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RECENT PROGRESS IN THE THEORY OF
RELATIVISTIC EFFECTS AND Q.E.D.
CORRECTIONS IN TWO AND THREE-ELECTRON
IONS
P. Indelicato
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C1, supplement au nO1, Tome 50, janvier 1989
RECENT PROGRESS IN THE THEORY OF RELATIVISTIC EFFECTS AND Q.E.D. CORRECTIONS IN TWO AND THREE-.ELECTRON IONS
P. INDELICATO
National Bureau of Standards, Gaithersburg MD 20899, U.S.A. and Laboratoire de Physique Atomique et Nucleaire, Universite Pierre et Marie Curie, F-75230 Paris Cedex 05, France
Abstract: In this paper we make a review of the most recent calculations of transitions energy in two and three- electron ions. Results of the different calculations are compared and checked against recent experimental results. We will compare results from G. Drake's unified theory (two-electron case only), from Relativistic Many Body Perturbation theory and from Multiconfiguration Dirac-Fock calculations. Among other points we will show experimental evidences that the retardation contribution has to be computed in Coulomb gauge, in agreement with theoretical derivations from Bethe and Salpeter equation. We will also discuss methods for approximation of two- electron radiative corrections (the so called self energy screening).
Resume: Nous prCsentons une xvue des calculs les plus rCcents d'tnergie de transition dans les ions B deux et trois Clectrons. Les rCsultas des diffkrents calculs sont compads entre eux et aux resultats expkrimentaux les plus rtkents. Les rCsultats de la thCorie unifiCe de.G. Drake (ions
A
deux Clectrons seulement), de la thCorie Relativiste des Perturbations B Plusieurs Corps, et de la mCthode Dirac-Fock Multiconfigurationnelle sont prCsentCs et comparks. Entre point on montrera que la contribution dl'interraction retard6 doit etre calculC en jauge de Coulomb, en accord avec les demonstrations faites B partir de 1'Cquation de Bethe et Salpeter. Les mCthodes pour l'tvaluation approximative des corrections radiativesA
deux electrons seront aussi p6sentkes.With the increasing availability of very precise high Z data for transition energies in few electron ions, it has become increasingly interesting to undertake fully relativistic calculations. Testing Quantum Electrodynamics in those systems is of particular interest since they are very often more accessible to experiments than corresponding one electron ions. The study of bound many electron systems in a relativistic model is still a challenge since such a problem has no simple exact Hamiltonian form. One has then to deal with many difficulties superimposed on the usual problem of computing correlations effects, and also to take care of radiative corrections. In this survey I will fist presents briefly the different methods which have been used to compute transition energies in two and three- electron systems, referring the interested reader to original literature for detailed description. I will then focus on a detailed comparison of the results of the differents method with experiment, this review being designed to show experimentalist how well the problem is understood and what advances have to be made, from an experimental point of view to trigger new progress in calculations.
JOURNAL DE PHYSIQUE
2 . h e o r e t i c a l foundations.
2.1. Perturbation exeansion versus Bound Stateseauations,
Quantum electrodynamics (QED) is the fundamental theory from which any sensible approach to high precision calculations of atomic properties has to be derived. As long as one is dealing with one electron in a (classical) external field the equivalence between measuring atomic properties and testing QED is easily established. One electron ions are an acceptable model for such a system, at least at the level of accuracy one is usually dealing with in medium to high Z ions. Such an equivalence is not at all evident when one is dealing with more than one electron. For example straightforward generalizations of non-relativistic method lead to difficulties which have been noticed as early as 1951 111. There is two equivalent ways to use QED to compute many-electron atomic properties. The first one uses the Gell-Man and Low theorem and exact one electron propagators in the (nuclear) external field to make a p e e b a t i o n expansion. The external field can be a pure Coulomb field and one can use hydrogen-like wavefunction as in /2//3/. One can also use some sort of nuclear model to get a more realistic nuclear potential, which can be combined with the use of a Hartree-Fock or Dirac-Fock potential and Hartree-Fock or Dirac-Fock orbitals as shown by J. Sapirstein 141. The latter method is a variant of the method developed in the seminal work of Kelly and is known as Relativistic Many Body Perturbation Theory. Both methods provide a rigorous framework for calculation of transitions energies in heavy ions or atoms including radiative corrections. However the practical difficulties to perform such calculations are formidables and to date I am not aware of any calculation of two-electron radiative corrections.
