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Experimental proof of configuration interaction on the hyperfine structure of the 57Fe atom

J. Dembczyński

To cite this version:

J. Dembczyński. Experimental proof of configuration interaction on the hyperfine structure of the

57Fe atom. Journal de Physique, 1980, 41 (2), pp.109-118. �10.1051/jphys:01980004102010900�. �jpa-

00209222�

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109

Experimental proof of configuration interaction on the hyperfine structure

of the 57Fe atom

J. Dembczy0144ski (*)

Institute of Physics, Technical University of Pozna0144, Poland (Rep le 13 juillet 1979, accepté le 27 septembre 1979)

Résumé.

2014

Les effets, sur la structure hyperfine d’un atome complexe, des interactions de configuration dépen-

dant du terme spectral considéré (SL-CI) sont déterminés pour la première fois avec précision. Pour l’atome de fer, les intégrales radiales évaluées à partir des données expérimentales corrigées des effets de second ordre

sont les suivantes (en unites atomiques).

Dans la configuration 3d7 4s

r-3 >01

=

3,913, r-3 >12

=

3,982 , r-3 > 10 = - 2,087 et [dP4s(r)/dr]20

=

37,0 .

Dans la configuration 3d6 4s2 .

r-3 >01 = 4,518, r-3 >12 = 4,594, r-3 >10 = - 0,722 .

Dans la notation de Bauche-Amoult, les valeurs des paramètres décrivant les effets du type SL-CI dans les confi- gurations dNs sont les suivantes :

x2 = x3

=

0,08 , x4

= -

0,062 , x7

=

0,068 , x11

= -

0,113 .

On trouve également :

x1=x5=x6=x8=x9=x12~0.

Pour rendre plus apparents les effets du type SL-CI, on introduit des paramètres 03B2, indépendants des effets rela- tivistes et des interactions de configuration ne dépendant pas du terme spectral considéré. Ces paramètres sont définis de la façon suivante :

03B2ksk1SL,S’L’

=

03B1ksk1SL/03B1S’L’.

Pour les termes 5F et 3F de la configuration 3d7 4s, ils prennent les valeurs suivantes : 03B2015F,3F

=

1,034(1) , 03B2125F,3F

=

1,02(1) et 03B2105F,3F(4s-electron)

=

0,944(5) .

Abstract.

2014

The effect of SL-dependent configuration interaction (SL-CI) on the hyperfine structure of a complex

atom is first determined accurately. The radial integrals for the iron atom evaluated from the experimental data

corrected to second-order perturbation theory are the following (in a.u.) : for the 3d7 4s configuration r-3 >01

=

3.913, r-3 >12

=

3.982, r-3 >10 = - 2.087 and [dP4s(r)/dr]20

=

37.0 ,

and for the 3d6 4s2 configuration

r-3 >01

=

4.518, r-3 >12

=

4.594 , r-3 >10 = - 0.722 .

The values of the parameters describing SL-CI in dNs configurations, according to Bauche-Arnoult notation,

are :

x2=x3=0.08, x4=-0.062, x7 = 0.068 , x11 = -0.113.

(*) Author’s address : Jerzy Dembczynski, Politechnika Poznanska, Instytut Fizyki, ul. Piotrowo 3, 60-965 Poznan, Poland.

J. Physique 41 (1980) 109-118 FEVR1ER 1980,

Classification

Physics Abstracts

35.10F 2013 31.20T

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004102010900

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It is also found that

To make the SL-CI apparent, a parameter fl independent of relativistic effects and of the SL-independent confi- guration interaction is introduced and defined as

The values obtained from the 5F and 3F multiplets of the 3d 74s configuration are the following :

1. Introduction.

-

Recent developments in expe- rimental methods, e.g. ABMR-LIRF [1], have enabled the measurement of the hyperfine structure (hfs)

of high lying metastable levels to be obtained to an

accuracy of within a few kHz. This provides new possibilities of proving the relativistic effects follow-

ing from the effective operator theory of Sandars

and Beck [2] as well as the configuration interaction effects on the (hfs) predicted by Bauche-Arnoult [3, 4].

