Mingtao Cao

Ping Tang

Zhibin Yao

Jun Sun

Chenjie Ding

Tao Peng

Shuwei Qiu

Haiwa Zhang

Bin Jialin Cai

Li, and other friends for helping me. Being able to work and study with a group of friends is my most precious experience and wealth.

Parents' support and encouragement are the strong motivation

Contents

Introduction

Bound state quantum electrodynamics (BSQED) can describe the structure of atoms and ions very accurately, and it is one of the most important cornerstones in modern physics. Highly charged ions and muonic atoms are generally considered to be the ideal candidates for studying BSQED of the strong-field regime. Yet there are still outstanding issues that require improved accuracy in both experimental and theoretical tests. Calculations are very difficult already at second order for the one-electron system. Considering accurately enough, the electron-electron interaction in two and more electron ions is even more difficult. For 50 years, many experiments have been performed to measure the energies of few-electron ions. The experiments have used laser-generated plasmas, beam-foil spectroscopy, highly-charged ion sources such as Electron-Beam Ion traps or Electron-cyclotron resonance ion sources, and storage rings.

Our group specializes in reference-free measurements, which provide improved tests. Machado et al. [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF] measured the width and transition energies in He-like 1s2p 1 P 1 → 1s 2 1 S 0 and Be-like 1s2s 2 2p 1 P 1 → 1s 2 2s 2 1 S 0 in argon ions. The experiment for measuring the transition energy of 1s2s 2 2p 1 P 1 → 1s 2 2s 2 1 S 0 is the first reference-free measurement with more than two-electrons. The group also measured 1s2s2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 transitions in lithiumlike sulfur and argon, and of the 1s2s2p 4 P 5/2 → 1s 2 2s 2 S 1/2 M2 transition in sulfur. The experiment used a double-crystal spectrometer (DCSs) connected to an Electron-Cyclotron Ion Source (ECRIS), without using any reference line. To validate the experiment and improve the theoretical model, we conducted theoretical research.

Muonic atoms are formed by a negatively charged muon that enters the orbit of an atom and decays by emitting X-rays. Detection of muonic X-rays is considered a sensitive and precise technique for determining the characteristics of the nucleus, such as charge distribution and deformation [START_REF] Engfer | Charge-distribution parameters, isotope shifts, isomer shifts, and magnetic hyperfine constants from muonic atoms[END_REF]. This method allows precision measurement of nuclear charge radii and has been an essential tool for more than 40 years, supplementing the knowledge of electron scattering experiments and isotope shifts by optical laser spectroscopy [START_REF] Fricke | Nuclear charge radii[END_REF]. There will be an upcoming experiment to perform muonic atom spectroscopy aiming at a precise measurement of the absolute nuclear charge radii of radioactive 226 Ra with a level of 0.2% relative precision at the Paul Scherrer Institute (PSI) [START_REF] Adamczak | Nuclear structure with radioactive muonic atoms[END_REF].

The experiment can map out the various level energies of muonic atoms by measuring the emitted X-ray photons. The nuclear charge radius can be extracted by comparing the measured energy level to a calculation result. Therefore, in addition to the need for experimental techniques, this experiment requires a complete theoretical method and extensive and accurate theoretical calculations.

Bergern et al. [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] measured low-lying transition energies of muonic 208 Pb with a v precision up to 11ppm using a Compton suppression spectrometer. They achieved the charge radius using the best-fit parameters and the fitted nuclear polarization values. We conduct a detailed analysis of muonic lead referencing many lines measured between 1s, 2s, 2p, 3p, 3d, 4f, 5f, 6f levels comparing the results with others and testing our theoretical model. Finally, we provide reliable theoretical values of 226 Ra and 248 Cm for the experiment. We perform multi-configuration Dirac-Fock (MCDF) research on three-electrons ions and muonic atoms by improving the MCDFGME code developed by Desclaux and Indelicato. We use the newest 2018 version of the program, which improves the finite size corrected self-energy, vacuum polarization to all order and effective model operator evaluation of the one-electron self-energy screening values for electronic and muonic atoms.

The outline of this manuscript is the following:

The first chapter starts with a background of the subject. The status of Quantum Electrodynamic (QED) tests, including electronic and muonic atoms, are presented. We also briefly describe muon-electron universality and the properties of muonic atoms.

The second chapter introduces the theoretical method of atomic calculations on one and three electrons. The multiconfiguration Dirac-Fock (MCDF) method, QED corrections, and nuclear effects are discussed. We take into account all one loop and the main two-loop QED contributions. We calculate for the first time the exact one-loop self-energy for muonic atoms with finite nuclear size contribution. We also mention all corrections which affect transition energy.

Several important characteristics of muonic atoms are emphasized in the third chapter.

In the fourth chapter, the computational procedure using the relativistic atomic structure code is explained in detail.

In the last two chapters, we present the discussion and analysis of the results. For Li-like atoms, we describe the results from the Welton model and effective operator models and compare the result between calculations and experiments. The effects of each contribution are also analyzed. Finally, we give the sources of theoretical errors and uncertainty. For muonic atoms, the best fit nuclear parameters that minimize the weighted theory-experiment discrepancy are found in the muonic lead with two-parameter Fermi model and three-parameter Gauss model. We also tabulate the energy levels in muonic radium and curium with all the radiative corrections.

Chapter 1 Background

Quantum electrodynamics (QED) is a foundation for modern physics as the first relativistic quantum field theory. It can be said to be the most rigorously tested part in the standard model. In the past few years, QED and relativistic multi-body problems have been undergoing significant progress. The purpose of this part is to present the current status of experiment and theory on QED tests, with tools of one to three electronic ions and muonic atoms. Accurate tests of QED are performed by comparing experimental data with theoretical calculations. Next, we briefly introduce the muon, which can be treated as a large mass electron, and muonic atom. At last, we list several widely used methods dealing with the multi-body problem.

Basic properties of muonic atoms 1.Properties of muons

In 1936, Anderson and Neddermeyer [START_REF] Anderson | Cloud chamber observations of cosmic rays at 4300 meters elevation and near sea-level[END_REF] discovered the muon, a fundamental particle, when studied cosmic rays using cloud chambers and Geiger counters. Some data relevant to the basic properties of muon are summarized in Table 1.1. The muon is a lepton and does not participate in the strong interaction, and its spin is 1/2. The charge of the muon is the same as the electron within about 2 ppm. The muon has the same electromagnetic properties as the electron but with greater mass (m µ ≈ 207m e ). The muonic magnetic moment can be obtained by measuring the precession of the muon spin directly in a magnetic field or the combination of muonium transition frequencies in a strong magnetic field. The magnetic moment of the muon is smaller than that of the electron, which makes the muon less sensitive to magnetic fields and the nuclear current. Its motion is mainly governed by its electronic charge interaction with nuclear charge, so at least in terms of the nuclear interaction, there is no requirement to provide great precision information on µ.

The lifetime of the muon is known very accurately (2.197134(77) × 10 -6 s) [START_REF] Trippe | Review of particle properties[END_REF]. This lifetime is determined entirely by the decay mode.

Since the mass and energy of a muon are much larger than the decay energy of common radioactive decay, a muon cannot be produced by radioactive decay. A muon can be generated by a hadron-involved nuclear reaction in a high-energy physics experiment performed on an accelerator. Besides, cosmic rays interact with the Earth's atmosphere The muon is a point particle obeying the Dirac equation and the usual quantum-electrodynamical refinements where applicable Weak interaction Universal Fermi (U -A) interaction to produce a large number of muons, which method is the only known natural source of muons. The weak interaction of the muon is two-fold, one is the decay of µ -→ e -+ν e +ν µ and another is captured by the nucleus (µ -+ P → N + ν µ ). This phenomenon limits the mean lifetime of muonic atoms.

In light atoms, free decay is important. While µ -captured by the nucleus is important, and the lifetime is around 0.08 µsec in heavy atoms. This seriously limits the possible measurements on the ground state of the muonic atom. In addition, this short lifetime seems to make the manufacture and the study of muonic atoms with two or few muons completely impractical. The validity of the QED basic theory is tested by using highprecision spectroscopy with µ-containing atoms.

Muonic atoms

A muonic atom is created when a negatively charged muon that enters the orbit of atom and decays a series of lower energy levels by emitting X rays. Eventually, the muon is absorbed by the nucleus.

The mass m of an orbital particle has a great influence on the atomic properties. In low-Z muonic atoms, muonic binding energies are on the order of several keV, and there is a strong overlap between the muonic wave function and that of the nucleus. This leads to a significant effect of the finite size of the nucleus on the atomic energy levels. These atoms provide an interesting opportunity to extract the properties of the nucleus with high precision.

In Table 1.2, we list the main features of muonic atoms comparing with the electronic atoms. The finite nuclear size effect plays a dominant role in muonic atoms. In heavy muonic atoms, the interaction of electric-quadrupole become of the same size as the finestructure splitting of 2p or 3d levels. Magnetic hyperfine splitting is much smaller as it dependents on the magnetic moment of muon (in inverse proportion to mass).

There should be noticed that in addition to the mass, the other differences between the observed interactions of the electronic and muonic atom is shown in Table 1.2.

The electrons only perceive the effects of the fully unipolar field Ze/r near the nucleus. The muon can feel the shielding effect from electrons if there are residual electrons. In fact, Table 1.2 -Some typical characteristics of muon and electron atoms [START_REF] Devons | Muonic Atoms[END_REF].

Dimensions

M,Z Approx,ratio, dependence muon/elec. many electrons are removed from the nucleus during the muon capture, so the screening effect of electrons is very small in muonic atoms.

Magnetic hyperfine structure (HFS) is on the size of several keV, usually two-time orders smaller than fine structures; while, this is large enough to cause internal transitions in N and O shell electrons. This, in turn, greatly enhances M 1 transitions between magnetic hyperfine states.

The low-lying orbits of muonic atoms penetrate the nucleus deeply, so the orbit 1s 1/2 is most sensitive to finite nuclear size effects. The electromagnetic interaction of the muon and the nucleus depends on the nuclear charge and current densities in these states.

Status of QED tests

A good understanding of the QED contribution is critical to physical precision testing. Determining the fundamental constants, for example, the fine structure constant, the Rydberg constant R ∞ and the mass ratio of electron and proton, also requires precise knowledge of QED contributions. The accurate testing of QED is obtained by comparing experimental results with theoretical predictions.

Few-electron ions a) Theory

High precision calculations of QED effects rely on the perturbation expansion in power of the expansion parameter α, which is the fine-structure constant α = 2πe 2 /hc. The individual terms of the perturbation sequence can be represented by Feynman diagrams, with the number of virtual photons indicated by the power of α. The starting point is the Dirac equation of the electrons in an external nuclear field. The interaction of the bound electrons with the electromagnetic field as a perturbation leads to radiation correction and electron-electron interactions. This interaction causes energy shifts and should be checked in the first order of m/M (m represents the electron mass and M the nuclear mass) and the first and second orders of α. Researchers have perfected the basic methods of Dirac, Pauli, Born, Fock, Wigner, Fermi, Feynman, Bethe, Schwinger, and others for calculating QED effects with more accuracy.

In light atoms, the earliest QED calculations were based on the expansion of Zα, which characterized the Coulomb interaction between nucleus and electrons. This calculation brings accurate results for hydrogen, helium, and other low-Z atoms. But it is not suitable for high-Z systems. Therefore, there is a need for a strict theory to describe the activity of electrons in a strong nuclear field before performing a comparison between theory and experiment. So, the calculations should be performed to all orders in the parameter Zα. This requires the development of the nonperturbative QED theory, which method is suitable for the calculations in highly charged ions.

The main difficulty is that the calculation of QED corrections on all orders in the parameter Zα requires a large numerical calculation. Mohr [START_REF] Mohr | Numerical Evaluation of the 1s1/2 State Radiative Level Shift[END_REF] reported a major achievement of the electron self-energy in H-like ions, which compiled by Mohr [START_REF] Mohr | Energy levels of hydrogen-like atoms predicted by quantum electrodynamics, 10≤ Z ≤ 40[END_REF] and Johnson and Soff [START_REF] Johnson | The lamb shift in hydrogen-like atoms, 1≤ Z ≤ 110[END_REF] as a standard reference for many years. Soon after, researcher calculated various corrections to the Lamb shift of H-like atoms including all orders vacuum-polarization in Zα [START_REF] Soff | Vacuum polarization in a strong external field[END_REF] and two-loop self-energy in Zα [START_REF] Yerokhin | One-loop self-energy correction to the 1s and 2s hyperfine splitting in H-like systems[END_REF][START_REF] Yerokhin | Two-loop self-energy correction in high-Z hydrogenlike ions[END_REF], the nuclear size effect to the selfenergy [START_REF] Mohr | Nuclear Size Correction to the Electron Self-Energy[END_REF], first order to all orders of nuclear recoil correction [START_REF] Artemyev | Relativistic nuclear recoil corrections to the energy levels of hydrogenlike and high-Z lithiumlike atoms in all orders in αZ[END_REF], even the difficult nuclear polarization correction [START_REF] Plunien | Erratum: Nuclear-polarization contribution to the Lamb shift in actinide nuclei[END_REF]. The two-loop effects to orders α 2 (Zα) 5 [12] and α 2 (Zα) 6 [START_REF] Pachucki | Two-loop Bethe-logarithm correction in hydrogenlike atoms[END_REF][START_REF] Jentschura | Nonrelativistic QED approach to the Lamb shift[END_REF] are further achievements with the Zα-expansion approach.

The first high-precision calculation in self-energy contributions was performed by Mohr [START_REF] Mohr | Numerical Evaluation of the 1s1/2 State Radiative Level Shift[END_REF] for the 1s level at medium and high-Z. It was extended to different n, l, j levels [START_REF] Indelicato | 6s and 8d state self-energy for hydrogen-like ions and new results on the self-energy screening[END_REF] and super-heavy elements [START_REF] Cheng | Self-energy corrections to the K-electron binding in heavy and superheavy atoms[END_REF]. This evaluation, in which the expansion in Z is only asymptotic, does not converge even for relatively small values of Z. Jentschura, Mohr and Soff [START_REF] Jentschura | Calculation of the electron selfenergy for low nuclear charge[END_REF] performed for 1 ≤ Z ≤ 5 using highly efficient resummation techniques. This resolved the long-standing inconsistency between the Zα-expansion approaches and numerical all-order calculations. Abnormally, large higher-order terms not included in the Zα-expansion lead to the disagreement. It is considered that in the 1S and 2S levels, this work eliminated another theoretical uncertainty of the hydrogen Lamb shift.

In recent years, QED test has also made significant progress based on the theory of predecessors. Yerokhin and Shabaev [START_REF] Yerokhin | Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms[END_REF] carried out a high-precision nonperturbative (in Zα) calculation of Lamb shift, where the nuclear recoil effect is considered in n = 1, and n = 2 states in light hydrogenic atoms. This method extended the calculation of fractional Z as low as 0.3 and improved the numerical accuracy by 2-3 orders of magnitude. The result is exactly consistent with the terms of Zα expansion. Also, the nonperturbative high-order remainder can be accurately identified. Yerokhin and Shabaev [START_REF] Yerokhin | Lamb Shift of N = 1 and N = 2 States of Hydrogen-Like Atoms, 1 ≤ Z ≤ 110[END_REF] systematically summed up the latest tabulation of the n = 1 and n = 2 level energies with the nuclear Z from 1 to 110 of H-like atoms, which is considered an ideal test QED effects for stretching the theory up to the utmost precision.

The study of one-electron ions is expected to be the best way to test QED corrections, which have been plotted in Fig. 1.1 including all the QED contributions and nuclear Fig. 1.1 Size of all the one-electron QED contributions to the Lyman α 1 transition energy as a function of Z. Figure from Indelicato P [START_REF] Indelicato | QED tests with highly charged ions[END_REF]. contributions as a function of Z for the Lyman α 1 transition. The calculations were performed using the 2018 version of the MDFGME program developed by Indelicato and Desclaux [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Desclaux | A relativistic multiconfiguration Dirac-Fock package[END_REF][START_REF] Indelicato | Projection operators in Multiconfiguration Dirac-Fock calculations. Application to the ground state of heliumlike ions[END_REF][START_REF] Desclaux | Computational approaches of relativistic models in quantum chemistry[END_REF][START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF].

For systems with more than one electron, QED corrections are much more difficult to perform because of the nuclear potential screened, which is caused by the additional electrons. The electron-electron interactions can be considered as a perturbation. However, the expansion in Zα cannot be used at high Z. As a result, the electron-electron interaction results in a perturbative-expansion factor of 1/Z. By calculating several terms of the expansion, the electron correlation and one-loop QED screening correction can be strictly described with an accuracy sufficient to deduce the two-loop Lamb shift, which has been presented in [START_REF] Yerokhin | Evaluation of the Two-Photon Exchange Graphs for the 2p 1/2 -2s Transition in Li-Like Ions[END_REF][START_REF] Yerokhin | Two-Photon Exchange Corrections to the 2p1/2-2s Transition Energy in Li-Like High-Z Ions[END_REF][START_REF] Sapirstein | Determination of the Two-Loop Lamb Shift in Lithiumlike Bismuth[END_REF].

Yerokhin, Indelicato and Shabaev [START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF] carried out all order calculations of the twoloop self-energy, especially for heavy Hydrogen-like ions. This calculation eliminated the largest theoretical uncertainty of 2p -2s transitions and produced the first experimental determination of the two-loop QED correction. Worth mentioning, the researchers performed systematic QED corrections to the lithium isoelectronic sequences with accurate numerical results. A strict QED evaluation was proposed for the first two terms of the expansion using zeroth-order approximation, instead of a local model potential for n = 2 states by Yerokin et al. [START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF]. The three-photon exchange correction was also included under the many-body perturbation theory (MBPT). Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] conducted a similar study of S-matrix calculations with all one and two-photon diagrams, and part of three-photon diagrams in lithium isoelectronic sequence using modified Furry representation. Lithium-like uranium provided the testing of first-order QED effects with a 0.2% accuracy level and second-order with a 6% accuracy level in α by Volotka et al. [START_REF] Volotka | Progress in quantum electrodynamics theory of highly charged ions[END_REF]. The nuclear polarization correction eventually becomes the limiting uncertainty in the theoretical accuracy of QED tests with HCI.

Researchers generally believe that the significant contributions of nuclear size effects, nuclear polarization correction, and nuclear recoil correction, which are not well known, affect the extraction of higher-order QED contribution information. Shabaev et al. [START_REF] Shabaev | Towards a Test of QED in Investigations of the Hyperfine Splitting in Heavy Ions[END_REF] proposed different scaling method of the QED effects and nuclear size corrections with Z and n. Nuclear deformation corrections are investigated on the level energies of Li-like U 89+ by Kozhedub et al. [START_REF] Kozhedub | Nuclear Deformation Effect on the Binding Energies in Heavy Ions[END_REF]. Consequently, the theoretical uncertainties of the 2s-2p 1/2 transition energy are obviously reduced.

For complex multi-electron ions, the mean-field approximation (like the Dirac-Fock approximation) is indispensable under the framework of relativistic quantum mechanics. The QED correction can be contained by the radiative potential [START_REF] Ginges | QED radiative corrections and manybody effects in atoms: vacuum polarization and binding energy shifts in alkali metals[END_REF]. Dirac-Coulomb or Dirac-Coulomb-Breit no-pair approximation is usually used for including relativistic effects [START_REF] Konovalova | Correlation, Breit, and QED effects in spectra of Mg-like ions[END_REF].

b) Experiment

Theoretical advances over the past few decades have been complemented by the highprecision spectroscopy experiments with highly charged ions. The experiments have used laser-generated plasmas, beam-foil spectroscopy, highly-charged ion sources like Electron-Beam Ion traps or Electron-cyclotron resonance ion sources, and storage rings in large facilities of HITRAP facility at GSI, GANIL in Caen, France, and GSI in Darmstadt, Germany.

Modern experiments studied the impact of the strong nuclear field on bound state QED via hydrogen-like neon up to uranium along with the entire periodic table at Fig. 1.2. At low-Z ions, the experimental measurements have reached a high degree of precision. The experimental results are in perfect agreement with the theoretical values. In the middle and high-Z ions, there seems not so optimistic due to the uncertainty of 2nd order QED effects. For example, the Lamb-shift measurements of the 1s state in H-like uranium has achieved the value of 460.2 eV±4.6 eV (about 1% uncertainty) by Gumberidze et al. [START_REF] Gumberidze | Quantum Electrodynamics at Strong Electric Fields: The Ground State Lamb Shift in Hydrogenlike Uranium[END_REF]. While the theoretical calculation result is 463.99 [START_REF] Lupton | Measurement of the Ground-State Lamb-Shift and Electron-Correlation Effects in Hydrogenlike and Heliumlike Uranium[END_REF] eV, with QED contribution of 265.2 eV, 2nd order QED of -1.26 [START_REF] Tschischgale | Absolute wavelength measurement of the Lyman-α transition of hydrogen-like silicon[END_REF] eV, and a comparably large correction of finite nuclear-size effects of 198.54(19) eV. These results provided testing of QED at the 2% level by combining theory and experiment in the strongest stationary electromagnetic fields. Beiersdorfer et al. [START_REF] Beiersdorfer | Measurement of the Two-Loop Lamb Shift in Lithiumlike U[Sup 89+[END_REF] provided the first test of two-loop Lamb shift for the 1s level in hydrogen-like U 89+ based on passive emission spectroscopy. This experimental value obtained with the weighted-average is consistent with the theoretical value. It will be expected to make some advances in these experiments, which will make them sensitive to 2nd order QED effects in non-perturbative methods. part-per-million (ppm) as a function of Z for one-and two-electron ions, separately. The experimental verifications of QED prediction have mainly concentrated on a few low-Z ions with high precision, while there is lack of some accurate measurements at medium and high-Z ions. Bruhns et al. [START_REF] Bruhns | Testing QED Screening and Two-Loop Contributions with He-Like Ions[END_REF] measured the line of 1s2p 1 P 1 -1s 2 1 S 0 in He-like Ar 16+ for a testing in two-photon QED corrections with a relative uncertainty of 2 × 10 6 . The experimental result was in good agreement with theoretical calculation, also with a subsequent 1.5 × 10 -6 measurement by Kubicek et al. [START_REF] Kubiçek | High-precision laser-assisted absolute determination of x-ray diffraction angles[END_REF]. However, there is a discrepancy of 3σ between the measurement of the w transition energy in helium-like titanium and advanced QED theory by Chantler et al. [START_REF] Chantler | Testing Three-Body Quantum Electrodynamics with Trapped Ti 20+ Ions: Evidence for a Z-dependent Divergence Between Experiment and Calculation[END_REF]. They noted that there is a general trend of Z-dependent divergence between QED predictions and average experimental data in the w line of helium-like isoelectronic sequence for Z > 20 at the level of about five standard errors. The evidence of the low transition energy predicted systematically suggests that there is an error term in the calculations, or the missing term is much larger than currently expected in the three-body QED calculation. Researchers have launched a heated discussion furthered by Epp [START_REF] Epp | Comment on "Testing Three-Body Quantum Electrodynamics with Trapped Ti 20+ Ions: Evidence for a Z-Dependent Divergence Between Experiment and Calculation[END_REF] and Chantler et al. [69]. It seems that the interpretation is not easy to find, and further research on experiments and theories is needed. Kubiček et al. [START_REF] Kubiçek | Transition energy measurements in hydrogenlike and heliumlike ions strongly supporting bound-state QED calculations[END_REF] present relative measurements of the resonant line in He-like Fe 24+ , which is in perfect agreement with the advanced BSQED predictions including screening QED terms. Another measurement on spectroscopy data of Fe 24+ by Rudolph et al. [START_REF] Rudolph | X-Ray Resonant Photoexcitation: Linewidths and Energies of K alpha Transitions in Highly Charged Fe Ions[END_REF] are also agreement with QED theory. These evidences refute the argument that there is an obvious deviation between experiment and advanced three-body QED theory.

In 2012, Amaro et al. [START_REF] Amaro | Absolute Measurement of the Relativistic Magnetic Dipole Transition Energy in Heliumlike Argon[END_REF] measured the magnetic dipole transition of 1s2s 3 S 1 -1s 2 1 S 0 in helium-like argon without any energy reference. The theoretical prediction from Artemyev et al. [START_REF] Artemyev | QED calculation of the n = 1 and n = 2 energy levels in He-like ions[END_REF] is 1.6σ smaller than this experimental value. In 2017, our group [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF] measured the transition of 1s2p 1 P 1 → 1s 2 1 S 0 in He-like argon ion and ransition of 1s2s 2 2p 1 P 1 → 1s 2 2s 2 1 S 0 in Be-like argon ion adopting the same experimental device: a DCS connected to an ECRIS, the "Source d'Ions Multichargés de Paris" (SIMPA) [START_REF] Gumberidze | Electronic temperatures, densities, and plasma xray emission of a 14.5 GHz electron-cyclotron resonance ion source[END_REF], jointly operated by the Laboratoire Kastler Brossel and the Institute des Nanosciences de Paris on the Université Pierre and Marie Curie campus.

For three-electron systems, a notable example, the 2s 1/2 -2p 1/2 transition energy of lithium-like U 91+ was measured about 280.645 [START_REF] Indelicato | QED tests with highly charged ions[END_REF] eV with an 0.005% uncertainty by Beiersdorfer et al. [65]. This result, which is consistant with the theoretical value of 280.71 [START_REF] Johnson | The lamb shift in hydrogen-like atoms, 1≤ Z ≤ 110[END_REF] eV, displays relative QED effects of up to 15% with second-order QED correction of 6%. Our group made reference-free measurements of 1s2s2p 2 P 1/2,3/2 -1s 2 2s 2 S 1/2 transitions in lithium-like sulfur and argon, and of the 1s2s2p 4 P 5/2 -1s 2 2s 2 S 1/2 M2 transition in sulfur using a double flat-crystal spectrometer connected to an ECRIS.

A higher accuracy of 0.6 ppm was obtained in 1s 2 2s 2 2p 2 P 3/2 -1s 2 2s 2 2p 2 P 1/2 transition of boron-like Ar 13+ ions by Mackel et al. [START_REF] Mäckel | Laser spectroscopy on forbidden transitions in trapped highly charged Ar 13+ ions[END_REF]. The theoretical wavelength obtained can be up to 441.261 [START_REF] Epp | Comment on "Testing Three-Body Quantum Electrodynamics with Trapped Ti 20+ Ions: Evidence for a Z-Dependent Divergence Between Experiment and Calculation[END_REF] (nm, air) by Artemyev et al. [START_REF] Artemyev | QED Calculation of the 2 p 3/2-2 p 1/2 Transition Energy in Boronlike Argon[END_REF] and consistent with the experimental value of 441.2559(1) (nm, air), with two orders of magnitude difference in this forbidden transition. This transition determined by QED and relativity effects is considered an excellent candidate for accurate QED tests since the two states, p 1/2 and p 3/2 , are the same in non-relativistic energies. It has recently been confirmed that the demonstration of sympathetic cooling of HCIs can significantly improve the accuracy of the experiment by Schmoger et al. [START_REF] Schmoger | Coulomb crystallization of highly charged ions[END_REF]. The accuracy of both experiment and theory urgently needs to be improved to better test QED effects.

Muonic atoms

In the early days of modern physics, a series of simple atoms or ions composed only of hydrogen and after helium. Today, there are other atoms under investigation, such as the muonic atom, which is the same with hydrogen-like of two-body atoms. The orbital is much closer to the nucleus in the muonic atom than in the conventional atom with electrons, and the muonic levels are more susceptible to the nuclear structure. As a result, the muonic atom provides a unique opportunity to study the nuclear properties with high precision by spectroscopic methods, which is an approved tool for determining the nuclear radius [78].

The transition frequencies and Lamb shift of 2p 1/2 -2s 1/2 in muonic hydrogen was measured with great accuracy by Antognini et al. [78]. The proton structural parameters were determined by the comparison with theoretical predictions. This theory is updated from the 2P -2S Lamb shift in the bound muonic hydrogen by Jentschura [START_REF] Jentschura | Lamb shift in muonic hydrogen-II. Analysis of the discrepancy of theory and experiment[END_REF], using full Coulomb with Uehling potential of vacuum polarization. Indelicato [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF] calculated the contributions to the Lamb shift at n = 2, fine structure interval and hyperfine structure of muonic hydrogen in the framework of nonrelativistic QED. A lot of contributions to the Lamb shift, fine structures, and hyperfine structures of muonic hydrogen have been evaluated in Refs. [START_REF] Pachucki | Theory of the Lamb shift in muonic hydrogen[END_REF][START_REF] Pachucki | Proton structure effects in muonic hydrogen[END_REF][START_REF] Veitia | Nuclear recoil effects in antiprotonic and muonic atoms[END_REF][START_REF] Eides | Theory of light hydrogenlike atoms[END_REF][START_REF] Borie | Lamb shift in muonic hydrogen[END_REF][START_REF] Martynenko | 2 S Hyperfine splitting of muonic hydrogen[END_REF]. Relativistic two-body effects of vacuum polarization shift and the nuclear charge distribution shape are also investigated. These radii play an important role in understanding the bound state QED of the atomic hydrogen spectrum. They also provide the information for testing quantum chromodynamics (QCD) in non-perturbative theory.

The proton radius puzzle has troubled the physics world for more than five years. The puzzles are that there exist 7 standard deviations between the experimental measurements in muonic atoms and normal atoms. This problem has led to abundant research by many groups in the world.

The discrepancy in muonic hydrogen energy of 0.42 meV is far outside the experimental uncertainty of ±0.01 meV. Finding this difference between theoretical calculations and experimental measurements is the main driving force for further development of physics, leading to new physics to solve these deviations. In this sense, the QED method is currently regarded as the most advanced basic theory and serves as a blueprint for other new quantum field theory.

Many theoretical research in heavy muonic atom has been evaluated [START_REF] Michel | Theoretical prediction of the fine and hyperfine structure of heavy muonic atoms[END_REF][START_REF] Dong | Correlation between muonic levels and nuclear structure in muonic atoms[END_REF][START_REF] Johnson | 241am and 243am Charge Distributions from Muonic X-Ray Spectroscopy and the Quadrupole Moment of the 240am Fission Isomer[END_REF][START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF]. Moreover, experimental measurement accuracy is increasing constantly [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF][START_REF] Kessler | Muonic x rays in lead isotopes[END_REF]. QED, recoil and nuclear polarization (NP) corrections limit the accuracy of muonic atom level energies. The calculation of the NP effect is difficult and imprecise. It is well-known to be the origin of the discrepancy between the experimental value and calculated value in the splitting energy of the 2p levels of Pb [START_REF] Yamazaki | Discrepancy between Theory and Experiment in Nuclear Polarization Corrections of Muonic Pb 208[END_REF][START_REF] Rinker | Nuclear polarization in muonic 204, 206, 207, 208Pb in the random-phase approximation[END_REF].

Indelicato [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF] evaluated the contributions of Lamb shift at n = 2 states, 2s hyperfine structure and fine structure interval in muonic hydrogen using accurate Dirac equation. Akihiro Haga et al. [START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF] reanalyzed muonic 90 Zr and 208 Pb atoms on nuclear polarization corrections using the full-electromagnetic nuclear response. They put forward to enhance the transverse nuclear polarization effects and considered newly established pygmy dipole resonances (PDR) employing collective models to reanalyze NP effects.

There are some experimental measurements for high-Z of muonic atoms, but the number of spectra measured is very small. The spectrum of the muonic 208 Pb is relatively complete and accurate. Bergern et al. [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] measured low-lying transition energies of muonic 208 Pb with a precision up to 11×10 -6 using a Compton suppression spectrometer. There is ongoing experiment to perform muonic atom spectroscopy aiming at the precise measurement of level energies of radioactive 226 Ra with 0.2% relative precision at the Paul Scherrer Institute (PSI) [START_REF] Antognini | Measurement of the charge radius of radium (Proposal for BVR 47)[END_REF].

Overview of methods

In order to achieve high precision calculation of atomic states, electron-electron correlation and QED contributions have been the focus of theoretical investigations. These effects can be evaluated using many methods, including the multiconfiguration Dirac-Fock method (MCDF), the relativistic configuration-interaction (RCI) method, relativistic many-body perturbation theory (RMBPT), the relativistic coupled-cluster method, S-Matrix method, and covariant-evolution-operator (CEO) method.

For establishing the correct relationship between multibody problems and QED method, it is necessary to begin with a no-pair Hamiltonian.

