One-body parameters

Sandars and Beck [10] have developed a theory which simplifies calculation and interpretation of relativistic hfs effects in many electron atoms. This theory yields three effective radial integrals for each open shell (l > 0) and for each multipole interaction, which should be handled as free adjustable parameters in order to take into account relativistic and configuration interaction effects. Therefore the following definitions of one-body radial parameters (if a κk nl , b κk nl are expressed in MHz) for the considered configurations have been assumed :

a κk nl = 2µ B h g I [ r -3 κk nl + I κk n0l0,n l ] = 2µ B h g I r -3 κk nl eff , κk = 01, 12, 10, b κk nl = = e 2 h Q[ r -3 κk nl + I κk n0l0,n l ] = e 2 h Q r -3 κk nl eff , κk = 02. (15)
In above definitions, r -3 κk nl are the relativistic HFS radial integrals, as was shown by Lindgren and Rosen [12], which can be calculated ab-initio theoretically, and I κk n0l0,n l are radial parameters represented configuration interaction effects which reduced to one-body [START_REF] Lindgren | Atomic Many-Body Theory[END_REF], commonly named "core polarization effects" [START_REF] Armstrong | Theory of the Hyperfine Structure of Free Atoms[END_REF].

The excitations of one electron from a closed shell n 0 l 4l0+2 0 to an empty shell n l are two-body hfs interactions which can be reduced to one-body interactions.

The radial parameter I κk n0l0,n l has been defined [START_REF] Armstrong | Theory of the Hyperfine Structure of Free Atoms[END_REF] as follows:

I κk n0l0,n l = - n0l0,n l r -3 κk n0l0,n l 2 t κk coef f (n 0 l 0 , n l ) t κk coef f (nl, nl) × 2δ(κ, 0) 2k + 1 (l 0 C k l )(l C k l)R k (n 0 l 0 nl, n l nl)/∆E(n 0 l 0 , n l ) + t (-1) 1+k+t l k l l 0 t l (l C t l )(l 0 C t l) ×R t (n 0 l 0 nl, nln l )/∆E(n 0 l 0 , n l ) . ( 16
)
where t κk coef f (nl, nl) is the angular part of the hfs operator t κk : nl t κk nl = t κk coef f (nl, nl) r -3 κk nl , ∆E is the (positive) energy difference between the relevant closed-and emptyshell orbitals, and n covers all empty shells including continuum.

The radial parameter determined from experimental data r -3 κk nl eff should be interpreted as the sum of r -3 κk nl and I κk n0l0,n l :

r -3 κk nl eff = r -3 κk nl + I κk n0l0,n l . ( 17
)
The relations between the one-body hfs parameters and the radial effective integrals are as follows [12] : are the sums of relativistic and configuration interaction effects contributions.

a κk 5d =

Two-body parameters

The effects of excitations of one electron from an open shell to an empty shell or from a closed shell to an open shell, are referred to as two-body hfs interactions (see Section 4).

The values of one-and two-body fine structure and hyperfine structure parameters (predicted by theory for the configuration system (5d+6s) 3 ) determined in our procedure, and also values of the radial integrals, are listed in Table 6. The ratio of the two-body parameters κk = 12 and κk = 01 was assumed to amount to 1. For the parameters containing electrostatic integrals of the ordert = 4 the ratio with respect to the corresponding t = 2 parameters was assumed to be equal to 0.65071 (from Hartree-Fock calculations). In the case of parameters with relative errors exceeding 100%, their values were set to 0, which resulted in improvement of the fit for the hfs constants in the least squares method.

Recalculation to radial one-configuration parameters

Each set of parameters for the models, (c-o) or (o-e), given in the Table 6, were determined in a totally independent way. In the constructed fine structure and hyperfine structure energy matrices the angular coefficients in the matrix elements originating from the first order perturbation theory are identical. However, the angular coefficients in the matrix elements originating from the second order perturbation theory are different for both models considered, but they involve the angular coefficients of the first order elements and form implicit linear dependencies. Thus a simultaneous use of the sets of parameters obtained in models (co) and (o-e) has to be excluded. If a linear dependence between two-body parameters vanishes, it indicates erroneous values of angular coefficients in the matrix elements; this was applied as a correctness test in construction of the energy matrix. A confirmation of the correctness of the model of description of atomic structure can also be provided by the values of two-bodyfs and hfs parameters, which should differ only by their signs. Thus the values of two-body parameters, given in Table 6 , should be understood as algebraic sums of contributions from excitations of the type "closed shell -open shell" or the type "open shell -empty shell".

The respective one-body fs and hfs radial parameters obtained within both models are different, since in each model they effectively "include" a different combination of the contributions from one and two-body interactions.

The relations allowing elimination of those contributions and determination of the radial one-body parameters characteristic of the individual configurations 5d 3 , 5d 2 6s and 5d6s 2 , forming the model space (5d+6s) 3 , are as follows:

for excitations "closed shell -open shell"(c-o): electrostatically correlated spin-orbit interaction:

