12 avril 2013 TUNGSTEN - W VI

Hardet Aimlain NGOMA Atomic spectroscopy

aimlain@hotmail.fr

Radiative properties of the

W5+ ion

Comparison of the values gotten by the use of the code of COWAN and those

of the picture of the MCDHF-EOL calculations and RCI of ab initio for W VI.

Drawn of the presented article by:

S Enzonga Yoca, P Palmeri, P Quinet, G Jumet and E´ Biémont

Département de Physique, Faculté des Sciences, Université Marien Ngouabi, BP 69 Brazzaville,

Congo.

Astrophysique et Spectroscopie, Université de Mons—UMONS, B-7000 Mons, Belgium.

IPNAS, Université de Liège, B-4000 Liège, Belgium.

Chimie Quantique et Photophysique, Université Libre de Bruxelles, B-1000 Brussels, Belgium.

E-mail: enzosat@yahoo.fr

①Adjustment of the average Energies and spin-orbits parameters

The radiative properties of the W5+ ion is used in the survey of two theoretical

independent approaches, i.e. the method Hartree-Fock with relativistic corrections

executed by the Cowan code and the multiconfigurationnel method of Dirac-Hartree-Fock

(MCDF) executed in the GRASP2K set. The interrelationships (interactions) valence heart is

studied in detail while comparing the models of the potential of core-polarization more a

correction of the operator of the dipole considered (HFR + CPOL) in a case and on the

configurations core-excited are included explicitly in the valance interaction configuration

of the expansion of the atomic state function (Migdalek[5] showed that the interaction

corresponds to the interaction of configuration). In general, a good agreement is found

between these two theoretical methods. Remarkably, the core-polarization effects

lengthen the life spans of 15% until 35% and even by a factor of 2 for the levels 5f. The

lifetimes of these two 5f levels are instituted to be a dependent and in particular sensitive

model, excluding the core-penetration effects, clearly requiring for it of the precise

measures.

Hardet Aimlain NGOMA |tungstène W VI 1

12 avril 2013 TUNGSTEN - W VI

Calculations

Tungsten is a heavy element of the periodic picture with z=74. The level fundamental of

the W VI is 5 with electron of valence revolving around an erbium-like of the ionic

core of its 68 electrons. Therefore, it is important to consider the effects at a time

relativistic and of interrelationship to describe this atomic structure correctly. As no

measurements are available in the literature, two independent theoretical methods that

take in consideration the two the effects have been used in this survey to calculate and to

basis the radiative parameters. These are described briefly in the following subdivisions.

Method Hartree-Fock with the relativistic corrections

In the Hartree-Fock method with the relativistic corrections (HFR) of Cowan (1981), a whole of

orbital is gotten for every electronic configuration while solving the Hartree-Fock equations

spherically for an averaged atom. The equations result from the variational principle application to

the configuration average energy. The relativistic corrections are included in this set of equations,

i.e. the Blume-Watson spin-orbits , mass-velocity and of a part calls Darwin terms. The Blume-

Watson spin-orbit term consists of the part of the Breit interaction can be reduced to an uniform

operator [25, 26]. The multiconfiguration Hamiltonian matrix is constructed and diagonalized in the

LSJM representation in the structure of the theory of Slater-Condo. Every element of the matrix is j ∏

a sum of products of Racah angular coefficients and radial integrals (of Slaters and the spin-orbit

integrals),i.e.:

, ,

< │H │ J > = ∑ . ∏ ∏

,The radial parameters , can be adjusted to adapt the levels of available experimental energies in

,an approach of least squares; are the angular coefficients. The values and the states gotten in

this approach (ab initio or semi-empiric) are used to calculate the length of wave and the probability

of transition for every possible transition. This technique has been modified (to see for example,

Quinet and al (1999)) to include the disruptions of the core-polarization effects and is known as HFR

method + CPOL.

HFR method and programs of computer of Cowan

Briefly, the organization of the continuation of programs used for the setting in work of the

HFR method is the following [1]:

RCN: value the radial wave-functions and the parameters of Slater.

RCNPOLB: is a RCN program including the polarization of the heart.

RCN2: calculate the integrals of configuration interaction, the electric and magnetic

multipolar integrals (E1, E2, M1 and M2).

RCN2POLB: is a RCN2 program including the polarization of the heart.

RCG: calculate the energy angular coefficients matrixes, constructs and diagonalize the

matrixes of energy while giving the vectors and clean values, calculate the oscillator

strengths and the transition probabilities.

RCE: conduct the adjustment of the clean values calculated to the experimental levels of

energy.