MCDF

Dirac-Fock approximation, verified as a natural and well-defined multi-body computing starting point, is based on the relativistic equivalent of the Hartree-Fock. Electrons are in the independent particle approximation, and their wave functions are treated in the Coulomb field of the nucleus and the spherical mean-field from the electrons. This method is a powerfully modern tool for relativistic atomic calculations based on the Dirac-Fock Hamiltonian and the potential approach. The calculations can be divided into two important parts. One is the Slater determinant computing all angular integrals including one-electron integrals, Breit operator, and Coulomb interaction. Another is the radial part being responsible for solving the given MCDF equations and calculating the large and small component of the wavefunctions to probe atomic properties. The foundational idea of this method is that wavefunctions are used to represent the atomic states. These functions are linear combinations of a number of configuration state functions, which are sums of products of single-electron Dirac orbitals. This method is developed by Grant and co-workers [START_REF] Grant | Relativistic Calculation of Atomic Structures[END_REF][START_REF] Dyall | GRASP: A general-purpose relativistic atomic structure program[END_REF], Desclaux [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF], Gorceix and Indelicato [START_REF] Gorceix | Effect of the complete Breit interaction on twoelectron ion energy levels[END_REF].

RMBPT

In relativistic many-body perturbation theory [START_REF] Lindgren | Atomic many-body theory[END_REF][START_REF] Shavitt | Many-body methods in chemistry and physics: MBPT and coupled-cluster theory[END_REF], the Coulomb term is replaced by a model of one-electron localized potential. This provides a complete set of singleelectron eigenfunctions, including continuous eigenfunctions. Then, Breit interaction and the difference between the Coulomb term and the model potential are treated as a perturbation expansion. This method is very efficient in considering virtual core excitations.

RCI

In relativistic configuration-interaction (RCI) method [START_REF] Kagawa | Relativistic configuration-interaction theory for atomic systems[END_REF], the total CI wavefunction Ψ(γJP ) can be written in a linear combination of configuration state functions(CSF) as

Ψ(JP M ) = n c n Φ n (γ n JP M ), (1.1)
with a total parity P , angular momentum J and angular momentum projection M . Here Φ(γ n JP M ) is a CSF and c n is an expansion coefficient. The matrix elements of the DCB Hamiltonian between the CSFs,

{H ns } ≡ { γ n P JM |H DCB | γ s P JM } . (1.2)
The RCI method deals with the secular equation for a Hamiltonian matrix with respect to CSF's. This method is very successful for the explicit treatment of valence-valence interactions. This method is often used in conjunction with other methods.

Relativistic coupled-cluster

The first few terms of relativistic many-body perturbation series are sufficient for highly charged ions but are less suitable for neutral atoms, where higher orders of corrections are not negligible. It is more complex for higher-order expressions. The direct perturbation studies rarely go beyond second-order corrections or the third-order energy correction. The most promising of methods is the coupled-cluster (CC) formalism [START_REF] Shavitt | Many-body methods in chemistry and physics: MBPT and coupled-cluster theory[END_REF]. This theory takes into account correlations to all orders as a perturbation in every level of particle-hole excitation. Size extensivity is important for accurate calculations of heavy elements with significant relativistic effects. This method has high-accuracy electron correlation and is widely used in non-relativistic atomic and molecular calculations [START_REF] Eliav | Open-shell relativistic coupled-cluster method with Dirac-Fock-Breit wave functions: Energies of the gold atom and its cation[END_REF].

S-Matrix

In S-matrix theory, developed by Sucher [START_REF] Sucher | S-matrix formalism for level-shift calculations[END_REF] and used by Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF], the energy levels of atoms or ions are related to matrix elements of an operator which evolves the atom or ion from t = -∞ to t = ∞,

S ,λ = T e -iλ dte -|t| H(t) , (1.3) through E = E 0 + ∆E, where ∆E = lim →0 i 2 lim λ→1 ∂ ∂λ ln Φ 0 |S ,λ | Φ 0 , (1.4)
and where E 0 represents the energy of the atom at times t = -∞ and t = ∞. The interaction Hamiltonian is suppressed by the adiabatic damping factor . For the Li-like isoelectronic sequence we are treating here, the state |Φ 0 can be represented by,

|Φ 0 = a † ν |0 c , (1.5)
where ν is a valence electron and |0 c is a filled helium-like core. The wave functions obeying the Dirac equation can be written as,

c α • p + (β -1)mc 2 + U (r) Ψ n ( x) = n Ψ n ( x). (1.6)
in a spherically symmetric potential U (r).

CEO method

One of the main problems in extending the energy-dependent perturbation theory to include electron correlation is that most methods have structures that are completely different from the energy-independent perturbation theory, which makes it difficult to exploit the latter's sophisticated methods. An available method of the CEO method developed by Lindgren Salomonson and coworkers [START_REF] Lindgren | The covariant-evolution-operator method in bound-state QED[END_REF][START_REF] Lindgren | Many-body procedure for energy-dependent perturbation: Merging many-body perturbation theory with QED[END_REF][START_REF] Lindgren | Combining Many-Body Perturbation and Quantum Electrodynamics[END_REF], has a structure which is quite similar to standard energy independent MBPT. In this way, the electron correlation can be processed for the first time as an energy-dependent interaction of any order of QED type. CEO, which represents the time evolution of the relativistic wave function or state vector, can be constructed.

Chapter 2

Context and principle

The goal of this chapter is to give a general introduction to the theoretical context of our calculations. A muon can be regarded as an electron with heavy mass, so in this chapter, we only introduce the case of electronic ions and the corresponding corrections. I will present the one-loop and two-loop QED corrections and the calculations of transition energies in one and three electron ions. We also present the improvements in vacuum polarization to all orders and an effective operator approach for self-energy screening corrections.

MCDF-procedure

The MCDF method [START_REF] Grant | Relativistic Calculation of Atomic Structures[END_REF][START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Gorceix | Effect of the complete Breit interaction on twoelectron ion energy levels[END_REF][START_REF] Dyall | GRASP: A general-purpose relativistic atomic structure program[END_REF] is intended to provide an approximate solution to the relativistic multibody problem, beyond the single-particle approximation. It is introduced as a direct extension of the Hartree-Fock method. This method is applied in our MCDFGME program by Desclaux [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF] and developed by Indelicato [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Indelicato | Projection operators in Multiconfiguration Dirac-Fock calculations. Application to the ground state of heliumlike ions[END_REF][START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF].

Dirac equations

For an N-electron atom, the relativistic total Dirac-Coulomb-Breit (DCB) Hamiltonian is approximated by

H = i H D (i) + i<j 1 r ij + H B (ij) , ( 2.1)
Here i, j = 1, . . . , N are the numbers of electrons, H D (i) is the Dirac Hamiltonian of one-particle,

H D (i) = cα α α i • p p p i + (β -1)mc 2 + V N (r) , ( 2.2)
Here, we make use of atomic units( =e=m=1;c = 1/α). p p p i represents the momentum operator, and α α α and β are the Dirac 4 × 4 matrices,

α α α i = 0 σ σ σ i σ σ σ i 0 , β = I 0 0 -I , ( 2.3)
where σ i represents the 2 × 2 Pauli matrices. I is the second-order unit matrix. V N (r) represents the nucleus Coulomb potential. This results in the Coulomb field being included in all orders when doing the evaluation of relevant quantities. The two-body part 1 r ij of Eq. (2.1) represents Coulomb repulsion, H B is the Breit parts (magnetic and retardation interaction) of the electron-electron interaction (we will introduce this part in section 2.3 below).

When evaluating the Dirac equation with finite-size nuclear corrections, QED effects and recoil effects, the energy of a given level is written [START_REF] Indelicato | QED tests with highly charged ions[END_REF]

E n,κ (Z, A) = E D n,κ (Z) + E (1) QED (n, κ, Z) + E (2) QED (n, κ, Z) + E N uc. n,κ (Z, A) + E Rec. n,κ (Z, M A ) = E D n,κ (Z) + α π (Zα) 4 n 3 F (1) n,κ (Zα) + α π (Zα) 4 n 3 F (2) n,κ (Zα) + E N uc. n,κ (Z, M A ) + E Rec. n,κ (Z, M A ) (2.4)
where E D n,κ (Z) represents the solution of the Dirac equation, E

QED (n, κ, Z) the first order of self-energy corrections and vacuum polarization corrections, and E

(2) QED (n, κ, Z) the summation of two-loop corrections. The finite nuclear correction is represented by

E N uc.
n,κ (Z, A), which can include nuclear polarization. The remaining correction E Rec. n,κ (Z, M A ) is the recoil effects. These contributions will be introduced separately.

Wave functions

The total wavefunction of N-electron can be obtained by, HΨ Π,J,M (r 1 , . . . , r m ) = E Π,J,M Ψ Π,J,M (r 1 , . . . , r m ) ,

(2.5)

The MCDF method is defined by the specific choice of function to solve Eq. (2.5) with the parity Π, total angular momentum J, and its projection M on the z axis of J z as a linear combination of configuration-state functions (CSFs),

|Ψ Π,J,M = N CF ν=1 c ν |νΠJM , ( 2.6)
where N CF is the number of configurations and c ν are the configurations mixing coefficient. The label ν is all other values that are explicitly required to define CSF . As with many self-consistent field methods, the starting point for constructing an N-electron wave function is the central field of single electron orbital. In the relativistic case, the spin-orbital coupling is explicitly introduced in the Dirac Hamiltonian, so each Dirac four-spinor can be expressed as,

Φ nκm (r, θ, ϕ) = 1 r P nκ (r) χ κm (θ, ϕ) iQ nκ (r) χ -κm (θ, ϕ) . (2.7)
These spinors are simultaneous eigenfunctions of the parity operator Π, the total angular momentum operator J 2 J 2 J 2 and its z-component. P nκ (r) and Q nκ (r) are large and small radial components of the wave function respectively, n is the principal quantum number, and the quantum number kappa is defined by,

κ = -l -1 if j = l + 1/2 l if j = l -1/2 . (2.8)
In this expression, l represents the quantum number of the orbital and the total angular momentum j is related of

j = |κ| -1/2.
(2.9)

The functions χ κm in Eq. (2.7) are the two component Pauli spherical spinors,

χ -κm (θ, ϕ) = σ=±1/2 lm -σσ|l 1/2 jm Y m-σ l (θ, ϕ) φ σ , ( 2.10)
which include a spherical harmonic,

φ 1/2 = 1 0 , φ -1/2 = 0 1 . (2.11)
For a N-electron system, a CSF is a linear combinations of Slater determinants of Dirac four-spinors,

|νΠJM = i=1 d i Φ i 1 (r 1 ) • • • Φ i N (r 1 ) . . . . . . . . . Φ i 1 (r N ) • • • Φ i N (r N )
.

(2.12)

The integro-differential equation can be reduced to [START_REF] Desclaux | A relativistic multiconfiguration Dirac-Fock package[END_REF]

d dr + κ i r -2 α + αV i (r) -αV i (r) d dr -κ i r P i (r) Q i (r) = α j λ i,j Q j (r) -P j (r) + X i Q (r) X i P (r) (2.13)
in the spherically symmetric potential, the sum of the direct Dirac-Fock potential and the nuclear potential represented by V i (r), and exchange potentials represented by X i Q (r) , X i P (r) . The Lagrange parameter is λ i,j .

Numerical methods

Most numerical methods for solving relativistic Hartree-Fock equations are derived from long-known non-relativistic cases [START_REF] Fischer | The Hartree-Fock Method for Atoms[END_REF]. Integral differential equations can be solved using an iterative process. Each step reduces the integral differential equation to a simple differential equation by considering the direct and exchange potential term as a given source function, which is calculated by the wave function obtained in the previous step. The iterative process then continues until a given precision is achieved between two successive iterations.

The pair of two first-order coupled differential equations in Eq.(2.13) is solved by a five-point predictor-corrector method [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Desclaux | A relativistic multiconfiguration Dirac-Fock package[END_REF][START_REF] Desclaux | Computational approaches of relativistic models in quantum chemistry[END_REF] for the one body system and by a mixed predictor-corrector and finite difference method for the many-electron system. A linear mesh step in the variable t is defined by,

t n = ln r n r 0 + ar n . (2.14)
with t n = t 0 + nh, and the first point of the mesh gives t 0 = ar 0 , corresponding to n = 0.

The expression can also be inverted by inverting equation to obtain,

r n = W ar 0 e tn a ,
dr n dt n = W ar 0 e tn a [1 + W (ar 0 e tn )]
.

(2.15)

This yield the defining equation for the W function of f (W ) = W (z)e W (z) , which represents the Lambert or the product logarithm function. The differential equation and wave function between 0 and r 0 are represented by a ten-term series expansion. Usually the first point is decided as r 0 = 10 -2 /Z and h = 0.025. We use the values as low as r 0 = 10 -7 /Z and h = 0.002 to get the best accuracy for muonic atoms. The first-order contribution to the eigenvalue is given by a mean value of an operator O [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF],

∆E O = ∞ 0 dr P (r) 2 + Q (r) 2 O(r) = r 0 0 dr P (r) 2 + Q (r) 2 O(r) + ∞ r 0 dt dr dt P (r) 2 + Q (r) 2 O(r).
(2.16)

This method has the advantage of being fast and accurate.

Nuclear models

Several possible models are provided for the nuclear potential. One is that the nucleus is considered to be a point charge. When it is mandatory to go beyond the point nucleus approximation for the inner shell of heavy atoms, a finite nuclear charge distribution model should be used instead of a pointlike nucleus. For example, a uniform spherical model with radius R can be assumed as the nuclear charge distribution. The nuclear potential is given by [START_REF] Desclaux | A relativistic multiconfiguration Dirac-Fock package[END_REF] V nuc (r) = -eZ

2R 0 3 -r 2 R 2 0 if r ≤ R 0 -eZ r if r > R 0 (2.17)
In the present work, a realistic description of nucleus charge distribution is given by the two-parameter Fermi model,

ρ N (R) = ρ 0 (1 + exp[(R -c)/a]) -1 , ( 2.18)
with normalization(uniform nuclear charge density),

ρ 0 = 3Ze 4πR 2 0 , ( 2.19)
where the half-density radius is c, and R 0 is the radius of the uniform model having the same rms radius as the Fermi model,

R 0 = 5 3 R 2 , (2.20)
Here t is the skin-thickness parameter, which indicates the distance over which the density falls from 90% to 10% of its central value. It is related to the Fermi parameters by the expression of t = 4a ln 3.

(2.21)

The t = 2.30 fm value is used in our calculations, and c is calculated by the formulas given in Ref. [START_REF] Parpia | Relativistic basis-set calculations for atoms with Fermi nuclei[END_REF]. The nuclear radius of all elements come from experimental values. Since the detailed charge distribution is a key parameter for very heavy atoms, the default option for our calculations is to use a uniform charged sphere for atomic numbers below 45 and the Fermi distribution for Z > 45.

To the lowest order, the nucleus finite mass effect to the energy can be roughly corrected by reducing the mass. The reduced mass can be expressed by the following formula,

µ = m e M nuc m e + M nuc . (2.22)
where M nuc is the mass of the nucleus.

Electron-electron interaction

The effective electron-electron interaction operator is derived from the Feynman diagram in Fig. 2.1 based on a single photon exchange approximation. This operator is gauge dependent, and the contribution to the energy is also gauge dependent [START_REF] Gorceix | Effect of the complete Breit interaction on twoelectron ion energy levels[END_REF]. The Coulomb gauge should be used to avoid introducing spurious effects.

Here, we will describe the three common parts of electron-electron interactions, namely Coulomb, magnetic and delayed interactions. The corresponding interactions are shown in Fig. 2.1(b), 2.1(c) and 2.1(d), respectively. The expression representing the operator of the interaction between electrons i and j in the Coulomb gauge [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] (formulas are given in atomic units) is

g ij = g Coul ij + g M ag ij + g Ret ij , (2.23)
where the Coulomb interaction is

g Coul ij = 1 r ij , ( 2.24)
The magnetic (Gaunt) interaction is

g M ag ij = - α 1 • α 2 α 1 • α 2 α 1 • α 2 r ij , ( 2.25)
The retardation operator is

g Ret ij = -α 1 • α 2 α 1 • α 2 α 1 • α 2 cos (ω ij r ij ) -1 r ij + (α 1 • ∂ 1 α 1 • ∂ 1 α 1 • ∂ 1 ) (α 2 • ∂ 2 α 2 • ∂ 2 α 2 • ∂ 2 ) cos (ω ij r ij ) -1 ω 2 ij r ij . (2.26)
The Breit operator is

H B (ij) = - α 1 • α 2 α 1 • α 2 α 1 • α 2 r ij cos (ω ij r ij ) + (α 1 • ∂ 1 α 1 • ∂ 1 α 1 • ∂ 1 ) (α 2 • ∂ 2 α 2 • ∂ 2 α 2 • ∂ 2 ) cos (ω ij r ij ) -1 ω 2 ij r ij . (2.27)
including the magnetic and the retardation interaction due to the finite value of the speed of light. The value of ω ij represents the photon exchanged energy, α i α i α i are the Dirac matrices and r ij represents the interelectronic distance. The zero-order energy is calculated by the Coulomb operator and a single configuration of the expansion from the wave function, which leads to a Coulomb contribution. We can obtain the first order of the magnetic and the retarded corrections by adopting a single-configuration wavefunction.

One difficulty is the gauge dependent of the electron-electron interaction. When the electron-electron operator is solved with the Dirac equation, the gauge invariance is lost. In QED calculations, the diagrams provide gauge-invariant results. Another difficulty is that the value of ω ij can be well defined in the independent particle approximation. However, it can't decide what value to be used when evaluating the interaction operator between the correlation orbitals. Therefore there is always a problem of using the magnetic and retardation operator to evaluate the correlation energy. Our MCDFGME code allows this, which can result in a very large contribution to the correlation energy at heavy elements. This method is described by Indelicato [START_REF] Indelicato | Multiconfiguration Dirac-Fock calculations of transition energies in two electron ions with 10 ≤ Z ≤ 92[END_REF]. The correlation energies are expanded in both expansions of 1/Z and Zα [START_REF] Layzer | Relativistic Z-dependent theory of many-electron atoms[END_REF], and the nonrelativistic correlation energy is expressed as [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] ∆E

N R corr = ∆E 0 + ∆E 1 1 Z + ∆E 2 1 Z 2 + ∆E 3 1 Z 3 + • • • . (2.28)
The exact nonrelativistic calculations for the ∆E 0 term come from Horak et al. [START_REF] Horak | Correlation Energies of the 1s 2 3l 2 L States of the Lithium Sequence[END_REF]. The higher-order terms contribution, which is much smaller, is fitted to the nonrelativistic limit of our MCDF calculations. This method is used for the calculations.

QED Corrections

Quantum Electrodynamics (QED) describes all possible events related to charged particles as the relativistic quantum field theory of electromagnetic forces. The model of the QED operator is divided into three parts,

V QED = V SE + V U ehl + V W K .
(2.29)

Here, V SE is the self-energy operator, V W K is the Wichmann-Kroll part of vacuum polarization and V U ehl is the Uëhling part of the vacuum polarization. The two parts of Uëhling and Wichmann-Kroll are local potentials. Therefore their prescription is quite forthright.

In low Z systems, the QED contributions are calculated according to the expansion in two small parameters α and Zα. The parameter Zα is not small for high-Z systems. Therefore, the calculations based on the parameter Zα cannot be used as an expansion and should be calculated to all orders in Zα. The developments of nonperturbative QED methods are performed by Indelicato [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF].

One-loop QED correction

Fig. 2.2 depicts the first-order (one-loop) radiative corrections in α by Feynman diagrams. In these diagrams, the double lines signify an electron propagating in the external Coulomb field of the nucleus, and the wavy line is the photon propagator. Part (a) is the so-called self-energy, in which a photon is emitted and absorbed again by the bound electron. Part (b) indicates the vacuum polarization in which the photon mediating the interaction between the bound electron and the nucleus creates an electron-positron pair. The virtual electron-positron pair causes a change of the Coulomb potential and thus leads to energy shifts of the bound electrons. The one-loop self-energy contribution can be expressed as

∆E SE = µ m e 3 α π (Zα) 4 n 3 F (Zα) m e c 2 , (2.30)
where F (Zα) is the dimensionless function,

F (Zα) =A 41 ln(Zα) -2 + A 40 + A 50 (Zα) + A 62 (Zα) 2 ln 2 (Zα) -2 + A 61 (Zα) 2 ln(Zα) -2 + G SE (Zα)(Zα) 2 .
(2.31)

where G SE (Zα) contains all remainder terms of higher-order expansion in Zα and the values of the coefficients A ij have been listed in Ref. [START_REF] Mohr | CODATA recommended values of the fundamental physical constants: 2014[END_REF]. These diagrams must be calculated to all orders in Zα for highly charged ions. Desiderio and Johnson [START_REF] Desiderio | Lamb shift and binding energies of K electrons in heavy atoms[END_REF] first performed the nonperturbative calculation to the self-energy contribution. The first high-precision evaluation to the self-energy correction is due to Mohr [START_REF] Mohr | Numerical Evaluation of the 1s1/2 State Radiative Level Shift[END_REF] for the 1s level at medium and high-Z. Jentschura et al. [START_REF] Jentschura | Calculation of the electron selfenergy for low nuclear charge[END_REF][START_REF] Jentschura | Electron self-energy for the K and L shells at low nuclear charge[END_REF] performed all-order calculations at low-Z, which are very difficult, because the correction of E (1) QED (n, κ, Z) from QED is formally of order α π mc 2 and so terms of order 1, Zα, (Zα) 2 and (Zα) 3 have to be cancelled, requiring very large accuracy. Indelicato P [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF] evaluate the all order vacuum polarization contribution by numerical solution from the Dirac equation. Fig. 2.3 gives Feynman diagrams related to the full vacuum polarization correction. The double line represents the wavefunction in the external Coulomb field of the nucleus, and the wavy line corresponds to the Coulomb photon propagator. The single circle indicates a free electron-positron pair. The gray circle is the interaction with the nucleus. The nucleus, with a charge Ze, is considered to be a static point particle.

The Uëhling part play an important role in the one-loop vacuum polarization correction [START_REF] Uehling | Polarization effects in the positron theory[END_REF]. The Uëhling approximation considers the virtual e + e -pair to leading order of the coupling constant Zα. The Uëhling potential, which corresponds to Fig. 2.3(a) and represents a leading contribution in the QED, is given by

V 11 (r) = - 2αZα 3mr ∞ 0 dr r ρ(r ) K 0 2m r -r -K 0 2m r + r , ( 2.32)
where

K 0 (x) = ∞ 1 dte -xt 1 t 3 + 1 2t 5 t 2 -1. (2.33)
is a modified Bessel function. When deducing the Uëhling potential, it is assumed that the virtual electrons and the positrons are free to propagate. The remaining part is called the Wichmann-Kroll corrections, which include all terms of higher-order Zα, ∼ (Zα) n with n ≥ 3. Fig. 2.3(b) is higher orders of the Wichmann and Kroll corrections, which represent the main order effect of the distortion of the electron and positron wavefunctions in the nuclear Coulomb field. The explicit expression for the α(Zα) 3 term by Wichmann and Kroll [START_REF] Wichmann | Vacuum polarization in a strong Coulomb field[END_REF] is,

V 13 (r) = α(Zα) 3 πr ∞ 0 dt e -2tr 1 t 4 - 1 12 π 2 t 2 -1 Θ(t -1) + t 0 dx t 2 -x 2 f (x) .
(2.34) in the Laplace transform of the charge density. For all r, V 13 (r) is repulsive, and the energy shift is positive, thereby reducing the binding due to the attractive Coulomb and Uëhling potentials. The evaluation of the Wichmann-Kroll contribution is not an easy task. The first nonperturbative calculations of the Wichmann-Kroll part were calculated by Soff and Mohr [START_REF] Soff | Vacuum polarization in a strong external field[END_REF] to all orders in Zα. The most accurate results of the vacuum-polarization diagram were obtained in Ref. [START_REF] Sapirstein | Vacuum polarization calculations for hydrogenlike and alkali-metal-like ions[END_REF].

V 21 (r) = - α 2 Zα mπr ∞ 0 dr r ρ(r ) L 0 2m r -r -L 0 2m r + r , ( 2.35)
for a spherically symmetric nuclear charge. The series expansion for small r was provided by Blomqvist [START_REF] Blomqvist | Vacuum polarization in exotic atoms[END_REF] and Mohr PJ et al. [START_REF] Mohr | QED Corrections in Heavy Atoms[END_REF] to evaluate the function L 0 with very good precision.

The term named "VP iteration" correspond to Fig. 2.5, which can be expressed by ( α π ) 2 (Zα) 2 and ( α π ) 3 (Zα) 2 respectively. The Uëhling potential can be easily put in the Dirac Eq. (2.13) when it is solved numerically. This is equivalent to getting an accurate solution by inserting any number of vacuum polarization. Since the vacuum polarization is included in the Dirac equation potential, the calculation of all the energy performed by using the numerical wavefunction as a disturbance includes the contribution of the higher-order. This represents that a wavefunction with a bound propagator with one or more vacuum polarization insertions can replace the original wave function. For instance, when the Källén and Sabry corrections are evaluated in this way, they include corrections of the type shown in Fig. 2.6. These corrections are part of the three-loop corrections from [START_REF] Ivanov | Second-order corrections to the wave function at the origin in muonic hydrogen and pionium[END_REF].

Two-loop QED correction

The two-loop QED effect is represented by

∆E QED2 = µ m e 3 α π 2 (Zα) 4 n 3 F QED2 (Zα) m e c 2 , ( 2.36)
The dimensionless function F QED2 (Zα) is represented as an expansion in the term of Zα and ln(Zα),

F QED2 (Zα) =B 40 + B 50 (Zα) + B 63 (Zα) 2 ln 3 (Zα) -2 + B 62 (Zα) 2 ln 2 (Zα) -2 + B 61 (Zα) 2 ln(Zα) -2 + B 60 (Zα) 2 + • • • .
(2.37)

E QED2 = E SESE + E V P V P + E SEV P + E S(V P )E + E KS , ( 2.38)
where the two-loop self-energy contribution E SESE is shown in Figs. 2.7(a)-2.7(c). Two kinds of Green function methods [START_REF] Shabaev | Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms[END_REF] can easily derive the corresponding formal expressions. The two-loop self-energy is taken from Refs. [START_REF] Yerokhin | Two-loop self-energy in the Lamb shift of the ground and excited states of hydrogenlike ions[END_REF][START_REF] Yerokhin | Two-loop self-energy correction in high-Z hydrogenlike ions[END_REF][START_REF] Yerokhin | Evaluation of the twoloop self-energy correction to the ground state energy of H-like ions to all orders in Z alpha[END_REF][START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF][START_REF] Yerokhin | Two-loop QED corrections in few-electron ions[END_REF][START_REF] Yerokhin | Two-loop QED corrections with closed fermion loops[END_REF][START_REF] Yerokhin | Two-loop self-energy for the ground state of medium-Z hydrogenlike ions[END_REF][START_REF] Yerokhin | The two-loop self-energy: diagrams in the coordinatemomentum representation[END_REF].

Ones have performed a complete evaluation of the two-loop self-energy contribution in the parameter Zα for ground states and excited states of hydrogenlike ions. Since this correction is a major factor in the uncertainty of the theoretical ground-state Lamb shift in these systems, numerical accuracy is increased by an order of magnitude. At the same time, the improvements in extension to lower Z has also been performed. The crossed and mixed diagrams of SEVP and S(VP)E shown in Figs. 2.7(d)-2.7(f) are obtained from Ref. [START_REF] Yerokhin | Two-loop QED corrections with closed fermion loops[END_REF]. In view of the first-order perturbation of vacuum polarization potential, the self-energy can be modified by the vacuum polarization in the Coulomb potential, where the SEVP correction can be obtained. It is the summation of the irreducible contributions, the reducible contributions, and the vertex contributions, which are given by perturbations of the reference-state wave functions, the binding energy, and the electron propagator, respectively. The insertion of vacuum polarization into the selfenergy photon line is called S(VP)E correction, where the leading part of Zα is obtained under the free-loop approximation. Since it is a difficult problem to perform all-order calculations of S(VP)E correction beyond the free-loop approximation, this correction is just within the free-loop approximation.

The two-loop vacuum-polarization corrections are obtained from Refs. [START_REF] Yerokhin | Two-loop QED corrections with closed fermion loops[END_REF][START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF], represented by the diagrams in Figs. 2.7(g)-2.7(i). In the evaluation, it can be split into two parts: the loop-after loop vacuum-polarization correction in Figs. 2.7(g), and Källén and Sabry correction in Figs. 2.7(h)-2.7(i). The VPVP correction is considered as the second-order correction of the one-loop VP potential of U V P ,

∆E V P V P,g = n =a a |U V P | n n |U V P | a ε n -ε a . (2.39)
The numerical calculation of the VPVP contribution is relatively simple. This can be performed by using a general method for solving VP potential. The sum of the Dirac spectrum is carried out by the method of dual-kinetic-balance basis set [START_REF] Eides | Coulomb line vacuum polarization corrections to Lamb shift of order α2 (Zα) 5m[END_REF]. The correction of the higher-order part (Zα) 5 can be identified by considering the (Zα) expansion. The correction of Källén and Sabry is written as

∆E KS V P V P,hi = a |V KS | a . (2.40)
in the free-loop approximation. The Källén and Sabry potential is given by the reference [START_REF] Fullerton | Accurate and efficient methods for the evaluation of vacuum-polarization potentials of order Z α and Z α 2[END_REF] in a spherically symmetric nuclear charge distribution.

The calculations of complete non-perturbative processes described by these Feynman diagrams are an extremely difficult task due to many complicated integrals involved in these terms. The leading order of the three-loop QED correction is known in the Zα expansion as α 3 (Zα) 4 π 3 n 3 . These contributions are very small and are only associated with very low Z ions. When performing calculations for atoms with more than three electrons, additional diagrams of three-body QED corrections, represented in Fig. 2.8, should be taken into account [START_REF] Mittleman | Structure of heavy atoms: Three-body potentials[END_REF].

Screened QED corrections

For few-electron atoms or ions, the self-energy correction and vacuum-polarization correction are disturbed by an external potential, and this perturbation is caused by electronelectron interaction. We call this perturbation screened QED correlation. The self-energy screening and vacuum-polarization screening corrections for a two-electron system are depicted in Fig. 2.9. Fig. 2.9 Feynman representing the self-energy screening diagrams and vacuumpolarization screening diagrams.

The screening effect is considered to be a small change in the external potential relative to the total Coulomb potential [START_REF] Indelicato | Coordinate-space approach to the bound-electron self-energy: Self-Energy screening calculation[END_REF],

V (x) = V C (x) + δV (x).
(2.41)

Then, the perturbation theory methods may be extended to employ an effective screened potential which replaces the Coulomb one for few-electron ions. However, this method is too complicated to be directly included in the Dirac-Coulomb-Breit calculations. In the past, these corrections have been evaluated using an effective-Z parameter, which was derived by comparing the mean radius of the orbital with the radius of the hydrogenlike orbital in the same n, κ.

a) Welton picture

Indelicato et al. [START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] proposed to use the Welton approximation [START_REF] Welton | Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field[END_REF], which can correctly represent the lowest order of the self-energy. Then a more general scheme where the screening electron effect is considered as a first-order perturbation to the self-energy by an effective potential, has been proposed to evaluate all possible self-energy screening contributions between 1s 1/2 , 2s 1/2 , 2p 1/2 and 2p 3/2 states by Indelicato and Mohr [START_REF] Indelicato | Coordinate-space approach to the bound-electron self-energy: Self-Energy screening calculation[END_REF] using the method proposed by Indelicato and Mohr [START_REF] Indelicato | Quantum electrodynamic effects in atomic structure[END_REF].

An effective potential must be used to correct the lowest order part of the one-electron self-energy in Zα for the two-electron effects by correcting the changes of electronic density. This potential can be derived by using a more physical prescription based upon Welton's [START_REF] Welton | Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field[END_REF] semiclassical arguments. In the Welton picture, self-energy is a perturbation in the classical trajectory of electrons due to fluctuations in the vacuum electromagnetic field. Mohr et al. [START_REF] Mohr | Nuclear Size Correction to the Electron Self-Energy[END_REF] had calculated exactly the hydrogenic self-energy corrected for finite nuclear size. This approximation for screened self-energy has been applied to the two or three electron systems.