ζ(nd N n s N ) = ζ(c -o) + 2N × D 0 (n 0 d5d, 5d5d)ζ n0d,5d + 2 7 × D 2 (n 0 d5d, 5d5d)ζ n0d,5d - 8 21 × D 4 (n 0 d5d, 5d5d)ζ n0d,5d - 2 5 × E 2 (5d6s, 6sn 0 d)ζ n0d,5d δ(N , 2), (19)
electrostatically correlated hfs interactions:

a 01 5d (nd N n s N ) = a 01 5d (c -o) + 2N × D 0 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) + 2 7 × D 2 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) - 8 25 × D 4 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) - 2 5 × E 2 (n 0 d6s, 6s5d)P 01 (n 0 d, 5d)δ(N , 2), (20)
a 12 5d (nd N n s N ) = a 12 5d (c -o) + 2N × D 0 (n 0 d5d, 5d5d)P 12 (n 0 d, 5d) - 6 49 × D 2 (n 0 d5d, 5d5d)P 12 (n 0 d, 5d) + 8 49 × D 4 (n 0 d5d, 5d5d)P 12 (n 0 d, 5d) - 2 5 × E 2 (n 0 d6s, 6s5d)P 12 (n 0 d, 5d)δ(N , 2), (21)
a 10 6s (nd N n s) = a 10 6s (c -o) -2N × D 0 (n 0 s5d, 5d6s)P 10 (n 0 s, 6s), (22)
b 02 5d (nd N n s N ) = b 02 5d (c -o) + 2N × D 0 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) - 62 49 × D 2 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) + 8 49 × D 4 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) - 2 5 × E 2 (n 0 d6s, 6s5d)P 02 (n 0 d, 5d)δ(N , 2), (23)
for excitations "open shell -empty shell"(o-e): electrostatically correlated spin-orbit interaction:

ζ(nd N n s N ) = ζ(o -e) -2(N -1) × D 0 (5d5d, 5dn d)ζ 5d,n d + 2 5 × E 2 (5d6s, 6sn d)ζ 5d,n d δ(N , 2), (24)
electrostatically correlated hfs interactions:

a 01 5d (nd N n s N ) = a 01 5d (o -e) -2(N -1) × D 0 (5d5d, 5dn d)P 01 (5d, n d) + 2 5 × E 2 (5d6s, 6sn d)P 01 (5d, n d)δ(N , 2), (25)
a 12 5d (nd N n s N ) = a 12 5d (o -e) -2(N -1) × D 0 (5d5d, 5dn d)P 12 (5d, n d) + 2 5 × E 2 (5d6s, 6sn d)P 12 (5d, n d)δ(N , 2), (26)
a 10 6s (nd N n s) = a 10 6s (o -e) + 2N × D 0 (5d6s, 5dn s)P 10 (6s, n s), ( 27
) b 02 5d (nd N n s N ) = b 02 5d (o -e) -2(N -1) × D 0 (5d5d, 5dn d)P 02 (5d, n d) + 2 5 × E 2 (5d6s, 6sn d)P 02 (5d, n d)δ(N , 2). ( 28
)
The values of one-configuration parameters, obtained for both excitation models with the use of the above relations, are compiled in Table 7. The hyperfine structure parametersa κk and b κk given in this table for individual configurations are the hfs parameters obtained with the use of Sandars and Beck theory [10]. It can be seen that the respective values of the parameters obtained within two excitation models are equal up to third decimal place. It can serve as another test of correctness of the model of description of atomic structure.

Discussion of configuration interaction effects on the hyperfine structure

The influence of an excitation of an electron from a closed shell to an empty shell (a second-order effect) on the hyperfine structure is referred to as "hfs core-polarization effect". This term referees to both n 0 s and n 0 p, n 0 d or n 0 f electrons [START_REF] Armstrong | Theory of the Hyperfine Structure of Free Atoms[END_REF] which fill the closed shells in the atom. According to Lindgren and Morrison [START_REF] Lindgren | Atomic Many-Body Theory[END_REF], the operators representing this effect are "pseudo-two-body operators" which can be reduced to the structure of one-body operators, and thus the corresponding radial parameters cause the effects of "screening or antiscreening" of the parameters of the first order perturbation theory in Sandars and Beck approach [10]. The most often discussed and analyzed effect is the one first described by Bauche and Judd, concerning the excitation of electrons from the closed shells n 0 s to the empty shells n s. In Sandars and Beck theory the operator s and the radial parameter a 10 nl (where l > 0) represent relativistic effects in the hyperfine structure. Thus, according to e.g. Feneuille and Armstrong [20], Armstrong [START_REF] Armstrong | Theory of the Hyperfine Structure of Free Atoms[END_REF] and Lindgren and Morrisson [START_REF] Lindgren | Atomic Many-Body Theory[END_REF], the above mentioned effects are inseparable and cannot be independently determined in the least-squares procedure. Therefore in our method a 10 nl = 0 for l > 0 is assumed, while the effects of ns core polarization, which are different in each of the configurations 5d 3 , 5d 2 6s and 5d6s 2 , are described as follows:

-in configuration n 0 s 2 5d 3 closed shells are n 0 = 1,2,. . .,5 and empty shells are n = 6,7,. . ., -in configuration n 0 s 2 5d 2 6s closed shells are the same as above, but empty shells begin with n = 7, and the shell 6s is an open shell, -in configuration n 0 s 2 5d6s 2 closed shells are from 1 to 6, an empty shells begin with 7s.

Thus in the considered space (5d+6s) 3 the core polarization effect should be described as follows:

-in configuration 5d 3 : with the use of parameter E 2 (n 0 s5d,5d6s) P 10 (n 0 s,6s) + E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s), (n 0 =1,2,. . .,5, n =7,8,. . .), -in configuration 5d 2 6s: with the use of parameter E 2 (n 0 s5d,5d6s) P 10 (n 0 s,6s) (n 0 =1,2,. . ., 5) representing the excitation from the closed shells n 0 s to the open shell 6s, as well as parameter E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s) representing the excitations from the five closed shells n 0 s to the empty shells n s (n =7,8,. . .)

-in configuration 5d6s 2 : with the use of parameter E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s), (n 0 =1,2,. . .,6, n =7,8,. . .).

