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MCDF CALCULATIONS OF TRANSITION ENERGIES IN URANIUM HELIUMLIKE IONS
P. Indelicato, O. Gorceix, J. Desclaux
To cite this version:
P. Indelicato, O. Gorceix, J. Desclaux. MCDF CALCULATIONS OF TRANSITION ENERGIES IN URANIUM HELIUMLIKE IONS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-293-C9-296.
�10.1051/jphyscol:1987952�. �jpa-00227370�
JOURNAL DE PHYSIQUE
Colloque C9, suppl6ment au n012, Tome 48, decembre 1987
MCDF CALCULATIONS OF TRANSITION ENERGIES IN URANIUM HELIUMLIKE IONS
P. INDELICATO(~), 0. GORCEIX and J.P. DESCLAUX"
Laboratoire de Physique Atomique e t Nucleaire, CNRS UA-771, U n i v e r s i t e P i e r r e e t Marie Curie, BP 3 4 6 , F-75229 P a r i s Cedex 05, France
* C E N Grenoble, DRF/Service de Physique, BP 85X, F-38041 Grenoble, France
RCsumC.
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Les resultats de calculs ab initio pour les energies de transition dans l'ion hCliumoTde de l'uranium sont donnCs. La mCthode MCDF a CtC utilisCe pour traiter efficacement des effets relativistes et des corrClations. L'importance des corrections radiatives pour ce systkme fortement relativiste est discutCe. Des effets de retard intenses sont prCdits par les calculs sur certains niveaux.Abstract.
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Ab initio calculation results are reported for transition energies in uranium heliumlike ions. The MCDF method is used to treat relativistic effects and correlations efficiently. Radiative corrections for this highly relativistic system are discussed. Important retarded interaction effects are also encountered for some levels.I Introduction
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The experimental study of Lyman and Balmer lines in uranium heliumlike ions will provide very important checks of fundamental theories (QED, Dirac equation, few-body problem) for electrons in the presence of a strong external field. Indeed, as relativistic effects behave as high powers of the atomic number Z, the availability of uranium ions (with Z=92) for transition energy and lifetime measurment allows an extension to the relativistic region compared to the present experimental state of art 111. Recently, Munger and Gould 121 performed a measure of the ls2p 3Po- 1 ~ 2 s 3S1 energy separation in uranium through a lifetime measurment. As will be understood from the following, the precision of that experiment (3%) in not sufficient to test accurately the treatment of higher order QED effects. In this paper, MCDF calculation results are reported and various contributions to the energy are separetely depicted. From the given data, the precision that have to be reached to check the validity of the treatment for anyone of these contributions is seen. Further informations about the theoretical aspects of their treatment are to be found in three preceding papers 131. In those publications, calculations for such a relativistic ion as the uranium ion were not given due to the then unexpected experimental advancement. A final accuracy of the order of 1 eV is expected for spectroscopic studies at the SISESR accelerator in Darmstadt (Briand J.P. private communication) so that contributions of this order of magnitude should be included in theoretical calculations. Beyond the wish to obtain a good agreement between theory and experiment stands the wish to descrirnine between distinct methods the proper one in the ultra relativistic domain. Measurments on hydrogenlike uranium ions will be performed in the same experiments to test the techniques of computation for QED effects in the presence of the Coulomb field of the nucleus. Such techniques are presented along with numerical data in the paper by Johnson and Soff 141. For such a system the one-electron Lamb shift calculations have to be refined; for example, effects such as the influence of finite nuclear size on radiative corrections have to be considered. For heliumlike ions, the difficulties associated with the Lamb shift evaluation are still present but one is also faced with additional difficulties associated with the two-electron interaction.I1 The MCDF approach and the radiative correction treatment
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In the present paper, we use the same schemes as in paper I and I1 131 for the treatment of electron-electron interaction in many-("present address : National Bureau of Standards, Gaithersburg, MD 20899, U.S.A.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987952
JOURNAL DE PHYSIQUE
electron ions. Let us recall,briefly, the sequence of our method. First, the energy and the wavefunction of a given level is computed by numerical resolution of the Dirac-Fock equations. That is to say, the relativistic self-consistent field approximation is used to obtain eigenvalues and eigenstates for the Schriidinger-like equation:
(1) ( hDl+hDz
+
g c ) l Y > = E I Y >where the hDi 's stand for the one-electron Dirac hamiltonian in the electrostatic potential U of the extended nucleus:
(2) hD = c a .p
+ p
c2-
e U(r)and gc for the electrostatic repulsion. Atomic units are used throughout this paper. In equation (I), the interaction is approximated by the instantaneous electric repulsion while that contribution is known to be stronger than the magnetic and retarded interactions by more than one order of magnitude (even for very heavy atomic systems).
