Relativistic ab initio calculations of isotope shifts

isotope shifts

Cédric Nazé

Service de Chimie Quantique et Photophysique

Thèse présentée en vue de l’obtention du titre de Docteur en Sciences de l’Ingénieur

Promoteur : Année académique

Prof. Michel Godefroid

^2012-2013

Acknowledgments

Je remercie sincèrement le professeur Michel Godefroid, mon promoteur de thèse, pour son enseignement. Après quatre (cinq?) années passées sous sa direction, je reste toujours impressionné par la justesse de ses conseils et la rigueur de ses analyses. S’il est vrai qu’en rentrant avec une question dans son bureau, on en ressort souvent avec trois autres, ces dernières sont toujours des plus utiles. Merci à lui pour sa patience, ses (nombreuses) relectures et cette agréable ambiance de travail qu’il fait régner dans son Service. J’aimerais pouvoir, ma carrière durant, aller travailler chaque matin le coeur aussi léger.

I thank warmly the professors Per Jönsson from the Malmö University and Ged- iminas Gaigalas from the Vilnius University for their help and their many advices in the realization of the various projects that constitute this PhD thesis. I also take the opportunity to thank Pavel Rynkun and Erikas Gaidamauskas, especially as regards to our work on the development of the tensorial form of the relativistic operator.

I thank Dr. Jiguang Li for his enthusiasm, all the interesting projects that he brought as well as his help and the great job he did in the realization of our common work. I wish you the best in Sweden.

I thank the professor Vladimir M. Shabaev for the wonderful particular course of 20 minutes that he gave me during the international conference ecam-data2010 in Vilnius. I think more than a year have been necessary to understand the two pages full of equations that he gave me that day. Nevertheless, the latter helped me a lot to obtain a better comprehension of the relativistic theory of mass shifts.

Je suis infiniment reconnaissant envers le Centre de Calcul de l’ULB/VUB, tout particulièrement envers Georges Destrée ainsi que Raphaël Leplae qui a pris sa relève. Merci à eux pour leur aide précieuse lors des premières installations et compilations de programmes sur “hydra”, le super-ordinateur de l’université.

Certains “jobs” seraient encore “queued” sans leurs conseils.

Je remercie, dans l’ordre d’apparition, le Prof. José-Paulo Santos pour son accueil à l’“Universidade Nova de Lisboa” au Portugal lors de mon séjour en octobre 2009 et le Prof. Paul Indelicato pour sa disponibilité et son aide lors de ma venue dans ses locaux en avril 2011. Merci à eux et au Prof. Jean-Paul Desclaux, que je n’ai jamais rencontré, mais dont les emails ont apporté une foule d’informations sur le code mcdf-gme et ses subtilités.

Merci au Prof. Vaeck et Prof. Godefroid pour m’avoir donné la possibilité de participer à des activités d’enseignement. J’ai également beaucoup aimé organiser et animer les Atelier Jeunes Ingénieurs sous l’égide du Prof. Art. Merci à lui pour son enthousiasme communicatif.

Je remercie l’ensemble du Service de Chimie Quantique et Photophysique de l’ULB pour son soutien et ses conseils lors des étapes préparatoires au financement de cette thèse.

Je remercie sincèrement mes collègues de bureau Simon et Maxence sans oublier ceux qui sont déjà partis: Thomas et Jérôme. Merci à ces deux derniers pour les bons moments passés à faire un tas de choses autres que de la physique et bonne chance

pour la suite, quelle qu’elle soit. Bonne continuation à Maxence qui risque de trouver le bureau bien grand, mais je suis sûr, qu’un jour, il y arrivera de la main gauche.

Enfin, compagnon de bureau du tout début à la toute fin, je ne sais pas comment remercier Simon sans trop l’encenser. En plus de tous les bons moments passés ensemble, nos discussions et ses conseils m’ont vraiment aidé tout au long de cette thèse. Merci à toi. Je ne sais pas si on construira un jour ensemble le chauffe-eau du futur, mais si c’était à faire, je n’hésiterais pas. A tous, j’ai également beaucoup aimé nos “diabloliques” séances pour “héroïquement” “civilizer” le monde.

Merci à Stéphane, pour ses approvisionnements sans limite en chocolat et son scepticisme de libre exaministe que j’aime beaucoup. Dans le bureau du-fond-dégénéré (leurs bureaux ne comportent aucune séparation), ce serait un crime de ne pas souligner la présence de Keevin pour son incroyable culture musico-cinématographique et ses vidéos matinales, de Clément pour sa capacité à mettre des mini-blagues dans, à peu près, tout ce qu’il dit, de Tomas pour nous avoir tant appris sur “comment survivre en cas d’attaque de Zombies” et ses concerts de saxo le soir au Service.

Merci à Xavier, Badr, Darius et Guillaume pour leur participation aux nombreuses discussions qui ont composés nos repas de midi.

De tous, tous grands mercis vont à Marie, Ella, Stéphane, Aurélien et Rémi pour leur patiente relecture.

Merci à Brigitte pour sa bonne humeur et son accueil chaleureux. Puisse-t’elle ne pas nous en vouloir pour les nombreuses heures où nous avons, sans vergogne, accaparé Michel.

Dans un tout autre domaine, merci à François, Jérôme, Laurent, Rémi, Si- mon et Thomas pour les moments de détentes sportives dans tous ces parc/bois/

forêts/terrils/salles de squash/de badminton qui ont croisés notre chemin.

Toute ma gratitude file également vers mes parents qui, malgré toutes les dif- ficultés rencontrées, m’ont toujours montré un formidable et inébranlable soutien!

Merci à Papa, pour tout le temps qu’il a consacré m’expliquer des maths. Sans eux, je serais bien incapable de dire s’il m’eût été possible de finir Polytechnique ou même de passer cette difficile première année.

Des milliards de mercis à Marie. Je le découvre tous les jours mais il faut bien avouer qu’elle est la plus formidable de toutes; je me demande si cette révélation n’est pas l’une des plus importantes et des plus chouettes de ces quatre années de recherche. Merci pour ton support dans le rythme (pas évident) de ces derniers mois de rédaction. Merci aussi à toi pour ton aide et tes patientes (re)(re)lectures.

Enfin, c’est quand les choses se terminent qu’on réalise leur importance, mais je remercie sincèrement le “Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture" de Belgique pour son support financier dans la réalisation de cette thèse.

This thesis yielded the following publications

Much of the content of this Thesis was published in the following series of papers:

• Tensorial form and matrix elements of the relativistic nuclear recoil operator E. Gaidamauskas, C. Nazé, P. Rynkun, G. Gaigalas, P. Jönsson and M. Gode- froid.