An alternative method consists in using integral bound state equations in the frame-work of the Dyson- Schwinger equations (for details see 151 and references therein) They provide an infinite hierarchy of equations, a fact which limits their usefulness, but as they enable one to sum infinite classes of diagrams they may provide a very powerful tool. The only equation of that type which has been widely used in the study of bound states is the Bethe and Salpeter equation /6/,/7/j8/ which is of particular importance in the investigation of recoil and hyperfine corrections in two body systems 191. For the problem in which we are interested here these equations provides a rigorous framework to get effective Hamiltonians for the many electron problem. The difficulty of using the Harniltonian approach in a relativistic many problem is due to the fact that a Hamiltonian equation can only have one time while in the relativistic many-body problem there is one time for each particle. Previous derivations of relativistic many body Hamiltonians, performed using QJ?,D in second quantization /10/,/11/j12/ are valid only in the single configuration Hartree-Fock approximation i.e., essentially in an independent particle approximation, and have mainLy been designed to show that the continuum dissolution problem described in /I/ does not occur if a correct effective interaction using projection operators on positive energy electron states is used. The Bethe and Salpeter equation has been used to show that reduction to a one-time effective Hamiltonian can be performed in two particle system with /13//14/ or without 1151 external field. These works however do not yet provide an effective electron- electron interaction operator beyond the independent particle approximation, and the fact that self-energy corrections cannot be'accounted for by effective potentials has not yet been addressed.
2.2. Kelativistic Manv-Body ~ e r t u r b a t i o n theorv.
been used to compute the energy of the ground state of two-electron ions up to second order in the Coulomb contribution 1161. In the lithium-like case It has been used to evaluate ls2 2s and ls2 2p level energy to third order in the Coulomb contribution and second order in the transvetse interaction 1171. The infinite sum over states is replaced by a sum over a finite pseudo-spectrum. The formulae for the third order contributions are rather involved and can been found in 1191. In this calculation the Breit interaction has been computed to second order. The Relativistic Pair Equation method has been used to evaluate the ground state energy of the helium isoelectronic sequence up to xenon 1181. Their results indeed compare very well with those of reference 1161 but are limited to the first half of the periodic table because of approximations made in the explicit determination of the projection operator. In the method of ref. /161J171 and /19/, there is no such limitation because the projection operator is very simply implemented by restricting all sums over states (which are finite and discrete by construction) to only those states which have the correct eigenvalue.
method
The Multiconfiguration Dirac-Fock method is the direct generalization of the non-relativistic multiconfiguration H a m - F o c k method to the many-electron Dirac Harniltonian. After the pioneering work of Y.K. Kim using analytic wavefunction, two codes have been developed using numerical solution of differential equations, by I.P. Grant and co-workers 1201, and by J.P. Desclaux 1211. Latest versions of both codes allow computation of the complete Breit interaction in Coulomb gauge as a first order perturbation, and even inclusion of the magnetic part of the interaction in the self-consistent field process /22/,/23/. Since then new methods have been developed to use analytic wavefunctions without being disturbed by variational collapse and spurious solutions 1241. In those cases the complete Breit interaction (including retardation) can be treated self-consistently. The fact that the Breit interaction can be used beyond first order approximation, contrary to earlier claims, has been adressed in many circumstances /24/J25/J26/. Grant's code has been used several times in the past to evaluate transition energies in two-electron ions 1271 but those early attempts do not have the precision of more recent calculation and will not be considered here.
As suggested in the introduction, use of the MCDF method introduces several problems, none of which has been completely solved at the present time. Sucher 1121 has shown the necessity to use projection operator. Mittelman /13/J14/ has shown that projection operators in the one-particle Hartree-Fock potential are the sensible choice for single determinant wavefunction. More recently this choice has been confumed I281 as the only one not leading to the reintroduction of spurious negative energy states. However none of this derivation addresses the problem of multi-determinant wavefunctions, nor does it justify the implementation of such projection operators as boundary conditions enforced during the calculation. The second problem is due to the non-locality of the Hartree- Fock potential, which leads to a strong gauge dependency of the contribution of the retarded interaction to the ion energy /26/. Such a problem has been addressed only in the case of two particles without an external potential I171 and I will show later in the discussion that experience supports the use of Coulomb Gauge. Finally the effective electron-electron interaction itself is a source of trouble, since it should not be used beyond the independent particle approximation (this fact should be confused with neither the possiblity to use the operator to order higher than one, nor to problems related to the use of projection operators). I have however evaluated the contribution of the retarded part of the interaction in the MCDF case (I will call refer to this contribution in what follows as "Retardation Correlations") for the purpose of comparison with other calculations.