Hitherto only two (hfs) experiments for the iron atom have been performed. Childs and Goodman [5], using the ABMR method, measured the (hfs) splitt- ings of the 5D1,2,3,4 levels of the ground configuration

3d’ 4s2. Measurements by ABRM-LIRF of the (hfs) splittings of levels belonging to two different meta- ’

stable multiplets 5F and 3F of the 3d’ 4s configuration

have been presented in a separate paper and reported

at an international conference [6]. The main topic

of this work is a theoretical analysis of the (hfs) experimental results for the 3d’ 4s configuration.

Both of the above mentioned experiments enable, for the first time, an exact analysis to be made of the configuration interaction on the hyperfine structure

of a complex atom. Extraction of the SL-dependent configuration interactions (SL-CI) from the experi-

mental radial integrals makes these integrals suitable

for a comparison with ab initio theoretical calculations.

In contradistinction to the experimental work, the number of theoretical papers containing calcula-

tions of the (hfs) of the iron atom is much greater.

The ab initio theoretical calculations by Rosen and Lindgren [7] bearing on the influence of relativistic effects on the (hfs) led to predictions concerning

the radial parameters r-3 )kskl. The calculations by Bauche-Arnoult, performed especially for the confi-

gurations 3dN 4s2 [3] and 3dN + 14s [4], enable the

participation of (SL-CI) by way of a number of addi- tional parameters to be considered. The Fermi contact term X for 5D has been calculated by several

authors. Watson and Freeman [8], as well as Bagus

and Liu [9], used unrestricted Hartree-Fock (UHF)

wave functions, whereas Kelly and Ron [10] applied

the many-body theory method. Some results of nonrelativistic Hartree-Fock calculations for the (hfs)

of the iron atom are also to be found in the work of

Fraga, Karwowski and Saxena [11].

2. Intermediate-coupling wave functions.

-

Since the intermediate-coupling eigenvectors represent one of the more serious problems to overcome in the

theoretical analysis of the hyperfine structure experi-

mental data, a parallel investigation of the fine struc- ture interactions of the first spectra of the transition metals was performed. The details will be published separately [12]. Below, the results for the FeI spec- trum are reported briefly. All states of the 3d6 4s2,

3d’ 4s and 3d8 mixed configurations were considered simultaneously and the 22 parameters were varied

iteratively to produce a best fit of the calculated level

energies to the 62 known optical levels associated with these configurations. Besides the Slater and

spin-orbit parameters, the parameters taking into

account two- and three-electron electrostatic interac- tions with distant configurations are included. On taking into consideration three-electron interactions, the mean error of the fit is reduced very strongly, to

21 cm-1.

It was observed [12] that inclusion of the three- electron parameters changes the values and reduces the

errors of the spin-orbit parameters, on which first of all the eigenvectors of intermediate-coupling are dependent.

The values obtained for the spin-orbit parameters

are the following :

Thus, the errors of the spin-orbit parameters are only 1.5 and 1. l ,%, respectively. Hence, the eigen-

vectors obtained from the above described fine struc- ture analysis should allow, according to the formula

given in ref. [13], a fit to be made of the theoretically predicted expressions for the magnetic-dipole inter-

action constants A to their experimental values to

within 30 kHz and 60 kHz, respectively, for the 3d 64s’ and 3d’ 4s configurations.

The next step of the improvement in accuracy of the

eigenvectors consisted in taking into consideration the two-electron magnetic interactions within each

configuration. The spin-spin and spin-other orbit

interactions within the 3d6 4s’ configuration were

included by way of two additional parameters M°(d, d) and M2(d, d), and within the 3d’ 4s confi-

guration by way of three additional parameters

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111

M°(d, d), M2(d, d) and M°(d, s), all of which are M arvin integrals [15] in the absence of electrostati-

cally correlated spin-orbit interactions [14].

To test the participation of two-electron magnetic

interactions in the fine structure splittings, several

fits were performed with different numbers of Marvin

integrals as free parameters. The best result of the

least-squares fit was obtained when both M° and M2 in the 3d’ 4s2 configuration were taken as free para- meters and when, in the 3d’ 4s configuration,

was assumed [16] and M°(d, s) was a free parameter.