Journal of Physics B: Atomic, Molecular and Optical Physics, 44, no. 17, page 175003 (2011).

• Mass- and field-shift isotope parameters for the 2s-2p resonance doublet of lithium-like ions

J. G. Li, C. Nazé, M. Godefroid, S. Fritzsche, G. Gaigalas, P. Indelicato, P. Jönsson

Physical Review A,86, no 2, page 022518 (2012).

• On the breakdown of the Dirac kinetic energy operator for estimating the iso- tope normal mass shift

J. G. Li, C. Nazé, M. Godefroid, G. Gaigalas and P. Jönsson European Physical Journal D, (accepted) 2012.

• ris: a new program for relativistic isotope shift calculations

C. Nazé, E. Gaidamauskas, G. Gaigalas, M. Godefroid and P. Jönsson Computer Physics Communications, (submitted) 2012.

• Isotope shifts of beryllium-, boron-, carbon- and nitrogen-like ions from rela- tivistic configuration interaction calculations

C. Nazé, S. Verdebout, P. Rynkun, P. Jönsson, G. Gaigalas and M. Godefroid Atomic Data and Nuclear Data Tables, (to be submitted) 2012.

• Theoretical study of issues in isotope shifts transition in barium C. Nazé, J. G. Li, M. Godefroid

(in preparation) 2012.

This thesis does not include my work on hyperfine structure, neither the common work of most of the Service de Chimie-Quantique et Photophysique. I refer the reader to:

• Hyperfine structures and Landé gj factors of beryllium-, boron-, carbon- and nitrogen-like ions from relativistic configuration interaction calculations S. Verdebout, C. Nazé, P. Rynkun, P. Jönsson, G. Gaigalas and M. Godefroid Atomic Data and Nuclear Data Tables, (to be submitted) (2012).

• From atoms to biomolecules: a fruitful perspective

E. Cauët, T. Carette, C. Lauzin, J. Li, J. Loreau, M. Delsaut, C. Nazé, S. Verdebout, S. Vranckx, M. Godefroid, J. Liévin and N. Vaeck.

Theoretica Chimica Acta,131, no. 8, pages 1–17, 2012.

Our program has been used for the following proceeding paper:

• Relativistic calculations of 1s²2s2p levels splitting in Be-like Kr

J. M. Sampaio, F. Parente, C. Nazé, M. Godefroid, P. Indelicato, J. P. Marques Proceedings of the HCI Conference, 2012.

AS active set . . . .22

ASF atomic state function . . . .22

CAS complete active space. . . .22

CI configuration-interaction . . . .26

CSF configuration state function. . . .20

DF Dirac-Fock . . . .24

DHF Dirac-Hartree-Fock . . . .20

ESR experimental storage ring . . . .57

EOL extended optimal level . . . .26

FS field shift . . . .5

HF Hartree-Fock . . . .22

HST Hubble Space Telescope . . . .8

IS isotope shift . . . .5

RMBPT relativistic many-body perturbation theory . . . .26

MCDHF multiconfiguration Dirac-Hartree-Fock . . . .22

mcdf-gme multiconfiguration Dirac-Fock-general matrix elements . . . .36

MCHF multiconfiguration Hartree-Fock . . . .106

MS mass shift . . . .4

NEC negative energy continuum . . . .24

NMS normal mass shift . . . .4

QED quantum electrodynamics . . . .8

RCI relativistic configuration-interaction . . . .26

RIS residual isotope shift . . . .53

rms root-mean-square . . . .46

r.m.e. reduced matrix element . . . .37

RSCF relativistic self-consistent field . . . .24

SCF self-consistent field . . . .24

SMS specific mass shift . . . .4

TSR test storage ring . . . .57

I Theoretical Considerations 1

1 Introduction 3

1.1 And there was quantum mechanics . . . 3

1.2 Isotope shift: presentation and interest . . . 4

1.2.1 Definition and origin . . . 4

1.2.2 First experimental evidences and attempts of modeling . . . . 5

1.2.3 Space/time variation of the fine-structure constant . . . 6

1.2.4 Nuclear physics . . . 7

1.2.5 Other fields of applications . . . 8

1.3 Units. . . 9

2 Relativistic atomic structure theory 11 2.1 Dirac equation for one particle systems . . . 12

2.2 Hydrogenic atom with a point charge nucleus . . . 15

2.2.1 The Sommerfeld formula. . . 15

2.2.2 Average values using virial relations . . . 17

2.3 The issues of complex atoms. . . 19

2.3.1 The Dirac-Hartree-Fock method . . . 20

2.3.2 The multiconfiguration Dirac-Hartree-Fock method . . . 22

2.3.3 The negative states . . . 24

2.3.4 Variants of the MCDHF methods . . . 25

3 Relativistic isotope shifts 27 3.1 Non-relativistic mass shift . . . 27

3.2 Relativistic mass shift in hydrogenic systems. . . 30

3.2.1 The Coulomb contribution for a hydrogenic point charge system 31 3.2.2 The one-transverse-photon contribution for a hydrogenic point charge system . . . 32

3.3 Relativistic mass shift in many-electrons systems . . . 36

3.3.1 Normal mass shift expectation value . . . 37

3.3.2 Specific mass shift expectation value . . . 39

3.3.3 Useful one-electron reduced matrix elements . . . 45

3.4 Field shift theory . . . 46

3.5 General isotope shift units and orders of magnitude . . . 48

3.5.1 Mass shift units . . . 49

3.5.2 Mass shift effect . . . 49

3.5.3 Field shift units . . . 50

3.5.4 Field shift effect . . . 51

3.5.5 Total line frequency shift . . . 51

3.6 Experimental isotope shifts . . . 52

3.6.1 King plots and variants . . . 52

3.6.2 Measurement of nuclear properties through isotope shifts. . . 54

4 Codes and implementation 59 4.1 The programsgrasp2K and mcdf-gme . . . 59

4.1.1 The grid . . . 60

4.1.2 Nuclear potential . . . 61

4.1.3 What happens near the origin? . . . 63

4.1.4 Treatment of the Breit interaction and the negative energy continuum . . . 64

4.1.5 The B-Splines . . . 65

4.1.6 Derivation and integration . . . 65

4.1.7 Isotope shift treatment . . . 66

4.2 Work related to themcdf-gme code . . . 66

4.2.1 Structure of the mass shift program . . . 66

4.2.2 Total probability density at the origin . . . 68

4.3 Work related to thegrasp2K code . . . 69

4.3.1 Description of the release . . . 69

4.3.2 Calculation of the angular coefficients . . . 70

4.3.3 Characteristics . . . 72

4.4 Does it work? . . . 74

4.4.1 Orbital rotation: a stringent test . . . 74

4.4.2 Hydrogen-like selenium and lithium-like systems using Dirac one-electron wave functions . . . 75

4.4.3 Numerical problems with the NMS in mcdf-gme . . . 78

II Applications 81 5 Theoretical treatment of electronic isotope shifts parameters 83 5.1 On the breakdown of the Dirac kinetic energy operator for estimating the normal mass shift . . . 83