For s orbitals, the self-energy screening correction deduced from Mohr's results leads to a relation [START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF],

δE ns SE = ns |∆U n (r)| ns DF ns |∆U n (r)| ns Hyd E Hyd SE,ns , (2.42)
Here, the subscript DF represents Dirac-Fock, Hyd stands for hydrogenlike wave functions, |ns is a radical solution of the Dirac-Fock equation, U n is the nuclear potential, and E Hyd SE,ns is the self-energy of one-particle. For p, d, . . . orbitals, the screening correction proportional to the square of the wave function is canceled at the origin, and the g-2 diagram provides the leading correction

δE nl≥3 SE = nl ≥ 3 |βα • E α • E α • E| nl ≥ 3 DF nl ≥ 3 |βα • E α • E α • E| nl ≥ 3 Hyd E Hyd SE,nl≥3 . (2.43)
In the above expression, E represents the nuclear electric field.

b) Model operator approach

The true self-energy operator being not local, it cannot be well approximated by a local potential. Shabaev and Tupitsyn [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF] evaluated the screened self-energy corrections based on the self-energy operator model method. In this approach, the screened selfenergy contributions were estimated by evaluating the total energy together with the effective operator contained in the Dirac-Fock or the Kohn-Sham equation and subtracting the related energy evaluated without this model operator and the self-energy contribution performed with the H-like wave functions. Shabaev et al. [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF][START_REF] Shabaev | QEDMOD: Fortran program for calculating the model Lamb-shift operator[END_REF] approximated the QED operator, which are easily included in any calculations based on the Dirac-Coulomb-Breit equation, using a sum of short-range local and nonlocal potentials. The effective self-energy operator V SE in Eq. (2.29) can be divided into two parts, the local part V SE loc and the nonlocal part V SE nloc ,

V SE = V SE loc + V SE nloc , (2.44)
where V SE loc is written as

V SE loc = κ V κ (r) Pκ . (2.45)
Here V κ (r) represents the radial part, and Pκ stands for the projector operator. The nonlocal operator V SE nloc , which approximates the part of the exact self-energy operator, can be defined by

V SE nloc = n i,j=1 |φ i D ij φ j | , ( 2.46)
where the matrix D is expressed as

D ij = n κ,l=1 (S t ) -1 iκ ψ κ | 1 2 (ε κ ) + (ε l ) -V SE loc |ψ l (S -1 ) lj . (2.47)
Here ψ i (r) is the hydrogen-like wavefunction, and φ i (r) is the model of projected basis functions. S iκ stands for the overlap matrix S iκ = φ i |ψ κ .

Relativistic recoil

The recoil parts and mass-dependent are usually composed of normal and specific mass shifts. This mass shift was first evaluated by Hughes and Eckart [START_REF] Hughes | The effect of the motion of the nucleus on the spectra of Li I and Li II[END_REF] for multi-electron atoms. The normal-mass-shift operator is given by

H RN M S = 1 2M i p p p 2 i - αZ r i α α α i + (α α α i • r r r i )r r r i r 2 i • p p p i , ( 2.48)
an the specific-mass-shift operator can be written as

H RSM S = 1 2M i =j p p p i • p p p j - αZ r i α α α i + (α α α i • r r r i )r r r i r 2 i • p p p j , (2.49)
The specific-mass part gives a small contribution, so it can be treated in perturbation. The total recoil operator is defined by the sum of Eq. (2.48) and (2.49), which can be deduced from the recoil Hamiltonian [START_REF] Shabaev | Relativistic nuclear recoil corrections to the energy levels of multicharged ions[END_REF][START_REF] Shabaev | QED theory of the nuclear recoil effect in atoms[END_REF][START_REF] Li | Mass-and field-shift isotope parameters for the 2 s-2 p resonance doublet of lithiumlike ions[END_REF][START_REF] Gaidamauskas | Tensorial form and matrix elements of the relativistic nuclear recoil operator[END_REF],

H RM S = 1 2M i,j p p p i • p p p j - αZ r i α α α i + (α α α i • r r r i )r r r i r 2 i • p p p j . (2.50)
Here M represents the nuclear mass, the momentum operator is described by p p p i , and α α α i is the Dirac matrix operator. The recoil correction is the expectation value of the recoil Hamiltonian on the Dirac wave function. The recoil correction is usually expressed by the form αm(Zα) n (m/M ) k . The higher-order relativistic corrections have been calculated by V. Shabaev [START_REF] Shabaev | QED theory of the nuclear recoil effect in atoms[END_REF]. For hydrogenlike atoms, Artemyev et al. [START_REF] Artemyev | Relativistic nuclear recoil corrections to the energy levels of hydrogenlike and high-Z lithiumlike atoms in all orders in αZ[END_REF] finally wrote the recoil contribution as, ∆E rr = ∆E (1) + ∆E (2) .

(2.51)

with

∆E (1) = m M (αZ) 2 2N 2 mc 2 , (2.52)
and

∆E (2) = m M (αZ) 5 πn 3 P (αZ)mc 2 . (2.53)
where,

N = n 2 -2(n -|κ|)(|κ| -γ), γ = κ 2 -(αZ) 2 , κ = (-1) j-l+1/2 (j + 1 2
).

(2.54)

Eq. (2.51) corresponds to the diagrams in Fig. 2.10. The function P (αZ) has been evaluated numerically. The nontrivial radiative-recoil correction can be calculated up to terms of order (αZ) 6 . These corrections are the sum of one-and two-electron contributions for high-Z few-electron atoms.

Nuclear polarization

Nuclear polarization (NP) corrections are due to the interactions between the atomic electron and virtual excitation states of the nucleus. The NP correction is very small, yet it sets the ultimate precision limition which can be used to test QED corrections in highly charged ions. The uncertainty of nuclear excitation spectrum limits the determination of nuclear polarization correction. The NP energy shift can be written as

∆E a = b =a b |V R | a 2 E a -E b . (2.55)
where the state |a represents an electron or muon state coupled to the nuclear ground state, and |b is the entire spectrum of electron or muon and excited nuclear states. A relativistic field approach of effective photon propagators specified for collective nuclear excitations is evaluated in Ref. [START_REF] Plunien | Nuclear polarization contribution to the Lamb shift in heavy atoms[END_REF][START_REF] Plunien | Nuclear Polarization in Heavy Atoms and Superheavy Quasiatoms[END_REF]. A relation between the NP correction to the Lamb shift and bound-electron g factor is given

E N P ≈ mc 2 j(j + 1) 3κ 2 δg N P , ( 2.56)
where E N P represents the NP contribution to the Lamb shift, which is deduced from the bound-electron g factor. The evaluation of the nuclear polarization contribution was carried out only for a few ions. We can get a conservative estimate [START_REF] Yerokhin | Lamb Shift of N = 1 and N = 2 States of Hydrogen-Like Atoms, 1 ≤ Z ≤ 110[END_REF],

E N P ≈ - 1 1000 E F N S , (2.57)
were E F N S represents the finite nuclear-size correction. Volotka and Plunien [START_REF] Volotka | Nuclear polarization study: New frontiers for tests of QED in heavy highly charged ions[END_REF] evaluated systematic investigation on one-and few-electron high-Z ions. They presented ∆E N P /∆E F S (10 -3 ) values of -0.430 for 1s, -0.431 for 2s, and -0.425 for 2p1/2 in H-like 208 Pb and -0.947 for 1s, -0.951 for 2s, and -0.941 for 2p1/2 in H-like 238 U, respectively. The ratio of the screened nuclear polarization and screened finite nuclear size contributions ∆E SN P /∆E SF N S (10 -3 ) are -0.430 for 1s 2 binding energy and, respectively, -0.432, -0.441, and -0.388 to the (1s 2 )2s, (1s 2 )2p1/2, and (1s 2 )2p3/2 ionization energies in Li-like 208 Pb. For 238 U, this ratio is -0.948 to the 1s 2 binding energy and -0.952 to (1s 2 )2s, -0.955 to (1s 2 )2p1/2, and -0.847 to (1s 2 )2p3/2 of ionization energies respectively. These ratios appear to behave rather similarly for all the considered electron states.

Nuclear deformation

For heavy nuclei, nuclear deformations should be considered. However, it is more difficult to evaluate the corrections connected to internal nuclear structure. We used the experimental nuclear charge radius. The important parameter for the atomic calculation is the contribution of deformation to the RMS radius. For a deformed nucleus, the half-density radius c can be written as

c(θ, φ) = c 0 [1 + β 20 Y 20 (θ, φ) + β 40 Y 40 (θ, φ)].
(2.58)

where β 20 and β 40 are the quadrupole deformation parameters, exclusively extracted the data of the muonic X-ray experiment. The effect of nuclear deformation on the RMS radius and energy shifts are researched by Johnson et al. [START_REF] Johnson | 241am and 243am Charge Distributions from Muonic X-Ray Spectroscopy and the Quadrupole Moment of the 240am Fission Isomer[END_REF], Zumbro et al. [START_REF] Zumbro | E2 and E4 Deformations in 232th and 239,240,242pu[END_REF]134] and Indelicato and Lindroth [START_REF] Indelicato | Relativistic Effects, Correlation, and QED Corrections on K α Transitions in Medium to Very Heavy Atoms[END_REF]. Kozhedub et al. [START_REF] Kozhedub | Nuclear Deformation Effect on the Binding Energies in Heavy Ions[END_REF] have been performed the calculations in hydrogenlike and lithiumlike uranium using the approximate formulas. The contribution from this effect is about 140 meV and 26 meV for the 1s state and 2s state in 238 U 91+ respectively, where nuclear deformation offers a 0.07% contribution. Therefore, in order to compute the accuracy of the nuclear size correction to 0.1%, it is necessary to take into account the nuclear deformation effect.

Auger shift

Another effect from a feature of these lithiumlike ion core-excited states is Auger shift since the initial level is degenerate with a continuum. The MCDF method is very inefficient to calculate the hole state of the inner shell caused by autoionization. Special methods should be used, for example, perturbation theory or complex scaling (CS) method, in order to calculate the inner-shell Auger shifts of these states. For instance, the CS method is suitable for computing light atoms [START_REF] Sajeev | Fock space multireference coupled cluster calculations based on an underlying bivariational self-consistent field on Auger and shape resonances[END_REF]. Unfortunately, this method is not suitable for relativistic calculations of heavy atoms because the computation takes a long time.

In the RMBPT method framework, the discretized Dirac-Fock basis sets can provide a simpler but less accurate method for estimating Auger shifts, which is described in Refs. [START_REF] Indelicato | Relativistic Effects, Correlation, and QED Corrections on K α Transitions in Medium to Very Heavy Atoms[END_REF][START_REF] Mooney | Precision Measurements of K and L Transitions in Xenon: Experiment and Theory for the K, L and M Levels[END_REF][START_REF] Indelicato | Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10≤ Z ≤ 100 and the determination of the 1s Lamb shift for heavy elements[END_REF][START_REF] Deslattes | X-Ray Transition Energies: New Approach to a Comprehensive Evaluation[END_REF]. They noticed that the 3s level is the most sensitive to Auger shift.

Research on the transition of heavy atomic inner shells has become one of the most promising tools for testing QED in strong Coulomb fields. The complexity of relativistic multibody problem up to 100 electrons can be found in Refs. [START_REF] Indelicato | Relativistic Effects, Correlation, and QED Corrections on K α Transitions in Medium to Very Heavy Atoms[END_REF][START_REF] Mooney | Precision Measurements of K and L Transitions in Xenon: Experiment and Theory for the K, L and M Levels[END_REF][START_REF] Indelicato | Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10≤ Z ≤ 100 and the determination of the 1s Lamb shift for heavy elements[END_REF][START_REF] Deslattes | X-Ray Transition Energies: New Approach to a Comprehensive Evaluation[END_REF]. These calculations are time-consuming, they are only executed on selected elements, and the results are interpolated to obtain values for other elements.

The Auger shift does not always have the same smooth Z-dependence as other manybody effects, because the mixing of the two other holes can increase dramatically as the energy difference between the sum of the energy of the holes and the original hole becomes smaller. Recently, Tupitsyn et al.

[131] used the RMBPT method in the Brillouin-Wigner form to calculate the Auger shifts on neutral uranium.

Radiative transitions

We consider radiative transitions of an atom or ion from state |i and |f by emission of a photon γ from the S-matrix theory by Cheng [START_REF] Cheng | Relativistic formulas for multipole-transition probabilities[END_REF].

In quantum electrodynamics, the Hamiltonian density is described as

H int = - 1 c j µ A µ , ( 2.59)
which represent the interaction between the electromagnetic field and the electron-positron field. Here A µ represents the operator of electromagnetic field and the operator of Dirac current density is described by j µ . An initial state of the system can be expressed by |J i ⊗|0 , in which |0 is the vacuum state of the photon field and a final state of the system can be expressed by |J f ⊗ |1γ , in which |1γ is a one-photon state.

The matrix element of transition becomes

S i→f = - i f, γ |H int | i = i c f |j µ | i γ |A µ | 0 d 4 x = - e 2π c 2 ωV ψf γ µ µ e -ikν xν ψ i d 4 x.
(2.60)

The unperturbed electron states ψ are given by

ψ(x) = u( x)e -iEt/ .
(2.61)

Here u( x) satisfies the single-particle Dirac equation in an external potential. We reduce the multipole transition matrix element to a sum of radial integrals. The radiative transition rate between an initial state described by i with a total angular momentum of J i and a final state described by f with total angular momentum of J f , is written by

A if = α 2 | J f , 1γ |j µ A µ | J i , 0 | 2 . (2.62)
Eq. ( 2.62) can also be rewritten as a sum of radial integrals weighted by angular and configuration coefficients according to averaging over angles. If confined to the multipole electric component λ of the electromagnetic field operator, we can obtain [132]

A λ if = K λ ∆E if N conf i n i =1 N conf f n f =1 N orb n i m i =1 N orb n f m f =1 c m i c m f T m i m f n i n f R m i m f λ n i n f 2 , ( 2.63)
with

K λ = 4πα(2λ + 1)e (λ + 1)(2J i + 1)h . ( 2.64)
Here m i and m f are the orbital designation numbers in the configuration designation numbers of n i and n f , and the initial state is represented by the subscript i, and the final state is represented by f , and T is the reduced angular coefficient of the 2 λ -pole electric operator. The c m represents configuration mixing coefficients, and the transition energy is denoted by ∆E if . The R represents a dimensionless radial matrix element written as

R m i m f λ n i n f = ∞ 0 dr (λ + 1)j λ (αωr)(P m i n i P m f n f + Q m i n i Q m f n f ) -j λ+1 (αωr)[(κ m f n f -κ m i n i -λ)Q m i n i P m f n f +(κ m f n f -κ m i n i + λ)P m i n i Q m f n f ] .
(2.65)

where P m and P n are the large radial components, Q m and Q n are the small radial components of the Dirac wavefunction, j λ is the spherical Bessel function, ω is the transition energy, and κ = (l -j)(2j + 1), in which l is the orbital angular momentum and j is total angular momentum respectively.

Auger transitions

For the initial state of 1s 1 2s 1 2p 1 , the Auger transition is considered a two-electron transition, in which an outer electron of 2s orbital jump into an inner hole of 1s orbital, and another outer electron of 2p orbital are simultaneously excited into a continuous state or bound state. Auger decay can be treated in time-dependent perturbation theory because the interaction between the core-hole state and Auger continuum is weak. The basic idea in non-radiative situations is to regard the emission of Auger electrons as a resonance in single photoionization, which is considered a complete scattering process,

A * = A + + e - A , (2.66)
This equation is represented by the transition amplitude [START_REF] Howat | Relaxation and final-state channel mixing in the Auger effect[END_REF][START_REF] Åberg | Theory of the Auger effect[END_REF]]

V f |H -E| U i = ν =1,2 ν |ν k||12 + |F | 1 |F | 2 k|1 k|2 + |1 |2 k |F | 1 k |F | 2 -ξ |1 |2 k|1 k|2 + µ =1,2 µ |1 µ |2 k||1µ k||2µ + |µ k|µ µ ||12 kµ ||12   .
(2.67)

where U i is the restricted Hartree-Fock solution of the initial state and V f is the final state. The notation ab||cd stands for

ab||cd = ab 1 r 12 cd -ab 1 r 12 dc , (2.68)
here the Fock-like operator

F = h + µ =1,2 µ ||µ . (2.69)
The coefficient ξ of the quadratic overlap term is written as

ξ = E - 1 2 µ =1,2 µ |h| µ + µ |F | µ . (2.70)
The corrections to the lowest-order Auger amplitude, k||12 , involve both one and twoelectron interaction matrix elements in addition to a scaling factor. The initial and final states must be optimised separately and the continuum orbital solved in the field of the final doubly ionised ion.

Chapter 3

Muonic atoms

We evaluate higher-order finite-size corrections, starting from accurate numerical evaluations of the Dirac equation in a Coulomb potential within the framework of nonperturbative methods of some QED contributions for muonic atoms. In this chapter, we mainly introduce the difference between muonic atoms and electronic atoms. They can be summarized by describing muons as "heavy electrons" with a mass of 207 times of ordinary electrons. Since the mass of the muon is large, it is 207 times closer to the nucleus than an electron in the same orbital. Especially for heavy nuclei, this leads to large nuclear size corrections and a strong dependence of the bound state energy of the muon on the current distribution and nuclear charge. There is also a large relativistic effect for muonic atoms.

The theory of the muon and the electron within finite-size mechanics is the same, and corrections come from the same sources, such as relativity, recoil, QED, and proton structure effects. However, the relative importance of the various effects is not the same. More importantly, there are different QED proportions in muonic atoms due to the influence of the electron vacuum polarization, which is dominant in the QED corrections.

It should be observed that the large differences between the observed interactions of muonic and electronic atoms are not all due to differences in masses.

Wave function

Since the mass of the muon is 207 times that of the electron, the bound muon is close to the nucleus and has a relationship of m µ /m e ≈ 207 with the bound electron. The radius of the muon is smaller than the Compton wavelength of the electron λ = /m e c by a ratio of m e /αm µ ≈ 137/207 (m µ and m e are the muon and electron mass, respectively). Fig. 3.1 outlines how the muon in the low states penetrates the nucleus. The dotted line indicates the nuclear charge distribution. It can be noted that the 1s wave penetrates deeply into the nucleus, the 2p and 3p waves penetrate very little, and 3d has almost no penetration. The 2s wave has the greatest overlap with the nucleus next to the ground state. For the 2s level, the average radius of the muon wave function is 2.6 times larger than the Compton wavelength, which is the scale of QED corrections. The sensitivity of the muonic atom test to the effects of nuclear size is determined by the overlap integral (P (r) 2 + Q(r) 2 )ρ(r)r 2 , where Q(r) and P (r) are small and large components of the radial w a v e f u n c t i o n st ot h eD i r a ce q u a t i o n ,r e s p e c t i v e l y .

. 2 N u c l e a rc h a r g ed i s t r i b u t i o n m o d e l s

T h e m u o n i ce n e r g y l e v e l sa r es e n s i t i v et ot h en u c l e a rs t r u c t u r e .I no r d e rt od e t e rm i n e t h es i z eo ft h en u c l e if r omt h e m u o n i ca t omt r a n s i t i o ne n e r g i e s ,t h e U n i f o rm , F e rm i a n d3 -p a r am e t e r G a u s s i a nn u c l e a rc h a r g ed i s t r i b u t i o n sa r eo f t e nu s e dt os o l v et h e D i r a c e q u a t i o n ,a n dt h en u c l e a rp a r am e t e r sa r ea d j u s t e dt or e p r o d u c et h et r a n s i t i o ne n e r g i e s . T h ea n a l y t i cd i s t r i b u t i o n sa r ep a r am e t e r i z e d , s o t h e yp r o v i d e t h e s am em e a n s q u a r e r a d i u s R .T h e m om e n to ft h ec h a r g ed i s t r i b u t i o nc a nb ee x p r e s s e da s

r n =4π ∞ 0 r 2+ n ρ ( r ) d r , ( 3 . 1 )
F o ra s p h e r i c a l l y s ymm e t r i c c h a r g ed i s t r i b u t i o n , t h en u c l e a r c h a r g ed i s t r i b u t i o nρ ( r )= ρ N ( r ) / ( Z e )c a nb en o rm a l i z e da s

ρ ( r ) d ( r )=4 π ∞ 0 r 2 ρ ( r ) d r=1, ( 3 . 2 )
w h e r et h e m e a ns q u a r er a d i u s i sR= r 2 .

The Coulomb potential of the nucleus may be expressed as We have described the nuclear charge distribution using the two-parameter Fermi distribution in Eq. (2.18). We also use the three-parameter Gauss function, which is defined as

V N (r) = - 4πe r r 0 duu 2 ρ N (u) -4πe ∞ r duuρ N (u). ( 3
ρ(r) = ρ 0 1 + ω r c 2   1 + e ( r 2 -c 2 ) t 2    -1 , (3.4)
where ω is a shape parameter.

Finite-size effect

The consequences of the large mass of the muon are obvious. Although the point charge is still good for electronic atoms, the approximation of point charges is very wrong for muonic atoms. The finite nuclear size effect is no longer correction and dominates in muonic atoms. This effect greatly reduces (two times) the binding energy of 1s in heavy atoms. It is well known that the energy shift caused by the finite size effect is proportional to the charge distribution moments. The perturbation calculation shows that in the low-Z element, s state energy shift is proportional to r 2 and the p state to r 4 . Because the 2p state of the muon in light element spend a very short time inside the nucleus, and this results in a very small energy shift at the 2p level. The leading order has a simple expression [START_REF] Mohr | CODATA recommended values of the fundamental physical constants: 2014[END_REF]

E = 2 3 µ r m µ n 3 m µ c 2 ZαR λ 2 δ 1,0 , (3.5)
where R is the root-mean-square charge radius of the nucleus, and λ is the muon Compton wavelength. In addition to this leading limited size contribution, there are various relativistic and QED corrections.

For electronic atoms, the self-energy correction is much larger than the vacuum polarization correction. However, for muons, the vacuum polarization correction is dominant due to the Uëhling potential at a shorter distance. The vacuum polarization potential can be added to the Dirac equation directly with the term of α(Zα) 2 m µ c 2 . In this way, the calculation includes all iterations of the Uëhling potential. The Uëhling correction can produce 2s -2p splittings bigger than 2p 3/2 -2p 1/2 . The finite-size effects are the most prominent feature in the low levels of heavy muonic atoms.

Fine-structure splitting

The complicated nuclear structure leads to the uncertainty of the muon levels in heavy nuclei. Thus the muonic atoms become one of the tools for studying electromagnetic properties of the nucleus. However, with the continuous improvement of experimental accuracy, there is a long-term difference between the theoretical and experimental aspects of fine structure splitting in muonic heavy atoms. One of the discrepancies is that the theoretical nuclear polarization energy shift gives the opposite contribution to the predicted values from the experiment. This discrepancy was first reported by Yamazaki et al. [START_REF] Yamazaki | Giant hyperfine anomaly between bound negative muon and Rh nucleus in Pd metal[END_REF][START_REF] Yamazaki | Discrepancy between Theory and Experiment in Nuclear Polarization Corrections of Muonic Pb 208[END_REF] in the ∆2p splitting energy of µ -Pd/Rh and 208 Pb of the muonic x-ray analysis. In that analysis, the experimental nuclear polarization correction was derived from a model-independent analysis of muonic x-ray and elastic electron scattering. The same difference is also indicated in the 2p level of the muonic 90 Zr [START_REF] Phan | Nuclear polarization in muonic Zr 90[END_REF]. In view of the fact that the theoretical calculations are in good agreement with the experimental measurements for µ -Ni/Co cases [START_REF] Freeman | Theory of hyperfine anomalies in muonic atoms[END_REF], so the discrepancy is supposed to be due to experimental factors in heavy muon nuclei. Later, Bergern et al. [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] got the values from the experiment to consist of the calculations by a hypothesis inversion in the nuclear polarization corrections. The other difference is in the splitting energy from 3p, which was found in the X-ray measurements of the muonic 208 Pb. The difference is 300-500 eV, which is of the same order of magnitude as the 2p splitting energy [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF]. Haga et al. [START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF] reanalyzed this difference by adding pygmy dipole resonances to the excitation spectrum, in which the final fit of 208 Pb has been greatly improved. There is the same difference in 90 Zr, where the nuclear excitation spectrum is not sensitive to nuclear polarization. They guess that the existing discrepancy might be caused by other effects in addition to the nuclear polarization.

Nuclear polarization

The nucleus and muon will produce virtual transitions to excited intermediate states due to the electromagnetic interaction between the muon and the nucleon. This effect results in an increase in the muon binding levels. Fig. 3.3 shows diagrams contributing to nuclear polarization in lowest order. The muon (heavy line) interacts with the nucleus (heavy line) in its state via the exchange of virtual photons (wavy lines). [START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF] We evaluate the nuclear polarization with the RURP code, which is used in the low states of the muonic atoms and developed by Rinker [START_REF] Rinker | Static and dynamic muonic-atom codes-MUON and RURP[END_REF]. This method provides a general computational framework for calculating nuclear excitation in the muonic spectrum, where the states include all virtual excitations of the nuclei and the muon. The nuclear polarization calculations are defined in terms of first-order perturbation theory in which exact sums over the entire muon spectrum and certain closure approximations for the nuclear spectrum. Since the properties (such as transition matrix elements, angular momentum, and energy) of only a few low-excited states of these nuclei are known, calculating this correction ∆E a (2.55) is complicated. Most excited states effects can only be estimated by charged particle scattering data or photonuclear reactions. The other is by using a summation rule or a specific theoretical nuclear model if they are available with sufficient accuracy. Due to the lack of experimental information and a lack of good theoretical models of these nuclei, this calculation is primarily based on the summation rules. Various sums contain energy weighting and rules (EWSR) [START_REF] Bohr | Nuclear Structure[END_REF][START_REF] Bohr | [END_REF],

f (E f -E 0 ) n B(EL; 0 → f ) = L(2L + 1) 2 4π ( c) 2 2M Z r 2L-2 (3.6)
The monopole for L = 0 of the sum rules is obtained approximately by setting B(E 0 ; 0 → f ) → 0 r 2 f 2 and L → 2 in Eq. (3.6). In the RURP code, this correction ∆B N P (L) has been calculated for the electric monopole with L = 0, dipole with L = 1, quadrupole with L = 2, octupole with L = 3, and L ≥ 3 excitations. The result can be seen as the Hamiltonian approximate diagonalization over the entire and infinite muon-nuclear spectrum. The determination of the nuclear-polarization correction is limited by uncertainties in the nuclear excitation spectrum. The present computation of nuclear polarization corrections corresponds to the ladder diagram of Fig. 3.3(a). Haga et al. [START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF] reanalyzed the low-lying µ --90 Zr and µ --208 Pb states by using nuclear polarization with the full-electromagnetic nuclear response. In their analysis, they employed collective models, which are three forms for the transition density of Tassie-Goldhaber-Teller (TGT) [START_REF] Tassie | A Model of Nuclear Shape Oscillations for g? Transitions and Electron Excitation[END_REF][START_REF] Bohr | Nuclear Structure[END_REF], Rinker (RIN) [START_REF] Rinker | Nuclear polarization in muonic 204, 206, 207, 208Pb in the random-phase approximation[END_REF], and Jensen-Steinwedel (JS) [START_REF] Bohr | Nuclear Structure[END_REF][START_REF] Plunien | Nuclear polarization in heavy atoms and superheavy quasiatoms[END_REF] models to avoid the transverse nuclear polarization correction being gauge-dependent. In addition to leading-order nuclear polarization corrections for the ladder in Fig. 3.3(a), they also considered the cross diagram in Fig. 3.3(b), seagull diagram in Fig. 3.3(c), and the nuclear polarization combined vacuum polarization in Fig. 3.3(d). Although the seagull diagram does not polarize the nucleus, it needs to be included as part of the "nuclear polarization" correction due to it playing an important role in retaining the gauge invariance of the nuclear polarization correction.

Effects of remaining electron

In heavy elements, not all atomic electrons are ionized by Auger transitions during the muonic cascade. Furthermore, empty electron orbits can be refilled during the cascade in solids or in high-pressure gases. The remaining electrons have an effect on the muon and the binding energy. In addition to the muon, we also solve the full Dirac-Fock equation with the electrons from K, L, and M shells, respectively. We define this effect in terms of δE = E µ+e -E µ . In the following sections, we will discuss the calculation results in detail.

One important source of uncertainty in this effect calculations is lack of information on the number of present electrons during the muonic cascade. The electronic K x-ray experiments in heavy muonic atoms have been performed by Schneuwly and Vogel [START_REF] Schneuwly | Electronic K x-ray energies in heavy muonic atoms[END_REF], which showed that the internal electron shells were refilled almost immediately during the muonic cascade. It is highly probable that all ten K and L electrons are present during the radiative transition between states within n ≤ 8 of the heavy elements [START_REF] Borie | The energy levels of muonic atoms[END_REF]. The K and L electrons are responsible for more than 95% of the effective electron density. Therefore, we will consider that the electrons refill K, L, M shells, separately. The effect of the remaining electron is inevitable for solid-state targets. Modern measurement techniques have been developed to completely ionize low-Z exotic atoms in gas targets [START_REF] Bacher | Relevance of ionization and electron refilling to the observation of the M1 transition (γM1: 2s-1s) in light muonic atoms[END_REF][START_REF] Lenz | A new determination of the mass of the charged pion[END_REF].

Chapter 4

Computational procedure

In this chapter, our purpose is to introduce the relativistic atomic structure program and computational methods used in our calculations.

Relativistic atomic structure program

Our present calculations are performed using MCDFGME, a general multiconfiguration Dirac-Fock code, which is programmed ab initio completely to include both relativistic and correlation contributions in the calculations related to theoretical atomic structure. This code has evolved from Desclaux [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF], and the new version is developed by Indelicato [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF]. It can be used to evaluate level energies, including one-loop QED corrections with all-order Uëhling contribution (see section 2.4.1), two-loop QED corrections (see section 2.4.2), QED screening corrections with Welton picture (see section a)) and effective operator (see section b)), and relativistic recoil corrections (see section 2.5). This code is very versatile with numerical methods (see section 2.1.3) and allows to modify mesh grid where the wavefunctions (see section 2.1.2) are tabulated and calculate various other properties. The electron-electron interaction (see section 2.3) is evaluated by the sum of the Breit and the Coulomb parts. MCDFGME allows for an option of including a full Breit operator (Eq. 2.27) or omitting the magnetic part (Eq. 2.25) or omitting the first term of retardation (Eq. 2.26) in the self-consistent field process. The higher-order retardation (Eq. 2.26) can be treated as the first-order perturbation. All calculations are solved using the Dirac-Coulomb-Breit Hamiltonian (see section 2.1.2) with a finite nucleus model using several alternative distributions, in which the default thickness parameter of 2.3 fm or setting any values can be adopted. The nuclear radiuses are used from the experimental results compiled by Angeli [185,186]. All calculations use the latest atomic mass tables in [START_REF] Audi | The AME2003 atomic mass evaluation:(II). Tables, graphs and references[END_REF] to replace the electronic mass with the reduced mass in the Dirac equation. The radiative transition probabilities (see section 2.9) are discussed by Cheng [START_REF] Cheng | Relativistic formulas for multipole-transition probabilities[END_REF]. The transition probabilities are calculated by using Dirac wavefunctions, which are obtained in a complete self-consistent process, including relaxation.

The atomic structure can be explained in an extended sense, because the program can handle not only electrons, but also several " exotic " particles that can be added to electrons, such as muons(spin 1/2 lepton), kaons(spin 0 meson), and so on. The Klein Gordon equation is solved instead of the Dirac equation when the "exotic" particle is a boson. The muon can be regarded as an electron with heavy mass, where the process to deal with the muonic atoms [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF] is the same as the Hydrogen-like systems. In addition, we take into account the nuclear polarization correction (see section 3.5) and the effects from remaining electrons (see section 3.6).