The determined values of those parameters are given in Table 6. With the use of relation P 10 = 63.6086 g I E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s) we can determine the radial integrals representing the electronic part of these interactions, which amount to:

-for excitations "closed shell n 0 s -empty shell n s" E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s) = 4(3) au, (n 0 =1,2,. . .,6, n =7,8,. . .) E 2 (n 0 s5d,5dn s) P 10 (n 0 s,n s) = -23(3) au, (n 0 =1,2,. . .,5, n =7,8,. . .) -for excitations "closed shell n 0 s -open shell 6s" E 2 (n 0 s5d,5d6s) P 10 (n 0 s,6s) = -19(2) au, (n 0 =1,2,. . .,5)

From the above we can conclude, that excitations from the closed shells n 0 s to an open or an empty shell 6s plays a dominant role. For instance it can be seen that in the case of configuration 5d 3 the contribution of the core polarization effect (resulting from the second order of perturbation theory) to the observed hyperfine splittings is comparable to the contribution of the electrons 5d, and in some cases the former even exceeds the latter. Moreover, one can see that in configuration 5d6s 2 , where the shell 6s is closed and thus the excitation considered is excluded, only a small core polarization effect is observed. A negative-valued contribution from the excitations 1s,2s,. . .,5s to 7s,8s,. . . is compensated by a positive contribution from the excitations of 6s electron to empty shells. This resembles the picture resulting from theoretical calculations performed by Ron and Kelly [43] for configurations 3d 6 4s 2 in a Fe atom, where also the contributions of the excitations of the electrons 1s, 2s, 3s to 5s shell and the succeeding ones are negative, while the contributions of the excitations of the electron 4s are positive. As already mentioned above, it is not possible to determine the value of the relativistic parametera 10 5d in the least squares procedure. We examined the behaviour of the values of the remaining hyperfine structure parameters in dependence on the value of the relativistic effect a 10 5d . As expected, changes of the values concerned only the parameters E 2 (n 0 s5d,5dn s) P 10 (n 0 =1,2,. . .,6, n =7,8,. . . and E 2 (n 0 s5d,5dn s) P 10 (n 0 =1,2,. . .,5, n =7,8,. . ., according to the linear dependence: E 2 (n 0 s5d, 5dn s)P 10 (n 0 s, n s) = -2.5a 10 5d + 181.310 n 0 = 1, . . . , 6; n = 7, 8, . . . E 2 (n 0 s5d, 5dn s)P 10 (n 0 s, n s) = -2.5a 10 5d -190.270 n 0 = 1, . . . , 5; n = 7, 8, . . . , (29) where, for the parameter a 10 5d we can assume the values from ab initio theoretical calculations, performed with the use of differential methods [12].

Moreover, assumption of different values of parameter a 10 5d had no influence on the value of residuum. From the relations ( 22) and ( 27), as well as Table 7 the following information about configuration interaction can be obtained: a 10 6s (c-o) -a 10 6s (o-e) = 4 D 0 (n 0 s5d,6s5d)P 10 (n 0 s,6s) + D 0 (5d6s,5dn s)P 10 (6s,n s)= 149 MHz Thus we can determine the value of the contribution described in the second order perturbation theory as:

D 0 (n 0 s5d,6s5d)P 10 (n 0 s,6s) + D 0 (5d6s,5dn s)P 10 (6s,n s) = 37 MHz It results from Table 7, that in all the cases considered a 12 5d < a 01 5d . This result differs from the one expected on the basis of the relativistic theory of the hyperfine structure [12,[START_REF] Olsson | [END_REF], where the expected relation for the 5d elements should be amounts to a 12 /a 01 > 1.25.

We can explain the results obtained within this work only if we assume, that the core polarization effects concerned with the excitations n 0 s → n d, n 0 p → n p,n f and n 0 d → n s,n d,n g are different, dependent on κk = 01, 12 or 02, which specify the parameters a κk of the hyperfine structure. In order to explain such an effect of core polarization the full definitions of one-body radial parameters given below are very helpful:

