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The gj factor of a bound electron and the hyperfine structure splitting in hydrogenlike ions
Abstract
The comparison between theory and experiment of the hyperfine structure splitting and the electronic gj factor in heavy highly charged ions provides a unique testing ground for quantum electrodynamics in the presence of strong electric and magnetic fields. A theoretical evaluation is presented of all quantum electrodynamical contributions to the ground-state hfs splitting in hydrogenlike and lithiumlike atoms as well as to the gj factor. Binding and nuclear effects are discussed as well. A comparison with the available experimental data is performed, and a detailed discussion of theoretical sources of uncertainty is included which is mainly due to insufficiently known nuclear properties.
... Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 10 15 V/cm for the innermost electrons [1]. Especially in few-electron highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. ...
... Highly charged ions are an interesting candidate for such tests as due to the strong interaction between the (few) electrons and the nucleus, these systems also show enhanced sensitivity for potential new physics [19]. In these few-electron systems, the electric field experienced by the remaining electrons can exceed 10 15 V/cm [1], hence the electronic wave function is perturbed strongly, resulting in modified properties, that can be measured and compared to theoretical predictions. Thus far, bound-state QED in high-Z highly charged ions has been probed most accurately by measurements of the Lamb shift [20,21]. ...
... The non-relativistic QED approach which treats the interaction between electron and nucleus perturbatively ( [37]) cannot be expected to give good results for Z = 50 because the expansion parameter of this perturbation series, Zα, is too large. Non-perturbative calculations for one-loop diagrams are well established [38,1], while the calculations for two-loop diagrams are only partially done [39,40,41]. The theory of the bound-electron g factor has been previously tested in lighter ions, with 28 Si 13+ being the heaviest hydrogenlike ion for which the g factor has been measured [6,27]. ...
Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond $10^{15}\,\text{V}/\text{cm}$ for the innermost electrons. Especially in few-electron highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. Consequently, their Lamb shifts were intensively studied in the last decades. Another approach is the measurement of $g$ factors in highly charged ions. However, so far, either experimental accuracy or small field strength in low-$Z$ ions limited the stringency of these QED tests. Here, we report on our high-precision, high-field test of QED in hydrogenlike $^{118}$Sn$^{49+}$. The highly charged ions were produced with the Heidelberg-EBIT (electron beam ion trap) and injected into the ALPHATRAP Penning-trap setup, where the bound-electron $g$ factor was measured with a precision of 0.5 parts-per-billion. For comparison, we present state-of-the-art theory calculations, which together test the underlying QED to about $0.012\,\%$, yielding a stringent test in the strong-field regime. With this measurement, we challenge the best tests via the Lamb shift and, with anticipated advances in the $g$-factor theory, surpass them by more than an order of magnitude.
... Analytical expressions for the FNS effect were presented in [25][26][27][28]. The FNS correction can also be calculated with a higher accuracy numerically by using the Fermi distribution as a model for nuclear charge density [29]. However, even this model does not describe any fine details of nuclear charge distributions that are unique for each nucleus. ...
... with the radius parameter c and the diffuseness parameter a = 2.3/(4 ln3) fm [29]. ...
... Other contributions to the g factor are summarized, e.g., in Refs. [5,29,46]. ...
A state-of-the-art approach for calculating the finite-nuclear-size correction to atomic energy levels and the bound-electron g factor is introduced and demonstrated for a series of highly charged hydrogenlike ions. First, self-consistent mean-field calculations based on the Skyrme-type nuclear interaction are employed in order to produce a realistic nuclear proton distribution. In the second step, the obtained nuclear charge density is used to construct the potential of an extended nucleus, and the Dirac equation is solved numerically. The ambiguity in the choice of a Skyrme parametrization is suppressed by fine-tuning of only one parameter of the Skyrme force in order to accurately reproduce the experimental values of nuclear radii in each particular case. The homogeneously-charged-sphere approximation, the two-parameter Fermi distribution, and experimental nuclear charge distributions are used for comparison with our approach, and the uncertainties of the presented calculations are estimated. In addition, suppression of the finite-nuclear-size effect for the specific differences of g factors is demonstrated.
... 1.1.2. This leads to values ranging from g J ≈ 2.00 for light ions such as 12 C 5+ decreasing down to g J ≈ 1.66 for 238 U 92+ [10]. ...
... bound electron g-factor 10 20 30 Relativistic values are calculated according to [10,11]. Highlighted are selected so far measured systems and the projected hydrogenlike lead 208 Pb 81+ , for which the expectation value reaches the Schwinger limit. ...
... bound electron g-factor 10 20 30 Relativistic values are calculated according to [10,11]. Highlighted are selected so far measured systems and the projected hydrogenlike lead 208 Pb 81+ , for which the expectation value reaches the Schwinger limit. ...
The cryogenic double Penning-trap experiment ALPHATRAP aims to test bound-state quantum electrodynamics (BS-QED) under extreme conditions by measuring the magnetic moment (g-factor) of electrons bound to the nucleus of heavy highly charged ions (HCIs). The bound electron g-factor is measured employing the double-trap technique which uses the continuous Stern-Gerlach effect (CSGE) for a nondestructive detection of the spin state of the ion. The result of this thesis is twofold. In order to improve the achievable precision of future measurements the implementation of sympathetic laser cooling is envisaged. For this purpose a laser system was integrated into the existing setup and laser cooling of 9Be+ was demonstrated for the first time at ALPHATRAP. For the axial temperature of a single beryllium ion an upper limit of 69(30) mK can be given. This demonstration paves the way for further developments towards sympathetic laser cooling of HCIs. Furthermore, the optical access to the Penning trap was used for high-precision laser spectroscopy of the magnetic dipole fine structure transition in the ground state of boronlike argon 40Ar13+ with an unsurpassed relative uncertainty of the absolute frequency of 9.4×10e−9. To this end, a novel spectroscopy scheme was demonstrated for the first time, which uses the CSGE and does not require a detection of fluorescence. This proof-of-principle method can be extended to other systems, opening up new possibilities to test BS-QED in the strongest electromagnetic fields by investigating the optical hyperfine structure in heavy HCI by means of laser spectroscopy.
... Also, our SE-WFs were used for the calculation of some SE corrections, presented already in the literature, such as the irreducible part of the SE correction to the g factor and the loop-after-loop (LAL) SE contribution to the Lamb shift. Our results for these corrections and comparison with earlier works [15][16][17] can be found in Sec. V. ...
... In Table I, we compare our results, obtained with Eq. (45), with the values given in Ref. [15]. We used Ref. [15] for comparison because in that work, just as in our case, calculations were performed for extended nuclei. ...
... In Table I, we compare our results, obtained with Eq. (45), with the values given in Ref. [15]. We used Ref. [15] for comparison because in that work, just as in our case, calculations were performed for extended nuclei. Typically, we found a three-to four-digit agreement both with the coordinate and the momentum space representations of our SE-WFs. ...
The procedure for the calculation of the self-energy-corrected wave function of the bound electron in the field of the nucleus is discussed. We present the related formulas and discuss the numerical difficulties and the methods used to overcome them. The results of the calculation are presented for a wide range of ions. Possible applications of the numerically obtained wave functions are discussed.
... Analytical expressions for the FNS effect were presented in [25][26][27][28]. The FNS correction can also be calculated with a higher accuracy numerically by using the Fermi distribution as a model for nuclear charge density [29]. However, even this model does not describe any fine details of nuclear charge distributions that are unique for each nucleus. ...
... with the radius parameter c and the diffuseness parameter a = (2.3/4 ln3) fm [29]; ...
... Other contributions to the g factor are summarized e.g. in Ref. [29,47,48]. ...
A state-of-the-art approach for calculating the finite nuclear size correction to atomic energy levels and the bound-electron $g$ factor is introduced and demonstrated for a series of highly charged hydrogen-like ions. Firstly, self-consistent mean-field calculations based on the Skyrme-type nuclear interaction are employed in order to produce a realistic nuclear proton distribution. In the second step, the obtained nuclear charge density is used to construct the potential of an extended nucleus, and the Dirac equation is solved numerically. The ambiguity in the choice of a Skyrme parametrization is supressed by fine-tuning of only one parameter of the Skyrme force in order to accurately reproduce the experimental values of nuclear radii in each particular case. The homogeneously charged sphere approximation, the two-parameter Fermi distribution and experimental nuclear charge distributions are used for comparison with our approach, and the uncertainties of the presented calculations are estimated. In addition, suppression of the finite nuclear size effect for the specific differences of $g$ factors is demonstrated.
... For a complete derivation of these results, and more details, the reader is referred to, e.g. [125,[127][128][129][130][131][132][133]. ...
... This expression is also sensitive to finite nuclear size corrections. A complete description of the different theoretical methods and contributions can be found in [130,474]. ...