The program provides the associated energy as well as the numerical representation of the one-electron orbitals constituting the total wave function. Present, the program can directly deal with any number of electronic systems including one electron, two electrons [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF][START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], three electrons [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], four electrons [START_REF] Indelicato | QED and relativistic corrections in superheavy elements[END_REF][START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF][START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], more electrons [START_REF] Indelicato | QED and relativistic corrections in superheavy elements[END_REF][START_REF] Marques | Relativistic multiconfiguration calculations of the 2s 2 2p 2 P 3/2 level lifetime along the boron isoelectronic sequence[END_REF] and also super-heavy elements [START_REF] Indelicato | QED and relativistic corrections in superheavy elements[END_REF][START_REF] Indelicato | Are MCDF calculations 101% correct in the super-heavy elements range?[END_REF][START_REF] Indelicato | Effects of the Breit interaction on the 1s binding energy of superheavy elements[END_REF]. From these results, the code can evaluates a variety of atomic proprieties such as the total energy of a given state either at the multiconfiguration or single configuration level, the Landé g-factor [START_REF] Indelicato | QED and relativistic corrections in superheavy elements[END_REF][START_REF] Marques | Ground-state Landé g factors for selected ions along the boron isoelectronic sequence[END_REF], the Auger transition probabilities [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF], the radiative transition probabilities [START_REF] Sampaio | Dirac-Fock calculations of K-, L-, and M-shell fluorescence and Coster-Kronig yields for Ne, Ar, Kr, Xe, Rn, and Uuo[END_REF], the hyperfine structure parameters [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF] and so on.

Computational methods

We used optimized levels method (OLs) to calculate the energy of each state and wave function. By using this technique, the orbitals of the initial and final states are not orthogonal because they have been individually optimized. Indelicato [START_REF] Indelicato | Correlation and Negative Continuum effects for the relativistic M1 transition in two-electron ions using the multiconfiguration Dirac-Fock method[END_REF][START_REF] Indelicato | Radiative de-excitation of the 1s22s3p 3P0 level in beryllium-like ions: A competition between a E2 and a two-electron one-photon E1 transitions[END_REF] use the formalism developed by Löwdin [START_REF] Löwdin | Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction[END_REF] to take into account this non-orthogonality effect.

The atomic states are defined as a linear combination of electronic configurations, and each electronic configuration is defined as a determinant with their occupation numbers. These configurations are listed in the form of LS coupling, using the labels of 1s, 2s, 2p, etc. The program will use the labels of 1s, 2s, 2p * , 2p, etc. to generate all jj configurations resulting from a given LS configuration. The * denotes j = l -1/2, while no * stands for the j = l + 1/2.

The configuration space is occupied by all single to triple excitations from the occupied orbitals to some active sets. In this way, active variational space size is increased from all occupied shells to all virtual orbitals up to maximum n = 5 and l = n -1. It customarily doesn't consider the type of the single excitation of nκ → n κ (κ represents Dirac angular number), that is the configurations of electron excited to an orbital, which has the same κ with the initial orbital. In principle, this should have a zero effect on energies in non-relativistic transitions (Brillouin's theorem [START_REF] Bauche | Remarks on Brillouin's theorem in the atomic variational approach[END_REF][START_REF] Godefroid | Brillouin's theorem for complex atomic configurations[END_REF]). But there would get the wrong non-relativistic limit if you do not include them, which was proved by Indelicato et al. [START_REF] Indelicato | Nonrelativistic limit of Dirac-Fock codes: The role of Brillouin configurations[END_REF] and Kim et al. [START_REF] Kim | Failure of multiconfiguration Dirac-Fock wave functions in the nonrelativistic limit[END_REF]. So, we include the Brillouin configurations in the calculations of single excitations. The excitation is layer-by-layer enhanced, which makes it easy to monitor the convergence of various physical quantities. In the present cases of 1s2s2p 2S+1 P J , each of the associated layers labeled n = 2, 3, 4, 5 can include s, p, d orbitals of up to 4f, 5g subshells. The program can excite much higher orbitals for other cases. The process of these shells will result in a lot of configurations. The MCDFGME code will automatically perform the generation of the multiple configuration expansions.

Convergence considerations

Wavefunctions

If W F i n and W F f n represent the initial and final values of the wavefunction respectively at the nth iteration, the estimation value of the normalized wavefunction for the (n + 1)th iteration is considered to be [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF]

W F i n+1 = δW F f n + (1 -δ)W F i n (4.1)
The coefficient δ remains fixed during the self-consistent field process, or it is determined for each orbital according to the following scheme. We take ∆W F n as the maximum difference between ∆W F i n and ∆W F f n , if ∆W F n-1 and ∆W F n have the same sign then δ is increased by 0.1. Otherwise it is decreased by the same amount provided that 0.1 ≤ δ ≤ 0.9.

(4.2)

We also can compare the differences between successive correlation energy values obtained in different active spaces to assess the energy convergence of the calculation. Energy convergence is kept within 0.01 eV in order to ensure convergence of the calculation.

Weights of the configurations

The LSJ level is given by [XLJ]#nLSm where X = 2S +1, L and J are the total angular momenta of the orbital, S is the total spin, n is the energy ordering (n = 1 correspond to the lowest energy of all degenerate terms of the selected LS configuration), and LS is derived from its configuration (the mth following them are listed in the input data). The convergence will be towards an eigenstate with the greatest weight for a given LSJ level.

For a multiconfiguration calculation, the weights W ν of the configurations are determined in the following way. We suppose that the nth step is performed using a set of coefficients W n ν . The diagonalization of the energy matrix provides a new set (W n ν ) d . The (n + 1)th step can be performed using W n+1 ν coefficients, which can be described as [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF],

W n+1 ν = λ(W n ν ) d + (1 -λ)W n ν . (4.3)
The damping factor λ is extremely important when a calculation converges to a high eigenvalue or excited state, rather than to the lowest eigenvalue (ground state).

Chapter 5

Results and discussion on Li-like ions

In the present chapter, we discuss diverse QED corrections, for example, the self-energy screening from the Welton model and effective operator model, respectively. We analyze the electron correlation of Coulomb, magnetic, retardation and higher-order retardation terms. Our calculations with Welton or effective operator methods are compared with other theoretical and experimental results.

Welton model and Effective operator model

Table 5.1 -The screened self-energy for the transition 1s 2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 and 1s 2 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 of Li-like ions (in eV). The Welton (Wel) and effective operator (Eff) results are obtained by Welton picture and effective operator model. Shabaev et al. performed the calculations using the method of Kohn-Sham (KS) [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF], Dirac-Fock (DF) [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF] and perturbation theory (PT) [START_REF] Yerokhin | Screened self-energy correction to the 2p 3/2-2s transition energy in Li-like ions[END_REF][START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF][START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] in the Kohn-Sham potential. The previous MCDFGME code used the Welton approximation (see section a)) to evaluate the self-energy screening correction. Recently, Indelicato has modified the code using the model operator method (see section b)) developed by the St. Petersburg group [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF][START_REF] Shabaev | QEDMOD: Fortran program for calculating the model Lamb-shift operator[END_REF].

1s 2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 1s 2 2p 2 P 3/2 →
The new code is covered in detail in the theoretical part of Chapter 4.

In Table 5.1, the self-energy screening energies for the transition 1s 2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 and 1s 2 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 of Li-like ions are presented and are compared with results of other calculations. The present results of the Welton (Wel) and effective (Eff) are obtained by Welton picture and effective model operator, respectively. In early works, Blundell's screened self-energy and vacuum polarization energies [START_REF] Blundell | Calculations of the Screened Self-Energy and Vacuum Polarization in Li-Like, Na-Like and Cu-Like Ions[END_REF] already included estimates of contributions of one-loop Lamb-shift added to the MBPT results, whose screened calculations start from the core-Hartree potential, but the vertexexchange terms were not calculated.

The St. Petersburg group [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF][START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF][START_REF] Yerokhin | Screened self-energy correction to the 2p 3/2-2s transition energy in Li-like ions[END_REF] considered the screened self-energy correction using the model of SE operator technique, which can be expressed as the Kohn-Sham (KS) or Dirac-Fock (DF) equation in total energy. The DF and KS potentials are constructed self-consistently in consideration of the valence state. The calculations by the perturbation theory (PT) with the Kohn-Sham potential is obtained by YS Kozhedub [START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF] using the CI method and J. Sapirstein [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] using the S-matrix method. The screened self-energy from our results fits well with St. Petersburg group for the iso-electronic sequence. This also verifies the accuracy of our calculations on screened self-energy. Our results obtained employing the model SE operator approach are in rough agreement with the results obtained with the perturbation theory. Perturbation theory values are a little smaller than DF values.

We can compare the different values as an example to evaluate the reliability of this technique for calculating self-energy screening. For example, the 2p 1/2 -2s transition in Li-like calcium from the St. Petersburg group [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF] has a QED value of 0.038 eV, and Ref. [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] has a QED value of 0.038 eV. The Welton technique provides 0.034 eV, and the implementation of effective operator technique achieves a result of 0.037 eV, which is closer to St. Petersburg's calculations. Therefore, for the effective operator and the Welton operator technique, we can evaluate them by assuming an uncertainty of 0.001 eV and 0.004 eV, respectively. The same procedure applied to the tungsten ion transition gives the following data, respectively, 1.010 eV from St. Petersburg [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF], 0.930 eV from Ref. [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF], 1.042 eV from the effective operator method, and 1.077 eV from the Welton method. While, uranium ion transition provides 2.019 eV for the Welton method, 1.887 eV for the effective operator method, 1.800 eV using St. Petersburg [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF] and 1.500 eV using Ref. [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF].

In Table 5.2, the self-energy screening energies for core-excited transitions 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 and 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2 of Li-like ions are also presented. The deviation between Welton method and effective operator method in valence-excited transitions is 0.2 eV at Li-like curium, while this deviation has reached 2 eV in core-excited transitions. muon pairs is listed in the Table 5.3 and 5.4, which is very small and even in the high-Z ions, and the order of magnitude is only 0.001 eV for the 1s 2 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transition. The Wichmann-Kroll part of the vacuum-polarization contribution is also evaluated from Ref. [START_REF] Wichmann | Vacuum polarization in a strong Coulomb field[END_REF].

The next effects are the screened QED contributions from other electrons. The evaluation of the screened QED corrections is introduced in Sec. 2.4.3 following [START_REF] Indelicato | Relativistic Effects, Correlation, and QED Corrections on K α Transitions in Medium to Very Heavy Atoms[END_REF][START_REF] Indelicato | Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10≤ Z ≤ 100 and the determination of the 1s Lamb shift for heavy elements[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF][START_REF] Indelicato | 6s and 8d state self-energy for hydrogen-like ions and new results on the self-energy screening[END_REF]. Here we compare the self-energy screening obtained using the Welton approximation [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF] and the approach from Ref. [START_REF] Shabaev | Model operator approach to the Lamb shift calculations in relativistic many-electron atoms[END_REF][START_REF] Shabaev | QEDMOD: Fortran program for calculating the model Lamb-shift operator[END_REF]. We have made a detailed discussion in Sec. 5.1 for this contribution.

Another correction that leads to the largest theoretical uncertainty of middle-and high-Z ions comes from nuclear recoil. The evaluation of the recoil effect is described in Sec. 2.5. This calculation significantly improves the accuracy of theoretical simulation of 2p j -2s transition energy for Li-like ions. The relativistic correction with the nuclear movement is treated as a relativistic-recoil correction. In the calculation, all orders in m/M and the lowest order leading term in Zα are considered. We obtain the calculations of normal mass shift and specific mass shift evaluating the reduced mass correction. Their contributions are presented in the Table 5.3 and Table 5.4 separately. We have observed that for the valence-excited transitions, the specific mass shift dominates in the recoil correlation, and the normal mass shift is equal to zero at low-Z ions. While for coreexcited transitions, the normal mass shift dominates the recoil correlation and the specific mass shift is equal to zero at high-Z ions. We also notice that the relativistic recoil is equal to zero in all the transitions in low-Z ions.

Finally, we account for the two-loop QED effects. They are called SEVP, VPVP, and S(VP)E contributions, which are evaluated in Sec. 2.4.2. Yerokhin, Indelicato and Shabaev [START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF] performed all-order calculations for the two-loop one electron QED corrections of n = 2 states for several ions with Z ≥ 60 at first. We used the values from this reference within the Z range covered. The remaining two-loop self-energy correction (the "SESE" subset), which is taken from Refs. [START_REF] Yerokhin | Two-loop self-energy in the Lamb shift of the ground and excited states of hydrogenlike ions[END_REF][START_REF] Yerokhin | Two-loop self-energy correction in high-Z hydrogenlike ions[END_REF][START_REF] Yerokhin | Evaluation of the twoloop self-energy correction to the ground state energy of H-like ions to all orders in Z alpha[END_REF][START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF][START_REF] Yerokhin | Two-loop QED corrections in few-electron ions[END_REF][START_REF] Yerokhin | Two-loop QED corrections with closed fermion loops[END_REF][START_REF] Yerokhin | Two-loop self-energy for the ground state of medium-Z hydrogenlike ions[END_REF][START_REF] Yerokhin | The two-loop self-energy: diagrams in the coordinatemomentum representation[END_REF]. The SEVP and S(VP)E corrections are obtained from Ref. [START_REF] Yerokhin | Two-loop QED corrections with closed fermion loops[END_REF]. The indicated error bars are due to the free-loop approximation used in evaluating the VPVP and S(VP)E subsets, which Ref. [START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF] gives the details. The Kallen and Sabry potential is also included, as described in Ref. [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF].

The current level of experimental accuracy requires a rigorous QED calculation for the two-photon exchange contribution in Li-like ions. At the same time, no strict QED calculations have been performed for more photon exchange contributions so far. For few-electron ions with high-Z, the evaluation of these contributions within the Breit approximation is usually not sufficient.

The nuclear polarization correction for Li-like thorium was 0.02 eV and Li-like uranium was 0.03 eV for 2p 1/2 -2s transition, which were studied by Plunien [START_REF] Plunien | Erratum: Nuclear-polarization contribution to the Lamb shift in actinide nuclei[END_REF] and by Nefiodov [START_REF] Nefiodov | Nuclear polarization effects in spectra of multicharged ions[END_REF].

Electron correlation

In a relativistic calculation, the correlation contribution is composed of the Breit part of the electron-electron interaction and pure Coulomb interaction between the electrons. The Breit interaction is separated into the magnetic interaction, the first term of the retardation and the higher-order retardation of the electromagnetic field. As already discussed in section 2.3, the total instantaneous

g ins = (1 -α 1 • α 2 α 1 • α 2 α 1 • α 2 )
/R and the complete retardation g ω can be added in the self-consistent Dirac-Fock method. This will improve the results of the heavy ions, where the retardation interaction is no longer a small perturbation compared to the Coulomb repulsion. While these interactions become small compared to the nuclear potential, the wave functions become more hydrogenic, as Z increases. In Tables .10, .11, .12, .13, .14, .15, .16 of Appendix A (6.5), we provide individual values of the Coulomb, magnetic, retardation and higher-order retardation correlations, which are obtained using a wavefunction built up from all configurations with n = 1, 2, 3, 4 and 5 for Z=10 to 96.

The retardation correlation is more intricate, so it leads to yet unsolved fundamental problems. The definition of ω (Equation. 2.26) is terms of one-electron energies. Koopmans' theorem can be used to provide estimates of single-particle energies for a single configuration, while there is no single-particle parameter being identified as an approximation one-electron energy in the case of MCDF.

We also plot the individual correlations of Coulomb and Breit electron-electron interac-

Fig. 5.5

The difference correlation from Coulomb, magnetic, retardation, and higher order retardation between 1s 2 2s 2 S 1/2 and 1s 2s 2p 2 P 3/2 line represents the higher-order retardation correlation of the Breit interaction. Correlation corrections of the magnetic interaction start being the same size as Coulomb correlation at Z=47 and to be ten times larger at Z=92 for the ground state of three-electron systems. In the excited state of 1s 2 2p 2 P J , J = 1/2, 3/2, there is a similar trend consistent with the two-electron systems [START_REF] Gorceix | MCDF studies of two electron ions I: Electron-electron interaction[END_REF]. Although the Coulomb correlation has always been larger than the magnetic correlation for the core-excited state in the isoelectronic sequence of Li-like ions, the magnetic correlation energy gives a sizable contribution(about -0.6 eV) at high-Z. The results of the higher-order correlation are estimated to be less than 0.1 eV due to their scaling as 1/Z compared to the leading correlation contribution, while at high-Z, its value is higher than 0.1 eV seeing Fig. 5.1. In Fig. 5.4 and Fig. 5.5, we also plot the individual contributions of Coulomb and Breit electron-electron interactions to transition energies for the 1s 2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 and 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transitions in Li-like ions. We note that the Coulomb correlation dominates the contributions and is consistent with the trend of total correlations in the valence-excited transition. However, for core-excited transition, the magnetic correlation energy is consistent with the trend of total correlations, because the magnetic correlation energy is so large in the middle-and high-Z ions.

Total transition energy and transition probability

We have evaluated the valence-excited transitions of the 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, core-excited transitions of the 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2, and Auger transitions of the 1s 2s 2p 2 P J → 1s 2 1 S 0 , J = 1/2, 3/2, 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2 in Li-like from neon (Z=10) to uranium (Z=92). For some of these transitions, we calculate to curium (Z=96) with convergence allowed. We perform these calculations using the multiconfiguration Dirac-Fock approach, which is implemented in the relativistic MCDF code (MCDFGME), written by Indelicato and Desclaux [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF]. We used the new version of the code, which takes into account the most-recent two-loop self-energy corrections calculations [START_REF] Yerokhin | Two-loop self-energy in the Lamb shift of the ground and excited states of hydrogenlike ions[END_REF], although their effects are very small here. A complete description of the approach and code can be found in Refs. [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Grant | Relativistic Calculation of Atomic Structures[END_REF][START_REF] Indelicato | Projection operators in Multiconfiguration Dirac-Fock calculations. Application to the ground state of heliumlike ions[END_REF]. This approach also evaluates specific and normal mass shifts after [START_REF] Shabaev | Mass corrections in a strong nuclear field[END_REF][START_REF] Shabaev | Relativistic nuclear recoil corrections to the energy levels of multicharged ions[END_REF][START_REF] Shabaev | QED theory of the nuclear recoil effect in atoms[END_REF] as described in [START_REF] Sampaio | Relativistic calculations of 1s 2 2s2p level splitting in Be-like Kr[END_REF]. All of the calculations are performed taking into account the finite nucleus with fermi model. The atomic masses are taken from tables in Ref. [START_REF] Audi | The AME2003 atomic mass evaluation:(II). Tables, graphs and references[END_REF] and the nuclear radii from [185, 186], respectively.

The most important feature of the MCDF code is to include a lot of electronic correlations with a limited number of configurations [START_REF] Santos | Relativistic 2s1/2 (L1) atomic subshell decay rates and fluorescence yields for Yb and Hg[END_REF][START_REF] Santos | Relativistic correlation correction to the binding energies of the ground configuration of beryllium-like, neon-like, magnesium-like and argonlike ions[END_REF][START_REF] Martins | Production and decay of sulfur excited species in an electron-cyclotron-resonance ion-source plasma[END_REF]. The calculations are rather difficult for the excited states 1s2s2p, when the 1s2s core acquires a 1 S 0 component. In this case, the off-diagonal Lagrange multiplier that is used to maintain the orthogonality between the two orbitals tend to become very small, and the 2s orbital tends to become identical to the 1s. Here, we give three sets of (1s)2(2p)1, (1s)1(2s)1(2p)1, and (2s)2(2p)1 as reference configurations and use the method of 2s as frozen orbital to solve the Dirac equation. The presence of the 1s 2 orbital can prevent the 2s electron to become 1s orbital. The self-consistent field process includes Uëhling potential and full Breit interaction. The projection operator has been taken into account [START_REF] Indelicato | Projection operators in Multiconfiguration Dirac-Fock calculations. Application to the ground state of heliumlike ions[END_REF] to avoid coupling with the negative energy continuum.

Here, for Li-like ions, the correlation contributions are obtained by taking into account all single-electron, double-electron and triple-electron excitations containing n = 1 and n = 2 electrons in the undisturbed configuration until n = 5. For 1s 2 2s 2 S 1/2 ground state, it is related to the 1463 configurations and for 1s 2s 2p 2 P 3/2 core-excited state to 2478 configurations. The convergence of the radiative and Auger transition energies for the transition of 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 in Li-like neon and uranium can be found in Table 5.5, which allows us to make a detailed estimate of the reliability of the MCDF calculation. We had to employ virtual orbitals to ensure the convergence of the MCDF calculation results.

1s

2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 transition - - - ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( } ( ) ( ) ( ) ( ) + ( ) + + ( ) ( )
+ + + ) ( )

Fig. 5.6 Comparison of the transition energy between theoretical values and the present MCDF calculations for the 1s

2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 transition.
The reference value is obtained with the DF Welton unperturbed method. Theoretical works: [START_REF] Gu | Energies of 1s22lq (1 ≤ q ≤ 8) states for Z ≤ 60 with a combined configuration interaction and many-body perturbation theory approach[END_REF] [START_REF] Gu | Energies of 1s22lq (1 ≤ q ≤ 8) states for Z ≤ 60 with a combined configuration interaction and many-body perturbation theory approach[END_REF], Indelicato and [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF], [START_REF] Kim | Resonance transition energies of Li-, Na-, and Cu-like ions[END_REF] [START_REF] Kim | Resonance transition energies of Li-, Na-, and Cu-like ions[END_REF], [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF] [START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF], [START_REF] Ynnerman | Manybody calculation of the 2p1/2,3/2-2s1/2 transition energies in Li-like 238U[END_REF] [START_REF] Ynnerman | Manybody calculation of the 2p1/2,3/2-2s1/2 transition energies in Li-like 238U[END_REF], Chen et al. (1995) [START_REF] Chen | Relativistic Configuration-Interaction Calculations for the n=2 States of Lithiumlike Ions[END_REF], [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF] [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], Yerokhin et al. (2000) [START_REF] Yerokhin | Two-Photon Exchange Corrections to the 2mathitp 1/2 -2mathits Transition Energy in Li-Like High-mathitZ Ions[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF], [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF] [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF], [START_REF] Blundell | Improved many-body perturbation theory calculations of the n= 2 states of lithiumlike uranium[END_REF] [START_REF] Blundell | Improved many-body perturbation theory calculations of the n= 2 states of lithiumlike uranium[END_REF], [START_REF] Cheng | Screened Lamb shift calculations for lithiumlike uranium, sodiumlike platinum, and copperlike gold[END_REF] [START_REF] Cheng | Screened Lamb shift calculations for lithiumlike uranium, sodiumlike platinum, and copperlike gold[END_REF], Yerokhin et al. (2007) [START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF], [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] [START_REF] Nakano | Resonant coherent excitation of the lithiumlike uranium ion: A scheme for heavy-ion spectroscopy[END_REF] In three-electron systems, the valence-excited transitions of 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 in Li-like ions can be well used to test QED effects and relativistic multi-body effects. A great deal of research has been done in both experiment and theory. For the experiment, generating a basic state 1s 2 2s 2 S 1/2 becomes possible in the heavy ion source such as EBIT. However, it is still impossible to generate and stimulate this state for one and two-electron ions so far, since the production rate of the heavy elements is low [START_REF] Marrs | Production and trapping of hydrogenlike and bare uranium ions in an electron beam ion trap[END_REF][START_REF] Marrs | Production of U92+ with an electron beam ion trap[END_REF][START_REF] Beiersdorfer | Search for 1s2s 3 S 1-1s2p 3 P 2 decay in U 9 0+[END_REF]. At the same time, the experiments have been able to measure the transition energies very accurately using low energy X-ray or deep UV spectroscopy. Therefore, researchers have more measurements with higher accuracy. Accurate measurements of the valence-excited transitions in three-electronic systems can also be used to measure and compare isotope shifts and to obtain the discrepancy of nuclear radii among different isotopes. For example, this was used for different isotopes of Nd 57+ [START_REF] Brandau | Isotope Shift in the Dielectronic Recombination of Three-Electron Nd 57+ A[END_REF].

Theoretical calculations for the valence-excited transitions have been performed in diverse methods. Johnson et al. [START_REF] Johnson | Many-body perturbation-theory calculations of energy levels along the lithium isoelectronic sequence[END_REF] carried out an MBPT calculation with second-and third-order correlation contributions, and without QED contributions. Whereafter, Seely [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF] made a systematic calculation (from Z=24 to 54) with approximate QED contributions, which are calculated using grasp package with MCDF method [START_REF] Dyall | GRASP: A general-purpose relativistic atomic structure program[END_REF]. Indelicato and Desclaux [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] performed an MCDF calculation using MCDFGME code developed by themselves, with the Welton method for screened self-energy corrections. At high-Z ions, these results look very large with lack of some mixed two-photon QED. Kim et al. [START_REF] Kim | Resonance transition energies of Li-, Na-, and Cu-like ions[END_REF] deduced relativistic correlation energies by comparing the MBPT method from Johnson et al. [START_REF] Johnson | Many-body perturbation-theory calculations of energy levels along the lithium isoelectronic sequence[END_REF] and DF results from Indelicato and Desclaux [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF]. The total transition energies were obtained by combining these correlation energies, DF energies, and approximate QED screening corrections. These results look consistent with Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] in low-and middle-Z ions. Ynnerman et al. [START_REF] Ynnerman | Manybody calculation of the 2p1/2,3/2-2s1/2 transition energies in Li-like 238U[END_REF] carried out the coupled-cluster singleand double-excited approximation with the inclusion of higher-order Breit contributions on Li-like U, which is about 0.15 eV lower than Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF]. Chen et al. [START_REF] Chen | Relativistic Configuration-Interaction Calculations for the n=2 States of Lithiumlike Ions[END_REF] also used the coupled-cluster method and solved the Dirac equation by using Bspline as a basis set to evaluate nuclear deformation and higher-order Breit contributions. Later, Johnson [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] extended this method to more ions and added third-order manybody perturbation theory correlations. Safronova and Shlyaptseva [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] employed the 1/Z perturbation theory method with QED and relativistic corrections for 9 ≤ Z ≤ 26. Their calculation results are about 0.07 eV smaller than others. St. Petersburg group [START_REF] Yerokhin | Two-Photon Exchange Corrections to the 2mathitp 1/2 -2mathits Transition Energy in Li-Like High-mathitZ Ions[END_REF][START_REF] Yerokhin | Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions[END_REF][START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF][START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF] successfully performed the calculations using screened self-energy corrections, two-loop QED corrections, two-photon exchange diagrams, and higher-order correlations. Yerokhin et al. [START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF] calculated the electronic-structure for 3 ≤ Z ≤ 92. Gu [START_REF] Gu | Energies of 1s22lq (1 ≤ q ≤ 8) states for Z ≤ 60 with a combined configuration interaction and many-body perturbation theory approach[END_REF] evaluated the level energies of 1s 2 nl states for ions with Z ≤ 60 combining CI and MBPT approach. Cheng et al. [START_REF] Cheng | Quantum Electrodynamic Corrections in High-Z Li-Like and Be-Like Ions[END_REF] made relativistic CI calculations of the transition 1s 2 2p 2 P 3/2 -1s 2 2s 2 S 1/2 on Li-like Bi, Th, and U using the Dirac-Kohn-Sham (DKS) potentials in QED calculations. Sapirstein and Cheng [START_REF] Sapirstein | Determination of the Two-Loop Lamb Shift in Lithiumlike Bismuth[END_REF] evaluated Li-like Bi with Furry or extended Furry representation QED. Finally, Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] performed the calculations for all 10 ≤ Z ≤ 100 using the S-matrix method together with the DKS po- [START_REF] Bosselmann | Measurements of 2s 2s1/2-2p 2p1/2,3/2 Transition Energies in Lithiumlike Heavy Ions. Ii. Experimental Results for Ag44 + and Discussion Along the Isoelectronic Series[END_REF], [START_REF] Widing | The lithium-like 2s 2S-2p 2P transition in solar flares[END_REF] [START_REF] Widing | The lithium-like 2s 2S-2p 2P transition in solar flares[END_REF], [START_REF] Träbert | Extreme ultraviolet spectra of highly charged Xe ions[END_REF] [START_REF] Träbert | Extreme ultraviolet spectra of highly charged Xe ions[END_REF], Denne et al. (1989) [START_REF] Denne | Berylliumlike Mo XXXIV and Lithiumlike Mo XL Observed in the JET Tokamak[END_REF], Bernhardt et al. ( 2015) [START_REF] Bernhardt | Electron-Ion Collision Spectroscopy: Lithium-Like Xenon Ions[END_REF], Denne et al. (1989) [START_REF] Denne | Spectrum Lines of Kr XXVIII-Kr XXXIV Observed in the JET Tokamak[END_REF], Feili et al. (2000) [START_REF] Feili | Measurements of 2s2 S1/2 -2p2 P1/2[END_REF], [START_REF] Kukla | Extreme-Ultraviolet Wavelength and Lifetime Measurements in Highly Ionized Krypton[END_REF] [START_REF] Kukla | Extreme-Ultraviolet Wavelength and Lifetime Measurements in Highly Ionized Krypton[END_REF], [START_REF] Denne | Spectral Lines of Highly-ionized Atoms for the Diagnostics of Fusion Plasmas[END_REF] [START_REF] Denne | Spectral Lines of Highly-ionized Atoms for the Diagnostics of Fusion Plasmas[END_REF], [START_REF] Madzunkov | QED effects in lithiumlike krypton[END_REF] [START_REF] Madzunkov | QED effects in lithiumlike krypton[END_REF], [START_REF] Martin | Désesquelles; «2s-2p transitions in heliumlike and lithiumlike krypton[END_REF] [START_REF] Martin | Measurements of 2 s -2 p Transition Wavelengths in Helium-and Lithium-Like Xenon[END_REF], [START_REF] Staude | Measurements of 2s 2s1/2-2p 2p3/2,1/2 Transition Energies in Lithiumlike Heavy Ions: Experiments and Results for Ni25 + and Zn27 +[END_REF] [START_REF] Staude | Measurements of 2s 2s1/2-2p 2p3/2,1/2 Transition Energies in Lithiumlike Heavy Ions: Experiments and Results for Ni25 + and Zn27 +[END_REF], [START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF] [START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF] [START_REF] Schweppe | Measurement of the Lamb Shift in Lithiumlike Uranium (U89+)[END_REF] [START_REF] Schweppe | Measurement of the Lamb Shift in Lithiumlike Uranium (U89+)[END_REF], [START_REF] Hinnov | Empirical Evaluation of Three-Electron Quantum-Electrodynamics Effects from Lithiumlike Resonance Lines of Elements Z=22-42 in the Tokamak Fusion Test Reactor and Joint European Torus Tokamaks[END_REF] [START_REF] Hinnov | Empirical Evaluation of Three-Electron Quantum-Electrodynamics Effects from Lithiumlike Resonance Lines of Elements Z=22-42 in the Tokamak Fusion Test Reactor and Joint European Torus Tokamaks[END_REF], [START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF] [START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF], [START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF] [START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF], [START_REF] Zacarias | Measurements of 2s-2p transition energies in helium-like and lithium-like nickel[END_REF] [START_REF] Zacarias | Measurements of 2s-2p transition energies in helium-like and lithium-like nickel[END_REF], [START_REF] Biedermann | Extreme ultraviolet spectroscopy of highly charged argon ions at the Berlin EBIT[END_REF] [START_REF] Brandau | Precise Determination of the 2s1/2-2p1/2 Splitting in Very Heavy Lithiumlike Ions Utilizing Dielectronic Recombination[END_REF] [START_REF] Brandau | Precise Determination of the 2s1/2-2p1/2 Splitting in Very Heavy Lithiumlike Ions Utilizing Dielectronic Recombination[END_REF], [START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF] [START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF], [START_REF] Silwal | Identification and plasma diagnostics study of extreme ultraviolet transitions in highly charged yttrium[END_REF] [START_REF] Silwal | Identification and plasma diagnostics study of extreme ultraviolet transitions in highly charged yttrium[END_REF], [START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF] [START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF], [START_REF] Reader | Extreme ultraviolet spectra and analysis of Δn = 0 transitions in highly charged barium[END_REF] [START_REF] Reader | Extreme ultraviolet spectra and analysis of Δn = 0 transitions in highly charged barium[END_REF].

- - - - - - - - ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) + + ( ) ( )
+ + + ) ( )
tentials. Currently, this result can be compared to our calculation results as a complete reference.