C1-242 JOURNAL DE PHYSIQUE
field process (SCF). This corresponds to the inclusion of "ladder" diagrams with a single transverse photon and several longitudinal photons exchanged between the electrons. Including the magnetic interaction in the SCF process would have yielded higher order diagrams ("ladder" diagrams with several transverse photons and several longitudinal photons exchanged between the electrons). Such diagrams could have a sizeable contribution at high Z ,
but I could not achieve convergence in such a case with enough configurations to get precise results.
'fie method I have used for MCDF calculation has been described in 1291 4301 and 1311 for two-electron ions and I321 for three-electron ions These calculations have now been extended to a wider range of Z, using Coulomb gauge for the evaluation of the retardation contribution. Nuclear radiis and masses are the same as in /17/J33/J34/. As in ref. 1311, non-relativistic electrostatic correlations are included exactly. In the helium-Sike case, mass polarization contributions from 1331 are included. Approximate screening correction is included for both two and three-electron ions following the method described in 1301, which we will loosely describe later as the "Welton" approximation. In the new calculation presented here, the one electron self-energy and the approximate two-electron contribution are corrected for finite nuclear size. Vacuum polarization potential of order a ( Z a ) (Uehling contril>ution), aZ(Za) ( M l t n and Sabry contribution) and a(Za)3 are used to provide scmned vaccum polarization corrections. Theses corrections are evaluated using first order perturbation theory with Dirac-Fock wavefunctions The use of Dirac-Fock wavefunction in place of hydrogenic ones should account for most of the screening of the vaccum polarisation, and it does also take care of the finite nuclear size correction For very high Z the modification of the Uehling potential due to the presence of other electrons could play a role and should be further investigated.
'Unfortunately there is no potential for the self-energy correction and one has to find other ways to compute the self-energy screening. The "Welton" approximation consists of using semi-classical arguments to derive an effective potential to correct the lowest order part in (Za) of the one electron self energy for two-electron effects /30/. Very recently Feldman and Fulton /34*/ ave derived an identical effective potential by considering low order diagrams derived from QED. However while such a justification increases the interest of this method, its still remains as a problem that the lowest order term in (Za) account for a very small part of the self-energy at high Z, a fact wich can lead to sizeable error in the uranium region.
Detailed results, for elements bemeen Z=10 and 94 (helium-like systems), and Z=15 to 92 (lithium-like systems) will be submited soon for publications
8.
Indelicato and J.P. Desclaux).2.4. start in^ from high wecision non-relativistic results: Drake's unified theorv,
Using a point of view completely different from the previous ones is Drake's unified theory/351,/36/. In this approach the necessity of having both a very good non-relativistic stiuting point (particularly at low Z), as well as
1
using IDirac equation for heavy ions is handled trough the formalism of the relativistic
z
expansion, which is in fact a1 2
double expansion in power of
z
and ( Z a ) /37/. Drake's method uses high precision non-relativistic variational3.1. Introduction.
Due to the complexity of the structure of the available calculations, it is really very difficult to trace out reason for differences in the final result. The MCDF method by itself is very complicated since it sums partially different classes of Feynman graphs in such a way that, depending on the size of the basis set, fractions of several orders are included at the same time (this fraction obviously decreasing when the order increases). Fractions of many coefficients of terms in (f$'Za)2n+2 are then included, which are not included in Drake's calculation. RMBPT on the other hand allows one compute exactly second and third order corrections in the Coulomb interaction, and first and second order in the magnetic and retarded interaction, which is again completely different of what the MCDF method can achieve. I will present the differences as figures scaled with Z to get the main Z dependency. In the case of three-electron ions I will only compare the results of calculations without many-electron radiative correction since I am to date the only one with an approximate way to evaluate such corrections (if one except the widely used effective Z method which is, in most cases, giving results in poorer agreement with experiment than unscreened ones).