5.1.1 Computational Method . . . 84

5.1.2 Results and Discussion . . . 85

5.2 On the field shift formalism . . . 91

5.3 The Breit interaction, the negative energy continuum and the influ- ence of the isotope choice . . . 96

5.3.1 Effects on mass shift . . . 96

5.3.2 Effects on field shift . . . 103

6 Lithium-like systems 105 6.1 Neutral lithium in the MCDHF approach . . . 105

6.2 Other lithium-like ions . . . 110

6.2.1 Electron correlation effects on isotope shifts . . . 110

6.2.2 Mass and Field Shifts Balance. . . 115

6.3 Isotope Shifts in ^150,142Nd⁵⁷⁺ . . . 117

7 Spectrum calculations 119 7.1 Principle of spectrum calculations. . . 119

7.2 Results and evaluation of data. . . 119

7.2.1 Beryllium-like systems . . . 119

7.2.2 Boron-like systems . . . 126

7.2.3 Carbon-like systems . . . 128

8 Isotope shifts in barium 131 8.1 Experiments and other calculations . . . 131

8.2 Computational procedure . . . 132

8.3 In the depths of isotope shifts of barium . . . 136

8.3.1 A large “choice” of nuclear radii . . . 136

8.3.2 Some well-known transitions. . . 137

8.4 The issues of the¹P₁^o− ³D_1,2 transitions . . . 143

III Conclusions 147 IV Appendix 153 A 155 A.1 Tensors . . . 155

A.1.1 Irreducible tensor operator. . . 155

A.1.2 Uncoupling formulas for r.m.e. . . 156

A.1.3 Non-relativistic angular matrix element . . . 156

A.1.4 Relativistic angular matrix element . . . 158

A.2 The relativistic quantum number κ . . . 158

A.2.1 ... and its various expressions . . . 159

A.2.2 Particular relations . . . 160

A.2.3 Selection rules in the relativistic reduced matrix element of the momentum operator . . . 161

A.3 The operator cσ·p . . . 162

A.4 Complementary developments . . . 162

Bibliography 163

Theoretical Considerations

Introduction

(relativity) [. . . ] gives rise to difficulties only when high-speed particles are involved and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions [. . . ]

P. A. M. Dirac (1929)

T

^he main topic of this thesis is the ab initio study of isotope shifts in atomic spectra. In the following chapters, we present, formalize, calculate and discuss these effects. The basic notions of the non-relativistic quantum theory are assumed to be known.

After a short history, this introductory chapter presents the framework of this thesis as well as the basic notions of isotope shifts and their use in other fields of physics. The last section is dedicated to the general units used in this work.

1.1 And there was quantum mechanics

It is always rewarding to linger somewhat over the History of Sciences and the one of quantum mechanics underwent an impressive development. It only required fifteen years to develop its mathematical basis and build a theory on which underpins all our current works.

In 1913, namely almost a hundred years ago, N. Bohr gave the first attempt to explain the existence of line spectra. Two years later, A. Sommerfeld proposed a correct formalism describing the electronic transition of an atom, taking into account the fine structure of hydrogenic atoms (see subsection 2.2.1 for further developments of this formula). It only took eight years for French aristocrat L. de Broglie to suggest, during his PhD thesis, the idea that a wave may be associated with any particle, bringing an answer to the “strange” wave behavior of electrons and atoms. Surprisingly enough, E. Schrödinger quickly digested such a “crazy”

idea. Just before Christmas 1925, Schrödinger went to Swiss Alps, leaving his wife in Vienna. He took along a copy of de Broglie PhD thesis and a former girlfriend (It seems that Schrödinger was an inveterate womanizer) [Hecht 1999]. Around twenty days later, when they came back, Schrödinger had formulated his famous equation!!

The latter describes the behavior of the de Broglie’s waves, allows the prediction of atomic properties such as energy levels of an atom and has become the cornerstone of the quantum mechanics theory.

At that time, things were still “simple”: the atoms were considered as composed only of electrons and protons and, except for the photon, no other particle was known; but then P. Dirac opened a lot of new doors. Perhaps the highlight of his creative streak was the 1928 publication of his equation for the electron [Dirac 1928a, Dirac 1928b]. Consistent with both quantum mechanics and special relativity, the

“Dirac equation” describes the behavior of elementary massive particles with half- integer spin (such as electrons). As its consequences, the electron spin appears naturally, the expression of the Sommerfeld fine structure formula is established and the magnetic moment of the electron is calculated! This brief history allows us to realize how fertile this period was and how fast things were moving.

1.2 Isotope shift: presentation and interest

1.2.1 Definition and origin

If an atom in a state E<sub>`</sub> receives a given quantity of energy it might be excited and pass to an upper energy state Eu. The frequency of a spectral line k connecting levels `↔u, withEu> E_` is,

ν_k= E_u−E_`

h . (1-1)

While all isotopes of a given element share the same number of protons, they differ by their number of neutrons so that the energy of a spectral line depends slightly on the isotope. In other words, for a given atom, the line frequency(1-1) will differ from one isotope to another. This effect has two physical origins: the mass shift and the field shift.

The mass shift (MS) The mass shift of the energy levels in an atom with nuclear massM is caused by the recoil motion of the atomic nucleus. The best explanation comes from King [King 1984]: each atomic level is described by an eigenfunction with eigenvalues of angular momentum and energy. The angular momentum of a level has a definite fixed value, so if the mass of the nucleus changes, by substituting one isotope for another, the energy of the level will have to change so that the angular momentum remains the same. This leads to the measurable mass effect that Ureyet al. [Urey 1932] observed for the first time.

In theoretical approach, since an atomic nucleus has a much bigger mass¹ than the electrons around it, calculations are often done assuming that the nucleus pos- sesses an infinite mass. This simplifies the resolution of equations and, in a first approach, gives good results. For one electron systems, the consideration of the nuclear mass gives rise to the so-called normal mass shift (NMS). If more than one electron is present, a second effect appears: the specific mass shift (SMS). Its behavior is hard to predict and leads to a lot of computational difficulties.

1From 1.672621777(74)×10⁻²⁷kg to around 4.88×10⁻²⁵kg for the heaviest nucleus known. The electron mass is9.10938291(40)×10⁻³¹kg.