... Theoretical contributions to the Landé g-factor and comparison with experiment for He, C and O. The value corresponding to [508,509] has been derived from equations (3), (53), (55) and (56) 0.000 000 000 012 01 −0.000 000 000 0281 −0.000 000 000 13 [484] Radiat Recoil −0.000 000 000 022 61 −0.000 000 000 0679 [130,471] Other corrections Nuclear polarizability 0.000 000 000 000 00 0.000 000 000 0000 [505] Nuclear susceptibilty 0.000 000 000 000 00 0.000 000 000 0000 [525] Weak interaction (Zα) 0 0.000 000 000 000 06 0.000 000 000 0001 [26,526] Hadronic effects (Zα) 0 0.000 000 000 003 47 0.000 000 000 0035 [527][528][529] [508,509] been studied [505] to see what limitations there could be in going to higher-Z. It was found that the nuclear polarization effects did set a limit of around 1×10 −9 for medium-Z and 1×10 −6 for the heaviest elements leading to the same kind of limitations to test QED or derive fundamental constants than for transition energies. ...
The current status of bound state quantum electrodynamics calculations of transition energies for few-electron ions is reviewed. Evaluation of one and two body QED correction is presented, as well as methods to evaluate many-body effects that cannot be evaluated with present-day QED calculations. Experimental methods, their evolution over time, as well as progress in accuracy are presented. A detailed, quantitative, comparison between theory and experiment is presented for transition energies in few-electron ions. In particular the impact of the nuclear size correction on the quality of QED tests as a function of the atomic number is discussed. The cases of hyperfine transition energies and of bound-electron Landé g-factor are also considered.
... This is the monopole contribution from the electric interaction in Eq. (2.9) and, if vacuum polarization is considered, the Uehling potential from Eq. (2.13). A Fermi type charge distribution [126] is used to model the monopole charge distribution as ...
... The normalization constant N is chosen such that the volume integral is equal to one, since the charge is already included in the fine-structure constant. It has been proven, that a = t/(4 log3), with t = 2.30 fm, is a good approximation for most of the nuclei [126]. The parameter c is then determined by demanding, that the charge radius squared ...
... For atomic electrons, there are two QED corrections of the order α, namely the selfenergy (SE) QED and the vacuum polarization (VP) correction [126], which usually contribute equally. For muons, however, the VP correction is much larger due to virtual electron-positron pairs, which are less suppressed due to the small electron-to-muon mass ratio [83]. ...
In this thesis, several aspects of nuclear structure effects and corrections from quantum electrodynamics (QED) in the spectra of hydrogen-like systems are investigated. The first part is concerned with the structure of bound states between a muon and an atomic nucleus, so-called muonic atoms. Here, precise calculations for transition energies and probabilities are presented, using state-of-the-art numerical methods. QED corrections, hyperfine interactions, and the interaction with atomic electrons were evaluated and finite nuclear size effects were incorporated non-perturbatively. Furthermore, new methods for the calculation of higher-order corrections for the hyperfine structure are presented, including a complete calculation of the second-order hyperfine structure and leading-order vacuum polarization corrections for extended electric quadrupole distributions inside the nucleus. In connection with recent x-ray spectroscopic measurements on muonic atoms, the nuclear quadrupole moment of Re-185 and Re-187 is extracted. The second part of this thesis is about the g factor of a bound electron and its dependence on the shape of the nuclear charge distribution. A numerical, non-perturbative approach for the calculation of the corresponding nuclear shape correction is presented and implications for the uncertainties of theoretical predictions are discussed. In particular, the model-uncertainty of the finite-nuclear-size correction to the g factor can be reduced due to the more realistic model of the nuclear charge distribution. Finally, calculations of finite-size and vacuum-polarization corrections to the g factor of a muon bound to a He-4 nucleus significantly contribute to the theoretical prediction on the 10^−9 uncertainty level. As shown in an earlier work, an experimental value of the same accuracy could give access to an improved value of the muon’s mass or magnetic moment anomaly.
... The most dominant nuclear structural effect, namely, the finite size effect is rather well understood [31,32]. It can be calculated by means of perturbation theory, yielding easily evaluable analytical formulas [31], or numerically in an all-order fashion by including the potential corresponding to the extended nucleus into the radial Dirac equation [32]. ...
... The most dominant nuclear structural effect, namely, the finite size effect is rather well understood [31,32]. It can be calculated by means of perturbation theory, yielding easily evaluable analytical formulas [31], or numerically in an all-order fashion by including the potential corresponding to the extended nucleus into the radial Dirac equation [32]. Formulas for higher-order corrections to the nuclear size contribution, such as the anomalous-magneticmoment correction and the third Zeemach momentum of the proton charge distribution, have been derived very recently in Ref. [33]. ...
... The new evaluation provides, in particular, a new value of the isotopic shift for Li-like calcium, which is in a significantly better agreement with the experimental result of Ref. [26]. The treatment of Ref. [52] has also been extended to B-like ions [53]. 5. Determination of fundamental constants g-factor measurements, in combination with theory [7,8,13,14,32,54,55], allowed an independent determination of the electron mass m e [56,57]. The most accurate value [58,59] of m e has been obtained from a recent measurement employing 12 C 5+ ions, improving on previous measurements by more than an order of magnitude. ...
Recent years have witnessed a remarkable improvement in the theoretical description of bound-electron g factors, paralleled with a quantum jump in the experimental accuracy in the investigation of these quantities. In the present article we give a brief summary of the latest developments, with emphasis on the influence of quantum electrodynamic and nuclear effects on the g factor of few-electron highly charged ions, and on the possible determination of fundamental constants.
... For the "free" (unbound) electron, the anomaly is due to effects of quantum electrodynamics (QED) alone, whereas in highly charged ions, it is due to a number of effects related to the binding situation, mainly due to relativistic effects, QED effects, and nuclear effects. [3] The magnetic moment of the unbound electron in units of the Bohr magneton has been measured to 2.7 × 10 −13 relative accuracy by spectroscopy in a Penning trap. [4,5] From this measurement, in combination with QED theory that relates the electron magnetic moment and the fine structure constant α, [6] the value of α has been determined to about four parts in 10 9 . ...
... [15,16] Also in the case of bound electrons, magnetic moment measurements are seen as valuable for an independent determination of the fine structure constant α. [17] Highly charged ions make an interesting object of study to this end, since the binding situation changes the value of the electron magnetic moment by up to about 20%, which is many orders of magnitude larger than the typical experimental uncertainty. The magnitude of this change roughly scales with the square of the nuclear charge state Z of the ion, [3,18] hence making measurements at high values of Z potentially interesting. Figure 1 shows a cartoon as the example of a high-Z hydrogenlike ion like Bi 82+ and the involved quantities: a single electron is bound to an atomic nucleus (Z = 83). ...
... [29] In highly charged ions, such electron magnetic moment measurements are possible in two complementary ways, either by the continuous Stern-Gerlach effect applied to ions with zero-spin nuclei, in full similarity to the measurements, [10][11][12][13][14][15][16] or by making use of the fact that in highly charged ions above a certain value of Z, the hyperfine structure of a few-electron ion becomes accessible for laser spectroscopy. [3] Here, we discuss this latter approach by the ARTEMIS experiment [31,32] with highly charged ions in a cryogenic Penning trap that is optimized for optical spectroscopy under large solid angles. [33,34] This experiment is located at the HITRAP facility [35] at GSI, Germany. ...
... The contributions to the bound-electron g-factor due to one-loop Feynman diagrams have been discussed in the literature in great detail, e.g. [57,[69][70][71]. ...
... The g-factor of an elementary fermionic particle is a dimensionless quantity which parametrizes its magnetic dipole moment according to the formula [69] µ ...
... They correspond to Feynman diagrams with closed loops [6]. Pure QED effects can be parametrized as a perturbation series in powers of the fine structure constant α, where the number of closed loops in the diagram corresponds to the power of α [6,17,19,20,69]: ...
In this thesis, the theory of the g-factor of bound electrons and muons is presented. For light muonic ions, we include one-loop self-energy as well as one- and two-loop vacuum polarization corrections with the interaction with the strong nuclear potential taken into account to all orders. Furthermore, we include effects due to nuclear structure and mass. We show that our theory for the bound-muon g-factor, combined with possible future bound-muon experiments, can be used to improve the accuracy of the muon mass by one order of magnitude. Alternatively, our approach constitutes an independent access to the controversial anomalous magnetic moment of the free muon. Furthermore, two-loop self-energy corrections to the bound-electron g-factor are investigated theoretically to all orders in the nuclear coupling strength parameter Z α (alpha). Formulas are derived in the framework of the two-time Green's function method, and the separation of divergences is performed by dimensional regularization. Our numerical evaluation by treating the nuclear Coulomb interaction in the intermediate-state propagators to zero and first order show that such two-loop terms are mandatory to take into account in stringent tests of quantum electrodynamics with the bound-electron g-factor, and in projected near-future determinations of fundamental constants.
... The electron g factor is a dimensionless constant relating the electron's magnetic moment to its spin. The energy splitting of the Zeeman levels in a magnetic field B is given by [23]. Here, ν L is the Larmor spin precession frequency of an electron, h is the Planck's constant, and ðe=m e Þ is the electron's charge-to-mass ratio. ...
... The QED binding corrections to the free-electron g factor are calculated in the α expansion and include one-, two-, and three-loop contributions [23]. The one-loop correction is calculated in all orders of Zα, whereas for the two-loop correction the perturbation series expansion has been calculated up to O À ðZαÞ 5 Á [69]. ...