3.2. Helium-like svstems.
To date, there is no complete and reliable calculation of two-electron Self-Energy to all order in (Za). However the correction of the leading term of the Self-Ener has been shown to be due to modification of the Bethe
gir
logarithm and to change in the electron density at the origin.
z
coefficients for the two-electron Bethe logarithm have been evaluated /38/J39/. Table I shows how the results obtained through this calculation compare with the "Welton" approximation results. This table shows that indeed both results are very close. However the "Welton" method does not provide screening for p states, for which I still have to rely on the ill-behaved effective Z method. This does not leads to any sizeable effect except in the 2113 range.In figure 3.2.1 and 3.2.2 are displayed the comparison between Drake's total energies and the MCDF ones. In figure 3.2.2 retardation correlations have been included. It can be seen than the inclusion of the retardation comtion reduces the difference between the two calculations However this contribution accounts only for a small fraction of the difference on level energies, while we will see later that for 2p-1s transition energies this conmibution represents most of the difference. The reason of a Z4 difference between the two calculations affecting mainly total energies is not clear and should be further investigated This difference is of 12 eV at Z=92 for the ground state. About one eV of this can be explained by the use in the MCDF calculation of a screened vaccuum polarization
Table I: Comparison between self energy screening for Is 2s 3 S ~ level
JOURNAL DE PHYSIQUE
Figure 3.2.1. Differences between MCDF values and Drake's values I331 of the total energy of several levels in helium-like systems.
MCDF with retardation correlations
Figure 3.2.2. Differences between MCDF values, including retardation correlations, and Drakes values 1331 of the total energy of several levels in helium-like systems.
calculation give absolute level energy. The energy as described by W.R Johnson et al. 1171, using their notation, is a sum E=(EO'El+Ez+E3+.
.
.)+(Bl+B2+..
.) where Eg is the Dirac-Fock Coulomb energy with a frozen core, El, the first order correction, is strictly 0, ipd Bi is the Breit contribution to ifh order. In this context electrostatic correlations will be (EO+E~+E~+E~-EDF) and magnetic ones (including retardation ones) will be (B~+BzBDF), DF denoting single configuration Dirac-Fock energies. The first thing one can notice when looking to Dirac-Fock data is that, even when all contributing configurations with n=1,2 and 3 orbitals are included, the electrostatic correlation energy is far from convergence. However as that is already a respectable number of configurations (-50) and of computer time (112 h for each level on a CYBER 205) it has been decided in this first attempt to stop there. A precise non-relativistic contribution to electrostatic correlation energy, which is essential to have a good accuracy at low Z, has been achieved by fitting ow MCHF results with a polynomial ini.
The zeroth order contribution is then replaced by the exact non realtivistic value /40/J41/ The results obtained through this procedure are compared with /#I71 results on fig= 3.3.1I
* Lindgren 2p-2sI
Figure 3.3.1. Electrostatic correlations computed using MCHF method, or RMBPT method (from ref1171 J,B&S).
MCHF
results have been shifted to have the proper limitfor Z+.This figure shows that results from 1171 have indeed the correct limit for Z+- and they are in excellent agreement with the non-relativistic MBPT results of Lindgren /421 for Z=3. This figure also shows that the coefficients of
(g,
n=1,2,3, obtained from the MCHF are still far from convergence. Again this demonstrates that1
JOURNAL DE PHYSIQUE 20 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 I -0.2000
-
-0.2500-, E -0.3000-, (eV) -0.3500-
-0.4000-
-0.4500- -0.5000--
.
-0.5500z
Figure 3.3.2. Comparison between relativistic contribution to electrostatic correlationsfrom MCDF or RMBPT /17/.
2 ~ 3 1 2 - 2 s
MCDF total electrostatic Correlations
..-
Jonhson, Blundell &MCDF total electrostatic Correlations
..-
Jonhson, Blundell 8Figure 3.3.3. Comparison between relativistic contribution to electrostatic correlations from MCDF or RMBPT 1171.
Correlations Jonhson, Blundell &
As demonstrated earlier for lhe ls2p 3P0 level of helium-lie systems 1311 it is found that relativistic effects provide a major contribution to electrostatic conelations at high Z, when 2s or 2p1/2 electrons are involved.
In figure 3.3.5 I have finally plotted the differences between total energies from both calculations. These differences are easily traced from differences in Breit correlations. Again I have introduced "retadation correlations" in the figure but unlike the case of the ground state of helium-like ions, this have a mixed effect, degrading agreement at low Z and improving it at high Z, with a crossing p i n t around Z=60. The structure of the Breit term being more complicated than the Coulomb one and it is not yet possible to explain the observed difference which scale roughly l i e 23.