The field shift (FS) The electric field produced by protons inside the nucleus does affect the electronic energy of the atom. This field is a function of the size and the shape of the charge distribution in the nucleus. If the number of neutrons present inside the nucleus changes, not only the volume varies but the spatial distribution of the protons is also affected; this is a response to the variation of neutrons density through the proton-neutron interaction. Therefore, the electric field perceived by the electrons changes and affects the electron’s behavior².

The total shift Firstly, for an isotope of mass numberA, with a given mass M and possessing a nuclear radius, the mass and field shift corrected energy level is

E_i^A=E0+E_MS^A +E_FS^A , (1-2) where E₀ is the energy of an atomic system with a point charge nucleus carrying an infinite mass. The MS generally dominates for light systems and is overtaken by FS for middle Z-systems. However, if these two effects equivalently dominate, predicting the signs and the result of the summation is generally a difficult problem.

A discussion about physical manifestations of these two shifts and typical orders of magnitude of an isotope shift (IS) is presented in section 3.5.

The line frequency isotope shift between isotopes Aand A⁰ δν_k^A,A0 ≡ν_k^A−ν_k^A0 = δEu^A,A0 −δE_`^A,A0

h , (1-3)

with the level isotope shift

δE_i^A,A0 =E_i^A−E_i^A0 (i=`, u), (1-4) can be, in accordance with the above discussion, split in

δν_k^A,A0 =δν_k,MS^A,A0 +δν_k,FS^A,A0 . (1-5) According to (1-3), an IS is said to be “normal” if the line frequency is higher for the heaviest isotope, i.e. ifδν^A,A0 >0, with A > A⁰ [Fricke 1995]. This terminology is natural when restricting the ISto the “normal” mass shift (see section 3.5).

1.2.2 First experimental evidences and attempts of modeling Isotope shifts began to be recognized and studied in 1922, when Bohr suggested the existence of the FS. At that time, the neutron was still unknown. Urey and his coworkers [Urey 1932] first found that each of the Balmer lines had a very weak companion on the short wavelength side. The wavelengths of the additional lines are in accordance with the values obtained from the Balmer formula when the Rydberg constant for a mass 2 is used instead of for a mass 1. They had discovered (what

2W. H. King [King 1984] considers that the denomination “Field Shift” is preferable to the denomination “Volume Shift” since the variation of the shape as well as the size of the nuclear charge distribution induces an isotope shift.

they baptized) the deuterium. But the first quantum mechanical treatments of such effects in many-electron atoms are due to Hughes and Eckart in 1930 [Hughes 1930]

for the mass shift and in 1932 for the field shift [Rosenthal 1932]. Vinti proposed in 1939 [Vinti 1939] a method to calculate better values for the mass shifts in the spectra of many-electron atoms. This method however did not come into its own until much later, when computers made it possible to work with Hartree-Fock and other more realistic wave functions. A breakthrough in the calculation of the mass shift (more specifically about the specific mass shift) was made by Mårtensson and Salomonson in 1982 [Mårtensson 1982] who added corrections to the Hartree-Fock calculation using many-body perturbation theory.

For a more complete genealogy of isotope shifts, the History-interested reader may take a look at the King’s book [King 1984] and at the review by Bauche and Champeau [Bauche 1976].

1.2.3 Space/time variation of the fine-structure constant

Are any of nature’s fundamental parameters truly constant? If not, how do they change? This are questions raised by some extensions to the Standard Model where the variation of fundamental constants is a possibility or even a necessity [Uzan 2011].

One must, however, distinguish what may be called constants with dimensions and those without. About this topic, the “trialogue” of Duff et al.[Duff 2002] is a funny discussion between three CERN physicists who explain their (diverging) points of view about the number of fundamental constants and what could be observed in cases of any variations. Through some thought-experiments (as Feynman’s com- municating spatial directions to an alien, what would happen if the speed of light changes?) they investigate what kind of question should be asked to this alien to ensure that what he observes is the same as we do. Most scientists now agree that dimensioned quantities are not unambiguously observables. Dimensionless param- eters, like the ratio of the proton and electron mass (me/mp), the fine structure constant (α=e²/4πε₀~c), the proton g-factor,. . . are combinations of dimensioned quantities whose units cancel out. Their variations represent fundamental and po- tentially observable effects.

Several experimental methods exist to probe the consistency (or not) of funda- mental constants (relative drift in atomic clocks, study of the Oklo natural reac- tor,. . . ) but the ones where isotope shifts may have some interest are the studies of quasar absorption systems. Quasars’s light may pass through gas clouds during their (long) travel to Earth, which may result in the absorption of some wavelengths.

The wavelength of a particular absorption line depends on the value ofα in the gas cloud at the time of absorption. As a consequence, by comparing the observed wave- length with what is observed in the laboratory measurement on Earth, it would be possible to witness a change in the α value.

Various systematic effects are troublesome for the quasar work. One of them is the isotopic abundance ratios in the gas clouds sampled in the quasar absorption spectra, because it may not match terrestrial abundances. It is thus hard to tell

if the observed shift is actually due to the α-variation or due to different isotope abundances.

A separate, although related, potential systematic effect is the differential iso- topic saturation. For many of the lines seen in quasar absorption spectra, the laboratory wavelengths are a weighted mean and the individual isotopes are not re- solved. If in the quasar’s light, the dominant isotope saturates, i.e. there is nothing left at that frequency, the rarer isotopes will have an increasing effect on the fitted centroid. The resulting shift could again mimic the variation of α. Currently, a list of the isotope shifts measurements that are needed in order to resolve systematic effects in the study exists [Berengut 2011].

This α-variation problem is becoming a hot topic. At the time we are writing these lines, measurements at the Very Large Telescope seem to indicate a spatial variation ofα[Webb 2011] although the existence of the Brout-Englert-Higgs boson would confirm the Standard Model [Chatrchyan 2012,Aad 2012].

1.2.4 Nuclear physics

The nuclear root-mean-square charge radiushr²i^1/2is a fundamental property of the atomic nucleus. As discussed in section3.6, two experimental methods are known in order to determine directly the nuclear mean-square charge radius: measurements of transition energies in muonic atoms and elastic electron scattering experiments.

When accelerated electrons are sent to a nucleus, the electrostatic field of the nucleus will cause a scattering of the electrons. The analysis of the differential cross section gives information about nuclear charge distributions. It also has the advantage of providing an absolute value of the root-mean-square radius instead of a difference between two isotopes radii. Frickeet al.[Fricke 1995] enumerate several limitations of this method. One will note that the accuracy of the method is limited [Pálffy 2010]

and that most experiments only work with stable nuclear targets.