Hardet Aimlain NGOMA |tungstène W VI 2

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Modeling of the core-polarization effects

Calculations retailed of atomic structures, for the atoms (or ions) heavy, must take in account the

relativistic effects and the interaction of configuration; this last induced the interrelationships of

valence-valence type and core-valence. To treat these effects, Migdaleks and Baylises [5] proposed

an approach in which biggest part of the valence-valence interrelationship is represented inside the

diagram of configuration interaction, while the interrelationship core-valence is represented

roughly by a model of type "core-polarization”. For an atom (or ion) with n electrons of valence, the

single electronic operator of this potential can write:

∑ , (1)

( )

Where:

is the dipolar polarizability of the heart and is the ray of cut. One generally takes for the

average value of calculated in the HFR approach for the orbital of the heart most external.

Besides, the interaction between the modified electric fields undergone by the valence electrons

leads to one term bielectronic:

.

= − ∑ , (2) ⁄ 〔( )( ) 〕

A supplementary correction, taking into account the core-penetration of the valence electrons, has

been proposed by Hameed and al [3, 4]; it is translated, in the setting of the present formalism, by

the addition of the term of core-penetration, to know,:

( ) ∫ ( ) (3)

To the integral ( ) (4) ∫ ⁄( )

In (2)

The inclusion of the core-polarization in the hamiltonian imposes, for the consistency in the

reasoning, of the modifications to the operator of the dipolar moment in the element of transition

matrix [39]. The radial dipolar integral

( ) ( ) (5) ∫

Must be to replace by:

( ) ( ) ( ) 〔 − 〕 ( ) − (6) ∫ ∫

These modifications, programmed in the code of Cowan [1] by Quinet and al [2], lead to the HFR+PC

approach.

Procedure of adjustment by least squares

The HFR approach / (HFR+PC) can be used either of manner ab initio is in a semi-empiric manner

(combined with an adjustment by least squares of the calculated clean Hamiltonian values to the

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experimental values of the energy levels). We used the HFR+PC method extensively for the

obtaining of our results. The quality of the adjustment procedure is determined by the standard

deviation, s, definite by [1]:

∑ ( )

= 〔 〕 (7)

Where :

• et are respectively the calculated energy levels and observed;

• is the number of levels adjusted;

• is the number of adjustable parameters ( , , , , , . . .).

+5 In the tabeau1 of the ATOMIC article", MOLECULAR AND OPTICAL PHYSICS" on the w the average

-1energies levels begin by 5 represent the fundamental level whose energy is 0,0cm . On the .

-1other hand in the calculation with the code of COWAN, it is 546630cm that wants to say

-1 5,46630cm , because the comma binds between the fourth and the fifth number. And the indicated

level represents the parity and the value of ( = 1.5). The related term is written between

parentheses (2d).

Calculation of the code of Cowan

rcn and rcn file ing11:

LS coupling

W6 5d 6d 7d 6s 7s 5g 6g 7g

-5.466300* -5.466 0.000000 0.800 5d j= 1.5 100.0 5d (2d) 2d 0.0 7d (2d) 2d 0.0

6d (2d) 2d

3.644200* 3.644 0.000000 1.200 5d j= 2.5 100.0 5d (2d) 2d 0.0 7d (2d) 2d 0.0

6d (2d) 2d

77.756600* 77.757 0.000000 2.002 6s j= 0.5 100.0 6s (1s) 2s 0.0 7s (1s) 2s

256.904600* 256.905 0.000000 0.800 6d j= 1.5 100.0 6d (2d) 2d 0.0 7d (2d) 2d 0.0

5d (2d) 2d

259.292100* 259.292 0.000000 1.200 6d j= 2.5 100.0 6d (2d) 2d 0.0 7d (2d) 2d 0.0

5d (2d) 2d

274.512000* 274.512 0.000000 2.002 7s j= 0.5 100.0 7s (1s) 2s 0.0 6s (1s) 2s

354.360500* 354.361 0.000000 0.800 7d j= 1.5 100.0 7d (2d) 2d 0.0 6d (2d) 2d 0.0

5d (2d) 2d

355.500000* 355.500 0.000000 1.200 7d j= 2.5 100.0 7d (2d) 2d 0.0 6d (2d) 2d

0.0 5d (2d) 2d

355.760400* 355.760 0.000000 0.889 5g j= 3.5 100.0 5g (1s) 2g 0.0 7g (1s) 2g

0.0 6g (1s) 2g

Hardet Aimlain NGOMA |tungstène W VI 4

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