The relativistic one-electron effects are mated exactly by means of the Dirac equation formalism. On the other hand, relativistic effects are to be introduced in a less rigorous way. The Bethe-Salpeter equation in an external field 151 being, in practice, unsolvable, an effective potential is introduced 161 to describe the interaction process. This approach is possible in Dirac-Fock scheme due to our ability to define one- electron states in that scheme 131. The effective potential has in Coulomb gauge the following expression:
4 -+
while, it has, in Lorentz gauge the other expression:
( 4 4
In expressions (3) and (4), o stands for the energy separation between the two interacting electrons or, equivalently, for the exchanged photon energy. In the semi-relativistic domain, only the two first even orders in o (or equivalently in l/c) were taken into account until recently. In expression (3), one recognizes then, apart from the electrostatic interaction, the usual instantaneous magnetic interaction gm and the second order electric retardation g, operators with:
and:
The sum of those two operators is known as the Breit term. Such effects as spin-other orbit and spin-s in interactions are introduced at that stage. In Lorentz gauge, the same decomposition is also feasibE and gives rise to the g. and g(2) operators denoting the same physical effects in that gauge. The expressions for g,, and g(9 are distinct. The authors have already discussed the problems associated with the choice of a particular in paper I and 111 131. The last term in equation (3), or in equation (4), describes higher order (in l/c) retardation effects in the electron-electron interaction. These terms are described by the operator:
(7) gho = go
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gc-
gm-
grcIn table 1, computed energies for various transitions in heliumlike uranium ion are given. To show the numerical importance of distinct contributions, we report successively:
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the Dirac-Fock eigenvalue for equation (1);-
the mean value of the magnetic interaction operator (5) on the Dirac-Fock wavefunction;-
the mean value of the retarded interaction (6) on the same wavefunction;-
the mean value of the higher order terms (7) on the same wavefunction.Going beyond the independent particle approximation, correlation energies are then calculated by performing MCDF calculations /7/. The eigenvalue obtained for equation (1) is lower than the one obtained in a Dirac-Fock calculation; the difference between the two values is thus designated in table I by the name "electrostatic correlations". We use the same developments as in paper I. The mean value of the magnetic operator g,,, is then calculated on the MCDF wavefunction. The mean value previously calculated with the Dirac-Fock wavefunction is substracted to the MCDF value giving the magnetic correlations reported in table I. For physical reasons already discussed, we avoid MCDF calculations for the retarded interaction.
Radiative corrections are included as a superposition of one-electron and two-electron effects.
Mohr's results 141 are reported for the self energy contribution. They correspond to the exact hydrogenlike first a order contribution for a pointlike nucleus. The influence of the extension of the nucleus on self- energy is then calculated and reported. In an heliumlike system, the self-energy of an electron is altered by the presence of a second electron. An estimation of the screening of the self-energy has to be calculated, it is obtained by a phenomenological approach based on the Welton's picture (see paper I1 131).
The vacuum polarisation contribution to a ( Za ) order is calculated by means of the ~ h l i n g potential mean valud on the Dirac-Fock wavefunction; the finite extension of the nucleus and the presence of another electron are thus taken into account automatically. We report then in table I the mean value of the a ( Za )3 order vacuum polarisation potential and the a* ( Za ) order contribution 171. At last, the reduced mass correction is given. All the discussed contributions are then added up to give the total energy for any level.