A detailed comparison between our works and other different calculations mentioned above is shown in Fig. 5.6 and Fig. 5.8. The references are single-configuration Dirac Fock (DF). In these figures, we present our calculations with Welton model, effective operator model, and different electronic correlations, respectively. We can notice that the results for lack of proper QED corrections from Seely [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF] are quite far away from other calculations. This difference gets more and more significant with the increase of Z. The results of different calculations are in good agreement in low-and middle-Z ions. Notably, the transition energies coming from Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] appear to scatter around our calculations in heavy lithium-like ions (Z=90 and 92) and the deviation is as large as 5.9 for the 1s [START_REF] Bombarda | Dielectronic satellite spectrum of heliumlike argon: A contribution to the physics of highly charged ions and plasma impurity transport[END_REF] to measure core-excited ions. Some middle-ions results comes from beam-foil excited measurements [START_REF] Bosselmann | Measurements of 2s 2s1/2-2p 2p1/2,3/2 Transition Energies in Lithiumlike Heavy Ions. Ii. Experimental Results for Ag44 + and Discussion Along the Isoelectronic Series[END_REF][START_REF] Feili | Measurements of 2s2 S1/2 -2p2 P1/2[END_REF][START_REF] Büttner | Measurement of the 2s2S12-2p2P32 wavelength in Li-like nickel[END_REF]. Some available results come from EBIT experiments [START_REF] Zhang | Measurement of the QED energy shift in the 1 s 2 2 p 3/ 2-1 s 2 2 s 1/ 2 x-ray transition in Li-like Pb 79+ 208[END_REF][START_REF] Beiersdorfer | Structure and Lamb shift of 2 s1/2-2 p3/2levels in lithiumlike Th87+through neonlike Th80+[END_REF][START_REF] Beiersdorfer | Measurement of the Two-Loop Lamb Shift in Lithiumlike U[Sup 89+[END_REF][START_REF] Beiersdorfer | Structure and Lamb shift of 2 s 1/2-2 p 3/2 levels in lithiumlike U 8 9+ through neonlike U 8 2+[END_REF] at high-Z ions. Recently, ECRIS connected with a vacuum double crystal spectrometer can provide reference-free measurements [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF]. There are some precise measurements in the middle-and high-Z ions, which are meaningfully compared with the theory. Our results, whether with Welton models or effective operator model generally agree well with the experiment, except for Z=90 and Z=92. The difference is as large as 0.53 eV with the experimental error bar of 0.152 eV at Z=90. Different electron correlation effects are also shown in all the figures corresponding to the Table . 10 and Table .11 in Appendix A 6.5. The comparison between DF calculations with MCDF values shows that the total correlation effects reduce the transition energies. At low-Z ions, the size of total correlation is within -0.2 eV and this contribution can reach -0.48 eV at high-Z ions in transition 1s 2 2p 2 P 1/2 -1s 2 2s 2 S 1/2 . While the correlation no longer causes the transition energy to decrease after Z=70 of transition 1s 2 2p 2 P 3/2 -1s 2 2s 2 S 1/2 . This is because the magnetic correlation, which increases the transition energies, begins to get larger than other correlations.

1s 2s 2p

2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 and 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2 transitions - - - ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) + ( ) + + ( ) ( ) + + + ( )

Fig. 5.10 Comparison of the transition energy between theoretical values and the present MCDF calculations for the 1s 2s 2p

2 P 1/2 → 1s 2 2s 2 S 1/2 transition.
The reference value is obtained with the new effective operator method. Theoretical works: Zhang et al. ( 2011) [START_REF] Li | Relativistic configuration interaction calculations on Kα x-ray satellites of krypton[END_REF], Gorayev et al. ( 2017) [START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF], [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF] [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF], [START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], [START_REF] Feng | Wavelengths, Oscillator Strengths and Radiative Transition Rates for K α Lines in Titanium X-Ray Spectra[END_REF] [START_REF] Feng | Wavelengths, Oscillator Strengths and Radiative Transition Rates for K α Lines in Titanium X-Ray Spectra[END_REF], [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF] [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF], [START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266], [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF], [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF], [START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF] [START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF], [START_REF] Shuqiang | Properties of the K α and K β x-ray transitions in CuXX through CuXXVIII[END_REF] [START_REF] Shuqiang | Properties of the K α and K β x-ray transitions in CuXX through CuXXVIII[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Whiteford | Excitation of Ar 15+ and Fe 23+ for Diagnostic Application to Fusion and Astrophysical Plasmas[END_REF] [START_REF] Whiteford | Excitation of Ar 15+ and Fe 23+ for Diagnostic Application to Fusion and Astrophysical Plasmas[END_REF], [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] - 2018) [START_REF] Machado | High-precision measurements of n = 2 → n = 1 transition energies and level widths in He-and Be-like argon ions[END_REF], [START_REF] Aglitskii | Observation in laser plasmas and identification of dielectron satellites of spectral lines of hydrogen-and helium-like ions of elements in the Na-V range[END_REF] [START_REF] Aglitskii | Observation in laser plasmas and identification of dielectron satellites of spectral lines of hydrogen-and helium-like ions of elements in the Na-V range[END_REF], [START_REF] Hsuan | Satellite spectra of heliumlike nickel[END_REF] [START_REF] Hsuan | Satellite spectra of heliumlike nickel[END_REF], [START_REF] Decaux | High-Resolution Measurement of the Kα Spectrum of Fe XXV-XVIII: New Spectral Diagnostics of Nonequilibrium Astrophysical Plasmas[END_REF] [START_REF] Decaux | High-Resolution Measurement of the Kα Spectrum of Fe XXV-XVIII: New Spectral Diagnostics of Nonequilibrium Astrophysical Plasmas[END_REF], [START_REF] Wargelin | Dielectronic Satellite Contributions to Ne Viii and Ne Ix K-Shell Spectra[END_REF] [START_REF] Wargelin | Dielectronic Satellite Contributions to Ne Viii and Ne Ix K-Shell Spectra[END_REF], [START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF] [START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF], [START_REF] Smith | Measurement of Doubly Excited Levels in Lithiumlike and Berylliumlike Cobalt[END_REF] [START_REF] Smith | Measurement of Doubly Excited Levels in Lithiumlike and Berylliumlike Cobalt[END_REF], [START_REF] Decaux | Modeling of high-resolution K α emission spectra from Fe XVIII through Fe XXIV[END_REF] [START_REF] Decaux | Modeling of high-resolution K α emission spectra from Fe XVIII through Fe XXIV[END_REF], [START_REF] Tarbutt | Wavelength Measurements of the Satellite Transitions to the N = 2 Resonance Lines of Helium-Like Argon[END_REF] [START_REF] Tarbutt | Wavelength Measurements of the Satellite Transitions to the N = 2 Resonance Lines of Helium-Like Argon[END_REF], [START_REF] Biedermann | Line Ratios and Wavelengths of Helium-Like Argon N=2 Satellite Transitions and Resonance Lines[END_REF] [START_REF] Biedermann | Line Ratios and Wavelengths of Helium-Like Argon N=2 Satellite Transitions and Resonance Lines[END_REF], [START_REF] Rice | X-ray observations of Ca 19 + , Ca 18 + and satellites from Alcator C-Mod tokamak plasmas[END_REF] [START_REF] Rice | X-ray observations of Ca 19 + , Ca 18 + and satellites from Alcator C-Mod tokamak plasmas[END_REF], [START_REF] Beiersdorfer | Dielectronic satellite spectrum of heliumlike vanadium[END_REF] [START_REF] Beiersdorfer | Dielectronic satellite spectrum of heliumlike vanadium[END_REF], [START_REF] Rudolph | X-Ray Resonant Photoexcitation: Linewidths and Energies of K alpha Transitions in Highly Charged Fe Ions[END_REF] [START_REF] Rudolph | X-Ray Resonant Photoexcitation: Linewidths and Energies of K alpha Transitions in Highly Charged Fe Ions[END_REF], [START_REF] Rice | X-ray observations of helium-like scandium from the Alcator C-Mod tokamak[END_REF] [START_REF] Rice | X-ray observations of helium-like scandium from the Alcator C-Mod tokamak[END_REF] Table 5.8 -Comparison of the transition energy between experiment and the present MCDF calculations using Welton model and effective operators, respectively, for the 1s 2s 2p The precise knowledge of core-excited is necessary to explain astrophysics spectrum, and the states benefit the prominent K-shell emission lines, which can be observed in the spectra of almost all types of cosmic x-ray sources [START_REF] George | X-ray reflection from cold matter in active galactic nuclei and X-ray binaries[END_REF]. Another important objective in the study of K-shell emission [START_REF] Beiersdorfer | Structure and Lamb shift of 2 s 1/2-2 p 3/2 levels in lithiumlike U 8 9+ through neonlike U 8 2+[END_REF][START_REF] Widmann | Studies of He-like krypton for use in determining electron and ion temperatures in very-high-temperature plasmas[END_REF] is its use in the thermal laboratory plasma diagnosis, especially those in magnetic confinement fusion studies. For the study of coreexcited states 1s 2s 2p, there are not much reference results in theory, especially in the experiment. The exact theoretical description and calculation to the transitions involving core-excited states are complicated by the following two aspects. The first is a huge contribution from the QED effects. In fact, QED contributions are the strongest for K shell electrons, so they are also strong for the K shell transition. The second can be summarized as core-excited states are mainly the autoionizing states. This represents an intensive mixing of reference state with a tight continuous continuum of a single excited state, for example consisting of a continuous closed core and electrons. In theoretical calculations, the interaction with the continuum cannot be accurately calculated and described [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF][START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF][START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF][START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF].

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) + ( ) + + ( ) ( ) + + + ( )

Fig. 5.11 Comparison of the transition energy between experiment and the present MCDF calculations, using effective operators for the 1s 2s 2p

- ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) + ( ) + + ( ) ( ) + + + ( )
Fig. 5.12 Comparison of the transition energy between theoretical values and the present MCDF calculations for the 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transition. The reference value is obtained with the new effective operator method. Theoretical works: Yerokhin and Surzhykov (2012) [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Two-loop self-energy in the Lamb shift of the ground and excited states of hydrogenlike ions[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF], Gorayev et al. ( 2017) [START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF], [START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF] [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF], [START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266], [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF], [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF], [START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF] [START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF], [START_REF] Shuqiang | Properties of the K α and K β x-ray transitions in CuXX through CuXXVIII[END_REF] [START_REF] Shuqiang | Properties of the K α and K β x-ray transitions in CuXX through CuXXVIII[END_REF], [START_REF] Dong | Correlation between muonic levels and nuclear structure in muonic atoms[END_REF] [START_REF] Li | Relativistic configuration interaction calculations on Kα x-ray satellites of krypton[END_REF], [START_REF] Whiteford | Excitation of Ar 15+ and Fe 23+ for Diagnostic Application to Fusion and Astrophysical Plasmas[END_REF] [START_REF] Whiteford | Excitation of Ar 15+ and Fe 23+ for Diagnostic Application to Fusion and Astrophysical Plasmas[END_REF], [START_REF] Feng | Wavelengths, Oscillator Strengths and Radiative Transition Rates for K α Lines in Titanium X-Ray Spectra[END_REF] [START_REF] Feng | Wavelengths, Oscillator Strengths and Radiative Transition Rates for K α Lines in Titanium X-Ray Spectra[END_REF], [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] Vainshtein and Safronova [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF] were the first to obtain core-excited states of Li-like ions for atomic numbers Z=4 to 34 within the 1/Z expansion method. Bhalla and Tunnell [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] calculated the excited electron configurations of lithiumlike argon without QED and Breit interaction. Later, Chen et al. [266,[START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] addressed 1s2s2p configuration of Li-like ions (13 ≤ Z ≤ 30) using MCDF approach and the Moller two-electron operator with a generalized Breit and QED corrections, which results show smaller than others. Later, Safronova and Shlyaptseva [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] calculated the inner-shell excitation for the nuclear charge Z=6-54 by using the 1/Z perturbation theory method with relativistic and first and second-order corrections. More recently, these calculations have been reevaluated by Gorayev et al. [START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF] for 6 ≤ Z ≤ 36 based on the same method by considering the first-order correction to the power of 1/Z. The results of 1/Z perturbation methods are still far from those of other methods. Then, Safronova and Shlyaptseva [START_REF] Safronova | Relativistic Many-Body Calculations of E1, E2, M1, and M2 Transitions Rates for the 1 S 2 L' 2 L' ' -1 S 2 2 L Lines in Li-Like Ions[END_REF] addressed RMBPT, including second-order Breit-Coulomb interactions. These states were addressed by Gang Jing group [START_REF] Shuqiang | Properties of the K α and K β x-ray transitions in CuXX through CuXXVIII[END_REF][START_REF] Feng | Wavelengths, Oscillator Strengths and Radiative Transition Rates for K α Lines in Titanium X-Ray Spectra[END_REF][START_REF] Li | Relativistic Configuration Interaction Calculations on K α X-Ray Satellites of Krypton[END_REF], who calculated Li-like-Cu, Ti, and Kr using GRASPVU within the MCDF approach. Santos et al. [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] calculated Li-like-praseodymium using MDFGME code developed by Desclaux and Indelicato [START_REF] Desclaux | A Multiconfiguration Relativistic Dirac-Fock Program[END_REF][START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF][START_REF] Indelicato | MCDF studies of two electron ions II: Radiative corrections and comparison with experiment[END_REF]. Yerokhin and Surzhykov [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF] calculated all n = 2 core-excited states with a relativistic CI method for lithiumlike ions starting from argon (Z=18) and ending with krypton (Z=36). Then, Yerokhin et al. [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF] used the same method to calculate energy levels of the 1s 2 2l and 1s2l2l states of ions along the lithium isoelectronic sequence from carbon to chlorine. In 2018, Yerokhin and Surzhykov [START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] supplemented new calculations of Li-like ions from argon to uranium using the CI method with two-loop QED and nuclear recoil in QED part. Marques et al.

[267] calculated Li-like Nd, Sm, Gd, Dy, Er, and Yb with Welton model using the same code with us.

We perform the comparisons between different calculations are shown in Fig. 5.10, 5.12 for the 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 transition with DF Welton model as references and Fig. 5.14, 5.16, 5.18 for the 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2 transitions with effective operator as references. Our calculations start from neon (Z=10) to uranium (Z=92) with missing some ions in core-excited transiton of 1s 2s 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 due to bad converge. The corresponding tables are also listed in Tables 5.8, 5.9, 5.11, 5.10. It is noticed that the results from 1/Z perturbation method [START_REF] Vainshtein | Wavelengths and Transition Probabilities of Satellites to Resonance Lines of H-and He-Like Ions[END_REF][START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF] are far away from the CI and MCDF methods and the size of difference can reach as large as 4 eV at Z=36. Another large difference comes from Chen et al. [266,[START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF], [START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266], [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF], [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF], Gorayev et al. ( 2017) [START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF], [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] Nilsen et al. [START_REF] Nilsen | Dielectronic Satellite Spectra for Helium-Like Ions[END_REF] with generalized Breit interaction and QED corrections, which are about 2 eV smaller than those of other calculations. Except for these deviations, other calculations generally agree with each other. We also give a detailed comparison with the CI calculations described by Yerokhin et al. [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF][START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF], which are considered accurate and complete evaluations to the core-excited states. Our calculations have a good agreement with the latest results of Yerokhin et al. [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF]. There is just a shift of 0.4 eV at high-Z ions, within the uncertainty of 0.5 eV for their theory error. We use the same method for the screened QED and the same nucleus size, expect for Z=90 with the difference of nuclear radius of 0.0002 fm. They also provided the uncertainty of 0.95 eV from the nuclear size at Z=90. Our calculations of transition 1s 2s 2p 4 P 5/2 → 1s 2 2s 2 S 1/2 compare with Yerokhin's results shown in Table 5.11, in which deviations are 2.6 eV at Z=50 and 1.9 eV at Z=59. This is because the convergence is not achieved at n=5, with errors of 0.56 eV and 4.7 eV, respectively.

The results of Welton model are almost identical to the results of the new effective operator method. However, at Z=50, the value of Welton model starts to move away from the effective operator method following the increase with Z. The results show that the effective operator model is in better agreement with the experimental results at high-Z ions for valence-excited transitions. But for core-excited transitions, we can't judge which model is better because there is no experiment as a reference in the high-Z ions.

In experiments, all measurements are concentrated between Z=10 and Z=30, except that only praseodymium ion has been measured with a large uncertainty of 8.5 eV using SuperEBIT. Aglitskii et al. [START_REF] Aglitskii | Observation in laser plasmas and identification of dielectron satellites of spectral lines of hydrogen-and helium-like ions of elements in the Na-V range[END_REF] give observations of Li-like Z=11-23 using laser plasmas. Schlesser et al. [START_REF] Schlesser | High-accuracy x-ray line standards in the 3-keV region[END_REF] obtained transition energies in Li-like Z=16-18 within the accuracy of 10 ppm. Li-like Fe is identified by Decaux et al. [START_REF] Decaux | High-Resolution Measurement of the Kα Spectrum of Fe XXV-XVIII: New Spectral Diagnostics of Nonequilibrium Astrophysical Plasmas[END_REF] using EBIT technique. The experimental values of several other elements of lithiumlike ions of Ar 15+ , Sc 18+ , V 20+ , Cr 21+ , and Mn 23+ from T.F.R. group et al. [START_REF] Tfr Group | Charge-dependent wavelength shifts and line intensities in the dielectronic satellite spectrum of heliumlike ions[END_REF], Sc 18+ from Rice et al. [START_REF] Rice | X-ray observations of helium-like scandium from the Alcator C-Mod tokamak[END_REF] with accurate to ±0.1m

• A and Fe 23+ from Rudolph et al. [START_REF] Rudolph | X-Ray Resonant Photoexcitation: Linewidths and Energies of K alpha Transitions in Highly Charged Fe Ions[END_REF] with the 60 meV energy error bar. For the forbidden M2 transition of 1s 2s 2p 4 P 5/2 → 1s 2 2s 2 S 1/2 , only Li-like Ar been measured by Dohmann et al. [START_REF] Dohmann | High resolution spectroscopy of prompt and metastable decaying levels in highly ionized argon, especially of the metastable 3 P 2-state of Ar 16+ and the 4 P 5/2-state of Ar 15+[END_REF][START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF]. Recently, our group measured this transition in both Li-like S and Ar with a double-flat crystal spectrometer without the use of any reference line. The transition energy measurements are performed with accuracies ranging from 2.3 ppm to 6.4 ppm depending on the element and line intensity. Our theoretical values are in agreement with our experimental value.

The comparison of core-excited transitions between the available experimental results and calculations are presented in Table 5.8, 5.9 and 5.11. There is a shift in some ions Yerokhin and Surzhykov (2012) [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF], [START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266], [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF], [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF] [START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF], Gorayev et al.

- - - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
(2017) [START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF], [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF] [START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF].

between measurements and calculations. Especially, the experimental values from Tarbutt et al. [START_REF] Tarbutt | Measurement of the ground-state Lamb shift of hydrogen-like Ti21+[END_REF] and Biedermann et al. [START_REF] Biedermann | Line Ratios and Wavelengths of Helium-Like Argon N=2 Satellite Transitions and Resonance Lines[END_REF], which are 3.8 eV and 3.3 eV larger than the theory calculations at transition 1s 2s 2p 4 P 3/2 → 1s 2 2s 2 S 1/2 of Ar 15+ ion.

In the low-and middle-Z ions, all theories are in good agreement with each other, except for 1/Z expansion method, which is far away from other theories. It can be seen that the present MCDF calculation results have a small shift with the values from the advanced RMBPT, RCI and S-matrix methods at high-Z ions due to different finite nuclear sizes, nuclear polarization and the errors from our calculation convergence. Comparing the current works with other calculations for the different contributions to the transition energy shows the importance of the QED screening correction for rigorous processing. The difference between Welton model and effective operator model becomes larger in core-excited states of 1s2s2p than 1s 2 2p at high-Z ions.

1s 2s 2p

4 P J → 1s 2 1 S 0 , J = 1/2, 3/2 transition
This thesis follows the technique described in the reference [START_REF] Howat | Relaxation and final-state channel mixing in the Auger effect[END_REF] and calculates the Auger transition energy (see section 2.10) at the 1s2l2l level by using the MCDFGME code. We combine this technique with a fully correlated wave function until n = 5. The table 5.5 gives the convergence of the transition energy. Table 5.12 gives the comparison of present works of the effective operator and Welton picture, respectively, with the CI method from Yerokhin et al. [START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] of 1s2l2l states to ground state in Li-like Z from 18 to 92. The results of Auger transitions show the same trend as the radiative transitions. The Welton models are still 1.2 eV higher than the effective operator model at Li-like Uranium. At high-Z ions, the deviation is reached 0.2 eV with the results of Yerokhin et al. [START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF].

Radiative transition probability

Radiative transition probabilities are evaluated using the multipole expansion of the matrix element, as mentioned in section 2.9, for Li-like system. The initial and final state orbitals are fully relaxed, we use final-state channel mixing and take into account the non-orthogonality between the fully relaxed orbitals in the initial and final state, following [START_REF] Indelicato | Radiative de-excitation of the 1s22s3p 3P0 level in beryllium-like ions: A competition between a E2 and a two-electron one-photon E1 transitions[END_REF][START_REF] Löwdin | Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction[END_REF].

--

.18 Comparison of the transition energy theoretical values and the present

MCDF calculations for the 1s 2s 2p 4 P 5/2 → 1s 2 2s 2 S 1/2 transition. The reference value is obtained with the new effective operator method. Theoretical works: [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266], [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF] [START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF], [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF] [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF].

The radiative transition probabilities are plotted in Fig. 5.20 and Fig. 5.21. The single-configuration(DF) and multi-configuration(MCDF) with n = 2, 3, 4, 5 are presented separately in the figures. We adopt singly, doubly, and triply excited configurations up to 5g orbitals in order to ensure the convergence of MCDF calculation results. The comparison of DF and MCDF shows that multi-configurations do not cause big shifts to the radiative transition probabilities.

Errors and uncertainty

The numerical accuracy of the MCDF method can be easily controlled by changing the self-consistency criteria and mesh size. For all results given here, the numerical error is less than 0.003 eV [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF]. When we increase the correlation orbitals n to the maximum, the convergence of the correlated energy is kept within 0.1 eV. From the comparison between our calculations and experiments, we have found that the effective operator model can improve the accuracy of the total transition energy by 0.002% for high-Z ions at valenceexcited transitions.

The uncertainties of the calculated transition energies come from the omitted high- 2002) [START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF], [START_REF] Dohmann | High resolution spectroscopy of prompt and metastable decaying levels in highly ionized argon, especially of the metastable 3 P 2-state of Ar 16+ and the 4 P 5/2-state of Ar 15+[END_REF] [START_REF] Dohmann | High resolution spectroscopy of prompt and metastable decaying levels in highly ionized argon, especially of the metastable 3 P 2-state of Ar 16+ and the 4 P 5/2-state of Ar 15+[END_REF].

order QED effects, for example, the incomplete three-and more-photon correlations, in addition to the uncertainty of the Dirac-Coulomb-Breit energy. At middle-to high-Z, the main sources of uncertainty are due to uncertainties in the nuclear radii. Chapter 6

Analysis for muonic atoms

The nuclear charge-density distribution can be determined by the experiments of electron scattering and muonic x-rays or simultaneous analyses of the two experimental data, which can improve the nuclear information significantly.

There will be an upcoming experiment to perform precise measurement of the absolute nuclear charge radii of radioactive 226 Ra with a level of 0.2% relative precision by muonic x-rays at the Paul Scherrer Institute(PSI). Therefore, in order to prepare theoretical data for muonic 226 Ra and 248 Cm, we first analyze muonic 208 Pb in detail to test the reliability of our theory.

This section gives theoretical calculations for the transition energies of heavy muonic atoms and analyzes the corresponding contributions. This calculation can be used to determine and further improve nuclear parameter values combined with experimental data. The calculations take into account the finite size effects and QED effects. In addition, the effects of the surrounding atomic electrons and nuclear polarization are also considered.

For the analysis of the nuclear charge distribution in the two-parameter Fermi model, the half density parameter c and surface thickness parameter t, are varied around 6.6 fm and 2.3 fm, respectively, until a minimum χ 2 fit between theoretical and experimental transition energies are obtained.

Energy levels

When the nuclear charge parameters are fitted to the experimental transition energy, it is necessary to evaluate the QED corrections and the nuclear polarization correction. Then we study a set of muonic energy levels with the finite nuclear size. The calculations include all one-loop and the main two-loop QED contributions. A description of the method is given in Ref. [START_REF] Indelicato | Nonperturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N=2 Lamb Shift and Hyperfine Structure[END_REF]. These energies can be evaluated by exact numerical solution of the Dirac equation, which includes the lowest order vacuum polarization potential of order α(Zα). Table 6.3 lists the energy levels and radiation corrections. The maximum relative size of QED corrections to the energies is about 0.6% at 1s level for muonic lead. As the level becomes higher, the QED contributions get smaller.

In particular, the vacuum polarization is caused by the virtual electron-positron pair, which modifies electric interaction between the nucleus and muon. The electron loop part ), which decreases the transition energies. This correction accounts for about 1% to 1.8% of the total vacuum polarization for these states listed in Table 6.3 for the muonic lead. We study the contributions of Kallen and Sabry (see section 2.4), a fourth-order potential of α 2 (Zα) using our numerical wave functions with good accuracy. We also consider the vacuum polarization due to the generation of the virtual muon positron pairs. The term named loop after loop Uëhling (see section 2.4) only accounts for 0.4% of the Uëhling correction.

Concerning the muonic self-energy correction, Indelicato and Mohr [START_REF]Self-energy corrections for muonic atoms[END_REF] have calculated the exact one-loop self-energy with the finite-size contribution in the framework of nonrelativistic prescription for the first time. Semianalytic expansion of F (Zα) as the RMS dependence is plotted in Fig. 6.1. They have extended the work described in Refs. [START_REF] Mohr | Nuclear Size Correction to the Electron Self-Energy[END_REF][START_REF] Beier | Influence of nuclear size on QED corrections in hydrogenlike heavy ions[END_REF][START_REF] Mohr | QED Corrections in Heavy Atoms[END_REF] to muonic atoms. In Table 6.1, we listed their calculation of self-energy corrections with leading order α (Zα) 4 and compared with the results of Akihiro Haga et al. [START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF]. In their estimation, self-energy correction can be divided into two contributions, namely a low-energy term with the nonrelativistic multipole expansion and a high-energy term with only the lowest order of the external field using the mean value method. The result from Indelicato and Mohr [START_REF]Self-energy corrections for muonic atoms[END_REF] is 0.1 keV higher than Akihiro Haga at 2p 1/2 level. The self-energy generally increases the energy level, which is around 3.2 keV for the lowest Table 6.2 -Nuclear polarization corrections for muonic 208 Pb. This work is calculated using RURP code from Rinker and Speth [START_REF] Rinker | Nuclear polarization in muonic 204, 206, 207, 208Pb in the random-phase approximation[END_REF]. All energies are in keV.

This work

Akihiro Haga [ 208 Pb. The higher-order α 2 (Zα) combined vacuum-polarization correction has been considered in their analysis. The relativistic recoil corrections were considered with order (Zα) 5 and to all orders in mu mp by performing the direct numerical evaluation for the Dirac equation, the same method as we do.

However, the largest theoretical uncertainty comes from the calculation of nuclear polarization. Akihiro Haga also reanalyzed the nuclear polarization in 90 Zr and 208 Pb. They used three forms of transition densities, including Jensen-Steinwedel (JS) [START_REF] Bohr | Nuclear Structure[END_REF][START_REF] Plunien | Nuclear polarization contribution to the Lamb shift in heavy atoms[END_REF][START_REF] Plunien | Nuclear Polarization in Heavy Atoms and Superheavy Quasiatoms[END_REF], Rinker (RIN) [START_REF] Rinker | Nuclear polarization in muonic 204, 206, 207, 208Pb in the random-phase approximation[END_REF] and Tassie-Goldhaber-Teller (TGT) [START_REF] Tassie | A Model of Nuclear Shape Oscillations for g? Transitions and Electron Excitation[END_REF][START_REF] Bohr | Nuclear Structure[END_REF] models, in which the calculation results of the JS model are reasonable. So we choose the JS result as a reference. We also calculate the densities using the program of RURP from Rinker and Speth [START_REF] Rinker | Nuclear polarization in muonic 204, 206, 207, 208Pb in the random-phase approximation[END_REF], which is described in detail in section 2.7. Bergern et al. [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] give a set of best-fit nuclear polarization correction values of the low-lying muonic level, which are very suitable for experimental transition energies. The comparison of our calculations and other two theoretical results are presented in Table 6.2. For 208 Pb, we obtain -3.924 keV for the 1s orbital. This is consistent with the RIN model value of -3.599 keV, and the JS model gave a lower value of -5.721 keV. Bergern also gives a value of -4.252 keV, which is lower than our result, including nuclear excited states of both continuous (high-lying) and discrete (low-lying). They fixed the 1s correction at its theoretical fit. In addition, the nuclear polarization values of the two 3d states are also fixed due to the small shifts. At the same time, the nuclear polarization of 2s level is shown of -0.964 keV, slightly above the estimated value of -0.816 keV, which is adjusted simultaneously for nuclear polarization in the 2p and 3p levels. The nuclear polarization effects decrease the energy levels as a relatively large correction.

However, with the improvement of measurement accuracy, there is a certain difference between the calculation result in the nuclear polarization correction and experimental value. The difference in nuclear polarization correction for the muonic x-ray is that this correction of p 1/2 is larger than the p 3/2 levels in theory predicts, while the analysis from experiments gives the opposite result. The consistency between the measured and calculated transition energies requires the magnitude of the corresponding nuclear polarization value to be inverted. Furthermore, the same phenomenon can be found in the 2p state, and the 3p state and is also presented in both papers of Haga and Bergem. The same kind of discrepancies are discovered not only in the muonic 208 Pb but also in the muonic 90 Zr. The difference of nuclear polarization energy shift is about 0.036 keV for the 2p level, whereas the difference is 0.2 keV for the 3p level. We find that in the fitting data of Bergem, there is the same magnitude of influence in the 3p and 2p splittings.

In view of the large theoretical uncertainties, the nuclear polarization calculations can only give orders of magnitude, not exact results.

Effects from remaining electrons

Effects caused by the electrons refilling the orbitals should also be taken into account. The static electron effect dominates these corrections between the muon and the nucleus. Table 6.4 shows the effects on the transition energy of the muon by the surrounding atomic electrons. Small results of electron effects show that muon is less constrained by the presence of the effect. Since the wave function of the 1s electron has the largest overlap with the muon, it contributes to the main effect. Table 6.4 -The effects from remaining electrons to the transition energies in muonic 208 Pb. For the superscript (1), only the 1s electrons are considered, while for (1+2), all electrons from the first and second shells are considered, for (1+2+3), all electrons from the first, second and third shells are considered. All energies are in keV. The effect of remaining electrons is a basic electromagnetic effect and easy to calculate in principle. The key issue comes from the lack of an in-depth understanding of the state of the electrons in the muonic atoms during the X-ray transitions. This Auger transitions may occur, leading to ionization and excitation, and thus an unknown resultant electronic configuration. In addition to relying on the X-ray and Auger transition rates, the atomic ionization state relies on the formation of atoms and the ability of this system to regain the lost electrons from the surrounding environment. As a result, it is difficult to judge how many electrons are filled in the electronic shells.

The effects of remaining electrons to the 1s to 5f energy levels for muonic 208 Pb with the muon and 2, 10, 18 electrons has been obtained in Table 6.4, separately. For example, the effects to the transition 5f 5/2 -3d3/2, 5f 7/2 -3d5/2, and 5f 5/2 -3d5/2 are, respectively, -0.172 keV, -0.173 keV, and -0.171 keV in row 10 and 11, and 12 when 18 electrons are included. The effects for three of the transitions are significant due to the large effect on level 5f. However, there is only -0.006 keV value on transition 2p1/2-1s1/2. It can be found that for low-level muonic states, the effects are very small. The effect from K-shell occupies approximately 80% in this total effect, each L shell electrons 14%, and each M shell only 4% in the muonic lead.

Best fit nuclear parameter values

Firstly, in Table 6.5, we list the transition energies for the uniform sphere and Gaussian models compared with the Fermi model with the same RMS radius. The value of the three-parameter Gaussian model is very close to the Fermi model. In contrast, the uniform model has a big difference, more than 50 keV for the 2p 1/2 -1s 1/2 transition, in which the 1s and 2s states are very sensitive to the internal part of nuclear charge distribution. As a first step, we select a few sets of meaningful nuclear parameters according to the RMS given by Angeli [185], and corresponding to some sets of calculated transition energies for two-parameter fermi and three gauss models. Next, we give a fit and analysis using these energies combined with the experimental data. Our fit includes 18 transition energies, namely nine M-lines, five L-lines, two K α -lines, and two transitions concerning 2s level. These data include 14 energy levels from 1s to 5f . We report on the nuclear parameters that minimize the weighted theory-experiment distance. This experiment [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] obtained an accuracy of up to 11 ppm, which is a factor of five improvements over the result of Kessler et al. [START_REF] Kessler | Muonic x rays in lead isotopes[END_REF].