In this way, taking into consideration these two- electron magnetic interactions, the rms error was

reduced to 17 cm -1. The values obtained for the spin- dependent parameters are (in cm-1) :

-

for the 3d’ 4s2 configuration

-

for the 3d’ 4s configuration

and

Some of the above values of Marvin integrals differ

from those calculated by Froese-Fischer [16] by the

Hartree-Fock method. This can point to the presence of correlated spin-orbit interactions with distant

configurations, as confirmed by the fit with Marvin

integrals taken as constant parameters with values

according to Hartree-Fock calculations. In this case

the fit was much worse and rms error was 26 cm -1.

The above should be noted especially, since also

configuration interactions on (hfs) identical in origin

exist which, in (hfs) splittings might be observed independently.

When carrying out the fine structure analysis of the

iron atom several slightly different sets of eigenvectors

were obtained. This gives the opportunity to observe

the influence of the eigenvectors used on the evaluated (hfs) parameters as well as on the interpretation of the (hfs) interactions.

Recapitulating, the following sets of eigenvectors

were prepared for the theoretical analysis of the experimental (hfs) data :

a) two-electron magnetic-interactions taken into

account according to eq. (1) (best fs fit), b) without the above interactions,

c) Marvin integral values constant and equal to

their values from Hartree-Fock calculations,

d) all five Marvin integrals dealt with as free para- meters.

Admixtures from other configurations to the 5 F,

3F and 5D multiplets of interest are very small. They

are below 4 x 10 - 4 % for the 5 F and 5 D. The largest

admixture to the 3F originates in a 3d’ 4s2 ’F multiplet

and amounts to about 0.7 %. Moreover, it has been observed that an expansion of the theoretical expres-

sions for the magnetic-dipole constants Aj (see chaptEr 3) over all states of the two configurations 3d’ 4s and 3d’ 4s’ had no essential influence on the

(hfs) fit and the obtained values for the (hfs) parame- ters. For this reason, it was decided to truncate (and, of

course, renormalize) the eigenvectors to states of a

single configuration. The truncated eigenvectors,

obtained in case a), for the states of interest, are given

in table I.

Table Ia.

-

SL-eigenvectors for the states (3F) 5F

and (3F) 3F of the iron atom.

"

Table lb.

-

SL-eigenvectors for the state 5D of the

iron atom.

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3. Theoretical expressions for the (hfs) constants. - Sandars and Beck [2] have constructed the effective operators required for the relativistic hyperfine inter-

action in a many-electron atom. Expressions for the

matrix elements of these (hfs) operators have been

given by Childs [ 17J for configurations of the classes 1N and 1N 1’, which include d6 s2 and d’ s. The 57Fe nucleus has a nuclear spin I

=

1 /2 ; hence, only (hfs) magnetic-dipole interactions exist.

Intermediate-coupling expressions for the (hfs)

constants A were calculated by computer program (1),

and the expressions obtained with the set a) of eigen-

vectors are given in table II.

The quantities a3d, a3d, a3a and a4 are dealt with

as free parameters. In addition, the value of any given

parameter may in principle depend on which confi-

guration is considered. For the sets b), c) and d)

of the eigenvectors, expressions similar to those given

in table II were also derived; however, only their

solutions will be given.

Table iI. - Theoretical intermediate-coupling ex- pressions for the magnetic-dipole hyperfine interaction constants A J

1 ’ ) The program was written by S. Bvttgenbach and M. Herschel, Institut fur Augewandte Physik, Universitat Bonn.

4. Proof of configuration interactions, and evalua- tion of the (hfs) parameters.

-

4.1 CONFIGURATION 3d’ 4s.

-

In the absence of SL-dependent configural

tion interactions [18], the values of the (hfs) para- meters aksk’ should be identical for all levels in one

configuration. Thus the expressions of table II should

allow a fit to their experimental value (hfs fit) accord- ing to the discussion in section 2, much better than to within 0.06 MHz. Greater divergences will point

to the presence of SL-dependent configuration inter-

actions (SL-CI), in which case new parameters

taking into account these effects must be introduced.