Transition 2 3P0+2 3S1 2 3P2+2 3S1 2 3Pl+l 'So 2 'P1+l 'So
Coulomb energy 154.90 455 1.59 96754.18 101 170.29
Magnetic energy 147.86 13.10 -323.34 -310.66
Retardation energy -5.33 -5.54 -3.67 3.62
Higher order retardation 0.07 -7.32 0.03 3.06
Electrostatic correlations -0.18 -0.06 1.07 1.17
Magnetic cornelations -0.33 -0.02 3.62 3.56
Self Energy -56.73 -57.46 -350.00 -350.73
Finite size effect on S E 1.01 1.19 6.24 6.41
SE screening 0.20 0.73 7.09 7.67
Vacuum Polarisation 13.76 16.39 90.77 93.36
VP to a ( Za l3 order -0.58 -0.75 -4.49 -4.66
VP to a2 ( Za ) order 0.1 1 0.13 0.7 1 0.73
Reduced mass correction 0.00 -0.01 -0.22 -0.23
Total energy 254.77 4511.96 96181.99 100623.59
Table I
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Contributions (in eV) to transition energies in heliumlike uranium ionI11 Results for heliumlike uranium ion
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The method described in the preceding paragraph is now applied to study transitions in heliumlike uranium ion. Results are given in Table I for the An=O ls2p 3p2+
1 ~ 2 s 3S1 transition and for the An=l ls2p 3.1P1+
1s2 ISO transitions. These later transitions have been extensively studied in heavy ion spectroscopy /I/ to obtain informations on relativistic effects onC9-296 JOURNAL DE PHYSIQUE
the ground state. Indeed, from numerical data, it is understood that correlations and radiative correction contributions are mostly those involving the final ground state because of their weaker influence on excited states. We observe that, for that relativistic system, magnetic correlations are three times stronger than electrostatic correlations. The magnetic correlations behave as 22 while the electrostatic correlations are roughly constant with Z (apart from a relativistic behaviour discussed in paper I). In other respects, we obsenre that, as mentioned in reference 141, the finite size of the nucleus affects the self energy in a non- negligible way. Of the same order of magnitude, the vacuum polarisation contribution to the a ( Zcr )3 order, the self energy screening and high order retardation terms are also to be taken into account if a comparison with experiment is to be made within a precision of a few eV. Concerning that retardation term, it is seen that transitions involving Is 2p 3P2 and 'PI states are the only one for which this contribution is important.
For the ume being, the only available experiment on U+9O/2/ 'has given an indirect measure of the ls2p 3Po
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1 ~ 2 s 3S1 energy separation. This experimental preliminary result (260 f 8 eV) is in good agreement with our numerical result 254.77 eV. The precision is still insufficient to check the validity of the theoretical treatment of relativistic effects (retardation through an effective potential scheme and relativistic correlations). The same holds true for checking the many-electron radiative correction treatment. Further experimental investigations with increased precision are thus expected to give hints on those puzzling problems.References
1 For example, transition energies have been measured in heliumlike krypton in the X-ray range by Indelicato P, Briand JP, Tavernier M and Liesen D, 2.Phys.D 1986 2 249, and in the UV range by Martin S, Buchet JP, Buchet-Poulizac MC, Denis A, Dmetta M, DCsesquelles J, Grandin JP, Husson X, Leclerc D and Lesteven E, to be published in Phys. Rev. A; in heliumlike xenon experiments have been performed in the X-ray range by Briand JP and al. (private communication) and in the UV range by Martin S, These d'Etat, Lyon 1987.
2 Munger CT and Gould H, Phys. Rev. Lett. 1986 57 2927.
3 Gorceix 0, Indelicato P and Desclaux JP, J. Phys. B 1987 20 639 (paper I), Indelicato P, Gorceix 0 and Desclaux JP, J. Phys. B 1987 20 651 (paper 11) and Gorceix 0 and Indelicato P, to be published in Phys. Rev. A (paper 111).
4 Johnson WR and Soff G At. Data Nucl. Data Tables 1985 33 405 and Mohr PJ, Ann.Phys.(N.Y.) 1974 88 26.
5 Salpeter EE and Bethe H Phys. Rev. 1951 84 1232 and Sucher J Phys. Rev. 1958 109 1010
6 Akhieser A and Bereztetski VB Quantum Electrodynamics 2nd ed. New-YorlZ John Wiley and Sons Inc. 1963.
7 Desclaux JPComput. Phys. Comm. 1975 9 31
8 Wichmann EH and Kroll NM Phys. Rev. 1956 101 843 and Kallen G and Sabry A K.Dan.Vidensk.Selsk.Mat.Fis.Med. 1955 29 17