We present the measurements of individual bound electron g factors of Ne209+ and Ne229+ on the relative level of 0.1 parts per billion. The comparison with theory represents the most stringent test of bound-state QED in strong electric fields. A dedicated mass measurement results in m(Ne20)=19.992 440 168 77(9) u, which improves the current literature value by a factor of 18, disagrees by 4 standard deviations, and represents the most precisely measured mass value in atomic mass units. Together, these measurements yield an electron mass on the relative level of 0.1 ppb with me=5.485 799 090 99(59)×10−4 u as well as a factor of seven improved m(Ne22)=21.991 385 098 2(26) u.
... Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 10 15 V cm −1 for the innermost electrons 1 . Especially in few-electron, highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. ...
... In these few-electron systems, the electric field experienced by the remaining electrons can exceed 10 15 V cm −1 (ref. 1), hence the electronic wavefunction is perturbed strongly, resulting in modified properties that can be measured and compared with theoretical predictions. So far, bound-state QED in high-Z highly charged ions has been investigated most accurately by measurements of the Lamb shift 20,21 . ...
Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 10¹⁵ V cm⁻¹ for the innermost electrons¹. Especially in few-electron, highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. Consequently, their Lamb shifts were intensively studied in the past several decades2,3. Another approach is the measurement of gyromagnetic factors (g factors) in highly charged ions4–7. However, so far, either experimental accuracy or small field strength in low-Z ions5,6 limited the stringency of these QED tests. Here we report on our high-precision, high-field test of QED in hydrogen-like ¹¹⁸Sn⁴⁹⁺. The highly charged ions were produced with the Heidelberg electron beam ion trap (EBIT)⁸ and injected into the ALPHATRAP Penning-trap setup⁹, in which the bound-electron g factor was measured with a precision of 0.5 parts per billion (ppb). For comparison, we present state-of-the-art theory calculations, which together test the underlying QED to about 0.012%, yielding a stringent test in the strong-field regime. With this measurement, we challenge the best tests by means of the Lamb shift and, with anticipated advances in the g-factor theory, surpass them by more than an order of magnitude.
... Highly charged atomic ions are sought-after study objects in several contexts [1], particularly with regard to precision measurements of x-ray, optical and microwave transitions in few-electron systems that are bound by the extreme electromagnetic fields of the atomic nucleus [2]. This area * Authors to whom any correspondence should be addressed. ...
... Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of research comprises measurements of fine-and hyperfinestructure energies and lifetimes [3], of the Lamb shift [4,5] and the bound-electron magnetic moments [6][7][8] as benchmarks of bound-state quantum electrodynamics [2,9,10], as well as spectroscopy in the framework of metrology [11] and the determination of fundamental constants [12,13]. Much of this effort involves charged-particle traps such as Penning traps [14] for long-term confinement and preparation of the desired species under well-defined conditions, usually in cryogenic surroundings and at extreme vacua. ...
We have built and operated a cryogenic Penning trap arrangement
that allows for the efficient production, selection, and long-term storage of highly
charged atomic ions. In close similarity to an electron-beam ion trap (EBIT)
it works by electron-impact ionisation of atoms inside a dedicated confinement
region. The electrons are produced by field emission at liquid-helium temperature
and are subsequently accelerated to the keV energy range. The electron beam is
reflected through the trap multiple times to increase the ionisation efficiency. We
show a characterisation of the system and measurements with argon and tungsten
ions up to Ar16+ and W27+, respectively.
... In the case of heavy, highly charged ions such as 208 Pb 81+ and 209 Bi 82+ , the fields are as high as E nuc ≈ 10 16 Vcm −1 and B nuc ≈ 10 7 T [9,10]. These high fields cannot be generated in the laboratories. ...
... 10: An illustration of the principle of evaporative cooling. When the trap depth is reduced, the hotter particles leave the trap, thereby leading to a lower equilibrium temperature of the remaining ensemble. ...
Penning traps open up unique experimental possibilities for mass spectrometry and spectroscopy of atomic ions with high precision. Two such experiments based on Penning traps are SHIPTRAP and ARTEMIS at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt. The ARTEMIS experiment is designed to measure the magnetic moment (g-factor) of an electron in heavy, highly charged ions at the 10−9 level, by the means of laser-microwave double-resonance spectroscopy. Such measurements represent highly stringent tests of QED in extreme fields. The ion of choice for the demonstration and development of the experimental methods is 40Ar13+, which can be produced internally in the trap, for later measurements 209Bi82+ is foreseen. For each of the measurements, the preparation of a cleaned and cooled cloud of ions in the spectroscopy trap is necessary. In this work, the system is optimised for non-destructive detection and cooling of the ions, and production, transport, cooling, selection and storage of the Ar13+ ions are systematically demonstrated. Precision mass measurements of the long-lived decay products of 225Ac and 223Ra are carried out at SHIPTRAP, using the phase-imaging ion-cyclotron-resonance technique. The masses of 221Fr, 219Rn, 213Bi, 211Pb, 209Pb, 207Tl and 207Pb are measured with a relative precision of 10−9, allowing an increase in the accuracy of other masses in this region. Some of these masses find direct input into the g-factor measurements, such as in the case of 209Bi, which is of interest to ARTEMIS. Furthermore, the doublet 205Tl / 205Pb, which is of great significance in nuclear astrophysics, is also linked to the measured masses.
... tion, the vacuum polarization (VP) correction [19], in muonic atoms the VP correction is by far the dominant one [2]. Therefore, the SE correction is much smaller than the leading VP correction and was previously calculated within a relatively simple mean-value evaluation method, suggested in Ref. [20] and later used in Refs. ...
... Here, a is the skin thickness and it is usually assumed to be a = 2.3 fm/4 log(3) [19,33]. The condition that V (r) has to be normalized to the nuclear charge Z defines normalization constant ρ 0 and the half-density radius c is chosen to reproduce the rms value. ...
The fully relativistic, rigorous QED calculations of the self-energy correction to the fine-structure levels of heavy muonic atoms are reported, including rigorous predictions for excited states. We discuss nuclear model and parameter dependence for this contribution as well as numerical convergence issues. The presented results mostly agree with previously reported estimations, with some exceptions, including ones used for the determination of the nuclear root-mean-square radii, and underline the importance of rigorous QED calculations for the theoretical prediction of the spectra of muonic atoms.
... In the second case, the BW effect directly contributes to the magnetic moment, and an estimation of its uncertainty is complicated due to the absence of direct many-body calculations of this effect for the rhenium nucleus. It is possible to employ the obtained results also as follows: if one uses the tabulated values [37] [47,83,84] and experimental values [41]. ...
... However, according to our study, this effect can be more important than the solvent effect which is often estimated. Updated values of the nuclear magnetic moments resolve the disagreement between theoretical predictions [47,83,84] and experimental values [41] for the hyperfine splittings of H-like rhenium ions. In addition to the nuclear magnetic moment values, we have also used H-like data for rhenium HFS constants to extract the universal parameter [15] of the nuclear magnetization distribution. ...
The refined values of the magnetic dipole moments of 185 Re and 187 Re nuclei are obtained. For this, we perform a combined relativistic coupled cluster and density-functional theory calculation of the shielding constant for the ReO − 4 anion. In this calculation, we explicitly include the effect of the finite nuclear magnetization distribution in the single-particle nuclear model using the Woods-Saxon potential for the valence nucleon. By combining the obtained value of the shielding constant σ = 4069(389) ppm with the available experimental nuclear magnetic resonance data we obtain the values: μ(185 Re) = 3.1567(3)(12)μ N , μ(187 Re) = 3.1891(3)(12)μ N , where the first uncertainty is the experimental one and the second is due to theory. The refined values of magnetic moments are in disagreement with the tabulated values, μ(185 Re) = 3.1871(3)μ N , μ(187 Re) = 3.2197(3)μ N , which were obtained using the shielding constant value calculated for the atomic cation Re 7+ rather than the molecular anion. The updated values of the nuclear magnetic moments resolve the disagreement between theoretical predictions of the hyperfine structure of H-like rhenium ions which were based on the tabulated magnetic moment values and available experimental measurements. Using these experimental data we also extract the value of the parameter of nuclear magnetization distribution introduced in [J. Chem. Phys. 153, 114114 (2020)], which is required to predict hyperfine structure constants for rhenium compounds.
... One of the important effects is the self-energy (SE) correction. Unlike the case of atomic electrons, where SE is comparable to another QED correction, the vacuum polarization (VP) correction [19], in muonic atoms the VP correction is by far the dominant one [2]. Therefore, the SE correction is much smaller than the leading VP correction and was previously calculated within a relatively simple mean-value evaluation method, suggested in Ref. [20] and later used in Refs. ...
... Here, a is the skin thickness and it is usually assumed to be a = 2.3 fm/4 log(3) [19,33]. The condition that V (r) has to be normalized to the nuclear charge Z defines normalization constant ρ 0 , and the half-density radius c is chosen to reproduce the rms value. ...