r -3 01 5d eff = 5d r -3 5d 01 + 5 n0=2 n 1 ∆E n p r -3 n 0 p 01 × 2 5 R 1 (n 0 p5d, 5dn p) - 6 35 R 3 (n 0 p5d, 5dn p) + 4 n0=3 n 1 ∆E n d r -3 5d 01 2R 0 (n 0 d5d, 5dn d) + 2 7 R 2 (n 0 d5d, 5dn d) - 8 21 R 4 (n 0 d5d, 5dn d) (30) r -3 12 5d eff = 5d r -3 5d 12 + 5 n0=1 n 1 ∆E n d r -3 n 0 s 12 2 5 R 2 (n 0 s5d, 5dn d) + 5 n0=2 n 1 ∆E n p r -3 n 0 p 12 14 25 R 1 (n 0 p5d, 5dn p) + 18 175 R 3 (n 0 p5d, 5dn p) + 5 n0=2 n 1 ∆E n f r -3 n 0 p 12 6 25 R 1 (n 0 p5d, 5dn f ) + 72 175 R 3 (n 0 p5d, 5dn f ) + 4 n0=3 n 1 ∆E n d r -3 5d 12 2R 0 (n 0 d5d, 5dn d) - 6 49 R 2 (n 0 d5d, 5dn d) + 8 49 R 4 (n 0 d5d, 5dn d) + 4 n0=3 n 1 ∆E n g r -3 5d 12 72 245 R 2 (n 0 d5d, 5dn g) + 20 49 R 4 (n 0 d5d, 5dn g) (31) r -3 02 5d eff = 5d r -3 5d 02 + 5 n0=1 n 1 ∆E n d r -3 n 0 s 02 - 4 5 R 2 (n 0 s5d, n d5d) + 2 5 R 2 (n 0 s5d, 5dn d) + 5 n0=2 n 1 ∆E n p r -3 n 0 p 02 - 24 25 R 2 (n 0 p5d, n p5d) + 14 25 R 1 (n 0 p5d, 5dn p) + 18 175 R 3 (n 0 p5d, 5dn p) + 5 n0=2 n 1 ∆E n f r -3 n 0 p 02 - 36 25 R 2 (n 0 p5d, n f 5d) + 6 25 R 1 (n 0 p5d, 5dn f ) + 72 175 R 3 (n 0 p5d, 5dn f ) + 4 n0=3 n 1 ∆E n d r -3 5d 02 - 8 7 R 2 (n 0 d5d, n d5d) + 2R 0 (n 0 d5d, 5dn d) - 6 49 R 2 (n 0 d5d, 5dn d) + 8 49 R 4 (n 0 d5d, 5dn d) + 4 n0=3 n 1 ∆E n g r -3 5d 02 - 72 35 R 2 (n 0 d5d, n g5d) + 72 245 R 2 (n 0 d5d, 5dn g) + 20 49 R 4 (n 0 d5d, 5dn g) . ( 32
)
When analyzing the above relations, we may assume, that in the caseκk = 12 a larger compensation of the relativistic effects by polarization effects is possible than in the caseκk = 01, which might explain the least squares fit results, where the following ratio was obtained: a 12 /a 01 ≈ 0.8. The authors of the papers [12,[START_REF] Olsson | [END_REF] noticed, that the experimental values for these ratios are reversed with respect to the theoretical ones. This is due to the influence of configurations interaction effects One of the aims of our work was a precise definition of the radial integrals determined on the basis of the experimental data, which makes the determination of their contributions to the constantsA and B possible. Comparison of such contributions, estimated from the experiment, with their counterparts calculatedabinitio theoretically would be definitely more interesting than the comparison given in Table 3, which is currently possible.

If the formalism describing the interaction between the electronic shell and the atomic nucleus were strictly correct, the ratios of two-body hfs radial parameters should be identical. On the basis of the data from Table 6 we obtain: D 0 (5d5d, 5dnd)P 01 (5d, nd)/D 0 (5d5d, 5dnd)P 02 (5d, nd) = 0.94, D 2 (5d5d, 5dnd)P 01 (5d, nd)/D 2 (5d5d, 5dnd)P 02 (5d, nd) = 0.24, E 2 (5d5d, 5dnd)P 01 (5d, nd)/E 2 (5d5d, 5dnd)P 02 (5d, nd) = 0.46, D 2 (5dnd, 5d6s)P 01 (5d, nd)/D 2 (5dnd, 5d6s)P 02 (5d, nd) = 0.77, E 2 (5dnd, 6s5d)P 01 (5d, nd)/E 2 (5dnd, 6s5d)P 02 (5d, nd) = 0.92. [START_REF] Furmann | Investigation of electron levels of free atoms and ions of lanthanum, praseodymium and europium with laser spectroscopy methods[END_REF] This indicates that the description of the configurations interaction within the frame of magnetic dipole or electric quadrupole interactions in the hyperfine structure is not fully correct.

Conclusions

On the basis of the analytical content of the tables in this work we tried to address the problems stated in Introduction. As can be seen from Table 2, application of the method of energy matrix diagonalization in the basis SLJF allows to describe the observed hyperfine splitting intervals with the use of the constantsA, B and C within the accuracy of the measurements performed with the double resonance method (ABMR-LIRF), which is of the order of a few kHz. Thus, on the basis of the precise experimental data it can be stated that the operator of the form given in Equation ( 1) correctly describes the partition of the observed hyperfine splitting into the interactions of the ranksK = 1, 2, 3.

In Table 3 all the data concerning the constants A and B, available in the literature, were compiled; this concerns both the experimentally determined and the ab initio theoretically calculated values. From the comparison of the columns 4 and 5, or 7 and 8, one can rather speak of severe discrepancies and a few accidental coincidences between the experimental and theab initio theoretical values. For instance, for the level at 13238 cm -1 three different values of the constant A were predicted: 86, -18 and 102 MHz, while the experimental value amounts to -19 MHz, and for the level at 9910 cm -1 the theoretical values are as follows 431, 471 and 333 MHz, in severe discrepancy with the experimental result of 559 MHz. It indicates a limited usefulness of the theoreticalab initio calculations for the understanding of the mechanism of interaction between the electronic shells and the atomic nucleus.

According to Sandars and Beck theory the magnetic dipole interactions (rankK = 1) may be divided into the suborders κk = 01, 12, 01, describing the interaction of the electronic shell with the magnetic dipole moment of the nucleus. In a similar way the interactions of the rankK = 2 can be divided into interactions of suborders κk = 02, 13, 11 with the electric quadrupole moment of the nucleus. As evident from Table 5, for the electrostatic quadrupole interaction the agreement between B corr and B calc is of the order of the accuracy of their determination (differences between B exp,corr [13] and B corr (this work) reach even 5.7 MHz). Thus we can assume that the operator

T (κk)2 e • T (2) n = e 4π 0 N i=1 -C 2 i r -3 02 + 3 10 U (13)2 i r -3 13 + U (11)2 i r -3 11 • T (2) n , ( 34
)
rather correctly describes the interaction between the electronic shells and the electric quadrupole moment of the nucleus. It should be stressed, that in our analysis 19 constants B determined with high precision from ABMR-LIRF experiments were applied. In the fit procedure 9 free parameters were used and the rms error amounted to 0.47 MHz. A drastically different picture emerges when we try to divide the interactions of the rank K = 1 into the contributions grouped according to the operators subranksκk = 01, 12, 01.