It is not yet clear if precisions of 0.6 eV in uranium can be reached experimentally, but that seems to be a reasonable limit for the accuracy of the non-radiative part of the energy to permit very sensitive tests of QED corrections in very heavy elements.
*
2 p * - 2 ~ MCDF-MBPT -0- 2p'-2s ret corr'.-
2p-2p* MCDF-MBPT a- 2p-2p' ret corr*
2p-2s MCDF-MBPT*
2p-2s ret corrFigure 3.3.5. Comparison between total MCDF energies and total
RMBPT
energies 117 for lithium-like systems. 2p* stank for 2~112.4. C-on with
4.1. Helium-like svstems.
4 . 1 . 1 Transitions t o the ground state.
Experimental data concerning these transitions are rather scarce, and their precision is not yet sufficient (except may be for a point at Z=l8) to derive completely meaningful conclusions from the comparison with theory. Differences between experiment and theory are plotted in figure 4.1.1 (2 3P1-1 ISO transition) and 4.1.2 (2 lP1- 1 1So transition). References for experimental work are quoted in the legends. For the 2 1P1-1 1So transition, previous Beam-Foil and recoil ions results have been recently completed with Tokarnak data measured with hydrogen-like lines as references. MCDF values are generally in closer agreement with experimental values than Drake's ones, but no legitimate conclusion can be drawn from such figures if one cannot get experimental results 'with three-fold improved accuracy. The xenon region is of particular interest to c o n f m or i n f i a possible
JOURNAL DE PHYSIQUE
-' f experimental precision scaled by P'3
'm- (Exp-Drake)/Z"3
Figure 4.1.1. Experiment-Theory for 2 3p1 I IS* transition energy. Experimental results are from 143/ (2=16), 1441 (2=18), I451 (Z=26), I461 (2=36) and 1471 (2=54).
-'
*
experimental precision scaled by T'3.O- (Exp-Drake)lZ"3
Figure 4.1.2. Experiment-Theory for 2 lP1-1 ISo transition energy. Experimental results are from 1#43/ (2=16), l3##44/ (Z=18), I481 (Z=19,21,22,23,24), I451 (Z=26), I461 (2=36) and I471 (Z=54).
4.1.2 An=O transitions.
is clearly an experimental c o n f i a t i o n of the theoretical work cited in paragraph 2.1, and to my knowledge it is the f i s t time this can be clearly confiied in the presence of an external field.
lS2p 3 P 2 - 1 ~ 2 ~ 3S1 Retardation in coulomb gauge
'.-
(exp-MCDF(Coulomb))l(Z"3)
-' f experimental uncertainty
-
-
scaled by 2-3a- (Exp-Drake)l(Ze3)
*
(exp-MCDF Ret Corr)lZ"3*
(exp-MCDF(Lorentz))/(Z**3)Figure 4.1.3. Difference between experiment and theory for 2 3P2-2 3S1 transition energy (retardation in Coulomb gauge is MCDF(Coulomb), retardation in Lorentz gauge is MCDF(Lorentz), MCDF Ret Corr is MCDF with Coulomb gauge and "retardation correlations"). References to experimental data can be found in 1311,1331 and I491
In the case of the 2 3pO-2 3S1 transition the agreement between the two calculations is not as good as in the previous case (figure 4.1.4.). While the use of the Coulomb gauge in the MCDF calculation has greatly improved agreement with both Drake's calculation and experiment there is a residual discrepancy which increases roughly like
z3.
It is very likely that this is due to the screening of the self energy of the 2~112 electron for which the "Welton" approximation cannot be used, and for which I still had to rely upon the old "Z-effective" scheme. The uranium value for this transition /SO/ which is not shown in the figure to get a more detailed yiew of the low Z region, is in perfect agreement with both calculations, but should be made 10 times more accurate to really discriminate between them. 2 3PO-2 3S1*
(exp-MCDF)I(Z"3) -' f experimental uncertainty-
' scaled by 2-3 .n- (Exp-Drake)/(Z"3)*
(exp-MCDF Ret Corr)l(Z"3)C1-250 JOURNAL DE PHYSlQUE
It is impossible to compare directly RMBPT results of ref 1171 with experiment since they do not include radiative corrections. I have then arbitrarily decided to add to them hydrogenic radiative corrections as well as screening corrections estimated with the "Welton method". Another possible choice would have been not to include the screening corrections. However it would have prevented a meaningful comparison, giving a difference between theory and experiment about 10 times larger than when the screening correction is included-which by itself is a rather good proof that the "Welton" approximation is giving a good estimate of this screening correction in the low to med Z range. Comparison between theory and experiment for the 2pln-2s transition is shown on figure 4.2.1. Both calculation compare very well with experiment, with possible discrepancies smaller than 4 10-7 23 eV up to 2=54 for the MCDF calculation. Again there is not enough data to determine if the lo discrepancy at Z=54 between the "corrected RMBPT value and the experimental one has to do with the RMBPT or with the "corrected or with the experimental value itself. 2p3/2-2s transition energy (figure 4.2.2) and 2p3n-2pln energy separation (figure 4.2.3) are also very well represented by both theoretical predictions.