The first fully relativistic, rigorous QED calculations of the self-energy correction to the fine-structure levels of heavy muonic atoms are reported. We discuss nuclear model and parameter dependence for this contribution as well as numerical convergence issues. The presented results show sizable disagreement with previously reported estimations, including ones used for the determination of the nuclear root-mean-square radii, and underline the importance of rigorous QED calculations for the theoretical prediction of the spectra of muonic atoms.
... It is possible to employ the obtained results also as follows. If one uses the tabulated values [37] [47,83,84] and experimental values [41]. ...
... However, according to our study, this effect can be more important than the solvent effect which is often estimated. Updated values of the nuclear magnetic moments resolve the disagreement between theoretical predictions [47,83,84] and experimental values [41] for the hyperfine splittings of H-like rhenium ions. In addition to the nuclear magnetic moment values, we have also used H-like data for rhenium HFS constants to extract the universal parameter [15] of the nuclear magnetization distribution. ...
The refined values of the magnetic dipole moments of $^{185}$Re and $^{187}$Re nuclei are obtained. For this, we perform a combined relativistic coupled cluster and density functional theory calculation of the shielding constant for the ReO$_4^-$ anion. In this calculation, we explicitly include the effect of the finite nuclear magnetization distribution in the single-particle nuclear model using the Woods-Saxon potential for the valence nucleon. By combining the obtained value of the shielding constant $\sigma=4069(389)$~ppm with the available experimental nuclear magnetic resonance data we obtain the values: $\mu(^{185}{\rm Re})=3.1567(3)(12) \mu_N, \mu(^{187}{\rm Re})=3.1891(3)(12) \mu_N$,
where the first uncertainty is the experimental one and the second is due to theory. The refined values of magnetic moments are in disagreement with the tabulated values, $\mu(^{185}{\rm Re})=3.1871(3) \mu_N, \mu(^{187}{\rm Re})=3.2197(3) \mu_N$, which were obtained using the shielding constant value calculated for the atomic cation Re$^{7+}$ rather than the molecular anion. The updated values of the nuclear magnetic moments resolve the disagreement between theoretical predictions of the hyperfine structure of H-like rhenium ions which were based on the tabulated magnetic moment values and available experimental measurements. Using these experimental data we also extract the value of the parameter of nuclear magnetization distribution introduced in [J. Chem. Phys. \textbf{153}, 114114 (2020)], which is required to predict hyperfine structure constants for rhenium compounds.
... In the rest of this section, I will give some examples of the expression of the energy shift for several cases and present the results. For a complete derivation of these results, and more details, the reader is referred to, e.g., [124,[126][127][128][129][130][131][132]. Figure 1: One-loop QED corrections. A: Self-energy. ...
... This expression is also sensitive to finite nuclear size corrections. A complete description of the different theoretical methods and contributions can be found in [129,474]. ...
The current status of bound state quantum electrodynamics calculations of transition energies for few-electron ions is reviewed. Evaluation of one and two body QED correction is presented, as well as methods to evaluate many-body effects that cannot beevaluated with present-day QED calculations. Experimental methods, their evolution over time, as well as progress in accuracy are presented. A detailed, quantitative, comparison between theory and experiment is presented for transition energies in few-electron ions. In particular the impact of the nuclear size correction on the quality of QED tests as a function of the atomic number is discussed.The cases of hyperfine transition energies and of bound-electron Land{\'e} $g$-factor are also considered.
... In addition to serving as a reliable mass reference, an improved atomic mass value of uranium-238 is also needed for the planned investigation of the magnetic moment, and with it the g-factor of the bound-electron of hydrogenlike uranium at the experiment ALPHATRAP [16,17]. Electron g factors of heavy, highly charged ions provide stringent tests of bound-state quantum electrodynamics (QED) in strong fields as the size of the QED contribution to the g factor increases with the proton number Z [18]. However, the precision of a g-factor measurement is directly limited by the knowledge of the mass of the ion of interest. ...
The atomic mass of uranium-238 has been determined to be 238.050787618(15) u, improving the literature uncertainty by two orders of magnitude. It is obtained from a measurement of the mass ratio of U47+238 and Xe26+132 ions with an uncertainty of 3.5×10−12. The measurement was carried out with the Penning-trap mass spectrometer Pentatrap and was accompanied by a calculation of the binding energies EU and EXe of the 47 and 26 missing electrons of the two highly charged ions, respectively. These binding energies were determined using an ab initio multiconfiguration Dirac-Hartree-Fock method to be EU=39927(10) eV and EXe=8971.2(21) eV. The new mass value will serve as a reference for high-precision mass measurements in the heavy mass region of the nuclear chart up to transuranium nuclides.
... Trident processes(tri) [55,56] are not considered in these simulations. This should be reasonable as the ratio of the electric field strength of the laser to that of the atomic nucleus at ionic Debye length is 10 3 (using average fields at a Bohr radius for Z=6 being ⟨E⟩ ∼ 4 × 10 14 V/m [57]), favouring pair creation by photon-laser interaction over photon-nuclear interaction. Also with the sub-micron target of ion density of (200/Z)n c , the pair creation probability due to the ionic nuclear field should be lower as also in Ref. [58]. ...
The impact of radiation reaction and Breit–Wheeler pair production on the acceleration of fully ionized carbon ions driven by an intense linearly polarized laser pulse has been investigated in the ultra-relativistic transparency regime. Against initial expectations, the radiation reaction and pair production at ultra-high laser intensities are found to enhance the energy gained by the ions. The electrons lose most of their transverse momentum, and the additionally produced pair plasma of Breit–Wheeler electrons and positrons co-streams in the forward direction as opposed to the existing electrons streaming at an angle above zero degree. We discuss how these observations could be explained by the changes in the phase velocity of the Buneman instability, which is known to aid ion acceleration in the breakout afterburner regime, by tapping the free energy in the relative electron and ion streams. We present evidence that these non-classical effects can further improve the highest carbon ion energies in this transparency regime.
... The g factor of hydrogen-like silicon (Z = 14) has been measured with a 5 × 10 −10 relative uncertainty [1,2], allowing to scrutinize bound-state QED theory (see, e.g., Refs. [3][4][5][6][7][8][9][10][11][12][13]). Two-loop radiative effects and shifts due to nuclear structure and recoil are observable in such measurements. ...
The hadronic vacuum polarization correction to the g factor of a bound electron is investigated theoretically. An effective hadronic Uehling potential obtained from measured cross sections of e−e+ annihilation into hadrons is employed to calculate g-factor corrections for low-lying hydrogenic levels. Analytical Dirac-Coulomb wave functions, as well as bound wave functions accounting for the finite nuclear radius are used. Closed formulas for the g-factor shift in the case of a point-like nucleus are derived. In heavy ions, such effects are found to be much larger than for the free-electron g factor.
... Nowadays, the g factor of H-like ions is measured with a relative accuracy of up to few parts in 10 11 [1][2][3][4][5][6]. These measurements combined with the theoretical studies [7][8][9][10][11][12][13][14][15][16][17][18][19] have led to the most accurate up-to-date value of the electron mass [5]. Present experimental techniques also allow for the g-factor measurements in few-electron ions [20][21][22][23][24][25][26] with the accuracy comparable to that for H-like ions. ...
We present the systematic QED treatment of the electron correlation effects on the $g$ factor of lithiumlike ions for the wide range of nuclear charge number $Z= 14$ -- $82$. The one- and two-photon exchange corrections are evaluated rigorously within the QED formalism. The electron-correlation contributions of the third and higher orders are accounted for within the Breit approximation employing the recursive perturbation theory. The calculations are performed in the framework of the extended Furry picture, i.e., with inclusion of the effective local screening potential in the zeroth-order approximation. In comparison to the previous theoretical calculations, the accuracy of the interelectronic-interaction contributions to the bound electron $g$ factor in lithiumlike ions is substantially improved.
... The study of FNS effects on the spectral properties of few-electron multicharged or muonic ions is of special interest as such effects could provide critical entries for a variety of fundamental research. Since the experimental [12][13][14][15][16][17][18][19][20] and theoretical [21][22][23][24][25][26] determination of atomic energy levels and the bound-electron g factors have been considerably improved in recent years, various applications of these quantities are available in the literature, which include a stringent test of the QED theory [27][28][29][30][31][32], determination of the fine-structure constant [33][34][35][36][37] and electron's mass [38][39][40], as well as tests for new physics beyond the standard model [41,42]. With the fast * jianli@jlu.edu.cn ...
The finite-nuclear-size (FNS) effect has a large contribution to atomic spectral properties, especially for heavy nuclei. By adopting the microscopic nuclear charge-density distributions obtained from the relativistic continuum Hartree-Bogoliubov (RCHB) theory, we systematically investigate the FNS corrections to atomic energy levels and bound-electron g factors of hydrogenlike ions with nuclear charge up to 118. The comparison of the present numerical calculations with the predictions from empirical nuclear charge models, the nonrelativistic Skyrme-Hartree-Fock calculations, and the results based on experimental charge densities indicate that both the nuclear charge radius and the detailed shape of the charge-density distribution play important roles in determining the FNS corrections. The variation of FNS corrections to energy levels and g factors with respect to the nuclear charge are investigated for the several lowest bound states of hydrogenlike ions. It is shown that they both increase by orders of magnitude with increasing nuclear charge, while the ratio between them has a relatively weak dependence on the nuclear charge. The FNS corrections to the s1/2 and p1/2 bound-state energies from the RCHB calculations are generally in good agreement with the analytical estimations by Shabaev [V. M. Shabaev, J. Phys. B 26, 1103 (1993)] based on the homogeneously charged sphere nuclear model, with the discrepancy indicating the distinct contribution of microscopic nuclear structure to the FNS effects.