It can be seen in Table 5, that the differences between A corr and A calc reach even 20 MHz in some cases; this results e.g. in the difference between the total observed hyperfine splitting of the level 4 F 9/2 and its calculated counterpart of ≈700 MHz. Such differences are very large and they can be detected even with the classical optical methods. In the case of the analysis of magnetic dipole interactions we had at our disposal altogether 35 constants, among them 21 very precise ones (obtained with the ABMR-LIRF method) and 14 of an accuracy 1-4 MHz, from the LIF method. The number of free parameters amounted to 13 and the rms error amounted to ca. 12.1 MHz. Use of all one-and two-body parameters predicted by the theory (also of those given in Table 6 with the values set to 0) did not remove the dicrepancies between the constants A, and it solely increased the rms error. Therefore for the parameters determined with an error considerably exceeding 100% in the final procedure the values 0 were assumed. The reason of the observed discrepancies could be the inaccurate eigenvectors' amplitudes, determined on the basis of the fine structure analysis. The source of such errors, as already mentioned in Introduction, might be e.g. assignment of an incorrect J value to the correct energy value. Thus in this work any doubts of this kind were removed due to investigations of the hyperfine structure with LIF method. Lack of data concerning the levels 5d 3 2 1 D J = 3/2, as well as of the confirmation of the energy of the levelJ = 5/2, had no considerable influence on the accuracy of the obtained values of the eigenvectors'amplitudes. Their quality is proved by the excellent agreement between the experimental and the calculated values ofg J factors, as clearly visible in Table 5. We also supposed, that the reason of the discrepancies may be perturbation of the hyperfine structure by spin orbit coupling with distant configurations. This assumption was also examined in detail in the way analogous to the case described in Section 4; the electrostatic interaction operatorG in equation ( 6) was replaced by the spin-orbit interaction operator s•l. The parameters describing the above coupling proved to exhibit linear dependence with the relativistic parameters resulting from Sandars and Beck theory [10]. It applies to both magnetic dipole and electric quadrupole hyperfine interactions. Thus we proved, in a different way, the theoretical considerations of Feneuille and Armstrong [20] about the impossibility of differentiation between the relativistic contributions and the contributions of configurations interactions solely on the basis of experimental data. In the case of the interactions K = 1 κk = 01, 12 and K = 2 κk = 02 relativistic and configurations interaction contributions are added to the nonrelativistic contribution. In the casesK = 1 κk = 10 and K = 2 κk = 13, 11, where the nonrelativistic contribution amounts to 0, both the effects: the relativistic effect and the configurations interaction effect contribute to the non-zero value of the effective parameter. In our procedure of the analysis of atomic structure we critically analyzed all the available experimental data. In the theoretical description of the atomic structure all possible contributions originating from the second order perturbation theory were taken unto account. They were independently described in the excitation models: (c-o) and (o-e). The consistency of the results obtained, as seen in Table 6, proves the correctness of the procedure and the formulae applied in construction of the energy matrix. Thus, on the basis of the results of this work, we may state, that the operator in the form

T (κk)1 e • T (1) n = µ 0 µ B 2π N i=1 l i r -3 01 - √ 10 s i C 2 i (1) r -3 12 + s i r -3 10 • T (1) n , (35)
does not fully describe the observed magnetic dipole interactionK = 1 in the atomic hyperfine structure, i.e. its partition into the contributions κk = 01, 12 i 10. In the near future we plan to undertake investigations of the model of magnetic dipole and magnetic octupole interactions in an atom, where a more complex structure of the nucleus is assumed than in hitherto existing models. For the full verification of the new model more precise measurements of the hyperfine splittings would be required, where the constants of magnetic octupole interaction could be determined with the relative accuracy of the order of 10%. The present accuracy of the determined C constants (see Table 4) does not allow to put any hypothesis concerning this interaction.

Table 1. Compilation of spectral lines used for measurement of the hyperfine structure constants of levels belonging to even parity configurations in lanthanum atom, performed with the method of LIF in a hollow cathode discharge. Spectroscopic description of the levels belonging to odd-parity configurations on the basis of [START_REF] Furmann | Investigation of electron levels of free atoms and ions of lanthanum, praseodymium and europium with laser spectroscopy methods[END_REF]. Value of J quantum number for the level with energy E=33799.

  95.4128 g I r -3 κk 5d eff κk = 01, 12, a 10 6s = 63.6086 g I r -3 10 6s eff = 63.6086 g I dP 6s (r) dr 2 r=0 , b κk 5d = 234.9624 Q r -3 κk 5d eff κk = 02, 11, 13, (18) where g I is the nuclear g I -factor expressed in nuclear magnetons (for the 139 La nucleus g I = +0.7931); the values of the parameter a κk and b κk are given in MHz, and those of r -3 κk eff and dP6s(r) dr 2 r=0 -in atomic units. r -3 κk eff and dP6s(r) dr 2 r=0

FineFigure 1 .