As is seen from table III, the divergences between the experimental and calculated A-constants are much greater when the 4-parameter least-squares fit/denoted by I in table III to the seven known A-constants is

Table III.

-

Results of least-squares fits of the theore- tical expressions for the A J constants to their experi- mental values for the ’F and 3F multiplets of Fe (in MHz).

performed. Accordingly, one is led to admit the pre-

sence of SL-CI. Bauche-Arnoult [4] predicted 13

additional parameters taking into consideration SL- CI on the 3dN 4s configuration.

One-electron excitation from an open shell to an

empty shell, and one-electron excitation from a

closed shell to an open shell producing these SL-CI

were parametrized by Bauche-Arnoult as follows :

Excitation

Excitation

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113

Excitation ;

Excitation

Excitation

Excitation i

Excitation

Excitation

These SL-CI contribute to the measured (hfs) parameters in the following way :

where, for the multiplets of interest, the d 4k are expressed as :

The first step in the investigation of the SL-CI was to

establish which of the above mentioned kinds of excitation play the primary role. For this purpose a

(hfs) fit was carried out, where a50’la 12 = 0.98 and a°F/a3F = 0.98 was assumed in accordance with relati- vistic Hartree-Fock calculations [7]. The above pro- cedure does not change the number of free para- meters (they were still four), but in the first place distinguishes a F from a3o’. It can be noted from table III that the fit (denoted by II) becomes incompa- rably better. Thus, from table IV and an analysis of

eqs. (16)-(20), it can be concluded that the value of

the parameter X4 has to differ strongly from zero

because

Therefore, in the next (hfs) fit, the parameter x4

was introduced by way of the dependences

where the coefficients at X4 follow from eqs. (16)-(20).

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Table IV.

-

Values y of (hfs) parameters for the 5 F

and 3F multiplets of the 3d’ 4s configuration of 57 Fe (the akskt are in MHz). Only the last digit in each value quoted is charged by standard deviation.

The results of the 5-parameter (hfs) fit are given in

tables III and IV. One n tes that a further reduction of the rms error is obtained.

The results of 3b ne fits permit the following

The results of t e last two fits permit the following

statements :

a) the excitation 3d -+ n"’ d (parameters x2, X3 and x4) contributes very substantially to L1si1 and

d 12 . SL

,

b) the contributions from the excitations 4s ----> n"’ d

(X8 and x,) and 4s - 3d (X 12) must be very small and

can be neglected;

c) the contributions from the excitations 3d - n"’ s

(xl) and 3d - n"’ g (X5 and x6) are not directly per-

ceptible, but then nothing suggests they should have

significant values.

Moreover, the parameters xl, X8, x9 and x12 involve the integrals (n"’ s I r-3 3d) or (n"’ d I r-3 4s) (see

eqs. (2)-(15)), whose values, according to Hartree-

Fock calculations [19], are very close to zero. Thus, contributions from the excitations mentioned above sub b) and c) are omitted in our forther considerations

as they are completely unnecessary.

It remained only to check the difference in SL-CI between the parameters a5’0 and a3l 0. For this purpose,

a 6-parameter (hfs) fit was performed in which the two Fermi contact terms for 4s unpaired electron were independent. The results of this calculation are also included in tables III and IV. It can be remarked that,

now, a (hfs) fit within the experimental errors of the

constants A is obtained.

In order to make sure that the above obtained very fine differences between the (hfs) parameters belong- ing to ’F and to 3F are not a result of the eigenvector

set used, the same calculations were repeated using

the eigenvector sets denoted by b), c) and d) in sec-

tion 2.