... which needs to be complemented with 1-to 5-loop QED binding corrections, as well as terms originating from the nucleus, The theoretical contributions to the zero-field hyperfine splitting can be represented as [33,34] ...
Helium-3 has nowadays become one of the most important candidates for studies in fundamental physics [1, 2, 3], nuclear and atomic structure [4, 5], magnetometry and metrology [6] as well as chemistry and medicine [7, 8]. In particular, $^3$He nuclear magnetic resonance (NMR) probes have been proposed as a new standard for absolute magnetometry [6, 9]. This requires a high-accuracy value for the $^3$He nuclear magnetic moment, which, however, has so far been determined only indirectly and with a relative precision of $12$ parts per billon (p.p.b.) [10,11]. Here we investigate the $^3$He$^+$ ground-state hyperfine structure in a Penning trap to directly measure the nuclear $g$-factor of $^3$He$^+$ $g'_I=-4.255\, 099\, 606\, 9(30)_{stat}(17)_{sys}$, the zero-field hyperfine splitting $E_{\rm HFS}^{\rm exp}=-8\, 665\, 649\, 865.77(26)_{stat}(1)_{sys}$ Hz and the bound electron $g$-factor $g_e^\text{exp}=-2.002\, 177\, 415\, 79(34)_{stat}(30)_{sys}$. The latter is consistent with our theoretical value $g_e^\text{theo}=-2.002\, 177\, 416\, 252\, 23(39)$ based on parameters and fundamental constants from [12]. Our measured value for the $^3$He$^+$ nuclear $g$-factor allows for the determination of the $g$-factor of the bare nucleus $g_I=-4.255\, 250\, 699\, 7(30)_{stat}(17)_{sys}(1)_{theo}$ via our accurate calculation of the diamagnetic shielding constant [13] $\sigma_{^3He^+}=0.000\,035\,507\,38(3)$. This constitutes the first direct calibration for $^3$He NMR probes and an improvement of the precision by one order of magnitude compared to previous indirect results. The measured zero-field hyperfine splitting improves the precision by two orders of magnitude compared to the previous most precise value [14] and enables us to determine the Zemach radius [15] to $r_Z=2.608(24)$ fm.
... The theoretical contributions to the zero-field hyperfine splitting can be represented as 33,34 , where the uncertainty is dominated by neglected higher order QED terms. This high accuracy, due to the low value of Zα and to suppressed nuclear effects, enables an accurate extraction of the unshielded nuclear g-factor from the measured shielded g-factor. ...
Helium-3 has nowadays become one of the most important candidates for studies in fundamental physics1–3, nuclear and atomic structure4,5, magnetometry and metrology6, as well as chemistry and medicine7,8. In particular, 3He nuclear magnetic resonance (NMR) probes have been proposed as a new standard for absolute magnetometry6,9. This requires a high-accuracy value for the 3He nuclear magnetic moment, which, however, has so far been determined only indirectly and with a relative precision of 12 parts per billon10,11. Here we investigate the 3He+ ground-state hyperfine structure in a Penning trap to directly measure the nuclear g-factor of 3He+gI′=−4.2550996069(30)stat(17)sys, the zero-field hyperfine splitting EHFSexp=−8,665,649,865.77(26)stat(1)sys Hz and the bound electron g-factor geexp=−2.00217741579(34)stat(30)sys. The latter is consistent with our theoretical value getheo=−2.00217741625223(39) based on parameters and fundamental constants from ref. 12. Our measured value for the 3He+ nuclear g-factor enables determination of the g-factor of the bare nucleus gI=−4.2552506997(30)stat(17)sys(1)theo via our accurate calculation of the diamagnetic shielding constant13σ3He+=0.00003550738(3). This constitutes a direct calibration for 3He NMR probes and an improvement of the precision by one order of magnitude compared to previous indirect results. The measured zero-field hyperfine splitting improves the precision by two orders of magnitude compared to the previous most precise value14 and enables us to determine the Zemach radius15 to rZ=2.608(24) fm. Measuring the hyperfine structure of a single helium-3 ion in a Penning trap enables direct measurement of the nuclear magnetic moment of helium-3 and provides the high accuracy needed for NMR-based magnetometry.
... Nowadays, the boundelectron g factor is measured with a relative accuracy of a few parts in 10 11 in H-like carbon and silicon ions [1][2][3]. These measurements, accompanied by impressive theoretical studies [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], have led to the most accurate up-to-date value of the electron mass [3,23,24]. Furthermore, the present experimental techniques also enable high-precision g-factor measurements in few-electron ions [25][26][27][28][29]. ...
The bound-electron g factor is a stringent tool for tests of the standard model and the search for new physics. The comparison between an experiment on the g factor of lithiumlike silicon and the two recent theoretical values revealed the discrepancies of 1.7σ [Glazov et al. Phys. Rev. Lett. 123, 173001 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.173001] and 5.2σ [Yerokhin et al. Phys. Rev. A 102, 022815 (2020)PLRAAN2469-992610.1103/PhysRevA.102.022815]. To identify the reason for this disagreement, we accomplish large-scale high-precision computation of the interelectronic-interaction and many-electron QED corrections. The calculations are performed within the extended Furry picture of QED, and the dependence of the final values on the choice of the binding potential is carefully analyzed. As a result, we significantly improve the agreement between the theory and experiment for the g factor of lithiumlike silicon. We also report the most accurate theoretical prediction to date for lithiumlike calcium, which perfectly agrees with the experimental value.
... Nowadays, the boundelectron g factor is measured with a relative accuracy of a few parts in 10 11 in H-like carbon and silicon ions [1][2][3]. These measurements, accompanied by impressive theoretical studies [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], have led to the most accurate up-to-date value of the electron mass [3,23,24]. Furthermore, the present experimental techniques also enable high-precision g-factor measurements in few-electron ions [25][26][27][28][29]. ...
The bound-electron g factor is a stringent tool for tests of the Standard Model and the search for new physics. The comparison between an experiment on the g factor of lithiumlike silicon and the two recent theoretical values revealed the discrepancies of $1.7\sigma$ [D. A. Glazov $\textit{et al}$., Phys. Rev. Lett. $\textbf{123}$, 173001 (2019)] and $5.2\sigma$ [V. A. Yerokhin $\textit{et al}$., Phys. Rev. A $\textbf{102}$, 022815 (2020)]. To identify the reason for this disagreement, we accomplish large-scale high-precision computation of the interelectronic-interaction and many-electron QED corrections. The calculations are performed within the extended Furry picture of QED, and the dependence of the final values on the choice of the binding potential is carefully analyzed. As a result, we significantly improve the agreement between the theory and experiment for the g factor of lithiumlike silicon. We also report the most accurate theoretical prediction to date for lithiumlike calcium, which perfectly agrees with the experimental value.
... This interest is rising partly due to the development of experiments on highly charged ions utilizing ion traps. The study of these systems also provides the opportunity to determine fundamental constants such as the mass of the electron [7,8]. Experiments on ions of high Z are of great interest from theoretical and experimental point of view and are to be performed in the nearest future. ...
Electron correlation effects on the bound-electron g -factor and hyperfine structure in highly charged ions are calculated by a combination of perturbation theory to leading order in 1/ Z , with Z being the atomic number, and a large-scale relativistic configuration interaction method. Our results for lithiumlike ions improve the accuracy of earlier calculations by one or two digits.
... Today, many more contributions resulting from different types of effects have been calculated. Radiative corrections due to interactions of the bound electron with itself by the emission and reabsorption of a virtual photon (the self-energy effect) have been calculated to one-loop order perturbatively in the nuclear coupling strength up to order (Zα) 5 [26][27][28][29][30][31] and to all orders [32][33][34][35][36]. Radiative corrections due to the interaction of the bound electron with the nucleus through the creation and annihilation of virtual electron-positron pairs (the vacuum polarization effect) have been calculated [11,[37][38][39][40][41][42][43]. The perturbative expansion of two-loop radiative corrections to order (Zα) 4 [30,[44][45][46] has been recently extended to order (Zα) 5 [47,48], which has improved the theoretical uncertainty for low nuclear charges Z, and first milestones have been achieved in the non-perturbative calculation of two-loop corrections to the bound-electron g factor of H-like ions [5,49]. ...