The difference δhr²i^A,A0 ≡ hr²i^A − hr²i^A0 between two isotopes of the same element is determined by Kα x-ray and optical isotope shifts. Indeed, IS measure- ments have a long tradition and are known as a valuable source of information on (changes in) the nuclear charge radii and distributions. Each of these methods requires accurate calculation and measurement of isotope shifts. It was recently shown that dielectronic recombination measurements can be used for accurately inferring changes in the nuclear mean-square charge radii of highly-charged lithium- like neodymium [Brandau 2008]. To make use of this method to derive information about the nuclear charge distribution for other elements and isotopes, accurate elec- tronic isotope shift parameters are required. This topic will be more widely explored in section3.6.

Any progress in the accuracy of calculations is of significant importance. Since a few years, some really high precision calculations on low-lying Z values [Yan 2008, Puchalski 2006], allow a good nuclear charge determination. The nuclear charge ra- dius of¹¹Li was determined for the first time by Sánchez et al.[Sánchez 2006] using high-precision laser spectroscopy. They discovered that the charge radius decreases

monotonically from ⁶Li to ⁹Li, and then increases from 2.217(35) to 2.467(37) fm for ¹¹Li. More recently, Nörtershäuser et al. [Nörtershäuser 2011], at GSI, Darm- stadt, determined changes in the mean-square nuclear charge radii along the lithium isotopic chain, using a combination of precise isotope shift measurements and the- oretical atomic structure calculations. Also with the help of ab initio quantum electrodynamics (QED) calculations of the⁴He-³He isotope shift, Cancio Pastor et al. extract the difference of the mean-square nuclear charge radii δhr²i of³He and

4He [Cancio Pastor 2012] but their results disagree with the ones of van Rooijet al.

[van Rooij 2011]. As they do not have a satisfactory explanation at present one can expect further developments on this element.

1.2.5 Other fields of applications

There are several other “applications” where the isotope shifts are needed. Concern- ing the chemical evolution of the Universe, isotope abundances away from Earth, can tell us something about how the solar system, and the Universe itself, evolved. But we are still not able to travel in Space to bring back samples, and the only way to get information is throughISmeasurements. If the observation is precise enough one can observe theIS. Since the Hubble Space Telescope (HST) was launched in 1990, there has been an increasing interest in isotope shifts and hyperfine structures among as- trophysicists. The high resolution of stellar spectra obtained with the spectrograph aboard theHSTmakes it possible to resolve the isotope shift in many spectral lines.

For instance, in aNature publication, Federmanet al.[Federman 1996] explain that they were able to measure the isotope shift for some transitions of boron. In the framework of stellar spectra, Kurucz cites several systematic errors introduced if one ignores the isotopic splitting [Kurucz 1993].

For most lines, however, these small structures cannot be completely resolved, and instead they shift and broaden the lines. This may lead to erroneous interpreta- tion of lines of spectral astrophysical interest, where, for example, the unresolvedIS can be misinterpreted as a Doppler broadening. Since there are not many laboratory experiments, reliable computational methods are important to support the analysis of astrophysical spectra. Unfortunately, many of the isotope shifts have not been measured in the laboratory, hence the need to calculate them.

The area of cryptography may also be interested in the determination of IS.

In a work published on the ArXiv site, Durt and Hermanne [Durt 2008] explain the possibility of using metastable excited nuclei to transmit the “key” between the famous Alice and Bob without allowing a spy (called Eve, for “eavesdropping”) to intercept it. If the lifetime of the radioactive isotope that is used to encode the fresh key is significantly longer than the transit time between them, Eve cannot know the key because the decay will most often occur only after the carrier has reached the receiver. When the “key” is produced, through collision with highly energetic particles, the atomic number Z and/or the mass numberA may change during the process; isotope shifts would then play a role.

As a final notable fact, the isotope shifts are also a source of investigation at the

molecular scale, as shown by the recent experiment [Haestad 2012] on D₂ and H₂ molecules.

1.3 Units

Before we start our analysis, it is convenient to introduce the atomic units (a.u.) in order to rid the equations of “unnecessary” physical constants. Atomic units are defined by requiring that the Bohr radiusa₀, the electron’s mass m_e, the electron’s chargee, and the Planck’s constant~, all have the value 1; on its side the permittivity of free spaceε0 is1/(4π). Units for other quantities can be readily worked out from these basic few. From the expression for the fine-structure constant (a dimensionless number)

α= e²

4πε₀~c = 1

137.035999074(44) , (1-6)

the unit of velocity vB=αcis defined. It implies that the velocity of light has the valuec = 1/α'137a.u. In our work, we have a regular use of the hartree, a unit of energy defined as 1E_h = ~²/(m_ea²₀) = e²/(4πε₀a₀). As an order of magnitude the reader should remember that it corresponds to two times the energy needed to extract the electron from the hydrogen atom in its ground state (with infinite and point charge nuclear mass, but this “detail” will be fully studied in next chapters).

This unit can also be related to the Rydberg’s constant R∞= α²mec

2h = ~²

a²₀2hcm_e = 10973731.568539(55) m⁻¹ (1-7) through the relation 1E_h = 2hcR∞. The latter will be useful to describe isotope shifts quantities (cf. section 3.5.1). All these quantities are written in table 1.1, with their values in System International units.

Table 1.1: Atomic units and respective value in SI units. The digits in parentheses are the one-standard-deviation uncertainties in the last digits. These values are taken from the most recentcodata [Mohr 2012].

Quantity Symbol SI values

Length a0 = 4πε0~²/mee² = 5.2917721092(17) ×10⁻¹¹ m

Mass m_e = 9.10938291(40) ×10⁻³¹ kg

Charge e = 1.602176565(35) ×10⁻¹⁹ A s

Angular momentum ~ = 1.054571726(47) ×10⁻³⁴ J s

Energy E_h=~²/m_ea²₀ = 4.35974434(19) ×10⁻¹⁸ J

Speed vB=αc = 2.18769126379(71) ×10⁶ m s⁻¹

Aside from atomic unit of mass (me), the unified atomic mass unit (u) is also regularly used³. It is equal to 1/12 times the mass of a free carbon 12 atom, at rest and in its ground state, 1u = 1/12 m(¹²C)=1.660538921(73) ×10⁻²⁷ kg

=10⁻³kg mol⁻¹/N_A whereN_A= 6.02214129(27)×10²³mol⁻¹ is the Avogadro con- stant [Mohr 2012].

3For the NIST, the abbreviation, amu is not an acceptable unit symbol for theunified atomic mass unit. The only allowed name is “unified atomic mass unit” and the only allowed symbol is u.

Relativistic atomic structure theory

I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.