The observed influence of the eigenvectors can be

summarized as follows :

i) Obtained from the 4-parameter (hfs) fit, accord- ing to the set of eigenvectors used, the values of a°1, al°(3d) and al°(4s) vary within 1 MHz and that of a 12 within 10 MHz about those given in the second

row of table IV. The rms errors were about 0.5 MHz

irrespective of the set of eigenvectors used.

ii) Introduction of the assumed a"la"

=

0.98

reduced the rms errors very strongly (by a factor of 10)

in each case. The values of the parameters akskt are comprised within 0.5 MHz about those given in the

third row of table IV. A difference of about 2 MHz between the parameters a F and a F was obtained in

each case. Thus, it is clear that the assumption of a°F = a°F caused a very considerable perturbation

in (I), since the coefficients of these parameters are much greater that those of the others.

iii) The deviations of a°1, alo(3d) and al°(4s) in the 5-parameter hfs fit were in the limits of 0.16 MHz,

0.6 MHz and 0.35 MHz, respectively, depending on

the eigenvector set used. The values of the parameter

a5F were 68.65 MHz, 72.75 MHz and 71.89 MHz, respectively, when the sets b), c) and d) were used.

From the above, the exceptionally strong influence of the used eigenvectors on the parameter a 12 is

proved. In 5-parameter fits, a dependence of the quality of the (hfs) fit on the quality of the fine struc-

ture fit was observed. The (hfs) fit obtained using the

set a) of eigenvectors was twice as good as that with

the set b), and 4 times better than with the sets c)

and d). Therefore, only the set a) of eigenvectors is

assumed as sufficiently accurate to perform the 6-parameter fit. The use of a5’0, a3’0 and al°(3d’-core)

as three free parameters in the (hfs) fit requires very accurate eigenvectors, since in the SL-limit all three contact term parameters are linearly dependent.

Only break down of the SL-coupling permitted their

determination from the measurement of (hfs) splitt- ings in two different multiplets.

4.2 CONFIGURATION 3d64s2.

-

The (hfs) constants

A of the levels 5D 1, 2,3,4 were measured by Childs and

Goodman [5]. A (hfs) fit of the theoretical expressions

obtained in this work to their experimental data was performed. Although the A constants had been

measured in four states, the J-dependences of the

coefficients of the parameters ao5’ and a SD were iden-

tical in the SL-limit and differed only very slightly

in intermediate coupling. Thus, the sum ao5’ + a’50

and a5D were used as free parameters. By means of these two parameters all four experimental constants

A were fitted to within 7 kHz. The parameter values obtained are :

According to the prediction of Bauche-Arnoult [4],

the ratio

should hold if 3d - n"’ g excitations are neglected.

Using the value 0.953 for the above ratio obtained

in this work (see Table IV) and eq. (25), the two

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115

remaining (hfs) parameters for the 5D multiplet were

determined as follows :

The sum a5o’ + a’50 in eq. (25) is almost identical with the value given in ref. [20], but asp is greater by about 0.62 MHz (- 0.8 %). Here, too, a depen-

dence of the (hfs) fit quality and the value obtained

for the parameter a12 on the eigenvector sets used was

observed. The eigenvectors calculated without taking

into consideration the three-electron interactions

,

permitted a (hfs) fit to within 77 kHz. By taking into

account these interactions, the rms error was reduced to 15 kHz, and additionally taking into consideration two-electron magnetic interactions allowed the expe- rimental results to be fitted to the calculated A- constants to within 7 kHz, which exceeds experimental

error only by an insignificant amount.

5. Comparison of the experimental results and the ab initio theoretical calculations.

-

The relations bet-

ween the (hfs) parameters and the radial integrals

are the following [7] :

where gI is the nuclear gI factor in nuclear magnetons

(for the 5’Fe nucleus gI = 0.180 [7]) ; the values of the parameters ak8k1 are in MHz, and those of ( r-3 )

and [dPns(r)/dr]o in atomic units.

The experimental value a’o for 1 > 0 is the sum of the relativistic effects and the core-polarization contri-

bution due to the Fermi contact term. The experi-

mental value ( r -3 >11 may therefore be written as

The relativistic contributions amount to only 1 2013 2 %

of the experimental value for 3d-shell metals. They

can be obtained from the theory of Sandars and Beck [2] by using the relation

where F1° is the relativistic correction factor (RCF).