In this thesis, quantum electrodynamic (QED) effects in few-electron highly charged ions are investigated. The interaction of the electron with the nucleus is taken into account in a nonperturbative manner. A versatile approach to accurately calculate self-energy corrections combining finite basis sets with analytical methods is presented. The approach is applicable to many-electron ions using the screening-potential approximation. The method is applied to calculate self-energy corrections to the energy level of the electron in the 4d3∕2 state of ¹³¹Xe¹⁷⁺ and to the excitation energy of the 4d → 4f excitation in ¹⁸⁷Re²⁹⁺. QED corrections to the g factor of lithiumlike and boronlike ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, derived in a QED framework, and most of the terms of the vacuum polarization effect are evaluated to all orders in the nuclear coupling strength Zα. Uncertainties resulting from nuclear size effects, numerical calculations, and uncalculated effects are discussed. Finally, a new approach to determine the fine-structure constant α using a weighted difference of the bound-electron g factor and energy in hydrogenlike systems is put forward. It is shown that nuclear structural effects are sufficiently well suppressed while sensitivity to α is enhanced in this weighted difference, as compared to the g factor.
... Penning-trap experiments employing the continuous Stern-Gerlach effect achieve high precision nowadays, and are advancing towards heavy ions, in which the effects of quantum electrodynamics (QED) are most relevant. The g factor of hydrogenlike silicon (Z = 14) has been determined with a 5 × 10 −10 fractional uncertainty [1,2], allowing one to scrutinize the bound-state QED theory (see, e.g., [3][4][5][6][7][8][9][10][11]). Recently, the evaluation of two-loop terms of order (Zα) 5 (with Z being the atomic number and α the fine-structure constant) has been finalized [12] (see also [13]), increasing the theoretical accuracy especially in the low-Z regime. ...
QED corrections to the g factor of Li-like and B-like ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, the self-energy effect, and most of the terms of the vacuum-polarization effect are evaluated to all orders in the nuclear coupling strength Zα. Uncertainties resulting from nuclear size effects, numerical computations, and uncalculated effects are discussed.
... It has been shown, that a = t/(4 log3), with t = 2.30 fm, is a good approximation for most of the nuclei [13]. Then, c and β are chosen such that the root-mean-square radius r rms of the distribution agrees with the literature value [1] and the quadrupole moment agrees with a given value, which is obtained by fitting to the experimental data as described in Sec. ...
The hyperfine splitting of the 5g→4f transitions in muonic Re185,187 has been measured using high resolution high purity germanium detectors and compared to state-of-the-art atomic theoretical predictions. The spectroscopic quadrupole moment has been extracted using modern fitting procedures and compared to the values available in literature obtained from muonic x rays of natural rhenium. The extracted values of the nuclear spectroscopic quadrupole moment are 2.07(5) b and 1.94(5) b, respectively for Re185 and Re187.
... The g factor of hydrogen-like silicon (Z = 14) has been determined with a 5 · 10 −10 fractional uncertainty [1,2], allowing to scrutinize bound-state QED theory (see e.g. [3][4][5][6][7][8][9][10][11]). Recently, the evaluation of two-loop terms of order (Zα) 5 (with Z being the atomic number and α the fine-structure constant) has been finalized [12] (see also [13]), increasing the theoretical accuracy especially in the low-Z regime. ...
QED corrections to the $g$ factor of Li-like and B-like ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, the self-energy effect, and most of the terms of the vacuum-polarization effect are evaluated to all orders in the nuclear coupling strength $Z\alpha$. Uncertainties resulting from nuclear size effects, numerical computations, and uncalculated effects are discussed.
... It has been shown, that a = t/(4 log3), with t = 2.30 fm, is a good approximation for most of the nuclei [13]. Then, c and β are chosen such that the root-mean-square radius r RMS of the distribution agrees with the literature value [1] and the quadrupole moment agrees with a given value, which is obtained by fitting to the experimental data as described in Section II E. The connection between the charge distribution of Eq. (5) and the spectroscopic quadrupole moment is ...
The hyperfine splitting of the 5g -> 4f transitions in muonic 185,187-Re has been measured using high resolution HPGe detectors and compared to state-of-the-art atomic theoretical predictions. The spectroscopic quadrupole moment has been extracted using modern fitting procedures and compared to the values available in literature obtained from muonic X rays of natural rhenium. The extracted values of the nuclear spectroscopic quadrupole moment are 2.07(5) barn and 1.94(5) barn, respectively for 185-Re and 187-Re. This work is part of a larger effort at the Paul Scherrer Institut towards the measurement of the nuclear charge radii of radioactive elements.
... [10,11,14]. Values for the parameters can be estimated by using a value a=2.3 fm/(4 ln 3), which has proved to be a sufficiently accurate value for most nuclei [33]. Then, c and β are chosen such that the quadrupole moment and RMS value of the distribution are in agreement with the literature values from [29,30]. ...
A method for precise calculation of the energy corrections due to second order electric quadrupole interactions, as well as mixed electric quadrupole-vacuum polarization in the framework of the dynamic hyperfine structure in heavy muonic atoms is presented. For this, a multipole expansion of the Uehling potential is performed. The approach is applicable for an arbitrary nuclear electric charge distribution. By performing these calculations for muonic Rhenium and Uranium using a deformed Fermi distribution, it is shown that both corrections contribute on a level presumably visible in upcoming experiments.
... Fast progress in the theoretical understanding and experimental precision of the bound-electron g factor (see e.g., [8][9][10][11][12][13][14][15][16] and references therein) has enabled the most accurate determination of the mass of the electron in Penning trap g-factor experiments by means of the continuous Stern-Gerlach effect [8,[17][18][19][20]. In this paper we put forward a similar method for the extraction of the mass of the muon by employing light muonic ions, by which we mean here bound systems solely consisting of a nucleus and a muon without further surrounding electrons. ...
A theoretical description of the g factor of a muon bound in a nuclear potential is presented. One-loop self-energy and multiloop vacuum polarization corrections are calculated, taking into account the interaction with the binding potential exactly. Nuclear effects on the bound-muon g factor are also evaluated. We put forward the measurement of the bound-muon g factor via the continuous Stern-Gerlach effect as an independent means to determine the muon’s magnetic moment anomaly and mass. The scheme presented enables the increase of the accuracy of the mass by more than an order of magnitude.
... In an external, homogeneous, and weak magnetic field B, the g factor of the bound electron is defined by the first order energy splitting δE due to the external field as the proportionality coefficient [21] ...
The theory of the g factor of an electron bound to a deformed nucleus is considered non-perturbatively and results are presented for a wide range of nuclei with charge numbers from Z=16 up to Z=98. We calculate the nuclear deformation correction to the bound electron g factor within a numerical approach and reveal a sizable difference compared to previous state-of-the-art analytical calculations. We also note particularly low values in the region of filled proton or neutron shells, and thus a reflection of the nuclear shell structure both in the charge and neutron number.
In special situations, optical and microwave spectroscopy can be combined to constitute a powerful tool that uses the high spectral resolution of the microwaves and the good detection properties of optical light. Here, we briefly discuss its application to the determination of magnetic moments of the nucleus and the electron bound in highly charged ions.
This chapter briefly reviews the measurements of magnetic moments that have been performed by application of the continuous Stern-Gerlach effect to a single particle confined in a Penning trap with a magnetic bottle.
We have conceived, built, and operated a cryogenic vacuum valve with opening and closing times as short as 50 ms that can be used in strong magnetic fields and across a broad range of duty cycles. It is used to seal a cryogenic Penning trap at liquid-helium temperature for long-term storage of highly charged ions in a vacuum better than 10−15 hPa from a room-temperature ion beamline at vacuum conditions around 10−9 hPa. It will significantly improve any experiment where a volume at the most extreme vacuum conditions must be temporarily connected to a less demanding vacuum during repeated experimental cycles. We describe the design of this valve and show measurements that characterize its main features.
We present the systematic QED treatment of the electron correlation effects on the g factor of lithiumlike ions for the wide range of nuclear charge number Z=14–82. The one- and two-photon exchange corrections are evaluated rigorously within the QED formalism. The electron-correlation contributions of the third and higher orders are accounted for within the Breit approximation employing the recursive perturbation theory. The calculations are performed in the framework of the extended Furry picture, i.e., with inclusion of the effective local screening potential in the zeroth-order approximation. In comparison to the previous theoretical calculations, the accuracy of the interelectronic-interaction contributions to the bound electron g factor in lithiumlike ions is substantially improved.
A method is proposed to determine the M1 nuclear transition amplitude and hence the lifetime of the "nuclear clock transition" between the low-lying (∼8 eV) first isomeric state and the ground state of ^{229}Th from a measurement of the ground-state g factor of few-electron ^{229}Th ions. As a tool, the effect of nuclear hyperfine mixing in highly charged ^{229}Th ions such as ^{229}Th^{89+} or ^{229}Th^{87+} is used. The ground-state-only g-factor measurement would also provide first experimental evidence of nuclear hyperfine mixing in atomic ions. Combining the measurements for H-, Li-, and B-like ^{229}Th ions has a potential to improve the initial result for a single charge state and to determine the nuclear magnetic moment to a higher accuracy than that of the currently accepted value. The calculations include relativistic, interelectronic-interaction, QED, and nuclear effects.