A. Einstein

T

^hischapter introduces the fundamental atomic problem from the point of view of the structure of an atomic state. The basic notions of the Schrödinger equa- tion are assumed to be known as well as the Pauli principle. In section 2.1, the Dirac equation for hydrogenic systems is presented, since it represents an impor- tant building block for calculations on more complex many-electron systems. The section 2.2 presents some basic properties, linked to the Dirac equation, that may appear useful in our further investigations. The section 2.3deals with the methods we used to solve atomic problems with many electrons.

As soon as an atom possesses a high nuclear charge, the innermost electrons possess a velocity close to that of the light. Relativistic quantum mechanics is then required for the description of atoms and molecules whenever their orbital electron probe regions of space with high potential energy near the atomic nuclei.

Quantitatively, the following interpretation can be made [Desclaux 1982]. Starting from an electron rotating around a point charge nucleus and assuming the mass of this nucleus as infinite bearing a positive electric charge of Z|e|, the energy ob- tained through the Schrödinger equation is−m_eZ²e⁴/(32π²ε²₀~²n²), or equivalently

−m_ec²(Zα)²/(2n²). In these relations Z is defined as the number of protons and nas the principal quantum number. According to the virial theorem, the energy is proportional to the kinetic energy of the electron. Consequently, in a non-relativistic approximation, the mean speed of the electron is aroundv∼Zαc/n. On this basis, considering speed aroundv/c∼0.1, the1s electrons becomes relativistic for nuclei withZ ≥14. If the criteria is v/c∼0.01, the electrons around nuclei withZ ≥40 are relativistic. Moreover, in order to describe the fine structure of atomic states from first principles, it is also necessary to treat the bound electrons relativistically.

Consistent with both quantum mechanics and special relativity, the Dirac equa- tion describes the behavior of elementary massive particles with half-integer spin (as electrons). It is presented in an exhaustive way in many books [Thaller 1993, Greiner 2000] and the XXIst century saw the first atomic books specifically ded- icated to the relativistic atomic structure [Johnson 2007, Grant 2007]! Therefore, the history and derivation of the Dirac equation will not be discussed here.

2.1 Dirac equation for one particle systems

The Dirac equation has many forms that are useful in different contexts. For free particles possessing any mass m, many already exist

1 c

∂

∂t+α₁ ∂

∂x+α₂ ∂

∂y +α₃ ∂

∂z +imcβ

ψ(x, t) = 0, or ic

−α₁ ∂

∂x−α2

∂

∂y −α3

∂

∂z −imcβ

ψ(x, t) = i∂ψ(x, t)

∂t ,

or

c(−iα·∇) +mc²β ψ(x, t) = i∂ψ(x, t)

∂t ,

or

c(α·p) +mc²β ψ(x, t) = i∂ψ(x, t)

∂t , (2-1) or with k= 1,2,3

n

∂0+α^k∂_k+imcβ o

ψ(x, t) = 0, or with µ= 0,1,2,3 {γ^µpµ−mc}ψ(x, t) = 0.

The relation (2-1) is kept since it is the closest to the Schrödinger equation and the most used in the atomic framework. The last compact and elegant form is the covariant notation, which is more convenient for demonstrating the covariance properties of the equation. These notations are closely related to the special theory of relativity: the gradient∂0 = ¹_c∂t^∂,pµ≡(i∂µ) = (i∂0, i∇)andγ^µare4×4matrices

γ⁰ =

I 0 0 −I

, γⁱ=γ⁰αⁱ=

0 σⁱ

−σⁱ 0

with i= 1,2,3.

The quantities αⁱ or α (in its vectorial notation) and β are 4×4 Dirac matrices [Grant 2006]:

α=

0 σ σ 0

, β =

I 0 0 −I

(=γ⁰), (2-2)

whereI is the unit matrix 2×2, and σ is defined by hermitian matrices σx=

0 1 1 0

, σy =

0 −i i 0

, σz =

1 0 0 −1

. (2-3)

These so-called Pauli matrices constitute, with the unitary matrix, a basis of the 2×2hermitian matrix, vectorial space. To describe a particle moving in an external electromagnetic field, the equation (2-1) is modified as

i∂ψ

∂t =

c α·(p−qA) +βmc²+qΦ ψ . (2-4) The equation (2-4) can be used for the more particular case of a particle in a spherically symmetric scalar potential by settingA= 0andqΦ =V(r)wherer =|r|

is the distance from the origin. The Dirac equation with the Coulomb potential describes the motion of an electron (whose mass is designed by me) in the field of

an atomic nucleus. This problem can be solved exactly and the solutions agree with the experiments in a nice way. We are now able to introduce the Hamiltonian that will be the fulcrum of most of incoming developments: the Dirac Hamiltonian

H_D =cα·p+c²β+V(r), (2-5)

wherer is the radial distance between the electron and the nucleus,V(r) is the po- tential energy. Furthermore, we will only deal with stationary states. Our eigenvalue problem is then reduced to

H_Dψ_nκm(r) =E_nκψ_nκm(r). (2-6) The Hamiltonian H_D is invariant under rotations and space-like reflections, which means that it commutes with the operators of total angular momentum and the parity. One can construct a common basis, the spherical spinors, using the Clebsh-Gordan definition. Let us suppose that we have two commuting angular momentum vectorsJ1 and J2 with |j₁, m1i and |j₂, m2i, the eigenstates of J₁² and J_1z and J₂² andJ_2z respectively. One constructs a basis |J Mi,

|J Mi=

j1

X

m1=−j₁ j2

X

m2=−j₂

hj₁, j₂;m₁, m₂|J, Mi |j₁, m₁i ⊗ |j₂, m₂i, (2-7) whose elements are eigenstates of J² and Jz with eigenvalues J(J + 1), and M, respectively. In our case, these elements, the spherical spinors Ω_jlmj, are formed by combining spherical harmonics Y_m^l

l(θ, φ), which are eigenstates of L² and L_z, and spinors φσ, which are eigenstates of S² and Sz. One can combine the orbital wave functions given by spherical harmonics and the spin wave functions to obtain eigenfunctions of J =L+S. If one uses the relation ml = mj −σ, the spherical spinors are

Ω_jlmj(θ, φ) =

1/2

X

σ=−1/2

hl,1/2;mj−σ, σ|j, m_jiY_m^l

j−σφσ , (2-8) that are also useful in their spinor form:

Ω_jlmj(θ, φ) =







hl,¹₂;mj −¹₂,+¹₂|j, m_ji Y_m^l

j−1/2(θ, φ) hl,¹₂;m_j +¹₂,−¹₂|j, m_ji Y_m^l

j+1/2(θ, φ)





. (2-9)

In these expressionsmj andσ =±1/2stand for the quantum number related toJz

and S_z respectively. Each of these eigenvectors satisfies the eigenvalue equations:

L²Ω_jlmj = l(l+ 1) Ω_jlmj, (2-10) S²Ω_jlmj = 3

4 Ω_jlmj, (2-11)

J²Ω_jlmj = j(j+ 1) Ω_jlmj, (2-12)

JzΩjlmj = mj Ωjlmj, (2-13)

where l = j±1/2. But H_D does not commute with L² and S² and the quantum numbersjandm_j are not sufficient to describe one state (each value ofjcorresponds to two non-relativistic states). This is why it is useful to introduce the operatorK:

K =β(2S·L+1) =β(σ·L+1) , (2-14) that commutes with H_D as well as with J [Mathur 2008]. Its application on an eigenvector gives

(σ·L+1) Ω_jlmj(θ, φ)≡κΩ_jlmj(θ, φ), (2-15) where the eigenvalue κ is











κ=− j+¹₂

, if l=j−¹₂ κ= + j+¹₂

, if l=j+¹₂ .