 1 Figure 1. Scheme of the atomic structure calculations

Table 2 :

 2 23[cm-1 ], that was changed from 5/2 to 3/2 on the basis of the investigations performed, has been marked with asterisk Comparison of the experimental and calculated hyperfine structure intervals

	No			Line	Lower level		Upper Level
		λ air [nm] kvac[cm -1 ] Designation Energy[cm -1 ]	J	Designation	Energy[cm -1 ]	J
	1	602.724	16586.75	5d 3 4 P	16617.30	1/2	4f 5d6s ( 1 D) 2 P	33204.05	3/2
	2	594.797	16807.81 5d 2 ( 1 S)6s 2 S	16991.42	1/2	4f 6s6d ( 3 D) 2 P	33799.23	*3/2
	3	587.231	17024.34 5d 2 ( 1 S)6s 2 S	16991.42	1/2	5d 2 ( 3 F)7p 4 F	34015.76	3/2
	4	662.503	15090.10	5d 3 2 P	20392.60	1/2	5d 6s7p ( 3 P) 4 P	35482.70	1/2
	5	607.036	16468.91	5d 3 4 P	16735.14	3/2	4f 5d6s ( 1 D) 2 P	33204.05	3/2
	6	569.189	17563.97	5d 3 4 P	16735.14	3/2	5d 2 ( 3 F)7p 2 D	34299.11	3/2
	7	633.431	15782.67	5d 3 2 D	18037.64	3/2	4f 6s6d ( 3 D) 2 P	33820.31	1/2
	8	573.069	17445.06	5d 3 2 D	18037.64	3/2	5d 6s7p ( 3 P) 4 P	35482.70	1/2
	9	630.369	15859.33	5d 3 2 P	21037.30	3/2	6s 2 ( 1 S)7p 2 P	36896.63	1/2
	10	597.412	16734.23	5d 3 2 P	21037.30	3/2	4f 5d 2 ( 3 F) 4 D	37771.53	5/2
	11	587.234	17024.27	5d 3 2 P	21037.30	3/2	4f 5d 2 ( 1 D) 2 D	38061.57	3/2
	12	656.023	15239.14	5d 3 2 D	18776.62	5/2	5d 2 ( 3 F)7p 4 F	34015.76	3/2
	13	654.440	15276.02	5d 3 2 F	21969.32	5/2	5d 6s7p ( 3 P) 2 F	37245.34	5/2
	14	634.251	15762.27	5d 3 2 F	21969.32	5/2	4f 5d 2 ( 3 F) 4 F	37731.59	5/2
	15	632.027	15817.72	5d 3 2 F	21969.32	5/2	4f 5d 2 ( 3 F) 4 P	37787.04	3/2
	16	629.808	15873.45	5d 3 2 F	21969.32	5/2	4f 5d 2 ( 3 F) 4 D	37842.77	7/2
	17	625.139	15992.02	5d 3 2 F	21969.32	5/2	4f 5d 2 ( 3 F) 4 F	37961.34	5/2
	18	624.306	16013.36	5d 3 2 F	21969.32	5/2	4f 5d 2 ( 1 D) 2 G	37982.68	7/2
	20	580.683	17216.31	5d 3 2 G	17023.36	7/2	4f 6s6d ( 3 D) 2 H	34239.61	9/2
	21	653.348	15301.54	5d 3 2 F	21943.80	7/2	5d 6s7p ( 3 P) 2 F	37245.34	5/2
	22	633.226	15787.79	5d 3 2 F	21943.80	7/2	4f 5d 2 ( 3 F) 4 F	37731.59	5/2
	23	623.312	16038.88	5d 3 2 F	21943.80	7/2	4f 5d 2 ( 1 D) 2 G	37982.68	7/2
	24	580.043	17235.31	5d 3 2 F	21943.80	7/2	4f 5d 2 ( 1 D) 2 F	39179.11	5/2
	25	584.677	17098.71	5d 3 2 G	17140.90	9/2	4f 6s6d ( 3 D) 2 H	34239.61	9/2
	26	584.491	17104.15	5d 3 2 G	17140.90	9/2	4f 6s6d ( 3 D) 2 H	34245.05	11/2
	27	627.820	15923.73	5d 3 2 H	18315.88	9/2	4f 6s6d ( 3 D) 2 H	34239.61	9/2
	28	627.605	15929.17	5d 3 2 H	18315.88	9/2	4f 6s6d ( 3 D) 2 H	34245.05	11/2
	29	590.252	15937.22	5d 3 2 H	18315.88	9/2	5d 2 ( 3 F)7p 2 F	35253.10	7/2
	30	583.453	17134.59	5d 3 2 H	18315.88	9/2	5d 2 ( 3 F)7p 4 F	35450.47	9/2
	31	627.410	15934.13	5d 3 2 H	18310.92	11/2	4f 6s6d ( 3 D) 2 H	34245.05	11/2
	32	583.284	17139.55	5d 3 2 H	18310.92	11/2	5d 2 ( 3 F)7p 4 F	35450.47	9/2
	[44] Ting Y 1957 Phys. Rev. 108 295
	[45] Beck D R 1997 Int. J. Quant. Chem. 65 555
	[46] Karacoban B and Özdemir L 2009 Acta Phys. Pol. A 115 864
					interval [MHz]
	Energy	J F ↔ F
	[cm -1 ]			observed	calculated	obs.-calc.
		0.00	3/2	3 ↔ 2	391.603(10) [44]	391.601		0.002

Table 3 :

 3 Experimental and corrected values of the hfs A and B constants [MHz] for the even parity energy levels of lanthanum atom. The values of A