When estimating the relativistic part of ( r- 3 >10

for 3d’-core, ( r-3 >nonrel. = 4.46 a.u. [16] and F1°(3d) = - 0.012 were assumed. This last value was

linearly interpolated from the RCF’s given by Lind-

gren and Rosen [7] for the 3d’ 4s configuration of VI

and the 3d’ 4s configuration of Col. From the above and eq. (29), r -3 >10 for 3d 7-core equals

-

0.057 a.u.

The relativistic contribution to the radial integral r-3 >10 of the 5D multiplet is assumed to be

-

0.048 a.u. after Rosen [21].

The relation between r-3 >10t."t obtained with the effective operator formalism and the Fermi contact term is [21] : :

Using the preceding dependences as well as the experimental values of the (hfs) parameters for the 5F and 3F multiplets given in table IV (6-parameter fit) and for the 5 D multiplet given by eqs. (25 ) and (26),

the radial integrals for these multiplets were calculated.

All values are given in table V.

Table Va.

-

Experimental and theoretical values of

radial integrals for the multiplets of the 3d’ 4s configu-

ration of Fe (all in atomic units).

Table Vb.

-

Experimental and theoretical values of

radial integrals for the multiplets of the 3d6 4s2 configu-

ration of Fe (in a.u.).

Before comparing the experimental radial integrals

to those calculated ab initio, the contribution from

configuration interactions of all kinds has to be extracted. For the Fe atom, the (hfs) splittings of

levels belonging to three different multiplets have

hitherto been measured. Therefore, the SL-dependent contributions of the configuration interactions SL-CI to the individual radial integrals could be determined.

According to the discussion of section 4.1, only

the excitation 3d - n"’ d causes substantial pertur-

bations of the radial integrals r-3 )ol and r-3 )12.

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Therefore, the relations (16)-(20) can be rewritten in

a simpler form as follows :

For the 5D multiplet, Bauche-Arnoult predicted

The radial integrals common to all levels of the

configuration can be expressed [22, 18] as :

where the correction A ksk13d-" takes into consideration excitation of one electron of a closed shell to an

empty shell and is independent of the term SL of

the configuration considered and, also, independent

of the number N of 3d electrons [22]. For transition

metals the corrections d 3d’ give the contributions from excitation n" p6 -+ n" p5 d" 1’, where n" = 2 and 3, n"’ denotes the empty shell, and I’

=

p or f. Using

the formula proposed by Judd [22], these corrections

can be written as follows :

where

Hence, the experimental ratio ( r - 3 ) sL ( r - 3 ) sL , usually denoted by Xsp, for the SF multiplet should

be interpreted as :

In the theoretical analysis of (hfs) experimental results

it is generally assumed without proof that, unless accidental cancellations take place, 4fl) should be

of the same order of magnitude as d 3d , that is 4 fl# = 039412 3d. However, as can be seen from eqs. (36)

and (37), it is very difficult to make any more precise

statement, since the integrals R t of these equations

are expected to be either positive or negative, depend- ing on n" and n"’.

The above statement will now be checked by a comparison with the experimental results.

On the basis of eqs. (31)-(35) and from the definition of the spin-orbit parameter C, the following relation

between the experimental data should be fulfilled :

This expression moreover takes into consideration the predicted configuration perturbations 4 §) of the

parameters 03B6, since they are identical to the 0394 01

The coefficient of X3 in eq. (39) is much smaller than that of X2, and it can also be expected that R(2) in

eq. (3) should be of the same order of magnitude as

R(4) in eq. (4); therefore, the assumption X2 = X3 has no essential influence on the results of further calculations. Putting into eqs. (38) and (39) the experimental value of the radial integrals from table V and the values given by eq. (1), we obtained

The value of the A 0’ correction can be estimated from fine structure investigation [22]. The excitation

n’,p6 -+ n" p’ n"’ p (see eq. (36)) produced exactly

the same screening affect on the spin-orbit parameter ,.

The divergence of the spin-orbit parameter calculated by the Hartree-Fock method and that observed in the fine structure splitting may have the same origin

as the correction d 3d .

’ Hence, using eq. (1) and the values ’calc. from

ref. [16], we have :

°

It is seen that the corrections d 3a and A 12 probably

do not vanish. With regard to eqs. (36) and (37) and

the above results, the fact that the relation 4 fl# = d 3a

holds experimentally is highly intriguing.