We report the 2018 self-consistent values of constants and conversion factors of physics and chemistry recommended by the Committee on Data of the International Science Council. The recommended values can also be found at physics.nist.gov/constants. The values are based on a least-squares adjustment that takes into account all theoretical and experimental data available through 31 December 2018. A discussion of the major improvements as well as inconsistencies within the data is given. The former include a decrease in the uncertainty of the dimensionless fine-structure constant and a nearly two orders of magnitude improvement of particle masses expressed in units of kg due to the transition to the revised International System of Units (SI) with an exact value for the Planck constant. Further, because the elementary charge, Boltzmann constant, and Avogadro constant also have exact values in the revised SI, many other constants are either exact or have significantly reduced uncertainties. Inconsistencies remain for the gravitational constant and the muon magnetic-moment anomaly. The proton charge radius puzzle has been partially resolved by improved measurements of hydrogen energy levels.
We report the 2018 self-consistent values of constants and conversion factors of physics and chemistry recommended by the Committee on Data of the International Science Council (CODATA). The recommended values can also be found at physics.nist.gov/constants. The values are based on a least-squares adjustment that takes into account all theoretical and experimental data available through 31 December 2018. A discussion of the major improvements as well as inconsistencies within the data is given. The former include a decrease in the uncertainty of the dimensionless fine-structure constant and a nearly two orders of magnitude improvement of particle masses expressed in units of kg due to the transition to the revised International System of Units (SI) with an exact value for the Planck constant. Further, because the elementary charge, Boltzmann constant, and Avogadro constant also have exact values in the revised SI, many other constants are either exact or have significantly reduced uncertainties. Inconsistencies remain for the gravitational constant and the muon magnetic-moment anomaly. The proton charge radius puzzle has been partially resolved by improved measurements of hydrogen energy levels.
A method is presented for precise calculation of the energy corrections due to second-order electric-quadrupole interactions, as well as mixed electric quadrupole-vacuum polarization in the framework of the dynamic hyperfine structure in heavy muonic atoms. For this, a multipole expansion of the Uehling potential is performed. The approach is applicable for an arbitrary nuclear electric charge distribution. By performing these calculations for muonic rhenium and uranium using a deformed Fermi distribution, it is shown that both corrections contribute on a level presumably visible in upcoming experiments.
research is aimed at using advanced corrosion identification and corrosion inhibition
technique to investigate the corrosion inhibition mechanism of Al Alloy (AA5052) in 1.0 M HCl
solution using Newboudia laevis leaf extract.
Place and Duration of Study: The research was carried out at Material Science Laboratory
Group, Physics/Electronic Department, Abia State Polytechnic, Aba Abia State.
Methodology: Extraction of the green leaf was done with reflux apparatus under constant
temperature. The 100% inhibitor solvent was diluted to the various inhibitor concentrations.
Gravimetric technique and electrochemical impedance spectroscopy (EIS) were employed to study
the corrosion and corrosion inhibitory behavior of the metal in the various environments. Scanning
Electron Microscope and OPTICAL emission microscopy were used to investigate the surface
structure of the corroded and inhibited metal alloy. The influence of the leaf extract on the
mechanical hardness of the Al alloy AA5052
The theory of the g factor of an electron bound to a deformed nucleus is considered nonperturbatively and results are presented for a wide range of nuclei with charge numbers from Z=16 up to Z=98. We calculate the nuclear deformation correction to the bound electron g factor within a numerical approach and reveal a sizable difference compared to previous state-of-the-art analytical calculations. We also note particularly low values in the region of filled proton or neutron shells, and thus a reflection of the nuclear shell structure both in the charge and neutron number.
The nuclear recoil effect on the g factor of highly charged Li-like ions is evaluated in the range Z=10–92. The calculations are performed by using 1/Z perturbation theory. The one-electron recoil contribution is evaluated within the fully relativistic approach with the wave functions which account approximately for the screening effects. The two-electron recoil contributions of the first and higher orders in 1/Z are calculated within the Breit approximation by using a four-component approach.
The nuclear recoil effect on the g factor of highly charged Li-like ions is evaluated in the range Z=10-92. The calculations are performed using the 1/Z perturbation theory. The one-electron recoil contribution is evaluated within the fully relativistic approach with the wave functions which account for the screening effects approximately. The two-electron recoil contributions of the first and higher orders in 1/Z are calculated within the Breit approximation using a four-component approach.
The measurement of the electronic g-factor of hydrogenic ions is a sensitive test of bound-state QED (Quantum Electrodynamics). The deviations of the g-factor of the bound electron from the free-electron value are mainly due to i) the relativistic binding energy correction -(Zα)2/3 and ii) the bound-state radiative correction α(Zα)2/4π. In the experiment a single hydrogenic ion is stored in the magnetic field of a Penning trap. The g-factor is measured by inducing spin-flip transitions with a microwave field. The magnetic field is calibrated measuring the cyclotron frequency of the stored ion. In the Penning trap the ion is detected electronically and cooled to 4° K through a superconducting resonance circuit connected to the trap electrodes.
We have performed a pure optical frequency measurement of the 2S-12D two-photon transitions in atomic hydrogen and deuterium. From a complete analysis taking into account this result and all other precise measurements (by ourselves and other authors), we deduce optimized values for the Rydberg constant, R∞ = 109737.31568516(84)cm-1 (relative uncertainty of 7.7×10-12) and for the 1S and 2S Lamb shifts L1S = 8172.837(22)MHz and L2S-2P = 1057.8446(29)MHz [respectively, L1S = 8183.966(22)MHz, and L2S-2P = 1059.2337(29)MHz for deuterium]. These are now the most accurate values available.
The single photon annihilation contributions for the positronium ground state hyperfine splitting are calculated analytically to order m{sub e}αⶠusing nonrelativistic QED. Based on intuitive physical arguments the same result can also be determined using results from previous calculations. Our result completes the hyperfine splitting calculation to order m{sub e}αⶠ. We compare the theoretical prediction with the most recent experimental measurement. {copyright} {ital 1997} {ital The American Physical Society}
This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
This paper gives the 1998 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. Further, it describes in detail the adjustment of the values of the subset of constants on which the complete 1998 set of recommended values is based. The 1998 set replaces its immediate predecessor recommended by CODATA in 1986. The new adjustment, which takes into account all of the data available through 31 December 1998, is a significant advance over its 1986 counterpart. The standard uncertainties (i.e., estimated standard deviations) of the new recommended values are in most cases about 1/5 to 1/12 and in some cases 1/160 times the standard uncertainties of the corresponding 1986 values. Moreover, in almost all cases the absolute values of the differences between the 1998 values and the corresponding 1986 values are less than twice the standard uncertainties of the 1986 values. The new set of recommended values is available on the World Wide Web at physics.nist.gov/constants.
Supercritical electromagnetic fields are predicted to lead to spontaneous emission of positrons in nuclear systems with Z > 173. A possible route to identify spontaneous positron creation is discussed. The radiative quantum electrodynamical corrections are calculated. Their contribution amounts to about one per cent of the electron binding energy in nearly critical systems. The formation of supercritical high-Z quasiatoms in heavy-ion collisions is investigated, and the use of δ-electron spectra as measurement tool for nuclear delay times and electron binding energies in superheavy quasiatoms is pointed out. Positron creation by dynamical processes and internal pair conversion is evaluated.
The theory of hyperfine-structure splitting in highly charged ions is reviewed. The available theoretical results for the ground-state hyperfine splitting in hydrogen-like and lithium-like ions are presented. It is shown that the Bohr Weisskopf effect, which mainly defines the uncertainty of the theoretical values, can be eliminated in a combination of the hyperfine splitting values of hydrogenlike and lithium-like ions. It allows one to make high precision predictions for the ground-state hyperfine splitting in lithium-like ions if the ground-state hyperfine splitting of the corresponding hydrogen-like ions is known from experiment. The transition probabilities between the hyperfine components are also presented.
Precise intrinsic quadrupole and hexadecapole moments of $^{233,\phantom{\rule{0ex}{0ex}}234,\phantom{\rule{0ex}{0ex}}235,\phantom{\rule{0ex}{0ex}}238}\mathrm{U}$ have been determined from muonic $K$, $L$, $M$, and $N$ x rays. For $^{233,\phantom{\rule{0ex}{0ex}}235}\mathrm{U}$ seven $E2$ matrix elements were independently determined. These $E2$ matrix elements are in good agreement with the adiabatic rotational model; this agreement is further improved if a correction for $\ensuremath{\Delta}K=1$ band mixing is included. The measured hexadecapole moments are in good agreement with shellcorrection calculations and Hartree-Fock calculations.
A study is carried out of the vacuum polarization in a strong Coulomb field. Radiative corrections are neglected. A perturbation calculation is avoided by making use of the explicit solutions of the Dirac equation in a Coulomb field. The Laplace transform of the polarization charge density times ${r}^{2}$ is found and used as a basis for further study. It is proved to be an analytic function of the strength of the inducing charge. It is verified that the first-order term in a power series expansion in the strength of the inducing charge just corresponds to the Uehling potential. The third-order term is studied in some detail. The leading term in the polarization potential close to the inducing charge and the space integral of the induced potential divided by $r$ are found to all orders in the strength of the inducing charge. Ambiguities are handled by a method corresponding to regularization.