The values of nuclear polarization effects and self-energy correction calculated by Haga are adopted directly. We have also calculated them respectively introduced in section 6.1. For the two-parameter Fermi model, we performed a polynomial fit comparing with the experimental values. When the parameters are r = 5.5057 fm, and t = 2.3919 fm, the m i n im umo ft h ec h i -s q u a r e i s χ 2 =1 1 6 0. 6 3f o r1 4d e g r e e so f f r e e d om . T h ee r r o rb a r sa r e r e r r o r =0. 0 0 3fma n dt e r r o r =0. 0 2 6fm . T h ee r r o r so b t a i n e d i nt h efi t t i n gc om ef r om t h ec om b i n a t i o no ft h e o r e t i c a lu n c e r t a i n t i e sa n de x p e r im e n t a le r r o rb a r s . T h efi to ft h e d a t ai sn o tg o o d . T h i sl e a d st od o u b t sa b o u tt h e m o d e ld e p e n d e n c ef r omt h ec h a r g e d i s t r i b u t i o no ft h en u c l e u s w h e no n et r i e st ofi tt h ed a t a . S o , w ec o n t i n u et ofi t w i t h t h r e eg a u s s m o d e l , i nw h i c ht h eRM Sr a d i u s i sd r a s t i c a l l y im p r o v e d . O u rfi tr e s u l t sa r e c=6. 1 7 7 8fm , t =2. 9 7 0 6fm ,a n d w=0. 3 7 9 0fm ,w i t h r=5. 5 0 3 1fm . T h i sa g r e e sw i t h t h ee x p e r im e n t a lr e s u l t( 5 . 5 0 3 1 ( 1 1 ) fm )g i v e nb yB e r g e r n ,b u tt h et r a n s i t i o ne n e r g i e sa r e s t i l ln o tc o n s i s t e n t w i t ht h ee x p e r im e n t . S o , w ec o n t i n u et oc o n s i d e rt h ee ff e c t sf r om 2a n d1 8r em a i n i n ge l e c t r o n s ,r e s p e c t i v e l y . T h et r a n s i t i o ne n e r g i e sh a v eb e e ng r e a t l y im p r o v e d f o rh i g h -s t a t e ,b u t i t i su s e l e s s f o r l o w -s t a t e .I th a sb e e n f o u n dt h a tt h e l a r g e s t d e v i a t i o nb e t w e e nt h ee x p e r im e n ta n dc a l c u l a t i o nv a l u e so c c u r s i nt h e K αt r a n s i t i o n .I n o u rc u r r e n tw o r k ,t h i sd e v i a t i o n r e a c h e d0 . 6k e V .A n o t h e rd i s c r e p a n c y i s0 . 4 5k e V f o r t h e 4 d-3 pt r a n s i t i o n s . Although the radius is improved from Gauss model fit, it does not provide any improvement for the transition energy. In addition, χ 2 in both fits turn out to be very large. One reason might be some important effects of nuclear polarization. In Table 6.6, we give the uncertainty of nuclear polarization between JS model and RIN model. The deviations of the calculation results from the two models are within the uncertainty of nuclear polarization at low-states. Another reason is that too many energy levels involved in the fitting fail to achieve convergence. In addition to nuclear polarization shifts, which must be applied to the muonic energy levels, other corrections of first-order vacuum polarization can be excluded, due to accurate calculation. The self-energy would affect the 2p levels by 0.7 keV, and the 3p levels by 0.2 keV, while nuclear polarization amounts to -2.2 keV and -1.2 keV, respectively. Regarding higher-order vacuum polarization corrections, these values are just 0.4 keV. In addition, it may be important to consider high-order corrections of nuclear polarization. P. Bergern et al. [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF] obtained charge moments and nuclear polarization from precise experimental measurements of muonic X-rays. The fit just using Fermi charge distribution as free parameters is very poor, in which the χ 2 is 187 for 208 Pb. If fitting includes more free variables of eight nuclear polarization parameters, 13 low-lying lines, and two charge parameters matching the experimental transition energies, the χ 2 value is reduced to 0.19 per degree of freedom. Finally, they obtained the RMS radius of r 2 1/2 = 5.5031 [START_REF] Soff | Vacuum polarization in a strong external field[END_REF] fm, which is consistent with the elastic electron scattering value 5.503(6) fm. However, the latest data is 5.5012 fm compiled by Angeli [185].

It is needed to recalculate the 2p and 3p splitting energy considering such large χ 2 value. In table 6.7, we compare the fine structure splitting(∆p) in the χ 2 minimum with the experimental results. We can notice that the calculated the fine structure splitting(∆p) of 2p and 3p are not within the scope of the experimental error bars. The discrepancies are less than 0.2 keV in the present calculation comparing with the experiment. The effects of two electrons from the 1s orbital are only 0.6 eV to 2p splitting and 1.5 eV to 3p splitting.

Although the present theoretical evaluation provides some large deviations, the nuclear radius is reproduced combining with the experimental results.

Energy levels of Muonic Radium and Curium

The charge radius of a nucleus is a fundamental parameter and important for understanding the strong interactions in the nucleus. Accurate measurements of the charge radius of Fig. 6.4 Self-energy with finite size correction in muonic curium. From Indelicato and Mohr(2018) [START_REF]Self-energy corrections for muonic atoms[END_REF].

radioactive atoms are essential. An upcoming measurement in PSI [START_REF] Antognini | Measurement of the charge radius of radium (Proposal for BVR 47)[END_REF] will be performed to measure atomic parity violation in single Ra + ion. There, various levels of radium can be drawn from the exact spectrum of the emitted muonic X rays. The extraction of charge radius is to be performed with the accuracy of 0.2%. The charge radius of 226 Ra, 248 Cm can be extracted by combining the transition energies from measurement with the calculation results as we test on lead.

With the upcoming experiments on high Z muonic radium and curium and the expected improvement in experimental precision, correct treatment is necessary of all relevant effects including QED corrections with vacuum polarization to leading order and second-order terms and finite-nuclear size corrected self-energy, nuclear polarization, deformation, nuclear excitations and so on. The precise theoretical calculation is very important for the extraction of nuclear parameters from future experiments.

We calculate muonic radium and curium using the same prescription of exact QED corrections, which were discussed above. Although there were some uncertainties for the nuclear polarization in the present analysis due to the uncertainties in the nuclear excitation spectrum, we can provide the correction as an order-of-magnitude estimation. In view of the precise calculations of the element lead, we can provide reliable values for the upcoming experiment.

All corrections causing the energy shifts are listed in Table 6.8 for muonic 226 Ra and 248 Cm levels with n ≤ 3. The χ 2 analysis with nuclear charge parameters will be performed after the experimental results are obtained.

Finite nuclear size

The measurements of the Lamb shift in light muonic atoms and the proton radius puzzle prompt ones to conduct a large number of new studies on muonic atoms. The accuracy of determining the nuclear radii depends on the precision of the experiments, and on the exactness of the QED calculations and nuclear structure evaluations.

In Table 6.9, the level energies for muonic 208 Pb, 226 Ra, and 248 Cm are shown with pointlike nucleus size, finite size, QED corrections. For heavy nuclei, the finite nuclear size correction can amount up to 50% in the total energy. The leading-order effects of Uëhling potential dominate more than 95% in the QED corrections. One can see that the finite size corrections are 100 times larger than QED corrections. It seems that such low-level transitions are suitable for extraction of the nuclear structure information, owing to the huge contribution from the nucleus.

Conclusions and Perspectives

In this thesis, we use the MCDF method to perform theoretical calculations, which include the valence-excited transitions of 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, core-excited transitions of 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 and 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2, and Auger transitions of 1s 2s 2p 2 P J → 1s 2 1 S 0 and 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2 in the lithium isoelectronic sequence from neon (Z=10) to uranium (Z=96). These energies are calculated by taking into account the QED effects and contributions from electronic correlation. In addition, radiative corrections, as well as all-order vacuum polarization and Breit corrections, are also included with fully optimized active set wave functions.

The Welton approximation and the effective operator methods are used for the selfenergy screening contributions. We note that the results from the two models are almost identical to each other at low-Z ions. However, from Z = 50, the value of the Welton model begins to move away from the effective operator method following the increase with Z. In the high-Z regions, the values of the Welton picture method are bigger than the effective operator method. The deviation reaches 0.2 eV at curium for valence-excited transitions, while this deviation has reached 2 eV in core-excited transitions. The effective operator model can improve the accuracy of the total transition energy by 0.002% for high-Z ions in valence-excited transitions. However, for core-excited transitions, we can't judge which model is better because there is no experiment as a reference in the heavy nuclear.

We also provide individual values of the Coulomb, Magnetic, Retardation, and higherorder Retardation correlations of electron-electron interaction correction. These correlations are obtained using a wavefunction built up from all configurations with single, double, and triple excitations to 5g orbitals. The Magnetic correlation corrections start being the same size as Coulomb correlation at Z=47 and being ten times larger than Coulomb part at Z=92 for the ground state of three-electron systems. The magnetic and retardation part can lead to a very large contribution to the correlation energy at high-Z. For example, the Magnetic correlation corrections are consistent with the trend of total correlations in core-excited transitions because the Magnetic parts dominate in the correlation contributions at middle-and high-Z ions.

It is very necessary to perform rigorous QED calculations. In the low-and middle-Z ions, diverse calculations are in good agreement with each other, except for 1/Z expansion method, which is far away from other theories. It can be seen that the present MCDF calculation results have a small shift with the values from the advanced RMBPT, RCI and S-matrix methods at high-Z ions due to different finite nuclear sizes, with or without nuclear polarization and the theoretical errors. The differences of many kinds of methods come from the uncertainty on the finite nuclear size and their evolution over time, 97 from whether some second-order QED corrections included or not, from what part of the electron-electron operator included in the correlation energy evaluation, and from Auger shift.

A number of measurements are performed using Tokamaks, ECRIS or astrophysics plasmas. Recently, our group measured He-, Li-and Be-like sulfur and argon ions with a double-flat crystal spectrometer without the use of any reference line. The transition energy measurements are performed with accuracies ranging from 2.3 ppm to 6.4 ppm depending on the element and line intensity. At low-and middle-Z, it clearly shows a generally good agreement between successive experiments and the most advanced theoretical calculations from the present work.

An upcoming experiment will be performed for muonic atom spectroscopy aiming at a precise measurement of the absolute nuclear charge radii of radioactive 226 Ra with a level of 0.2% relative precision at PSI. Extensive theoretical calculations are needed to deduce nuclear size. The muonic transition energies are also calculated numerically using Dirac wave functions with constant self-energy, including the QED contributions and finite nuclear size. Firstly, we give a detailed test on the muonic lead due to many experimental lines with high accuracy, double-magic property, and having a well-know radius. Muonic energy levels are highly sensitive to nuclear charge distribution due to large overlap with nuclear.

Nuclear parameters that minimize the weighted theory-experiment distance are found in the muonic 208 Pb with two-parameter Fermi model and three-parameter Gauss model, including 18 transition lines. We obtain r=5.5057 fm in the Fermi model and r=5.5031 in Gauss model by polynomial-fitting with large chi-square values. Many energy levels involved in the fitting fail to achieve convergence. Gauss model agrees with the experimental result (5.5031(11) fm) given by Bergern [START_REF] Bergem | Nuclear polarization and charge moments of Pb 208 from muonic x rays[END_REF], and disagrees with the adjustment result (5.5012(13) fm) compiled by Angeli et al. [186]. Although the nuclear radius is improved from Gauss model fit, it did not provide any improvement for the transition energy.

One reason is that the effects caused by the electrons refilling the orbitals should be also taken into account. The effect from K-shell occupies approximately 80% in this total effect, each L shell electrons 14%, and each M shell only 4% in the muonic lead. The transition energies have been greatly improved by adding this effect for high-state, but it is useless for low-state.

Another reason is the large uncertainty of nuclear polarization due to the nuclear excitation spectrum. The deviations of the calculation results from the two models are within the uncertainty of nuclear polarization at low-states.

For tansition energies of heavy nuclei, the finite nuclear size corrections can amount up to 50%, which are more than 100 times larger than QED corrections. It seems that such low-levels are suitable for the extraction of nuclear structure information, owing to the huge contribution from the nucleus.

Finally, we also perform the calculations of the muonic 226 Ra and 248 Cm using the same prescription of exact QED corrections. The precise theoretical calculations are very important for the extraction of nuclear parameters from future experiments. We provide a rigorous QED calculation, except that the nuclear polarizations give an orderof-magnitude estimation.

Perspectives

The next step, I will continue to complete the calculations of the Auger transitions of 1s 2s 2p 2 P 1/2 → 1s 2 1 S 0 and 1s 2s 2p 4 P 5/2 → 1s 2 1 S 0 , and it is difficult to achieve convergence for some free wavefunctions at n = 5. In some ions, numerical problems prevent us from doing calculations for a sufficiently large basis set, especially for the core-excited levels such as 1s 2s 2p. It is hard to correlate those core excited states as the correlation wavefunctions need to have a part at the 1s distance and a part at the 2s and 2p distance at the same time. We plan to place constraints on the energy with the way that the orbitals are generated, for example from the 1s 2 2p in place of 1s2s2p when it changes the list of Brillouin configurations. The Auger rate is performed by using the Aberg method with multichannel exit. Normally all possible wave functions are equivalent, and the energy is the same at the end. Yet, it could happen that it is not true for the Auger rate, and the extra matrix elements could change the cross-section when the wavefunction rotates between those equivalent states. The radiative rates should be immune to such problems.

Before, one had to use the same configuration generation methods for the initial and final state in our MCDFGME code. So, one would enter for the 1s 2 double excitation and single ones including Brillouin. CF Fischer showed that normally, one should only use of 1s 2 + 2s 2 + 2p 2 + 3s 2 + ... due to symmetries but also get with singles of 1s2s, 2s3s, 2p3p and so forth. One can get rid of those by doing local unitary transformations in the Hamiltonian, which preserve the norm. In the version of 2019v2, Indelicato has been changed the code so that one can use the "build" option for generating configurations automatically for one state and the "given" one for the other configuration, which allows entering the configurations by hand.

In view of the fact that nuclear polarization has high uncertainty. Next, I will conduct the fitting of nuclear polarization corrections by combining experimental results. The best-fit nuclear polarization will be very suitable for the experimental transition energies. therefore, I will try to deduce nuclear size using the three-parameter Fermi model in the muonic lead, because the three-parameter model can provide good improvements for the nuclear radius. Once the experiment of 226 Ra is completed, we can perform a fitting by extensive theoretical calculations with different nuclear parameters deriving the nuclear size of 226 Ra and 248 Cm.

It is very difficult to conduct a full QED evaluation beyond the two-photon exchange, which requires the development of some methods being able to contain higher orders. Another difficulty with the use of all-order methods lies in the photon energy of ω ij exchanged between two electrons when the interaction operators are calculated in the correlation orbitals. Our MCDFGME code can perform it with the help of introduction the diagonal Lagrange multipliers, and this can result in a large contribution to the correlation contributions of magnetic and retardation part at high-Z.

The experiments have been performed in He-, Li-and Be-like sulfur and argon ions by our group using a double-crystal spectrometer from ions produced in an ECRIS without the use of any reference line. Based on the experimental results, I would like to continue to study four and five electrons systems to analyze the QED contributions and nuclear effects. As the number of electrons increases, the characters of QED approximate become more and more challenging to evaluate.

The uncertainty of the nucleus, including finite nuclear size correction, nuclear polarization, and nuclear deformation limit the understanding of QED theory. Therefore, for a long-term goal, there is clearly a need to improve our knowledge of nuclear structures and their interactions with electrons. This may require combining the measurements of " exotic " particles, for example, muonic atoms, pionic atoms, and so on. Another important issue is the need to improve the accuracy of constants, including electron mass and the fine structure constant. L'énergie de corrélation est évaluée en prenant en compte les excitations simples, doubles et triples de la configuration non perturbée vers les niveaux virtuels avec un nombre quantique principal n = 5. Pour l'état fondamental 1s 2 2s 2 S 1/2 cela conduit à 1463 configurations et pour les états 1s 2s 2p 2 P 3/2 à 2478 configurations. Toutes les orbitales ont étés totalement relaxées. Cela a permis d'obtenir une convergence des energies vérifiée en comparant leur évolution en fonction du niveau virtuel maximal utilisé dans le calcul.

La précision de l'écrantage de la self-énergie est évaluée en comparant les résultats obtenu en utilisant m'approximation de Welton à ceux obtenus par la méthode de l'opérateur effectif développée récemment par l'équipe de Saint Petersbourg. Les résultats du modèle de Welton sont en bon accords avec ceux de méthode d'opérateur effectif pour les ions jusqu'à Z = 50, mais au delà les valeurs du modèle de Welton commencent à s'éloigner de celles de la méthode de l'opérateur effectif. Dans la région de Z élevé, la valeur de la méthode d'image Welton est supérieure à celle de la méthode de Saint Petersbourg L'écart atteint un maximum de 0.13 eV à Z = 92 pour les transitions 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 et 1.6 eV dans les transitions à partir des niveaux excités en couches internes (1s 2s 2p 2S+1 P J ).

Les corrections de corrélation dues à l'interaction magnétique commencent à être de la même taille que la corrélation de Coulomb à partir de Z = 47 et sont dix fois plus grandes à Z = 92 pour l'état fondamental des ions lithiumoïdes. Pour les transitions à partir des niveaux 1s 2s 2p 2S+1 P J , les énergies de corrélation magnétique dominent l'énergie de corrélation, et définissent donc l'évolution des correlations pour des numéros 109 atomiques moyens et élevés.

Les résultats de nos calculs sont comparés à l'ensembles des résultats expérimentaux existants et à l'ensemble des valeurs théoriques connues. Pour les éléments de numéro atomique faible et moyen, la comparaison montre une concordance généralement bonne entre les expériences et les calculs théoriques les plus avancés, et en particulier ceux présentés dans cette thèse. Les énergies de transition théoriques comportant les corrections d'électrodynamique quantique montrent une certaine dispersion pour les éléments lourds, liées en partie à l'utilisation de paramètres nucléaires un peu différents. L'écart est cependant inférieur à 0.2 eV pour Z ≈ 90 pour les transitions à partir de niveaux excités en couches internes.

Dans une deuxième partie, nous étudions les effets d'électrodynamique quantique et de taille finie du noyau pour le plomb muonique. L'objectif de cette étude est de préparer l'analyse des résultats pour le radium et le curium qui sont en train d'être mesurés à l'Institut Paul Scherrer pour mesurer le rayon de charge du noyau. Les énergies des transitions muoniques sont calculées avec le programme MCDFGME. Ces calculs incluent la self-énergie, la polarisation du vide (potentiel de Uehling) à tous les ordres, le terme de Whichmann et Kroll, et certaines corrections d'ordre deux d'électrodynamique quantique (potentiel de Källèn et Sabry). La plus grandes incertitude théorique provient de la polarisation nucléaire dont nous ne pouvons fournir que l' ordre de grandeur. La correction de taille finie du noyau a été faite avec deux modèles différents (Fermi et Gauss à trois paramètres).

Nus avons déterminé les paramètres nucléaires qui minimisent l'écart pondéré entre la théorie et l'expérience pour l'ensemble des énergies des transitions mesurées pour le modèle de Fermi à deux paramètres et le modèle de Gauss à trois paramètres. Nous obtenons ainsi 5.5057 fm pour le rayon quadratique moyen sphérique et 2.3919 fm pour le paramètre d'épaisseur dans le modèle de Fermi et r = 5.5031 dans le modèle de Gauss. Le modèle de Gauss à trois paramètre ne donne cependant pas d'amélioration sur l'accord théorie-expérience.

Pour permettre d'analyser les expériences à venir sur le 226 Ra et le 248 Cm muoniques nous avons également effectué des calculs des énergies de transitions correspondantes en utilisant les mêmes corrections d'électrodynamiques quantiques que pour le plomb. Un calcul théorique précis est en effet essentiel pour permettre l'extraction des paramètres nucléaires à partir de ces expériences futures.

1 . 1

 11 Basic properties of muonic atoms . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Properties of muons . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Muonic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Status of QED tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Few-electron ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . b) Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Muonic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Context and principle 2.1 MCDF-procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Electron-electron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 QED Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 One-loop QED correction . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Two-loop QED correction . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Screened QED corrections . . . . . . . . . . . . . . . . . . . . . . . a) Welton picture . . . . . . . . . . . . . . . . . . . . . . . . b) Model operator approach . . . . . . . . . . . . . . . . . . 2.5 Relativistic recoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Nuclear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Nuclear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Auger shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Radiative transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Auger transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Ze 2 (R nucl ) 2 r 3 Z 4 m 3 m 3 µ m 3 e( 207 ) 3 Finem 2 c 2 r 3 Z 4 m mµ me 207 Nuclear magnetic-dipole HFS µ nucl µ lepton r 3 Z 3

 2332073320733 -structure splitting,2p Z a•σ a•σ a•σ e 2 2

Fig. 1 .Fig. 1 . 2

 112 Fig. 1.2 Comparison between theory (Yerokhin and Shabaev 2015 [20]) and experiment for the Lyman α 1 line in hydrogen-like ions. Figure from Indelicato P [15]. Experiments: Kubiček et al. (2014)[32], Tschischgale et al. (2002)[33], Armour et al. (1980)[34], Tavernier et al. (1985)[35], Briand et al. (1983)[36], Beyer et al. (1985)[37], Beyer et al. (1994)[38], Lupton et al. (1994)[39], Beyer et al. (1995)[40], Marmar et al. (1986)[41], Briand et al. (1989)[42], Stöhlker et al. (1992)[43], Beyer et al. (1991)[44], Briand et al. (1990)[45], Stöhlker et al. (1993)[46], Stöhlker et al. (2000)[47], Widmann et al. (2000)[48], Tarbutt and Silver (2002)[49], Chantler et al. (2007)[50], Chantler et al. (2009)[51], Thorn et al. (2009)[52], Gillaspy et al. (2010)[53], Kraft-Bermuth et al. (2017)[54], Källne et al. (1984)[55], Deslattes et al. (1985)[56], Richard et al. (1984)[57], Hölzer et al. (1998)[58], Silver et al. (1987)[59], Gumberidze et al. (2005)[60], Schleinkofer et al. (1982)[61], Beyer et al. (1993)[62], Gassner et al. (2018)[START_REF] Gassner | Wavelength-dispersive spectroscopy in the hard x-ray regime of a heavy highlycharged ion: The 1s Lamb shift in hydrogen-like gold[END_REF]

Fig. 1 . 3

 13 Fig. 1.3 Histogram of the number of measurements for a given accuracy in part per million and the atomic number Z for Lyα 1 and Lyα 2 . Figure from Indelicato P [15].

Fig. 1 . 4

 14 Fig. 1.4 Histogram of the number of measurements for a given accuracy in part per million and the atomic number Z for two-electron ions. Figure from Indelicato P [15].

Fig. 2 . 1

 21 Fig. 2.1 Feynman diagrams for the electron-electron interaction. The double lines represent the Coulomb-bound electron propagator. The wavy lines represent the virtual photon propagator.

Fig. 2 . 2

 22 Fig. 2.2 Feynman diagrams representing: one-loop self-energy (a) and vacuumpolarization (b) radiative corrections. The double line indicates an electron propagating in the Coulomb field of the nucleus and the wavy line signifies the photon propagator

Fig. 2 . 3

 23 Fig. 2.3 Feynman diagrams corresponding to the full vacuum polarization contribution and expansion in Zα. One-loop vacuum polarization of the free electron propagator(single line). (a): Uëhling potential. (b): Wichmann and Kroll correction. Figure from Indelicato P [157].

Fig. 2 . 4

 24 Fig. 2.4 Feynman diagrams included in the Källén and Sabry V 21 (r) potential. Figure from Indelicato P [157].

Fig. 2 .

 2 Fig. 2.4 corresponds to the Källén and Sabry potential with a fourth-order potential. The expression of this potential is

Fig. 2 . 5

 25 Fig. 2.5 Feynman diagrams obtained when the Uëhling potential is added to the nuclear potential in the Dirac equation. Figure from Indelicato P [157].

Fig. 2 . 6

 26 Fig. 2.6 Lower-order Feynman diagrams included in the Källén and Sabry V 21 (r) potential, when the Uëhling potential is included in the differential equation. Figure from Indelicato P [157].

Fig. 2 .

 2 Fig. 2.7 shows the two-loop (second-order) QED corrections in α 2 by Feynman diagrams. The two-loop QED contribution can be divided,

Fig. 2 . 7

 27 Fig. 2.7 Feynman representing of QED: two-loop diagrams. (a), (b), (c): twoloop self-energy. (d), (e), (f): SE-VP correction. (j): S(VP)E correction. (h), (i): Källén and Sabry correction. (g): loop after loop vacuum polarization. Double line indicates an electron propagating in the Coulomb nuclear field.

Fig. 2 . 8

 28 Fig. 2.8 Three-electron interaction in QED.

Fig. 2 .

 2 Fig. 2.10 Feynman diagrams representing the relativistic recoil correction. The heavy double line indicates the proton wave function or propagator.

  l e a r c h a r g e d i s t r i b u t i o p t e r3 . M u o n i ca t om s F i g .3 . 1 M u o n w a v ef u n c t i o n ( s o l i dl i n e s )f o rr e l a t i v e l yl o w -l y i n gs t a t e s ,c omp a r e dt ot h en u c l e a rc h a r g ed i s t r i b u t i o n s ( d a s h e d l i n e ) f r omt h eF e rm i m o d e l i n m u o n i c 2 0 8 P b .

Fig. 3 . 2

 32 Fig. 3.2 Nuclear charge distributions of Uniform model, Fermi model and threeparameters Gauss model in 208 Pb for a mean spherical radius of 5.5057f m.

Fig. 3 . 3

 33 Fig. 3.3 Feynman diagrams representing the lowest-order nuclear polarization effect to the muon binding energy. (a) ladder diagram, (b) cross diagram, (c) seagull diagram, (d) NP-VP diagram.The muon (heavy line) interacts with the nucleus (heavy line) in its state via the exchange of virtual photons (wavy lines).[START_REF] Haga | Reanalysis of muonic Zr 90 and Pb 208 atoms[END_REF]

Fig. 5 . 7

 57 Fig. 5.7 Comparison of the transition energy between experiment and the present MCDF calculations, using effective operators for the 1s 2 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 transition. Experiments: Bosselman et al.(1999)[START_REF] Bosselmann | Measurements of 2s 2s1/2-2p 2p1/2,3/2 Transition Energies in Lithiumlike Heavy Ions. Ii. Experimental Results for Ag44 + and Discussion Along the Isoelectronic Series[END_REF],[START_REF] Widing | The lithium-like 2s 2S-2p 2P transition in solar flares[END_REF][START_REF] Widing | The lithium-like 2s 2S-2p 2P transition in solar flares[END_REF],[START_REF] Träbert | Extreme ultraviolet spectra of highly charged Xe ions[END_REF][START_REF] Träbert | Extreme ultraviolet spectra of highly charged Xe ions[END_REF],Denne et al. (1989)[START_REF] Denne | Berylliumlike Mo XXXIV and Lithiumlike Mo XL Observed in the JET Tokamak[END_REF],[START_REF] Bernhardt | Electron-Ion Collision Spectroscopy: Lithium-Like Xenon Ions[END_REF][START_REF] Bernhardt | Electron-Ion Collision Spectroscopy: Lithium-Like Xenon Ions[END_REF],Denne et al. (1989)[START_REF] Denne | Spectrum Lines of Kr XXVIII-Kr XXXIV Observed in the JET Tokamak[END_REF],Feili et al. (2000)[START_REF] Feili | Measurements of 2s2 S1/2 -2p2 P1/2[END_REF],[START_REF] Kukla | Extreme-Ultraviolet Wavelength and Lifetime Measurements in Highly Ionized Krypton[END_REF][START_REF] Kukla | Extreme-Ultraviolet Wavelength and Lifetime Measurements in Highly Ionized Krypton[END_REF],[START_REF] Denne | Spectral Lines of Highly-ionized Atoms for the Diagnostics of Fusion Plasmas[END_REF][START_REF] Denne | Spectral Lines of Highly-ionized Atoms for the Diagnostics of Fusion Plasmas[END_REF],[START_REF] Madzunkov | QED effects in lithiumlike krypton[END_REF][START_REF] Madzunkov | QED effects in lithiumlike krypton[END_REF],[START_REF] Martin | Désesquelles; «2s-2p transitions in heliumlike and lithiumlike krypton[END_REF][START_REF] Martin | Measurements of 2 s -2 p Transition Wavelengths in Helium-and Lithium-Like Xenon[END_REF],[START_REF] Staude | Measurements of 2s 2s1/2-2p 2p3/2,1/2 Transition Energies in Lithiumlike Heavy Ions: Experiments and Results for Ni25 + and Zn27 +[END_REF][START_REF] Staude | Measurements of 2s 2s1/2-2p 2p3/2,1/2 Transition Energies in Lithiumlike Heavy Ions: Experiments and Results for Ni25 + and Zn27 +[END_REF],[START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF][START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF] [START_REF] Schweppe | Measurement of the Lamb Shift in Lithiumlike Uranium (U89+)[END_REF][START_REF] Schweppe | Measurement of the Lamb Shift in Lithiumlike Uranium (U89+)[END_REF],[START_REF] Hinnov | Empirical Evaluation of Three-Electron Quantum-Electrodynamics Effects from Lithiumlike Resonance Lines of Elements Z=22-42 in the Tokamak Fusion Test Reactor and Joint European Torus Tokamaks[END_REF][START_REF] Hinnov | Empirical Evaluation of Three-Electron Quantum-Electrodynamics Effects from Lithiumlike Resonance Lines of Elements Z=22-42 in the Tokamak Fusion Test Reactor and Joint European Torus Tokamaks[END_REF],[START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF][START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF],[START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF][START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF],[START_REF] Zacarias | Measurements of 2s-2p transition energies in helium-like and lithium-like nickel[END_REF][START_REF] Zacarias | Measurements of 2s-2p transition energies in helium-like and lithium-like nickel[END_REF],[START_REF] Biedermann | Extreme ultraviolet spectroscopy of highly charged argon ions at the Berlin EBIT[END_REF] [START_REF] Brandau | Precise Determination of the 2s1/2-2p1/2 Splitting in Very Heavy Lithiumlike Ions Utilizing Dielectronic Recombination[END_REF][START_REF] Brandau | Precise Determination of the 2s1/2-2p1/2 Splitting in Very Heavy Lithiumlike Ions Utilizing Dielectronic Recombination[END_REF],[START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF][START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF],[START_REF] Silwal | Identification and plasma diagnostics study of extreme ultraviolet transitions in highly charged yttrium[END_REF][START_REF] Silwal | Identification and plasma diagnostics study of extreme ultraviolet transitions in highly charged yttrium[END_REF],[START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF][START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF],[START_REF] Reader | Extreme ultraviolet spectra and analysis of Δn = 0 transitions in highly charged barium[END_REF][START_REF] Reader | Extreme ultraviolet spectra and analysis of Δn = 0 transitions in highly charged barium[END_REF].

  (eV) Beiersdorfer et al. (2005) Nikolic et al. (2004) Bosselman et al. (1999) Widing and Purcell Denne et al. (1989) Denne et al. (1989) Brown et al. (1987) Martin et al. (1990) Podpaly et al. (2014) Kukla et al. (2005) Denne and Hinnov (1987) Staude et al. (1998) Hinnov et al. (1989) Beiersdorfer et al. (1993) Beiersdorfer et al. (1995) Podpaly et al. (2009) Clementson et al. (2011) Zhang et al. (2008) Bernhardt et al. (2015) Peacock et al. (1984) Suckewer et al. (1980) Kieslich et al. (2004) Knize (1991) Beiersdorfer et al. (1998) Bockasten et al. (1963) Edlén (1983) Lestinsky et al. (2008) Coulomb corr. (BSC) Coul.+Mag. corr. (BSC) Coul.+Mag.+Ret. (Breit) corr. (BSC) Coul.+Mag.+Ret.+H.O. Ret. corr. (BSC) Eff. Oper. Eff. Oper. no Ret. corr.