To conclude our considerations of the radial

integrals ( r-3 >" and r -3 >12, their values common to all multiplets in each configuration of interest

were calculated using eqs. (31)-(34) and (40). The

results are given in table V.

Likewise it was possible to determine, to second-

order perturbation theory, corrections to the Fermi contact terms of the unpaired 4s-electron in the ’F and 3F multiplets.

By definition [23], the contact terms for the 3dN-core

are :

where

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117

Also,

Here, summation extends over n" = 1,2 and 3 only,

and d" is the same in (42) and (43).

Therefore, according to eq. (8), the difference in

core-polarization effects between the 3d6 4s2 configu-

ration and the 3d’-core gives exactly the contribution from the excitation 3d’ 4s -+ 3d’ rt" s to the Fermi contact term of the unpaired 4s-electron, i.e.

The above result is positive as was to be expected,

since it is simultaneously the contribution from exci- tation of the paired 4s-electron of the 3d’ 4s2 configu-

ration to an empty shell. Using eqs. (16), (21), (22)

and (28) and the suitable values from tables IV and V, the first-order value as well as the second-order corrections for the unpaired 4s-electron were obtained

as :

and

Fraga, Karwowski and Saxena [11] have performed

nonrelativistic Hartree-Fock calculations for many elements. Their value for the Fermi contact term of the unpaired 4s-electron in the 3d’ 4s configuration

of the iron atom is 32.36 a.u. The RCF factor

Flo = 1.096 2 [21]. Therefore, the theoretically pre- dicted value of this parameter taking into conside- ration the relativistic effect is :

Table V includes the radial integrals evaluated

from measurements of the (hfs) splittings and results of ab initio theoretical calculations available from the literature. The experimental values of the non-

relativistic radial integrals r - 3 > in both configu-

rations of interest are by about 10 % smaller than those calculated by the Hartree-Fock method. This can be

explained by the presence of radial excitation from the inner closed shells 2p and 3p to an empty shell,

as already mentioned.

The value obtained for the contact term x5D is in rather good agreement with recent calculations [9,10].

On taking into account the second-order correction

originating in SL-CI, the experimental and calculated

Hartree-Fock values of the Fermi contact term of the

unpaired 4s-electron show quite good agreement.

6. Conclusion.

-

In spite of the fact that the states of interest are to 99.62-99.87 % pure-SL states,

evaluation of the (hfs) parameters akskl in pure-SL

coupling has no meaning, as can be seen from tables III and IV. For the precise theoretical analysis of the

measured (hfs) splittings the eigenvectors of inter-

mediate coupling are one of the more serious problems

to be overcome. Very careful preparation of these eigenvectors the contributions from SL-CI to the individual (hfs) parameters of the multiplets of

interest to be determined. The difference in a" 1 parameters measured in 5F 2,3,4,5 and in 3F2,3,4 is

about 3 %, and has a predominant influence on the

results of the (hfs) fit. The SL-CI enhances the Fermi contact term of the unpaired 4s-electron by about

22.4 and 29.7 %, respectively, in the 5F and 3F multiplets of the 3d’ 4s configuration of the iron atom. As shown, the parameter a 12 is very sensitive to the accuracy of the eigenvector set used; therefore,

so is the parameter a

=

ao’la 12 generally used for making apparent the configuration interaction. None- theless, the SL-independent configuration interaction, which needs not be identical in the parameters aUl 1 and a12, as well as the relativistic effects, also have an influence on the value of the parameter a. Therefore, it might be reasonable to define other parameters to render apparent the SL-dependent configuration inter-

action. At present, when (hfs) experimental results

for a greater number of multiplets of one configuration

become available, a parameter p defined as

where 4-" and abt[I denote the (hfs) parameters

belonging to two different multiplets, could fulfil this role. The parameter thus defined is independent

of the RCF and SL-independent configuration inter-

action. In the absence of SL-CI, the value of fl is obviously equal to unity. The values of this new

parameter obtained from the ’F and 3F multiplets

for the 3d’ 4s configuration of the iron atom are the

following :

and

The proof, by the (hfs) analysis performed, that the

SL-CI parameters X2 and X3 differ considerably from

.