(2-16)

The value ofκdetermines both the angular momentajand orbitall. Consequently, the more compact notation Ωκm = Ωjlm can be used. The many forms of the quantum number κ and its operator will be useful later. For this reason, they are the subjects of a whole sectionA.2in the appendix. The conventions used in labeling Dirac 4-spinors in spherical symmetry are given in the table 2.1. One will notice that each state is completely defined by κ.

Table 2.1: Correspondence between spectroscopic notation and the angular momen- tum quantum numbers l,el= 2j−l,j and κ.

Orbitals s_1/2 p_1/2 p_3/2 d_3/2 d_5/2

s p*/p p d*/d d

l 0 1 1 2 2

el 1 0 2 1 3 j 1/2 1/2 3/2 3/2 5/2

κ −1 +1 −2 +2 −3

Parity + − − + +

As the Dirac Hamiltonian does not change under space-like reflexions, the par- ity operator P that maps r → −r may also produce good quantum numbers. In spherical coordinates, the operator P transformsφ→φ+π and θ→π−θ. Under a parity transformation, P Y_m^l(θ, φ) =Y_m^l(π−θ, φ+π) = (−1)^lY_m^l(θ, φ). From the action of the 4-components operator P =βP on the 4-rank spinor, it follows that the spherical spinors are eigenfunctions of P having eigenvalues Π = (−1)^l. The two spinors Ωκm(θ, φ) and Ω−κm(θ, φ), corresponding to the same value of j, have values of l differing by one unit and, therefore, have opposite parity (as observed table 2.1).

Finally, in a spherical coordinate system, eigenstates of (2-6) have the 4-spinor structure

ψnκm(r) = 1 r

P_nκ(r)Ω_κm(θ, φ) iQnκ(r)Ω−κm(θ, φ)

, (2-17)

P_nκ(r) and Q_nκ(r) are the large and small component radial wave functions and Ωκm(θ, φ)is the two-dimensional vector harmonic.

To conclude this section, let us remember that the spherical spinors satisfy the orthogonality relations:

Z π 0

sinθ dθ Z 2π

0

dφΩ^†_κ0m⁰(θ, φ)Ωκm(θ, φ) =δ_κ⁰_κδ_m⁰_m. (2-18)

2.2 Hydrogenic atom with a point charge nucleus

The study of the hydrogenic atom is almost an academic case but it allows to in- troduce several properties. The effects of relativity in the absence of screening by orbital electrons, although interesting, do not bring any real information related to our topic, so they will only be enumerated. More interesting will be the energy as- sociated to the eigenvalue problem presented hereafter and some expectation values calculated with hydrogenic wave functions.

2.2.1 The Sommerfeld formula

The eigenvalue problem that permits to obtain the binding energy of an electron in a static Coulomb field, V_c=−Z/r is

0 = (H_D−E_nκ)ψ_nκm(r).

With the expression of the wave function (2-17) (and a property of the operator σ·p; presented in the appendixA.3and extremely well described in [Grant 2007]), it can be formulated as:





c²−Enκ+Vc(r) −c _dr^d −^κ_r c _dr^d +^κ_r

−c²−Enκ+Vc(r)









Pnκ(r) Qnκ(r)



= 0. (2-19) Solving these eigenvalue equations, the energy levels of the bound states are

E_nκ = c²

q

1 +_{(γ+n−|κ|)}^(αZ)2 2

=c² n_r+γ

p(γ+nr)²+ (αZ)² =c²n_r+γ N

= c² r

1−(αZ)² N² =c²

s

1− (αZ)²

(γ+n− |κ|)²+ (αZ)² , (2-20) where

γ² =κ²−(αZ)² , (2-21)

and nr, the radial quantum number, represents the number of nodes of the radial function P_nκ(r),

n=nr+|κ| with nr= 0,1,2, . . . (2-22) The radial function Q_nκ(r) has the same number of nodes asP_nκ(r)if κis negative and possesses an additional node when κ is positive. The value N is called the apparent principal quantum number,

N =

(n_r+γ)²+ (αZ)² 1/2

=

(n²−2n_r(|κ| −γ) 1/2

. (2-23) It plays the same role as its non-relativistic analogue nin the energy formula and, if c tends to infinity N has the same value as the principal quantum number n.

These rules for the nodes will be useful in designing a numerical eigenvalue routine for solving the Dirac equation.

Efforts have been made in (2-20) to write this expression in all possible combi- nations that can be encountered in the literature. Moreover, if one prefers removing the rest energy of the electrons, as some authors like to define, without using atomic units,ε_nκ=E_nκ−c² (sinceε_nκ tends towards the non-relativistic value in the limit c→ ∞), giving birth to another way to express the energy [Shabaev 2002],

εnκ = c²





1 q

1 +_{(γ+n−|κ|)}^(αZ)2 2

−1



=c² r

1−(αZ)² N² −1

!

(2-24)

= −c²(αZ)² 2ν²

2 1 + (αZ/ν)²+p

1 + (αZ/ν)² , where

ν = n+p

(j+ 1/2)²−(αZ)²−(j+ 1/2),

= n+γ− |κ|=nr+γ .

The relation (2-20)can be expanded in powers of αZ : Enκ= c²

|{z}

Rest energy of electron

−Z² 2n²

| {z }

Balmer results, non -relativistic Coulomb

-field binding energy

−α²Z⁴ 2n³

1

|κ|− 3 4n

| {z }

Lowest relativistic corrections:

the single non-relativistic energy is split into several

fine structure levels

+O(Zα)⁶

(2-25) where |κ| = j+ 1/2. Here, we would like to make a slight digression on the for- mula (2-25). At first glance, it seems to correspond exactly with Sommerfeld’s investigations that date back to 1916. However, at that time, Sommerfeld was

not aware of the spin-orbit interaction and even less of the spin. Several people talk about a “coincidence”. Heisenberg himself calls it a “miracle”; but in 2004, i.e.

extremely recently compared to the age of the Sommerfeld formula, Granovski˘ı ex- presses his doubts and explains this “coincidence” by charging Sommerfeld with a mistake [Granovski˘ı 2004]. For him, Sommerfeld erroneously replaced the square of the orbital momentum L² by L²_z and since the quantization proposed by Bohr (L = n~) actually corresponds to the values of |κ| = 1,2, . . . the good physical answer was found.