							B calc		44.853			54.197		14.108			13.377		18.981			32.010			28.146				36.184			-34.246		42.191
							Bcorr		44.776			54.168		13.983			13.505		18.840			31.980			28.295				37.066			-34.793		41.896			-6.964
	th and B th are taken from	Childs and Nielsen (MCDF method) [13]( a ), from Beck (MCDF and RCI method) [45]( b )	and from Karacoban (MCHF method) [46]( c ). The experimental values are taken from	Ting [44]( d ), Childs and Goodman [26]( e ), Childs and Nielsen [13]( a ), Furmann et all	[28,29]( f ) and from our measurements. The values of Acorr, Bcorr and A calc , B calc are	obtained from our calculations	A th Aexp Acorr A calc B th Bexp Bexp,corr		111.23 a 141.1959 d 141.1961 144 29.59 a 44.781 d 44.781 d	129.7 b 29.9 b	143.27 c 27.57 c	235.64 a 182.1706 d 182.1702 163 32.21 a 54.213 d 54.213 d	184.10 c 37.57 c	-398.55 a -480.312 e -480.208 -501 14.84 a 15.082 e 14.2 e	-414.7 b -480.292 a 10.7 b 15.188 a 16.343 a	-322.87 c 11.66 c	250.19 a 300.583 e 300.643 321 11.41 a 10.873 e 14.0 e	246.44 c 11.62 c 7.800 a	392.78 a 462.868 e 462.891 480 15.45 a 17.925 e 19.3 e	411.3 b 17.9 b 16.102 a	379.86 c 16.08 c	451.32 a 489.534 e 489.533 505 26.17 a 32.180 e 31.9 e	440 b 24.2 b 31.535 a	410.92 c 24.22 c	271.92 a 304.372 e 304.376 308 20.04 a 28.091 e 27.8 e	209.15 c 21.26 c 28.289 a	1913.46 a 2460.119 e 2460.229 2464	2137.07 c 2460.161 a	844.26 a 930.0 e 929.666 918 28.58 a 42.1 e 37.2 e	842.b b 929.618 a 25.0 b 37.221 a 34.723 a	822.32 c 23.27 c	731.87 a 802.8 e 802.166 795 -36.09 a -23.9 e -40 e	604.60 c 802.172 a -17.17 c -34.186 a -36.365 a	-12.80 a -197.066 e -197.065 -212 29.02 a 40.677 e 41.4 e	-170.7 b -197.064 a 37.5 b 40.754 a 40.349 a	-11.01 c 27.46 c	-321.39 a -424.7 e -422.419 -415 -0.85 a -11.3 e -13 e
							Energy Designation	[cm -1 ]	0.00 5d6s 2 2 D 3/2			1053.16 5d6s 2 2 D 5/2		2668.19 5d 2 6s 4 F 3/2			3010.00 5d 2 6s 4 F 5/2		3494.53 5d 2 6s 4 F 7/2			4121.57 5d 2 6s 4 F 9/2			7011.91 5d 2 6s 2 F 5/2		7231.41 5d 2 6s 4 P 1/2		7490.52 5d 2 6s 4 P 3/2			7679.94 5d 2 6s 4 P 5/2		8052.16 5d 2 6s 2 F 7/2			8446.04 5d 2 6s 2 D 3/2

Table 4 .

 4 Corrected values of the hfs C constant [kHz] for the even parity energy levels of lanthanum atom.

	Energy	Designation	Ccorr	Ccorr	Ref.
	[cm -1 ]		(Ref.)	(this work)
	0.00	5d6s 2 2 D 3/2	0.15(44)	0.2(1.3) [44]
	1053.16	5d6s 2 2 D 5/2	-0.6(1.0)	-0.2(3.0) [44]
	3010.00	5d 2 6s 4 F 5/2	2(3)	-0.9(2.1) [26]
	3494.53 5d 2 6s 4 F 7/2	-2(2)	-2.8(2.4) [26]
	4121.57 5d 2 6s 4 F 9/2	3(4)	2.5(2.9) [26]
	7011.91 5d 2 6s 2 F 5/2	-2(3)	-2.8(2.1) [26]
	7679.94 5d 2 6s 4 P 5/2		-0.7(2.1)
	8052.16 5d 2 6s 2 F 7/2		-0.5(2.4)
	9919.82 5d 2 6s 2 G 9/2		6.8(2.9)
	9960.90	5d 2 6s 2 G 7/2		-0.2(2.4)
	12787.40 5d 3 4 F 5/2		0.3(1.1)
	13238.32 5d 3 4 F 7/2		-0.5(2.4)
	13747.28 5d 3 4 F 9/2		-0.1(2.9)

Table 5 :

 5 Comparison of the experimental and calculated energy values [cm -1

	] and hfs A

Table 6 .