zero confirms automatically the supposition made

in section 2 concerning the existence of correlated

spin-orbit interaction on the fine structure of the iron atom. This follows from the fact that both effects, the one observed on the fine structure and the other

on the hyperfine structure have the same origin,

namely the excitation 3d - n"’ d. An independent

(11)

determination of the parameters X2 and X3 directly from fine structure fit will be attempted in the near

future. It should be noted that the SL-dependent

contributions to the Sternheimer shielding or anti- shielding factor R [18] are expressed by parameters

xi identical with those discussed in sections 4 and 5.

Thus, from the investigation of magnetic-dipole (hfs) interaction, the SL-dependent part of the Stern- heimer factor R can be obtained.

Acknowledgments.

-

The author thanks the Alexan- der von Humboldt Foundation in the Federal Repu-

blic of Germany for the fellowship during which

this work was performed.

The author also wishes to acknowledge Prof. Pen-

selin and his coworkers Ertmer, Dicke and Johann at the Institut fur Angewandte Physik der Universitat Bonn for suggesting the problem and their very

helpful discussions.

References [1] ERTMER, W., HOFER, B., Z. Phys. A 276 (1976) 9.

[2] SANDARS, P. G. H., BECK, J., Proc. R. Soc. (London) A 289 (1965) 97.

[3] BAUCHE-ARNOULT, C., Proc. R. Soc. (London) A 322 (1971)

361.

[4] BAUCHE-ARNOULT, C., J. Physique 34 (1973) 301.

[5] CHILDS, W. J., GOODMAN, L. S., Phys. Rev. 148 (1966) 74.

[6] DEMBCZY0143SKI, J., ERTMER, W., JOHANN, U., PENSELIN, S.

and STINNER, P., Sixth International Conference on

Atomic Physics-Riga 1978, submitted to Z. Phys. A.

[7] LINDGREN, I., ROSÉN, A., Case Stud. Atom. Phys. 4 (1974) 250.

[8] WATSON, R. E., FREEMAN, A. J., Phys. Rev. 123 (1961) 2027.

[9] BAGUS, P. S., LIU, B., Phys. Rev. 148 (1966) 79.

[10] KELLY, H., RON, A., Phys. Rev. A 2 (1970) 1261.

[11] FRAGA, S., KARWOWSKI, J. and SAXENA, K. M. S., Handbook

of Atomic Data (Elsevier Scientific Publishing Company, Amsterdam) 1976.

[12] DEMBCZY0143SKI, J., Submitted to Physica.

[13] DEMBCZY0143SKI, J., ERTMER, W., JOHANN, U., PENSELIN, S.

and STINNER, P., Z. Phys. A 291 (1979) 207.

[14] RAJNAK, K., WYBOURNE, B. G., Phys. Rev. 134A (1964) 596;

PASTERNAK, A., GOLDSCHMIDT, Z. B., Phys. Rev. A 6 (1972) 55.

[15] MARVIN, H. H., Phys. Rev. 71 (1947) 102.

[16] FROESE-FISCHER, Ch., Can. J. Phys. 49 (1971) 1205.

[17] CHILDS, W. J., Case Stud. Atom. Phys. 3 (1973) 215.

CHILDS, W. J., Phys. Rev. A 2 (1970) 316.

[18] ARMSTRONG, L., Jr., Theory of the Hyperfine Structure of Free

Atoms (Wiley-Interscience, New York) 1971.

[19] STERNHEIMER, R. M., Phys. Rev. A 6 (1972) 1702.

[20] CHILDS, W. J., Phys. Rev. 160 (1967) 9.

[21] ROSÉN, A., Physica Scripta 8 (1973) 154.

[22] JUDD, B. R., Proc. Phys. Soc. 82 (1963) 874.

[23] BAUCHE, J., JUDD, B. R., Proc. Phys. Soc. 83 (1964) 145.

[24] FROESE-FISCHER, Ch., The Hartree-Fock Method for Atoms

(Wiley-Interscience, New York) 1977.

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