5 .
C m l u s i m andDrosDects.
In this paper I have presented in details theoretical transition energies in few electron ions computed using three different methods. I have shown that all three methods, which are completely ab initio, are able to reproduce quite well all experimental data. In this respect all three methods are very successful but none of them is without drawbacks. First off all they all need to be tested around Z=92 with high precision, since there is indication from high precision X ray measurement of K a transitions that a possible problem arise above Z=83 in the comparison between theory and experiment 1541. Drake's unified method is limited to two-electron systems, and does not provide a way to correct high order in (Za) in the self energy for two-electron effects. The latter criticism is also m e for the "Welton approximation" I have used in the MCDF calculations. The main problem with the MCDF method is its slow convergence, and while it can reach in principle any precision by extending the basis set enough, it may be so cumbersome in practice (convergence problems, excessive computer time.. .) than one has to stop much below the precision one is hoping to achieve. This method still lacks a true many body expression for the retarded part of the electron-electron interaction. The RMBPT which is the newest technique.has provided an impressive set of high precision results. It would be very interesting to get third order results for helium-like system, and for system with two open shells (excited states of helium-like system for example). While one of its main attractions is that it can provide a rigorous framework for self-energy screening calculations, it has still to be proven to be feasible in practice. At any rate we have now several reliable, precise way to compute the non radiative part of the energy of two and three-electron ions In order to have more precise values for heavy elements it would be unteresting to redo the MCDF calculations, including the magnetic interaction in the self-consistent field process. However numerical problems have prevented us doing so for big enough basis sets. That should be worked out in the future.
" f experimental precision scaled by P ' 3
.n- (Exp-MCDF_RC)IZ"3
Figure 4.2.1. Comparison between experimental energies, MCDF energies and RMBPT energies 11 71 with radiative corrections from the MCDF calculation for the 2~112-2s transition in lithium-like ions. Transition energies up to 2=22 are from ref. 1511, transition energies for 2=24,26,28,29,32,34 are from rqf. 152. Transition energy for Z=36 is from ref. 1531 and B Denne (private communication
1988). Data for 2=54 are from S Martin (private communication 1988).
-' i experimental precision Scaled By 2-3
.a- (Exp-MCDF-RC)/Z"3
JOURNAL DE PHYSIQUE 2p 312-2p 112
--
f experimental precision Scaled by 2-3 .U- (Exp-MCDF_RC)/Z"3 -& (Exp-MBPT)IZ"3 -2.OE-06 -( 15 20 25 30 35 40 45 50 55 zFigure 4.2.3. Comparison between experimental energies , MCDF energies and RMBPT energies I171 with radiative corrections from the MCDF calculation for the 2p3/2-2plI2 separation in lithium-like ions. Experimental transition energies are from the same refrences as in figure 42.1.
I wish to thank Dr. R. D. Deslattes, and Dr. Y. K. Kim for many helpful discussions. The new features of MCDF code I have used throughout the work presented here has been developed by, or with the help of J.P. Desclaux during his stay at the 1.T.P in Santa Barbara last spring. I am really indebted for his continuous support and help. I have also to thank Pr W. Johnson and Pr G.W.F. Drake for very helpful explanation about their works.
Most of the MCDF calculations presented in this work have been done using the CS2 computer facility at the National Bureau of standards.
Je tiens il remercier le Directeur du secteur Mathtmatiques et Physique de Base du CNRS et le Dr R.D. Deslattes pour le financement de mon stjour au N.B.S.
Correspondence address: National Bureau of standards, building 221, room A141, Gaithersburg MD20899 USA.
Permanent address: Laboratoire de physique Atomique et Nuclkaire, Unit6 associ6e au CNRS no 771, Universitk P&M Curie, 4 place de Jussieu, F75231 PARIS CEDEX 05 France.
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