The radial wave functions associated with these eigenvalues are also analytically available. They can be found in the books of Grant and Johnson [Grant 2007, Johnson 2007]¹. Much less simple than their non-relativistic analogues, their general forms require the use of gamma functions and confluent hypergeometric functions that can be found in [Abramowitz 1972].

2.2.2 Average values using virial relations

Unlike in the non-relativistic theory, some expectation values, such as the radial moments, are not easy to obtain. The virial relations for the Dirac equation, derived by Shabaev [Shabaev 2003] in an amazingly clear way, give us the key to derive these values. But as often, the author uses the relativistic units (~= c = 1), which are a source of confusion (in such units, the coulombian potential V_c is expressed as V_c=−αZ/r)². Hereafter, we only present the three expectation values that will be helpful for the developments of the mass corrections considered in the next chapter:

A^s ≡ hnκm|r^s|nκmi= Z ∞

0

r^s P_n,κ² +Q²_n,κ

dr , (2-26)

B^s ≡ hnκm|βr^s|nκmi= Z ∞

0

r^s P_n,κ² −Q²_n,κ

dr , (2-27)

and the much less used relation

C^s ≡ hnκm|ir^s(α·e_r)β|nκmi

= Z ∞

0

Z 1

r Pn,κΩ^∗_κ,m,−iQ_n,κΩ^∗_−κ,m

× (ir^s)

0 −σ_r σ_r 0

1 r

Pn,κΩκ,m

iQ_n,κΩ−κ,m

r²drdΩ

= 2 Z ∞

0

r^s(P_nκQ_nκ)dr , (2-28)

wheresis an integer,er is the radial unitary vector er=r/r andσr=σ·er. The orthogonality relation (2-18)requires conditions onκ and mthat are not displayed since the above relations concern diagonal matrix elements only.

1The reader should be warned that in Grant’s book, nr is to be replaced by (nr!) in the denominator of theNEκterm, that follows the equation (3.3.23).

2Indeed, as~=c= 1, it comes directlyα=e²/(4πε0).

The radial momentA⁻¹ can be found in the Handbook of Atomic Molecular and Optical Physics [Grant 2006]

A⁻¹ =hr⁻¹i = Zγn+ (|κ| −γ)|κ|

N³γ

= Zγ(

nr

z }| { n− |κ|) +κ²

N³γ . (2-29)

Concerning the radial moment r⁻²

, there are several equivalent writings, see [Burke 1967, Kobus 1987]. The easiest to use in such analytical treatment is the one of Shabaev [Shabaev 2003]

A⁻² = hr⁻²i=Z² 2κ[2κ(γ+nr)−N]

N⁴(4γ²−1)γ . (2-30)

We finally present three last matrix elements whose interest will appear in sec- tion 3.2.

B⁰ = hβi= Enκ

mc² = nr+γ

N , (2-31)

B⁻¹ = hβr⁻¹i= mc² αZ

1− E_nκ² m²c⁴

, (2-32)

C⁻¹ = hi(α·er)βr⁻¹i= (αZ)²κ

N³γ mc² , (2-33)

C⁻² = 2(αZ)³{2κ(γ+nr)−N}m²c⁴

N⁴(4γ²−1)γ . (2-34)

As a consequence, the matrix element of the Coulomb potential, that is here excep- tionally written in SI units, is

hV_ci = h−Zr⁻¹i=−Z²

mec²α²

z }| { e²

4πε0a0

γn_r+κ²

N³γ (2-35)

hV_c²i = hZ²r⁻²i=Z⁴ e²

4πε0a0

2

| {z }

(mec²α²)²

2κ(2κ(γ+nr)−N)

N⁴(4γ²−1)γ . (2-36)

The braces in the latter expressions, although useless in atomic units, help to make the link with others and help to understand the developments of section 3.2.

2.3 The issues of complex atoms

If more than one electron is present in the system, an additional term, arising from the interaction between electrons, must be added in the Dirac-Coulomb Hamiltonian describing the problem

H_DC =

N

X

i=1

cα_i·p_i+ (β_i−1)c²+V(r_i) +

N

X

i<j

V(i, j). (2-37) The first braced term is the one-electron HamiltonianH_D (2-5)where the rest mass of the electron has been removed; it is built with the one-electron Coulomb potential V(ri) and the Dirac kinetic energy operator:

T_i =cα_i·p_i+ (β_i−1)c² . (2-38) The term V(i, j) represents the interaction energy of electrons i and j; it can be constructed by considering only the Coulomb operator or the latter, corrected with the Breit interaction.

As well explained by Mann and Johnson [Mann 1971], the Breit interaction is a correction to the Coulomb repulsion between two electrons due to the exchange of a transverse photon. The electron-electron interaction is evaluated with the operator

Vij = 1 rij

higher-order retardation

z }| {

−αi·αj

rij

cos

ωijrij

c

+c²(α· ∇)_i(α· ∇)_jcos ^ωij_c^rij

−1 ω_ij²r_ij (2-39) ' 1

r_ij −αi·αj

r_ij

| {z }

Gaunt

+αi·αj

2r_ij −(αi·rij)(αi·rij) 2r_ij³

| {z }

Breit retardation

, (2-40)

wherer_ij =|r_i−r_j|is the interelectronic distance andω_ij is the energy of the photon exchanged between the electrons that is taken to be the difference in the diagonal Lagrange multipliers εi and εj associated with the subshell radial wave functions.

The first term in Eq. (2-39) represents the regular Coulomb interaction. Higher- order retardation terms of (2-39) constitute the so-called, “frequency-dependent”

Breit operator. The approximation (2-40) is its static form in which the second term is the magnetic (Gaunt) interaction, and the rest is the Breit retardation. Its influence on the isotope shift parameter is studied in chapter5.

The resulting many-body Hamiltonian is called the Dirac-Coulomb Hamiltonian.

The study of the bound states of the atomic system consists in solving the eigenvalue problem

H_DCΨ =EΨ, (2-41)

whose solutions are square-integrable. Such equation can no longer be solved ana- lytically and all the challenge consists in determining an approximate wave function for anN-electron atom.