Fig. 5 . 9

 59 Fig. 5.9 Comparison of the transition energy between experiment and the present MCDF calculations, using effective operators for the 1s 2 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transition. Experiments: Beiersdorfer et al. (2005)[65], Nikolic et al. (2004)[223], Bosselman et al. (1999)[194], Widing and Purcell (1976)[195], Denne et al. (1989)[197], Denne et al. (1989)[199], Brown et al. (1987)[224], Martin et al. (1990)[204], Podpaly et al. (2014)[221], Kukla et al. (2005)[201], Denne and Hinnov (1987)[202], Staude et al. (1998)[205], Hinnov et al. (1989)[208], Beiersdorfer et al. (1993)[276], Beiersdorfer et al. (1995)[220], Podpaly et al. (2009)[222], Clementson et al. (2011)[225], Zhang et al. (2008)[227], Bernhardt et al. (2015)[START_REF] Bernhardt | Electron-Ion Collision Spectroscopy: Lithium-Like Xenon Ions[END_REF],[START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF][START_REF] Peacock | Highly-ionized atoms in fusion research plasmas[END_REF],[START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF][START_REF] Suckewer | Tracer element injection into PDX tokamak for spectral line identification and localized doppler temperature measurement[END_REF],[START_REF] Kieslich | Determination of the 2 s-2 p excitation energy of lithiumlike scandium using dielectronic recombination[END_REF][START_REF] Kieslich | Determination of the 2 s-2 p excitation energy of lithiumlike scandium using dielectronic recombination[END_REF],[START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF][START_REF] Knize | Measurement of QED Effects in Z=24 to 34 Lithium-Like Ions[END_REF],[START_REF] Beiersdorfer | Measurement of QED and Hyperfine Splitting in the 2 s 1/2-2 p 3/2 X-Ray Transition in Li-like B 209 i 8 0+[END_REF][START_REF] Beiersdorfer | Measurement of QED and Hyperfine Splitting in the 2 s 1/2-2 p 3/2 X-Ray Transition in Li-like B 209 i 8 0+[END_REF],[START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF][START_REF] Bockasten | The Spectra of Highly Ionized Light Elements in a High Temperature Plasma[END_REF],[START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF][START_REF] Edlén | Comparison of Theoretical and Experimental Level Values of the N = 2 Complex in Ions Isoelectronic with Li, Be, O and F[END_REF],[START_REF] Lestinsky | Screened radiative corrections from hyperfine-split dielectronic resonances in lithiumlike scandium[END_REF][START_REF] Lestinsky | Screened radiative corrections from hyperfine-split dielectronic resonances in lithiumlike scandium[END_REF]

  (eV) Schlesser et al. (2013) Suleiman et al. (1994) Machado et al. (2018) Bitter et al. (1985) T.F.R. group et al. (1985) T.F.R. group et al. (1985) Aglitskii et al. (1974) Hsuan et al. (1987) Decaux et al. (1997) Wargelin et al. (2001) Smith et al. (1995) Decaux et al. (2003) Tarbutt et al. (2001) Beiersdorfer et al. (2002) Biedermann et al. (2003) Rice et al. (2014) Beiersdorfer et al. (1991) Träbert et al. (1979)

Fig. 5 .

 5 Fig. 5.13 Comparison of the transition energy between experiment and the present MCDF calculations, using effective operators for the 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transition. Experiments: Schlesser et al. (2013)[228], Suleiman et al. (1994)[241], Machado et al. (2018)[1], Bitter et al. (1985)[229], T.F.R. group et al. (1985)[273], Aglitskii et al. (1974)[269], Hsuan et al. (1987)[230], Decaux et al. (1997)[270], Wargelin et al. (2001)[231], Smith et al. (1995)[232], Decaux et al. (2003)[233], Tarbutt et al. (2001)[234], Beiersdorfer et al. (2002)[272], Biedermann et al. (2003)[235], Rice et al. (2014)[236], Beiersdorfer et al. (1991)[237], Träbert et al. (1979)[242], Thorn et al. (2008)[243], Rudolph et al. (2013)[START_REF] Rudolph | X-Ray Resonant Photoexcitation: Linewidths and Energies of K alpha Transitions in Highly Charged Fe Ions[END_REF]

Fig. 5 .

 5 Fig. 5.14 Comparison of the transition energy between theoretical values from others and the MCDF calculations of the present work for the 1s 2s 2p 4 P 1/2 → 1s 2 2s 2 S 1/2 transitions. Theoretical works:[START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF][START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF],[START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF][START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF],[START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF],[START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF][START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF],[START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266],[START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF][START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF],[START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF][START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF],Gorayev et al. (2017)[START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF],[START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF][START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF]

Fig. 5 .

 5 Fig. 5.15 Comparison of the transition energy between experimental values and the MCDF calculation of the present work for the 1s 2s 2p 4 P 1/2 → 1s 2 2s 2 S 1/2 transitions. The MCDF value with effective operator for the self-energy is used. Experiments are from: Beiersdorfer et al. (2002)[272], Decaux et al. (1997)[START_REF] Decaux | High-Resolution Measurement of the Kα Spectrum of Fe XXV-XVIII: New Spectral Diagnostics of Nonequilibrium Astrophysical Plasmas[END_REF]

Fig. 5 .

 5 Fig.5.[START_REF] Cheng | Self-energy corrections to the K-electron binding in heavy and superheavy atoms[END_REF] Comparison of the transition energy theoretical values and the present MCDF calculations for the 1s 2s 2p 4 P 3/2 → 1s 2 2s 2 S 1/2 transition. The reference value is obtained with the new effective operator method. Theoretical works:[START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF][START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF],[START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF][START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF],[START_REF] Marques | Theoretical determination of K X-ray transition energy and probability values for highly charged (He-through B-like) Nd, Sm, Gd, Dy, Er, and Yb ions[END_REF],[START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF][START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF],[START_REF] Chen | Relativistic Auger and x-ray emission rates of the 1 s 2 s 2 p configuration of Li-like ions[END_REF][266],[START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF][START_REF] Bhalla | Theoretical lifetimes, transition energies, fluorescence yields, and nonradiative branching ratios for highly excited states of lithium-like argon[END_REF],[START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF][START_REF] Chen | Dielectronic satellite spectra for He-like ions[END_REF],Gorayev et al. (2017)[START_REF] Goryaev | Atomic Data for Doubly-Excited States 2lnl of He-Like Ions and 1s2lnl of Li-Like Ions with Z = 6˘36 and N = 2, 3[END_REF],[START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF][START_REF] Santos | Modeling Praseodymium K X-Ray Lines in an Electron Beam Ion Trap[END_REF].

Fig. 5 .

 5 Fig. 5.17 Comparison of the transition energy experimental values and the MCDF calculation of the present work for the 1s 2s 2p 4 P 3/2 → 1s 2 2s 2 S 1/2 transitions. The MCDF value with effective operator for the self-energy is used. Experiments are from: Beiersdorfer et al. (2002)[272], Tarbutt et al. (2001)[234], Decaux et al. (1997)[270], Biedermann et al. (2003)[235].

Fig. 5 .

 5 Fig. 5.19 Comparison of the transition energy between experimental values and the MCDF calculation of the present work for the 1s 2s 2p 4 P 5/2 → 1s 2 2s 2 S 1/2 transitions. The MCDF value with effective operator for the self-energy is used. Experiments are from: Machado et al. (2019),[START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF][START_REF] Beiersdorfer | Identification of the 1 s 2 s 2 p 4 P 5/2-1 s 2 2 s 2 S 1/2 magnetic quadrupole inner-shell satellite line in the Ar 1 6+ K-shell x-ray spectrum[END_REF],[START_REF] Dohmann | High resolution spectroscopy of prompt and metastable decaying levels in highly ionized argon, especially of the metastable 3 P 2-state of Ar 16+ and the 4 P 5/2-state of Ar 15+[END_REF][START_REF] Dohmann | High resolution spectroscopy of prompt and metastable decaying levels in highly ionized argon, especially of the metastable 3 P 2-state of Ar 16+ and the 4 P 5/2-state of Ar 15+[END_REF].

  u s t em e n t s T h i s w o r k 2 0 8 P b c h a r g e r a d i um ( Fm ) Y e a r C h a p t e r6 . A n a l y s i sf o r m u o n i ca t om s F i g .6 . 2 T h et r e n do fn u c l e a rc h a r g er a d i u sf r ome x p e r im e n t s i nt h ep a s t5 0 y e a r sa n dc om p a r e dw i t ho u rc a l c u l a t i o nd a t ao n m u o n i c l e a d . E l e c t r o ns c a t t e r i n g : V a n N i f t r i ke ta l . (19 6 6 ) [ 3 1 0 ] , B e l l i c a r de ta l . ( 1 9 6 7 ) [ 3 0 9 ] , E u t e n e u e re ta l .[ 3 1 4 ] ,F r i c k e e ta l . ( 1 9 9 5 ) [ 3 1 7 ] , W a s o w i c z ( 2 0 0 9 ) [ 3 1 6 ] . M u o n i cx -r a y : A c k e re ta l . ( 1 9 6 5 ) [3 1 1 ] ,A n d e r s o ne ta l . ( 1 9 6 6 ) [ 3 1 2 ] ,A n d e r s o ne t a l . ( 1 9 6 9 ) [ 3 1 3 ] , K e s s l e re ta l . ( 1 9 7 5 ) [ 3 1 5 ] ,E u t e n e u e re ta l . ( 1 9 7 8 ) [ 3 1 4 ] ,B e r g e r n e ta l . ( 1 9 8 8 ) [ 5 ] . A d j u s t em e n t s : A n g e l i ( 2 0 0 4 ) [1 8 5 ] ,A n g e l ie ta l .( 2 0 1 3 ) [ 1 8 6 ] .

Table 1 .

 1 1 -Properties of the Muon[START_REF] Devons | Muonic Atoms[END_REF].

	Mass	206.768277(24)m e
	Charge	e µ = e e
	Spin	1/2
	Magnetic moment(µ µ /µ p ) µ = e 2mµc (-g), g = 2(1 + α 2π + 0(α 2 ) + • • • )
	Free decay lifetime(10 -6 s) 2.19695(6)
		2.197078(73)
	Decay mode	e -+ νe + ν µ
	Wave equation

Table 5 .

 5 2 -The screened self-energy for the transition 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 and 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2 of Li-like ions (in eV). The Welton (Wel) and effective operater (Eff) results are obtained by Welton picture and effective operator model.

	1s 2 2s 2 S 1/2

Table 5 .

 5 [START_REF] Adamczak | Nuclear structure with radioactive muonic atoms[END_REF] -Individual contributions to the 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transitions energies on selected lithiumlike ions, as evaluated with the MCDFGME code. All energies are given in eV.

	Contribution	Z=10	Z=26	Z=56	Z=79	Z=92
	Coulomb+ Uelhing	908.056 6671.882 33102.521 70584.687 101002.176
	Magnetic	-0.332	-6.585	-67.854	-203.327	-340.466
	Retardation	0.011	0.237	0.627	0.488	0.200
	Higher-order Retardation	0.000	0.009	0.113	0.175	0.123
	Self-energy	-0.156	-4.259	-56.915	-192.355	-346.156
	Sef-energy screening	0.021	0.248	1.935	5.608	9.811
	Uelhing (muon pairs)	0.000	0.000	0.001	0.005	0.015
	Electronic density Uelhing	0.000	-0.004	-0.049	-0.174	-0.336
	Wichmann and Kroll	0.000	-0.002	-0.198	-1.638	-4.455
	Källèn and Sabry	0.000	0.003	0.067	0.318	0.696
	Two-loop self-energy	0.000	0.003	0.121	0.667	1.556
	SEVP	0.000	-0.001	-0.060	-0.421	-1.129
	S(VP)E	0.000	0.000	-0.011	-0.056	-0.135
	Normal mass shift	-0.025	-0.065	-0.123	-0.164	-0.165
	Specific mass shift	0.003	0.011	0.006	0.002	0.000
	Relativistic Recoil	0.000	-0.001	-0.021	-0.074	-0.126
	Coulomb correlation	0.572	0.452	0.236	-0.063	-0.345
	Magnetic correlation	0.038	0.290	1.705	3.850	5.748
	Retardation correlation	-0.011	-0.081	-0.434	-0.930	-1.357
	Higher-order ret. Corr.	0.000	0.001	0.011	-0.043	-0.166
	Total	908.178 6662.136 32981.676 70196.554 100325.490

s 2p 2 P 3/2 1s 2 2s 2 S 1/2 correlation

	( ) ( ) Correlation energy (eV)	------( ) S -s s P / --s P / 20 40 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Correlation energy (eV) 1s 2s 2p 2 P 3/2 correlation s S / 60 Z Coul. corr. Mag. corr. Ret. corr. 20 40 60 80 -2 0 2 4 6 Z 1s 2 Coul. corr. Mag. corr. Ret. corr.	80
		-
	Fig. 5.2 Correlation from Coulomb, magnetic, retardation, and higher order Fig. 5.4 The difference correlation from Coulomb, magnetic, retardation, and
	retardation on excited state of 1s 2 2p 2 P 1/2 higher order retardation between 1s 2 2s 2 S 1/2 and 1s 2 2p 2 P 1/2

/ Fig.

5

.1 Correlation from Coulomb, magnetic, retardation, and higher order retardation part on ground state of 1s 2 2s 2 S 1/2 tions to level energies for the ground state of 1s 2 2p 2 S 1/2 , the excited state of 1s 2 2p 2 P 1/2 and core-excited state of 1s 2s 2p 2 P 3/2 in Li-like ions in Fig.

5

.1, Fig. 5.2 and Fig. 5.3, H.O. ret. corr.

Fig. 5.3 Correlation from Coulomb, magnetic, retardation, and higher order retardation on excited state of 1s 2s 2p 2 P 3/2 respectively. The blue line indicates the Coulomb correlation, the red line represents the magnetic correlation, the green line stands for the retardation correlation, and the pink H.o. ret. corr. Total corr.

Table 5 .

 5 5 -Convergence of radiative and Auger transition energies for the transition of 1s 2s 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 in Li-like neon and uranium. All energies are in eV.

				Radiative				Auger
			Welton	Model operator		Welton	Model operator
	Max.n	10	92	10	92	10	92	10	92
	DF	907.58 100323.28 907.58 100321.61 668.63 67486.95 668.63 67485.39
	2	908.22 100325.51 908.22 100323.83 669.63 67490.54 669.63 67488.99
	3	908.23 100326.81 908.23 100325.16 669.26 67490.30 669.26 67488.75
	4	908.21 100327.02 908.21 100325.38 669.16 67490.30 669.16 67488.75
	5	908.18 100327.10 908.19 100325.47 669.11 67490.30 669.11 67488.79

Table 5 .

 5 6 -Comparison of the transition energy between experiment and the present MCDF calculations using Welton model and effective operators, respectively, for the 1s 2 2p 2 P 1/2 → 1s 2 1 S 0 transition. The other theory is from Sapirstein and Cheng[START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF].

		Energies are in eV.
			1s 2 2p 2 P 1/2 → 1s 2 1 S 0
	Z Experiment	Error	Diff. Welt. Diff. Effec. Diff.Th.	Ref.
	10	15.88870 0.00025	-0.00688	0.00015 -0.00110	Edlén (1983) [209]
	10	15.88881 0.00020	-0.00699	0.00004 -0.00121	Bockasten et al. (1963) [210]
	11	17.86141 0.00037	-0.00117	-0.00241 -0.00081	Edlén (1983) [209]
	11	17.86140 0.00100	-0.00116	-0.00240 -0.00080	Nikolic et al. (2004) [223]

. Our calculations: Coulomb corr.: Coulomb correlation; Mag.corr: magnetic correlation; Ret.: retardation; H.O.Ret.: higher-order retardation; Eff.Oper.:Effective Operator; DF:single-configuration Dirac Fock model.

Table 5 .

 5

		1s 2 2p 2 P 3/2 → 1s 2 1 S 0
	Z Experiment	Error	Diff. Welt. Diff. Effec. Diff.Th.	Ref.
	16.09330 0.00010	-0.00100	0.00667 -0.00050	Bockasten et al. (1963) [210]
	16.09315 0.00035	-0.00085	0.00682 -0.00035	Edlén (1983) [209]
	18.18761 0.00053	0.01931	0.01041 -0.00091	Edlén (1983) [209]
	18.18700 0.00100	0.01992	0.01102 -0.00030	Nikolic et al. (2004) [223]
	20.33228 0.00067	0.00121	-0.00596 -0.00078	Widing and Purcell [195]
	20.33180 0.00053	0.00169	-0.00548 -0.00030	Edlén (1983) [209]
	22.54132 0.00070	0.00106	-0.00418 -0.00102	Edlén (1983) [209]
	24.82514 0.00099	0.00124	0.00399 0.00066	Widing and Purcell [195]
	24.82635 0.00053	0.00003	0.00278 -0.00055	Edlén (1983) [209]
	27.2050	0.0021	-0.0015	0.0010	-0.0024	Edlén (1983) [209]
	29.6847	0.0014	-0.0058	-0.0180	0.0003	Widing and Purcell [195]
	29.6863	0.0011	-0.0073	-0.0196	-0.0013	Edlén (1983) [209]
	32.2891	0.0021	0.0074	0.0051	0.0016	Edlén (1983) [209]
	35.0357	0.0020	0.0049	0.0059	0.0014	Widing and Purcell [195]
	35.03803 0.00061	0.00249	0.00354 -0.00093	Peacock et al. (1984) [216]
	35.0369	0.0012	0.0036	0.0046	0.0002	Edlén (1983) [209]
	35.03160 0.00059	0.00892	0.00997 0.00550 Biedermann et al. (2007) [300]
	41.0286	0.0027	-0.0032	-0.0006	-0.0044	Widing and Purcell [195]
	41.0261	0.0014	-0.0007	0.0018	-0.0019	Edlén (1983) [209]

7 -Comparison of the transition energy between experiment and the present MCDF calculations using Welton model and effective operators, respectively, for the 1s 2 2p 2 P 3/2 → 1s 2 1 S 0 transition. The theory is from Sapirstein and Cheng

[START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF]

. Energies are in eV.

  2 2p 2 P 3/2 -1s 2 2s 2 S 1/2 transition. All experimental results, error bars, and discrepancy with theory are presented in Table5.6 and Table5.7, in which the other theory is from Sapirstein and Cheng[START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF]. Various techniques have been used in experimental development.Widing and Purcell [195] provided accurate astrophysical observations in solar flares. The TFR tokamak was used by T.F.R. group et al.(1985)

Table 5 .

 5 [START_REF] Mohr | Energy levels of hydrogen-like atoms predicted by quantum electrodynamics, 10≤ Z ≤ 40[END_REF] -Comparison of the transition energy between experiment and the present MCDF calculations using Welton model and effective operators, respectively, for the 1s 2s 2p 2 P 3/2 → 1s 2 1 S 0 transition. The theory is from Yerokhin et al.

	2017,2018)[193,

2 P 1/2 → 1s 2 1 S 0 transition. The theory is from Yerokhin et al. (

Table 5 .

 5 2s 2p 4 P 1/2 → 1s 2 2s 2 S 1/2 1s 2s 2p 4 P 3/2 → 1s 2 2s 2 S 1/2 1s 2s 2p 4 P 3/2 → 1s 2 2s 2 S 1/2

			Difference		Difference		Difference
	Z	Effec.	Welt.	Th.	Effec.	Welt.	Th.	Effec.	Welt.	Th.
		895.951 -0.009	0.009	895.999 -0.010	0.009	896.144 -0.010	0.010
		1097.672 -0.002	0.012	1097.752 -0.002	0.012	1097.978 -0.002	0.013
		1319.942 -0.009	0.009	1320.069 -0.009	0.009	1320.406 -0.009	0.011
		1562.795 -0.004	0.012	1562.983 -0.004	0.011	1563.470 -0.004	0.013
		1826.265 -0.006	0.007	1826.535 -0.006	0.007	1827.216 -0.006	0.009
		2110.392 -0.006	0.006	2110.766 -0.006	0.006	2111.695 -0.006	0.009
		2415.215 -0.004	-0.003	2415.719 -0.004 -0.003	2416.963 -0.004	0.000
		2740.768 -0.005	0.001	2741.430 -0.005	0.000	2743.066 -0.005	0.003
		3087.086 -0.006	0.084	3087.938 -0.006	0.084	3090.055 -0.006	0.086
		3454.258 -0.006	0.087	3455.334 -0.006	0.087	3458.039 -0.006	0.089
		3842.321 -0.007	0.089	3843.658 -0.007	0.088	3847.072 -0.007	0.090
		4251.334 -0.008	0.090	4252.969 -0.008	0.090	4257.233 -0.008	0.093
		4681.347 -0.009	0.091	4683.321 -0.009	0.092	4688.596 -0.009	0.096
		5132.428 -0.010	0.094	5134.779 -0.010	0.094	5141.250 -0.010	0.097
		5604.639 -0.010	0.094	5607.405 -0.010	0.094	5615.284 -0.010	0.099
		6098.055 -0.011	0.096	6101.274 -0.011	0.095	6110.797 -0.011	0.101
		6612.786 -0.012	0.055	6616.493 -0.012	0.055	6627.931 -0.012	0.059
		7148.830 0.310	0.055	7153.057 -0.013	0.055	7166.710 -0.013	0.060

10 -Comparison of the transition energy between the present MCDF calculations using Welton model and effective operators and other calculation, for the 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2, 5/2 transition. The other theory is from

[START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF]

[START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF]

. 1s

Table 5 .

 5 [START_REF] Soff | Vacuum polarization in a strong external field[END_REF] -Comparison of the transition energy between experiment and the present MCDF calculations using Welton model and effective operators, respectively, for the 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2, 5/2 transition. The other theory is from[START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF][START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF][START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF].

		1s 2s 2p 4 P 1/2 → 1s 2 1 S 0
	Z Experiment Error Diff. Welt. Diff. Effec. Diff. OtherTh.	Ref.
	3087.31	0.11	-0.23	-0.22	-0.14 Beiersdorfer et al. (2002)[272]
	6612.9	1.1	-0.2	-0.2	-0.1	Decaux et al. (1997)[270]
		1s 2s 2p 4 P 3/2 → 1s 2 1 S 0
	3088.15	0.10	-0.22	-0.21	-0.13 Beiersdorfer et al. (2002)[272]
	3091.783 0.154	-3.850	-3.845	-3.760	Tarbutt et al. (2001)[234]
	3091.228 0.308	-3.295	-3.290	-3.205 Biedermann et al. (2003)[235]
	6616.3	1.1	0.2	0.2	0.3	Decaux et al. (1997)[270]
		1s 2s 2p 4 P 5/2 → 1s 2 1 S 0
	2416.997 0.011	-0.038	-0.034	-0.034	Machado et al. (2019)
	3090.25	0.12	-0.20	-0.19	-0.11 Beiersdorfer et al. (2002)[272]
	3091	2	-1	-1	-1	Dohmann et al. (1978)[271]

Table 5 .

 5 [START_REF] Pachucki | Complete two-loop binding correction to the Lamb shift[END_REF] -Comparision of Auger transition energy between present works with effective operator and Welton picture, respectively and the CI method from Yerokhin et al.[START_REF] Yerokhin | Energy Levels of Core-Excited 1 s 2 l 2 l States in Lithium-Like Ions: Argon to Uranium[END_REF] of 1s2l2l states in Li-like Z from 18 to 92. Transition energies are in eV.

	1s 2s 2p 2 P 3/2 → 1s 2 1 S 0	1s 2s 2p 4 P 3/2 → 1s 2 1 S 0
	Z Effective operator	Welton	CI theory Effective operator	Welton	CI theory
	2195.743	2195.740	2195.755	2169.606	2169.603	2169.634
	2448.966	2448.963	2448.981	2420.836	2420.832	2420.865
	6170.608	6170.605	6170.643	6109.834	6109.830	6109.868
	12275.674 12275.688 12275.705	12115.312 12115.329 12115.327
	20712.650 20712.713 20712.654	20293.599 20293.669 20293.571
	27779.406 27779.548 27779.369	27035.487 27035.675 27035.424
	40867.826 40868.244 40867.719	39263.962 39264.411 39263.812
	52981.238 52982.066 52981.031	50282.792 50283.708 50282.587
	64031.991 64033.316 64031.738	60083.145 60084.225 60082.985
	67488.787 67490.297 67488.863	63099.171 63100.396 63099.367
	1s 2s 2p 4 P 1/2 → 1s 2 1 S 0
	Z Effective operator	Welton	CI theory
	2168.755	2168.751	2168.782
	2419.760	2419.757	2419.789
	6103.901	6103.896	6103.936
	12102.830 12102.844 12102.848
	20279.109 20279.183 20279.081
	27023.181 27023.354 27023.124
	39261.483 39261.933 39261.363

Table 6 .

 6 1 -Self-energy corrections for muonic208 Pb. All energies are in keV.

								0.15
		0.20
								0.14
		0.15
	F(82α)	0.10					F(82α)	0.13
		0.05						0.12
		0.00
		5.0	5.2	5.4	5.6	5.8	6.0	5.0	5.2	5.4	5.6	5.8
				RMS (fm)						RMS (fm)
			1s 1/2	2s 1/2	3s 1/2				2p 1/2	3p 1/2
		Fig. 6.1 Self-energy with finite size correction in muonic lead. From Indelicato
		and Mohr(2018). [304]
			Indelicato and Mohr(2018) [304]				Akihiro Haga [89]
			self-energy with finite size	leading self-energy higher-order self-energy total self-energy
	1s1/2		3.249			3.220			0.153	3.373
	2s1/2		0.634			0.696			0.025	0.721
	2p1/2		0.457			0.348			0.065	0.413
	3p1/2		0.223
	3p1/2		0.169			0.149			0.021	0.170
	causes a considerable correction to the energy levels. The correction is called Wichmann
	and Kroll correction (see section 2.4

Table 6 .

 6 3 -Corrections to muonic energy levels in208 Pb with r =5.5057 fm. All energies are in keV.

	208 Pb	1s1/2	2s1/2	2p1/2	2p3/2
	Dirac energy	-10526.651 -3581.248 -4781.507 -4599.336
	Uelhing (electron loop)	-67.121	-19.342	-32.273	-29.740
	Uelhing (muon loop)	-0.490	-0.122	-0.023	-0.005
	Loop after loop Uelhing	-0.084	-0.039	-0.094	-0.083
	Wichmann and Kroll	1.014	0.347	0.482	0.447
	K¨all `en and Sabry (electrons)	-0.554	-0.150	-0.252	-0.230
	Recoil 1	-0.290	-0.033	-0.060	-0.055
	Relat. Recoil	3.523	0.568	-0.054	-0.054
	Self-energy [89]	3.373	0.721	0.413	0.707
	Nuclear polarization [89]	-5.721	-0.930	-2.178	-2.214
		3p1/2	3p3/2	3d3/2	3d5/2
	Dirac energy	-2128.299 -2081.349 -2162.376 -2120.312
	Uelhing (electron loop)	-10.754	-10.222	-10.493	-9.839
	Uelhing (muon loop)	-0.008	-0.002	0.000	0.000
	Loop after loop Uelhing	-0.027	-0.026	-0.034	-0.030
	Wichmann and Kroll	0.195	0.186	0.202	0.193
	K¨all `en and Sabry (electrons)	-0.082	-0.077	-0.076	-0.071
	Recoil 1	-0.012	-0.011	-0.012	-0.012
	Relat. Recoil	-0.015	-0.015	-0.003	-0.003
	Self-energy [89]	0.170	0.244	-0.036	0.057
	Nuclear polarization [89]	-0.929	-1.179	-0.280	-0.038

Table 6 .

 6 5 -Muonic208 Pb transition energies (keV) in Fermi, Uniform and three parameter Gauss models with the same r =5.5057fm. Diff. represent the differences of Uniform and Gauss models with Fermi model. The Gauss parameter (c,t,w)=(6.1819,2.9071,0.38).

		Fermi model Uniform model	Diff.	Gaussian 3 model Diff.
	2p3/2-1s1/2	5962.475	5905.583 -56.892	5959.467 -3.008
	2p1/2-1s1/2	5777.495	5722.397 -55.098	5774.642 -2.853
	3d3/2-2p1/2	2642.414	2644.953	2.539	2641.860 -0.554
	3d5/2-2p3/2	2500.487	2504.914	4.427	2500.084 -0.403
	3d3/2-2p3/2	2457.434	2461.767	4.333	2457.035 -0.398
	3p3/2-2s1/2	1507.801	1491.239 -16.562	1507.369 -0.432
	3p1/2-2s1/2	1460.492	1444.738 -15.753	1460.100 -0.392
	2s1/2-2p1/2	1215.276	1233.434 18.157	1215.303 0.027
	2s1/2-2p3/2	1030.296	1050.248 19.952	1030.478 0.182
	5f5/2-3d3/2	1404.877	1405.190	0.313	1404.884 0.007
	5f7/2-3d5/2	1366.535	1366.755	0.220	1366.547 0.011
	5f5/2-3d5/2	1361.824	1362.043	0.219	1361.835 0.011
	4f5/2-3d3/2	972.029	972.341	0.312	972.034 0.005
	4f7/2-3d5/2	938.150	938.370	0.220	938.159 0.010
	4f5/2-3d5/2	928.976	929.195	0.219	928.985 0.009
	4d3/2-3p1/2	921.419	921.690	0.270	921.239 -0.181
	4d5/2-3p3/2	891.831	892.958	1.127	891.687 -0.144
	4d3/2-3p3/2	874.110	875.189	1.079	873.969 -0.141

Table 6 .

 6 6 -Muonic 208 Pb transition energies(keV) with the minimum value of χ 2 of two-parameter Fermi and three-parameter gauss models. The superscript (1) represents the effects of 2 electrons, and superscript (1+2+3) represents 18 electrons. The nuclear polarization uncertainties are from JS and RIN models. The Fermi model is evaluated with r =5.5057 fm, and t =2.3950 fm. The Three parameter gauss model is evaluated with c =6.1778 fm, t =2.9706 fm, and w =0.3790 fm, corresponding to r =5.5031 fm .

	Uncertainty	JS-RIN	-1.564	-1.534	-0.547	-0.562	-0.517	-0.054	-0.080	-0.269	-0.239	-0.041	0.004	0.004	-0.041	0.004	0.004	-0.239	-0.265	-0.265
	Experiment Fermi Fermi+∆E(1) Fermi+∆E(1+2+3) Gaussian	Energy Error Theory Diff. Theory Diff. Theory Diff. Theory Diff.	2p3/2-1s1/2 5962.854 0.09 5962.475 -0.379 5962.470 -0.384 5962.468 -0.386 5962.428 -0.426	2p1/2-1s1/2 5778.058 0.1 5777.495 -0.563 5777.490 -0.568 5777.489 -0.569 5777.451 -0.607	3d3/2-2p1/2 2642.332 0.03 2642.414 0.082 2642.395 0.063 2642.392 0.060 2642.421 0.089	3d5/2-2p3/2 2500.59 0.03 2500.487 -0.103 2500.467 -0.123 2500.463 -0.127 2500.499 -0.091	3d3/2-2p3/2 2457.569 0.07 2457.434 -0.135 2457.416 -0.153 2457.412 -0.157 2457.444 -0.125	3p3/2-2s1/2 1507.754 0.05 1507.801 0.047 1507.778 0.024 1507.774 0.020 1507.830 0.076	3p1/2-2s1/2 1460.558 0.032 1460.492 -0.066 1460.471 -0.087 1460.467 -0.091 1460.522 -0.036	2s1/2-2p1/2 1215.33 0.03 1215.276 -0.054 1215.267 -0.063 1215.265 -0.065 1215.273 -0.057	2s1/2-2p3/2 1030.543 0.027 1030.296 -0.247 1030.287 -0.256 1030.285 -0.258 1030.296 -0.247	5f5/2-3d3/2 1404.659 0.02 1404.877 0.218 1404.734 0.075 1404.705 0.046 1404.897 0.238	5f7/2-3d5/2 1366.347 0.019 1366.535 0.188 1366.391 0.044 1366.362 0.015 1366.554 0.207	5f5/2-3d5/2 1361.748 0.25 1361.824 0.076 1361.682 -0.066 1361.653 -0.095 1361.842 0.094	4f5/2-3d3/2 971.974 0.017 972.029 0.055 971.987 0.013 971.979 0.005 972.047 0.073	4f7/2-3d5/2 938.096 0.018 938.150 0.054 938.108 0.012 938.100 0.004 938.167 0.071	4f5/2-3d5/2 928.883 0.014 928.976 0.093 928.936 0.053 928.928 0.045 928.992 0.109	4d3/2-3p1/2 920.959 0.028 921.419 0.460 921.370 0.411 921.360 0.401 921.413 0.454	4d5/2-3p3/2 891.383 0.022 891.831 0.448 891.803 0.420 891.797 0.414 891.825 0.442	4d3/2-3p3/2 873.761 0.063 874.110 0.349 874.063 0.302 874.053 0.292 874.104 0.343

Table 6 .