 6 One-and two-body radial parameters[MHz] for the model space (5d + 6s)3 (n 0 denote electrons of closed shells, n denote electrons excited to empty shells) closed shells → open shells open shells → empty shells Spin-orbit and electrostatically correlated spin-orbit interactions D 4 (n 0 d5d, 5d5d)ζ n 0 d,5d 36.707 D 4 (5d5d, 5dn d)ζ 5d,n d -36.710 E 2 (5d6s, 6sn 0 d)ζ n 0 d,5d 86.837 (28.) E 2 (5d6s, 6sn d)ζ 5d,n d -86.844 (28.) D 2 (5d6s, 5dn 0 d)ζ n 0 d,5d -137.127 (63.) D 2 (5d6s, 5dn d)ζ 5d,n d 137.141 (63.) E 2 (5d6s, n 0 d5d)ζ n 0 d,5d -125.740 (61.) E 2 (5d6s, n d5d)ζ 5d,n d 125.729 (61.) D 2 (5d5d, n 0 d6s)ζ n 0 d,5d -79.676 (63.) D 2 (5d5d, n d6s)ζ 5d,n d 79.691 (63.) D 2 (n 0 d5d, 6s6s)ζ n 0 d,5d 0 D 2 (5dn d, 6s6s)ζ 5d,n d 0 D 0 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) -18.039 (4.0) D 0 (5d5d, 5dn d)P 01 (5d, n d) 18.039 (4.0) D 2 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) -2.717 (12.) D 2 (5d5d, 5dn d)P 01 (5d, n d) 2.717 (12.) D 4 (n 0 d5d, 5d5d)P 01 (n 0 d, 5d) -1.807 D 4 (5d5d, 5dn d)P 01 (5d, n d) 1.807 E 2 (n 0 s5d, 5d6s)P 10 (n 0 s, 6s) -918.597 (76.) E 2 (n 0 s5d, 5dn s)P 10 (n 0 s, n s) -190.249 (70.) E 2 (n 0 s5d, 5dn s)P 10 (n 0 s, n s) -1108.850 (16.) E 2 (n 0 d6s, 6s5d)P 01 (n 0 d, 5d) 34.275 (17.) E 2 (5d6s, 6sn d)P 01 (5d, n d) -34.279 (17.) D 2 (n 0 s5d, 5d5d)P 12 (n 0 s, 5d) 0 D 2 (5d5d, 5dn s)P 12 (5d, n s) 0 D 0 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) -19.206 (7.1) D 0 (5d5d, 5dn d)P 02 (5d, n d) 19.206 (7.1) D 2 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) 11.207 (3.5) D 2 (5d5d, 5dn d)P 02 (5d, n d) -11.206 (3.5) D 4 (n 0 d5d, 5d5d)P 02 (n 0 d, 5d) 7.292 D 4 (5d5d, 5dn d)P 02 (5d, n d) -7.292 E 2 (n 0 d6s, 6s5d)P 02 (n 0 d, 5d) 73.450 (2.1) E 2 (5d6s, 6sn d)P 02 (5d, n d) -73.449 (2.1) D 2 (n 0 s5d, 5d5d)P 02 (n 0 s, 5d) 0 D 2 (5d5d, 5dn s)P 02 (5d, n s) 0

	ζ 5d	596.425 (30.) ζ 5d	525.024 (20.)
	D 0 (n 0 d5d, 5d5d)ζ n 0 d,5d	-36.770 (5.4)	D 0 (5d5d, 5dn d)ζ 5d,n d	36.771 (5.4)
	D 2 (n 0 d5d, 5d5d)ζ n 0 d,5d	56.410 (28.)	D 2 (5d5d, 5dn d)ζ 5d,n d	-56.414 (28.)
	Magnetic-dipole hfs interactions
	a 01 5d	218.970 (20.)	a 01 5d	182.791 (12.)
	a 12 5d	175.074 (22.)	a 12 5d	139.040 (17.)
	a 10 6s	3897.652 (45.)	a 10 6s	3749.025 (74.)
	a 12 5d,6s	123.967 (36.) a 12 5d,6s	65.497 (36.)
	D 2 (n 0 d5d, 5d6s)P 01 (n 0 d, 5d)	-204.689 (66.)	D 2 (5dn d, 5d6s)P 01 (5d, n d)	204.683 (66.)
	E 2 (n 0 d5d, 6s5d)P 01 (n 0 d, 5d)	-52.146 (84.)	E 2 (5dn d, 6s5d)P 01 (5d, n d)	52.146 (84.)
	D 2 (n 0 s5d, 5d5d)P 10 (n 0 s, 6s)	1936.122 (130.) D 2 (5d5d, 5dn s)P 10 (6s, n s)	-1936.118 (130.)
	D 2 (n 0 s5d, 6s6s)P 01 (n 0 d, 5d)	0	D 2 (5dn s, 6s6s)P 01 (5d, n d)	0
	Electric-quadrupole hfs interactions
	b 02 5d	231.511 (4.1) b 02 5d	180.111 (1.7)
	b 13 5d	32.039 (3.1) b 13 5d	32.039 (3.1)
	b 11 5d	-4.061 (1.6)	b 11 5d	-4.061 (1.6)
	b 02 5d,6s	-2.874 (8.5)	b 12 5d,6s	-35.858 (1.3)
	D 2 (n 0 d5d, 5d6s)P 02 (n 0 d, 5d)	-2.674 (12.)	D 2 (5dn d, 5d6s)P 02 (5d, n d)	2.670 (12.)
	E 2 (n 0 d5d, 6s5d)P 02 (n 0 d, 5d)	56.384 (18.)	E 2 (5dn d, 6s5d)P 02 (5d, n d)	-56.388 (18.)
	D 2 (n 0 s5d, 6s6s)P 02 (n 0 d, 5d)	0	D 2 (5dn s, 6s6s)P 02 (5d, n d)	0

Table 7 .

 7 Values of the one-body hyperfine structure parameters [MHz] and effective radial integrals [a.u.] obtained from the experimental data

			Configurations
	Parameter		(5d+6s) 3	5d 6s 2	5d 2 6s	5d 3
	spin-orbit :
	ζ 5d	(c-o)	596.425	490.260	451.455	377.915
		(o-e)	525.024	490.260	451.482 377.940
	magnetic-dipole :
	a 01 5d	(c-o)	218.970	168.971	146.604	110.526
		(o-e)	182.791	169.079	146.713	110.635
	a 12 5d	(c-o)	175.074 122.432	99.716	65.446
		(o-e)	139.040	122.435	99.721	65.451
	a 10 6s	(c-o)	3897.652		3897.652
	˙r-3 ¸01 5d eff ˙r-3 ¸12 5d eff ˙r-3 ¸10 6s eff	(o-e)	3749.025	2.226 1.614	3749.025 1.932 1.316 77.060	1.456 0.865
	electric-quadrupole :
	b 02 5d	(c-o)	231.511	150.730	141.697	103.285
	˙r-3 ¸02 5d eff	(o-e)	180.111	150.731 3.208	141.699 3.015	103.287 2.198

Acknowledgments

This work has been supported by Polish Ministry of Science and Higher Education under the project N519 033 32/4065.