 6 [START_REF] Devons | Muonic Atoms[END_REF] -Theoretical and experimental fine structure splitting(∆p) energies in muonic208 Pb(keV).

	208 Pb level This work		Akihiro Haga [89]			P. Bergern [5]
			TGT model RIN model JS model	Fit	Best fit	Exp.
	∆2p	184.981	184.858	184.846	184.829 185.112 184.776 184.788(27)
	∆3p	47.308	47.231	47.208	47.225	47.531	47.196	47.197(45)

Table 6 .

 6 9 -The level energies on muonic208 Pb,226 Ra and248 Cm of the pointlike nucleus, finite nuclear size and QED effects are presented. The nuclear radius of 5.5057 fm for208 Pb, 5.7211 fm for226 Ra and 5.8687 fm for248 Cm are adopted in the Fermi-charge distribution. All energies are in keV.

		State Pointlike Finite size QED
	208 Pb	1s1/2 -20992.35 -10479.64 -67.23
		2s1/2 -5385.41	-3573.09	-19.31
		2p1/2 -5385.41	-4780.70	-32.16
		2p3/2 -4837.24	-4598.27	-29.61
		3p1/2 -2329.93	-2127.99	-10.68
		3p3/2 -2166.53	-2091.57	-10.14
		3d3/2 -2166.53	-2162.41	-10.40
		3d5/2 -2121.97	-2120.24	-9.75
		4d3/2 -1216.32	-1213.85	-4.49
		4d5/2 -1197.37	-1196.33	-4.25
		4f5/2 -1197.37	-1197.37	-3.71
		4f7/2 -1188.30	-1188.28	-3.61
		5f5/2	-766.38	-766.37	-1.86
		5f7/2	-761.71	-761.70	-1.81
	226 Ra	1s1/2 -24652.01 -11380.31 -71.37
		2s1/2 -6354.17	-3993.56	-21.55
		2p1/2 -6354.17	-5441.30	-36.60
		2p3/2 -5591.71	-5233.38	-33.84
		3p1/2 -2734.89	-2431.88	-12.32
		3p3/2 -2507.49	-2379.06	-11.75
		3d3/2 -2507.49	-2499.64	-12.57
		3d5/2 -2447.78	-2444.50	-11.70
		4d3/2 -1407.30	-1402.63	-5.45
		4d5/2 -1381.89	-1379.91	-5.13
		4f5/2 -1381.89	-1381.88	-4.53
		4f7/2 -1369.80	-1369.78	-4.38
		5f5/2	-884.49	-884.48	-2.28
		5f7/2	-878.27	-878.25	-2.21
	248 Cm 1s1/2 -12804.68 -12713.03 -78.27
		2s1/2 -4623.00	-4605.30	-25.01
		2p1/2 -6384.85	-6382.15	-42.98
		2p3/2 -6136.73	-6134.73	-39.86
		3p1/2 -2868.82	-2867.80	-14.75
		3p3/2 -2806.42	-2805.69	-14.11
		3d3/2 -2989.49	-2989.57	-15.83
		3d5/2 -2913.25	-2913.14	-14.62
		4d3/2 -1676.60	-1676.63	-6.91
		4d5/2 -1645.54	-1645.49	-6.47
		4f5/2 -1649.46	-1649.48	-5.77
		4f7/2 -1632.27	-1632.25	-5.55
		5f5/2 -1055.76	-1055.76	-2.93
		5f7/2 -1046.91	-1046.90	-2.83

Table . 10

 . -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2 2p 2 P 1/2 → 1s 2 1 S 0 transition. Energies are in eV.1s 2 2p 2 P 1/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr.Table.11 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2 2p 2 P 3/2 → 1s 2 1 S 0 transition. Energies are in eV. -0.12570 Table.12 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2s 2p 2 P 1/2 → 1s 2 1 S 0 transition. Energies are in eV. 1s 2s 2p 2 P 1/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr.Table.13 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2s 2p 2 P 3/2 → 1s 2 1 S 0 transition. Energies are in eV. 1s 2s 2p 2 P 3/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr. Table .14 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2s 2p 4 P 1/2 → 1s 2 1 S 0 transition. Energies are in eV. 1s 2s 2p 4 P 1/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr. Table .15 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2s 2p 4 P 3/2 → 1s 2 1 S 0 transition. Energies are in eV. 1s 2s 2p 4 P 3/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr. Table .16 -Correlations from Coulomb, magnetic, retardation, and higher order retardation part for the 1s 2s 2p 4 P 5/2 → 1s 2 1 S 0 transition. Energies are in eV. 1s 2s 2p 4 P 5/2 → 1s 2 1 S 0 Z Coulomb corr. Magnetic corr. Retardation corr. Higher-order Retardation corr. Total corr.Dans cette thèse, nous utilisons la méthode multi-configuration Dirac-Fock (MCDF) pour effectuer un calcul précis des énergies des transitions entre les niveaux de structure fine et le niveau fondamental 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 des ions de la sequence isoélectronique du lithium. Nous évaluons également l'énergie des transitions à partir de niveaux excités en couche internes 1s 2s 2p 2 P 1/2 → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 et 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2 dans la même séquence isoélectronique. Nous avons aussi évalué les énergies des transitions Auger 1s 2s 2p 2 P J → 1s 2 1 S 0 et 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2 correspondantes. Ces calculs ont effectués pour des ions de numéro atomique 10 ≤ Z ≤ 96. Nous avons utilisé la version 2018 du code relativiste MCDF (MCDFGME), écrit par Jean-Paul Desclaux et Paul Indelicato. Les énergies ci-dessus sont calculées en prenant en compte les effets d'électrodynamique quantique du premier ordre et la corrélation inter-électronique, en incluant les effets magnétiques et de retard. L'écrantage de la self-énergie et les corrections radiatives du deuxième ordre sont également incluses.

	-0.20933 0.5729 0.43963 1.00349 1.00366	0.01396 0.01901 0.20552 0.31606 0.15606 0.05651	0.01057 -0.00201 -0.05271 -0.08859 -0.03785 -0.01358	0.01717 0.00215 0.00065 0.00143 0.00027 0.00003	-0.16763 -0.10655 0.72636 0.66853 1.12197 1.04662
	-0.22477 -0.12592 0.57875 0.42724 1.00209 0.99784	0.01428 0.02049 0.23033 0.34343 0.17449 0.06710	0.01407 -0.00219 -0.05844 -0.09618 -0.04228 -0.01614	0.02138 0.00248 0.0008 0.00165 0.00034 0.00017	-0.17504 -0.10514 0.75144 0.67614 1.13464 1.04897
	-0.23319 -0.12608 0.56642 0.41493 0.99721 1.00099	0.01411 0.02200 0.2502 0.37249 0.19393 0.07949	0.01615 -0.00236 -0.06323 -0.10417 -0.04696 -0.01910	0.02346 0.00281 0.00094 0.00188 0.00028 0.00007	-0.17947 -0.10363 0.75433 0.68513 1.14446 1.06145
	-0.24687 -0.12622 0.56206 0.40293 1.00228 1.00104	0.01369 0.02352 0.27582 0.40322 0.21821 0.09309	0.01965 -0.00251 -0.0692 -0.11254 -0.05279 -0.02234	0.02683 0.00315 0.00112 0.00215 0.00051 0.00015	-0.18670 -0.10206 0.7698 0.69576 1.16821 1.07194
	-0.25173 -0.12637 0.55425 0.39142 0.99773 1.00164	0.01345 0.02525 0.30031 0.43560 0.23817 0.10775	0.02095 -0.00272 -0.07489 -0.12127 -0.05757 -0.02587	0.02791 0.00354 0.00133 0.00243 0.00064 0.00019	-0.18942 -0.10030 0.781 0.70818 1.17897 1.08371
	-0.26188 -0.12651 0.54686 0.38056 0.99827 1.00772	0.01276 0.02703 0.32583 0.46961 0.26478 0.12548	0.02373 -0.00294 -0.08075 -0.13034 -0.06391 -0.03007	0.03010 0.00398 -0.095 0.00274 0.00077 0.00015	-0.19529 -0.09844 0.69694 0.72257 1.19991 1.10328
	-0.27269 -0.11324 0.54047 0.37048 0.99368 1.00244	0.01189 -0.10832 0.35279 0.50516 0.28735 0.14022	0.02674 0.03094 -0.08685 -0.13974 -0.06929 -0.03359	0.03245 0.00518 0.0018 0.00306 0.00092 0.00022	-0.20161 -0.18544 0.80821 0.73896 1.21266 1.10929
	-0.28411 -0.12727 0.52713 0.36120 0.99208 1.00165	0.01056 0.03989 0.37947 0.54226 0.31533 0.15844	0.03012 -0.00467 -0.09322 -0.14945 -0.07595 -0.03794	0.03438 0.00692 0.00213 0.00343 0.00109 0.00027	-0.20905 -0.08513 0.81551 0.75744 1.23255 1.12242
	-0.29628 -0.12797 0.51655 0.35279 0.98827 0.99909 0.99685	0.00890 0.05403 0.40792 0.58090 0.34215 0.17715 0.04623	0.03385 -0.00679 -0.09982 -0.15950 -0.08231 -0.04238 -0.01098	0.03656 0.00981 0.00246 0.00377 0.00129 0.00035 0.00005	-0.21697 -0.07092 0.82711 0.77796 1.24940 1.13421 1.03215
	-0.30919 -0.12858 0.51352 0.34520 0.98403 0.99265 1.00419	0.00678 0.06444 0.4388 0.62074 0.37009 0.19687 0.05721	0.03793 -0.00848 -0.10663 -0.16971 -0.08892 -0.04706 -0.01358	0.03869 0.01166 0.00289 0.00418 0.32460 0.00029 0.00004	-0.22579 -0.06096 0.84858 0.80041 1.58980 1.14275 1.04786
	-0.33748 -0.12964 0.49712 0.32641 0.97943 0.99567 0.99866	0.00090 0.08080 0.46699 0.73993 0.39915 0.22143 0.06797	0.04739 -0.01127 -0.11358 -0.20126 -0.09580 -0.05290 -0.01615	0.04235 0.01415 0.00326 0.00508 0.00174 0.00052 0.00017	-0.24684 -0.04596 0.85379 0.87016 1.28452 1.16472 1.05065
	-0.37812 -0.04766 0.49593 0.31238 0.97444 0.98846 1.00219	-0.01051 -0.52743 0.50048 0.88814 0.42935 0.24166 0.08053	0.06209 0.13382 -0.12075 -0.23687 -0.10293 -0.05770 -0.01912	0.04740 0.02345 0.00375 0.00682 0.00203 0.00065 0.00008	-0.27914 -0.41782 0.87941 0.97047 1.30289 1.17307 1.06368
	-0.40568 -0.13156 0.48813 0.29055 0.96910 0.98563 1.00276	-0.02036 0.10619 0.53227 1.14694 0.46068 0.26852 0.09433	0.07283 -0.01589 -0.128 -0.30021 -0.11031 -0.06403 -0.02238	0.05000 0.01720 0.00431 0.00904 0.00235 0.00079 0.00016	-0.30321 -0.02406 0.89671 1.14632 1.32182 1.19091 1.07487
	-0.41556 -0.08807 0.47341 0.27539 0.96342 0.97686 1.00404	-0.02408 -0.21102 0.56394 1.31926 0.49315 0.29136 0.10921	0.07674 0.05958 -0.1356 -0.34193 -0.11797 -0.06942 -0.02590	0.05131 0.02268 0.00473 0.01011 0.00271 0.00093 0.00019	-0.31159 -0.21683 0.90648 1.26283 1.34131 1.19973 1.08754
	-0.49445 0.02379 0.44873 0.25060 0.95742 0.97019 1.01104	-0.06123 -0.98759 0.59409 1.56980 0.52677 0.31956 0.12721	0.11030 0.24207 -0.14355 -0.40172 -0.12587 -0.07605 -0.03014	0.05500 0.04664 0.00523 0.01098 0.00310 0.00112 0.00015	-0.39038 -0.67509 0.9045 1.42966 1.36142 1.21482 1.10826
	-0.52122 0.08081 0.41141 0.24596 0.95114 0.96029 1.00696	-0.07542 -1.36798 0.69578 1.65246 0.56150 0.34662 0.14222	0.12201 0.33041 -0.16807 -0.42205 -0.13404 -0.08239 -0.03367	0.06097 0.06655 0.00717 0.00939 0.00352 0.00132 0.00021	-0.41366 -0.89021 0.94629 1.48576 1.38212 1.22584 1.11572
	-0.11596 -0.54984 0.10864 0.37174 0.21037 0.94455 0.94889 1.00776	0.00031 -0.09230 -1.54773 0.8024 1.92125 0.59743 0.37480 0.16077	0.00013 0.13509 0.37175 -0.19451 -0.48517 -0.14247 -0.08899 -0.03805	0.00005 0.06319 0.08012 0.00921 0.00983 0.00400 0.00153 0.00027	-0.11547 -0.44386 -0.98722 0.98884 1.65628 1.40351 1.23623 1.13075
	-0.11522 -0.58072 0.10018 0.32994 0.20100 0.93773 0.93600 1.00724	0.00090 -0.11199 -1.47894 0.99416 1.99728 0.63448 0.40410 0.17985	0.00005 0.14961 0.35554 -0.24111 -0.50327 -0.15115 -0.09583 -0.04253	0.00003 0.05864 0.08281 0.01269 0.00915 0.00446 0.00179 0.00033	-0.11424 -0.48446 -0.94041 1.09568 1.70416 1.42552 1.24606 1.14489
	-0.12617 0.21280 0.30598 0.18135 0.93066 0.92159 1.00340	0.00014 -2.26070 1.12078 2.15493 0.67271 0.43457 0.19998	0.00028 0.53465 -0.27176 -0.54049 -0.16012 -0.10292 -0.04727	0.00017 0.13185 0.01508 0.00782 0.00499 0.00208 0.00028	-0.12558 -1.38140 1.17008 1.80361 1.44824 1.25532 1.15639
	-0.12718 -0.14339 0.26432 0.15924 0.90834 0.90569 1.00967	0.00011 0.22779 1.29309 2.31808 0.79424 0.46622 0.22502	0.00037 -0.04039 -0.31573 -0.57791 -0.18859 -0.11027 -0.05317	0.00007 0.02235 0.01702 0.00586 0.00669 0.00241 0.00052	-0.12663 0.06636 1.2587 1.90527 1.52068 1.26405 1.18204
	-0.13044 -0.15006 0.24183 0.13555 0.88476 0.88842 1.00650	0.00023 0.28904 1.53853 2.49381 0.92605 0.49908 0.24580	0.00044 -0.05350 -0.37485 -0.62015 -0.21948 -0.11789 -0.05805	0.00016 0.02081 0.02046 0.00237 0.00851 0.00278 0.00063	-0.12961 0.10629 1.42597 2.01158 1.59984 1.27239 1.19488
	-0.13241 -0.15485 0.55319 0.12322 0.84390 0.86983 1.00862	0.00030 0.33231 1.83706 2.67831 1.16817 0.53320 0.27329	0.00050 -0.06292 -0.44671 -0.66234 -0.27639 -0.12579 -0.06448	0.00020 0.01771 0.02446 -0.00077 0.01169 0.00318 0.00076	-0.13141 0.13225 1.968 2.13842 1.74737 1.28042 1.21819
	-0.12821 -0.15665 0.08286 0.81910 0.85014 1.00581	0.00237 0.34807 2.86398 1.32645 0.56856 0.29684	0.00012 -0.06639 -0.70461 -0.31381 -0.13395 -0.06998	0.00017 0.01671 -0.00581 0.01341 0.00361 0.00091	-0.12555 0.14174 2.23642 1.84515 1.28836 1.23358
	-0.13468 -0.17147 0.02259 0.79403 0.82941 1.00621	0.00037 0.47821 3.27612 1.54879 0.60527 0.32583	0.00072 -0.09542 -0.79971 -0.36765 -0.14241 -0.07677	0.00023 0.00065 -0.01868 0.01525 0.00410 0.00108	-0.13336 0.21197 2.48032 1.99042 1.29637 1.25635
	-0.13595 -0.17644 -0.06314 0.76984 0.80792 1.00460	0.00075 0.52267 3.85034 1.67112 0.64329 0.35380	0.00074 -0.10547 -0.93043 -0.39623 -0.15115 -0.08329	0.00030 -0.00548 -0.04296 0.01558 0.00456 0.00126	-0.13416 0.23528 2.81381 2.06031 1.30462 1.27637
	-0.13823 -0.18195 -0.12045 0.76122 0.78577 1.00276	0.00051 0.57128 4.23200 1.83832 0.68271 0.38296	0.00094 -0.11651 -1.01700 -0.44063 -0.16022 -0.09009	0.00037 -0.01376 -0.06292 0.01562 0.00512 0.31959	-0.13641 0.25906 3.03163 2.17453 1.31338 1.61522
	-0.14357 -0.18787 -0.14060 0.73665 0.71736 1.00074	0.00018 0.62418 4.36606 1.92320 0.80934 0.41336	0.00115 -0.12857 -1.04732 -0.45699 -0.18926 -0.09716	0.00032 -0.03104 -0.07014 0.01523 0.00685 0.00172	-0.14192 0.27670 3.10800 2.21809 1.34429 1.31866
	-0.13858 -0.29607 0.73614 0.64890 0.99850	0.00362 5.41070 2.03318 0.94884 0.44500	0.00047 -1.28220 -0.48820 -0.22117 -0.10452	0.00058 -0.14054 0.01459 0.00868 0.00198	-0.13391 3.69189 2.29571 1.38525 1.34096
	-0.14277 -0.34523 0.72251 0.54073 0.99606	0.00160 5.74780 2.16443 1.21047 0.47788	0.00111 -1.35747 -0.52122 -0.28071 -0.11215	0.00071 -0.16585 0.01322 0.01175 0.00230	-0.13935 3.87925 2.37894 1.48224 1.36409
	-0.14147 0.57222 -0.39672 0.69496 0.48032 1.15748	0.00510 0.03844 6.10532 2.31993 1.38538 0.52393	0.00048 -0.01052 -1.43704 -0.55665 -0.32029 -0.12536	0.00085 0.00005 -0.19532 0.01091 0.01335 -0.00086	-0.13504 0.60019 4.07624 2.46915 1.55876 1.55519
	-0.14492 0.57047 -0.45047 0.65809 0.40417 1.12431	0.00348 0.04747 6.48492 2.49446 1.64038 0.55822	0.00104 -0.01299 -1.52120 -0.59418 -0.37765 -0.13267	0.00102 0.00003 -0.23561 0.00836 0.01483 -0.00340	-0.13938 0.60498 4.27764 2.56673 1.68173 1.54646
	-0.14495 0.55695 0.66567 0.36744 1.12217	0.00612 0.05603 2.60791 1.77753 0.59576	0.00063 -0.01541 -0.62973 -0.40835 -0.14147	0.00121 0.00017 0.00403 0.01505 -0.00254	-0.13699 0.59774 2.64788 1.75167 1.57392
	-0.14685 0.55315 0.64366 0.31327 1.11944	0.00656 0.06617 2.90323 1.99603 0.63465	0.00074 -0.01819 -0.70817 -0.45703 -0.15059	0.00142 0.00007 -0.00526 0.01484 -0.00089	-0.13813 0.60120 2.83346 1.86711 1.60260
	-0.14876 0.54691 0.61590 0.29531 1.11608	0.00689 0.07729 3.29932 2.07240 0.67480	0.00090 -0.02129 -0.81485 -0.47397 -0.15996	0.00165 0.00015 -0.02520 0.01437 0.00153	-0.13932 0.60306 3.07517 1.90811 1.63245
	-0.15070 0.59072 0.54163 0.56769 0.25937 0.97689	0.00726 0.03861 0.08923 3.60592 2.23057 0.70259	0.00107 -0.01034 -0.02461 -0.88690 -0.50898 -0.16409	0.00191 0.00005 0.00020 -0.04196 0.01325 0.00490	-0.14046 0.61904 0.60645 3.24475 1.99421 1.52029
	-0.15269 0.59159 0.54225 0.60405 0.22315 0.96773	0.00765 0.04775 0.10409 3.62239 2.39640 0.83362	0.00125 -0.01276 -0.02867 -0.90708 -0.54557 -0.19435	0.00223 0.00003 0.00016 -0.05044 0.01172 0.00671	-0.14156 0.62661 0.61783 3.26892 2.08570 1.61371
	-0.15469 0.58107 0.53155 0.56731 0.18645 0.94954	0.00794 0.05645 0.11558 4.27759 2.57012 0.97587	0.00148 -0.01512 -0.03199 -1.08989 -0.58374 -0.22685	0.00257 0.00017 0.00021 -0.10600 0.00900 0.00780	-0.14270 0.62257 0.61535 3.64901 2.18183 1.70636
	-0.15675 0.58106 0.52562 0.55097 0.14880 1.05356	0.00832 0.06679 0.13042 4.49031 2.75246 1.26753	0.00169 -0.01783 -0.03615 -1.14756 -0.62362 -0.29709	0.00297 0.00007 0.00027 -0.12555 0.00565 0.00678	-0.14377 0.63009 0.62016 3.76817 2.28329 2.03078
	-0.15888 0.57878 0.51801 0.99610 0.10949 1.04086	0.00874 0.07807 0.14556 0.04504 2.94446 1.73101	0.00190 -0.02079 -0.04045 -0.01097 -0.66547 -0.40644	0.00339 0.00016 0.00035 0.00005 0.00123 0.15326	-0.14485 0.63622 0.62347 1.03022 2.38971 2.51869
	-0.16103 0.57013 0.50666 1.00322 0.02870 0.98556	0.00906 0.08956 0.16157 0.05572 3.35522 1.86297	0.00216 -0.02384 -0.04504 -0.01356 -0.75464 -0.43397	0.00385 0.00019 0.00029 0.00003 -0.01086 0.03276	-0.14596 0.63604 0.62348 1.04541 2.61842 2.44732
	-0.16326 0.584 0.50493 0.99743 -0.07582	0.00943 0.10548 0.18234 0.06614 3.92043	0.00244 -0.02786 -0.05083 -0.01613 -0.87646	0.00438 0.00016 0.00053 0.00018 -0.03370	-0.14701 0.66178 0.63697 1.04762 2.93445
	-0.16553 0.57856 0.49320 1.00064 -0.13967	0.00977 0.11762 0.19877 0.07832 4.28881	0.00276 -0.03101 -0.05564 -0.01907 -0.95541	0.00489 0.00022 0.00064 0.00007 -0.05293	-0.14811 0.66539 0.63697 1.05996 3.14080
	-0.16789 0.57919 0.48617 1.00084 -0.16440	0.01014 0.13328 0.22193 0.09171 4.42018	0.00307 -0.03498 -0.06209 -0.02232 -0.98323	0.00550 0.00027 0.00077 0.00015 -0.05991	-0.14918 0.67776 0.64678 1.07038 3.21264
	-0.17535 0.57862 0.47367 1.00168 -0.33147	0.01112 0.14921 0.24123 0.10612 5.41211	0.00420 -0.03894 -0.06768 -0.02582 -1.19302	0.00748 0.00035 0.00091 0.00019 -0.12820	-0.15255 0.68924 0.64813 1.08217 3.75942
	-0.18357 0.57279 0.46396 1.00814 -0.37925	0.01214 0.16601 0.26622 0.12361 5.72331	0.00550 -0.04314 -0.07466 -0.03002 -1.25864	0.00977 0.00028 0.00108 0.00016 -0.15260	-0.15616 0.69594 0.65660 1.10189 3.93282
	-0.19894 0.57781 0.45195 1.00345 0.99652 -0.41781	0.01344 0.18784 0.29040 0.13810 0.04566 6.03865	0.00838 -0.04846 -0.08145 -0.03353 -0.01097 -1.32650	0.01420 0.00052 0.00125 0.00021 0.00006 -0.17937	-0.16292 0.71771 0.66215 1.10823 1.03127 4.11497

(Zα)[START_REF] Engfer | Charge-distribution parameters, isotope shifts, isomer shifts, and magnetic hyperfine constants from muonic atoms[END_REF]

Acknowledgments

This thesis was carried out at Laboratoire Kastler Brossel of It was financed by the China Scholarship Council, a Chinese Ministry of Education's non-profit organization within the Chinese government (File No.201606240047). I give my gratitude to the financial support of the China Scholarship Council (CSC).

First, I owe the greatest thanks to my supervisor Paul Indelicato who is very patient and knowledgeable. .D. student. They are responsible for supervising my doctoral research process.

Thanks to Sichuan University for giving me the opportunity to go abroad to continue research. I appreciate my master thesis supervisor, Gang Jiang who introduced me to step into the road of scientific exploration. Special thanks to Professor Kening Huang for his recommendation, whose help set me on the road to study in France.

QED contribution

We now mainly discuss the calculation results for radiation corrections. Next, we consider the one-loop QED corrections, which are the largest effects. They are determined by the self-energy and vacuum-polarization corrections and are presently well described. We present the details in Sec. 2.4.1. The one-electron self-energy is taken from the work of Mohr and co-workers [START_REF] Mohr | Numerical Evaluation of the 1s1/2 State Radiative Level Shift[END_REF][START_REF] Mohr | Self-energy correction to one-electron energy levels in a strong Coulomb field[END_REF][START_REF] Mohr | Self-energy of excited states in a strong Coulomb field[END_REF][START_REF] Indelicato | Coordinate space approach to the one-electron self-energy[END_REF][START_REF] Le Bigot | QED self-energy contribution to highly-excited atomic states[END_REF], and corrected for finite nuclear size. The Uëhling part of the vacuum-polarization contribution, which is the exact contribution of the Uëhling potential with Dirac wave functions including the finite nuclear size, is calculated in the present works from Ref. [START_REF] Uehling | Polarization effects in the positron theory[END_REF]. The expression of the Uëhling potential relates to the charge density. Here, we calculate a correction in which the nuclear charge density is replaced by the charge density of the electron. This correction is very small, except for very high-Z ions. The vacuum polarization due to the creation of virtual 2007) [START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF], [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF], [START_REF] Gu | Energies of 1s22lq (1 ≤ q ≤ 8) states for Z ≤ 60 with a combined configuration interaction and many-body perturbation theory approach[END_REF] [START_REF] Gu | Energies of 1s22lq (1 ≤ q ≤ 8) states for Z ≤ 60 with a combined configuration interaction and many-body perturbation theory approach[END_REF], [START_REF] Cheng | Quantum Electrodynamic Corrections in High-Z Li-Like and Be-Like Ions[END_REF] [START_REF] Cheng | Quantum Electrodynamic Corrections in High-Z Li-Like and Be-Like Ions[END_REF], Indelicato and [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF] [START_REF] Indelicato | Multiconfiguration Dirac-Fock Calculations of Transition Energies with QED Corrections in Three-Electron Ions[END_REF], [START_REF] Kim | Resonance transition energies of Li-, Na-, and Cu-like ions[END_REF] [START_REF] Kim | Resonance transition energies of Li-, Na-, and Cu-like ions[END_REF], [START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF] [START_REF] Kozhedub | Relativistic Recoil, Electron-Correlation, and QED Effects on the 2pj-2s Transition Energies in Li-Like Ions[END_REF], [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF] [START_REF] Safronova | Inner-shell excitation energy and autoionization rates for Li-, Be-, B-like ions with Z = 6-54[END_REF], [START_REF] Chen | Relativistic Configuration-Interaction Calculations for the n=2 States of Lithiumlike Ions[END_REF] [START_REF] Chen | Relativistic Configuration-Interaction Calculations for the n=2 States of Lithiumlike Ions[END_REF], [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF] [START_REF] Seely | QED contributions to the 2p-2s transitions in highly charged Lilike ions[END_REF], [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF], [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF] [START_REF] Yerokhin | Relativistic Configuration-Interaction Calculation of Energy Levels of Core-Excited States in Lithiumlike Ions: Argon through Krypton[END_REF], [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF] [START_REF] Yerokhin | Relativistic configurationinteraction calculations of the energy levels of the 1 s 2 2 l and 1 s 2 l 2 l states in lithiumlike ions: Carbon through chlorine[END_REF], [START_REF] Santos | Multiconfiguration Dirac-Fock calculation of transition energies in highly ionized bismuth, thorium, and uranium[END_REF] [START_REF] Santos | Multiconfiguration Dirac-Fock calculation of transition energies in highly ionized bismuth, thorium, and uranium[END_REF], [START_REF] Sapirstein | Determination of the Two-Loop Lamb Shift in Lithiumlike Bismuth[END_REF] [START_REF] Sapirstein | Determination of the Two-Loop Lamb Shift in Lithiumlike Bismuth[END_REF], Yerokhin et al. (2007) [START_REF] Yerokhin | QED Treatment of Electron Correlation in Li-Like Ions[END_REF], [START_REF] Blundell | Improved many-body perturbation theory calculations of the n= 2 states of lithiumlike uranium[END_REF] [START_REF] Blundell | Improved many-body perturbation theory calculations of the n= 2 states of lithiumlike uranium[END_REF], [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] [START_REF] Johnson | Transition Rates for Lithium-Like Ions, Sodium-Like Ions, and Neutral Alkali-Metal Atoms[END_REF] 0.52 eV at Z=90 for the 1s 2 2p 2 P 3/2 -1s 2 2s 2 S 1/2 transition. Sapirstein and Cheng [START_REF] Sapirstein | S-Matrix Calculations of Energy Levels of the Lithium Isoelectronic Sequence[END_REF] also added the nuclear polarization effects [START_REF] Plunien | Erratum: Nuclear-polarization contribution to the Lamb shift in actinide nuclei[END_REF], which can lead to a shift of 0.02 eV at Z=90 and 0.03 eV at Z=92. They used the root-mean-square nuclear radii from Johnson and Soff [START_REF] Johnson | The lamb shift in hydrogen-like atoms, 1≤ Z ≤ 110[END_REF], except thorium (Z=90) from Ref. [START_REF] Zumbro | E2 and E4 Deformations in 232th and 239,240,242pu[END_REF] and uranium (Z=92) from Ref. [134]. Our works are from the tabulation of Angeli [185,186]. The finite nuclear sizes are uncertain, and they can actually result in a large difference. The deviation reduces to 0.36 eV at Z=90 when we used the same radii with them. Different calculation results are affected by the same limitations from nuclear size, with or without second-order QED contributions and which electron correlation involved. All of these contributions can lead to differences in a few eV. The comparison of total transition energies are carried out between our calculations and experimental works in Fig. 5.7 for the 1s 2 2p 2 P 1/2 -1s 2 2s 2 S 1/2 transition and Fig.

Chapter 6. Analysis for muonic atoms

Subject : Quantum Electrodynamic and nuclear effects in Li-like ions and muonic atoms

Abstract: We have performed multi-configuration Dirac-Fock calculations of the transition energies in the transitions of 1s 2 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, core-excited transitions of 1s 2s 2p 2 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2 and 1s 2s 2p 4 P J → 1s 2 2s 2 S 1/2 , J = 1/2, 3/2, 5/2, and Auger transition energies of 1s 2s 2p 2 P J → 1s 2 1 S 0 and 1s 2s 2p 4 P J → 1s 2 1 S 0 , J = 1/2, 3/2 in the lithium isoelectronic sequence for low-to high-Z ions (Z=10-96). Quantum-electrodynamic corrections with effective operator model and electronic correlation corrections are included in Dirac wave functions with finite nuclear size. All-order Breit and vacuum polarization corrections are also included in the calculation, with a fully optimized active set wavefunction. Extensive comparisons between existing theorical results and experiment are performed. We also evaluate nuclear parameters that minimize the weighted theory-experiment discrepancy in muonic 208 Pb with two-parameter Fermi and three-parameter Gauss models. We obtain r=5.5057 fm in the Fermi model and r=5.5031 in Gauss model by polynomial-fitting. We also analyze the individual contributions of muonic 226 Ra and 248 Cm using the same prescription of exact QED correction for the upcoming experiments.

Keywords : MCDF method, QED, Transition energy, Muonic atoms, Nuclear radii