The proton-size correction to the hyperfine structure in the ground state of atomic hydrogen is re-examined. It is shown by means of dispersion relations that this correction can be expressed as an integral over experimentally measurable cross sections for electron-proton scattering. This clarifies the physical nature of the correction, puts it on a rigorous basis and lends support to previous analyses. In the absence of experimental data, we give some theoretical estimates for the correction. They agree with previous estimates, and therefore we cannot explain the present experimental value for the hyperfine splitting. We discuss some possible implications of this disagreement and suggest some experiments which would clarify the situation.
The theory of the ground state hyperfine splitting in hydrogen-like lead and lithium-like bismuth is discussed. A method for calculation of the Bohr-Weisskopf correction in lithium-like Bi, based on using the experimental result for the ground state hyperfine splitting in hydrogen-like Bi, is proposed. The ground state hyperfine splitting in 209Bi80+ is calculated to be ΔE = 0.7969(2) eV.
The Tables summarize experimental results from muonic atom transition energies, nuclear charge parameters from elastic electron scattering, and K x-ray isotope shifts in so far as they provide information on nuclear ground-state charge radii. Numerous experimental results for optical isotope shifts have been published elsewhere; for eight elements the relevant information is condensed ("projected") here to one optical line per element. A model-independent analysis which combines data from all three experimental methods is applied to these elements and is presented as an illustration of the improved accuracy for the rms radii and Barrett radii which result from this analysis.
Spontaneous capture of free cooling electrons by bare high-Z ions leads to the emission of X rays which have been observed for the first time. The distinctive features of this X-ray source for precision spectroscopy of high-Z few-electron systems are demonstrated. They enabled a determination of the 1s Lamb shift of hydrogen-like Au78+ (212±15 eV) to an accuracy of 7%. This is more precise than any other measurement previously reported for a very heavy ion.
A new analytic calculation of radiative-recoil corrections to muonium ground-state hyperfine splitting induced by electron line insertions is performed. The starting point of this calculation is presented by the Fried-Yennie gauge expression for the electron line factor. The final result confirms the one obtained previously from the apparently different expression in the Feynman gauge and removes the slight discrepancy which existed in the literature between the calculations in different gauges.
Precise intrinsic quadropole and hexadecapole moments of 232Th and 239,240,242Pu have been determined from analysis of muonic K,L, and M X-rays. The quadropole and hexadecapole moments are in satisfactory agreement with both Hartree-Fock and shell-correction calculations. For 239Pu, nine E2 matrix elements were independently determined. These E2 matrix elements are in good agreement with the adiabatic rotational model.
X-rays are emitted with the radiative recombination of free electrons in an electron cooler of a heavyion storage ring. Due to a small width of the X-ray lines, an observation angle close to 0° and an accurate determination of the ion velocity, the ground-state Lambshift of hydrogenlike uranium (470 ± 16) eV could be measured to an accuracy of 3.4%. A re-evaluation of a measurement of the 1s
1/2 Lambshift in hydrogenlike gold gave a new value of (202.3 ± 7.9) eV as compared to the former value of (212 ± 15) eV. The results are in excellent agreement with QED calculations and are more precise than any other measurements previously reported for a high-Z, hydrogenlike ion.
The use of bound-state wave functions in calculations in positron theory is justified by the introduction of a new representation, in a certain sense intermediate between the Heisenberg and interaction representations. In the bound-state representation the definition of a stable vacuum state is possible only for a restricted class of external fields. Some attention is given to the problem of vacuum polarization, and it is shown that a very simple procedure accomplishes the charge renormalization with sufficient accuracy to be of use in certain scattering problems. The application to the scattering of radiation is discussed in some detail, in order to show the relation between the different points of view that may be adopted in problems of the coherent scattering by a bound electron and the "Delbrück scattering" by virtual electron pairs.
The detailed account of analytic calculation of radiative-recoil correction to muonium hyperfine splitting, induced by electron-line radiative insertions, is presented. The consideration is performed in the framework of the effective two-particle formalism. A good deal of attention is paid to the problem of the divergence cancellation and the selection of graphs, relevant to radiative-recoil corrections. The analysis is greatly facilitated by use of the Fried-Yennie gauge for radiative photons. The obtained set of graphs turns out to be gauge-invariant and actual calculations are performed in the Feynman gauge. The main technical tricks, with the help of which we have effectively utilized the existence in the problem of the small parameter-mass ratio and managed to perform all calculations in the analytic form are described. The main intermediate results, as well as the final answer, {delta}E{sub rr} = ({alpha}({Zeta}{alpha})/{pi}{sup 2})(m/M)E{sub F}(6{zeta}(3) + 3{pi}{sup 2} In 2 + {pi}{sup 2}/2 + 17/8), are also presented.
Corrections to the positronium energy levels of order {alpha}{sup 6} due to photon exchanges are calculated in the effective Hamiltonian approach. The quoted results are valid for all S states and arbitrary mass ratios. We further present implications on the comparison of theory and experiment. {copyright} {ital 1997} {ital The American Physical Society}
The relativistic two-body equation of Bethe and Salpeter is derived from field theory. It is shown that the Feynman two-body kernel may be written as a sum of wave functions over the states of the system. These wave functions depend exponentially on the energies of the states to which they correspond and therefore provide a means of calculating energy levels of bound states.
Wavelengths of the fine-structure transitions 2sthinsp²Sâââ2pthinsp²Pââ and 2sthinsp²Sâââ2pthinsp²Pââ in lithiumlike Ni{sup 25+} and Zn{sup 27+} have been measured using beam-foil excitation and grazing-incidence spectroscopy. The respective transition wavelengths of 142.461±0.006thinspâ« and of 216.061±0.011thinspâ« for Zn{sup 27+} have not been measured so far and present the most precise values in lithiumlike heavy ions measured by beam-foil spectroscopy. The Ni{sup 25+} results are in very good agreement with data from grazing-incidence spectroscopy at tokamaks. The measurement precision for both ions corresponds to less than or equal to 1% of the quantum electrodynamic contributions to the transition energies. This provides sensitivity to two-photon exchange processes for the screening of the electron self-energy. {copyright} {ital 1998} {ital The American Physical Society}
The first direct observation of a hyperfine splitting in the optical regime is reported. The wavelength of the [ital M]1 transition between the [ital E]=4 and [ital F]=5 hyperfine levels of the ground state of hydrogenlike [sup 209]Bi[sup 82+] was measured to be [lambda][sub 0]=243.87(4) nm by detection of laser induced fluorescence at the heavy-ion storage ring ESR at GSL. In addition, the lifetime of the laser excited [ital F]=5 sublevel was determined to be [tau][sub 0]=0.351(16) ms. The method can be applied to a number of other nuclei, and should allow a novel test of ED corrections in the previously unexplored combination of strong magnetic and electric fields in highly charged ions.
Using continuous-wave excitation to eliminate the problems inherent with pulsed laser measurements of nonlinear transitions, we have measured the 1[sup 3][ital S][sub 1-]2[sup 3][ital S][sub 1] interval in positronium (Ps) to be 1 233 607 216.4[plus minus]3.2 MHz. The quoted 2.6 ppb (parts per 10[sup 9]) uncertainty is primarily due to the determination of the Ps resonance relative to the Te[sub 2] reference line, with a 1.5 ppb contribution from a recent calibration of the Te[sub 2] line relative to the hydrogen 1[ital S]-2[ital S] transition. The uncertainty corresponds to 3.5[times]10[sup [minus]5] of the [alpha][sup 2][ital R][sub [infinity]] QED contribution to the 1[sup 3][ital S][sub 1-]2[sup 3][ital S][sub 1] interval. Our measurement is sufficiently accurate to provide a test of the as yet uncalculated [alpha][sup 4][ital R][sub [infinity]] QED corrections.
We present an analytic calculation of the O(mff 6 ) recoil and radiative recoil corrections to energy levels of positronium nS states and their hyperfine splitting. A complete analytic formula valid to O(mff 6 ) is given for the spectrum of S states. Technical aspects of the calculation are discussed in detail. Theoretical predictions are given for various energy intervals and compared with experimental results. PACS numbers: 36.10.Dr, 06.20.Jr, 12.20.Ds, 31.30.Jv e-mail: czar@quark.phy.bnl.gov y e-mail: melnikov@particle.physik.uni-karlsruhe.de z e-mail: yelkhovsky@inp.nsk.su 1 I. INTRODUCTION Spectroscopy of positronium (Ps) provides a sensitive test of bound state theory based on the Quantum Electrodynamics (QED). Because of the small mass of electron and positron, the effects of strong and weak interactions are negligible compared with the accuracy of present experiments. For this reason positronium represents a unique system which can, in principle, be described with ...
Historically, the spin magnetic moment of the electron µ
e
or its gvalue g
e
has played a central role in modern physics, dating from its discovery in atomic optical spectroscopy and its subsequent incorporation in the Dirac theory of the electron, which predicted the value g
e
= 2 The experimental discovery in atomic microwave spectroscopy that g
e
was larger than 2 by a multiplicative factor of about 1 part in 103, g
e
=2.00238(10), together with the discovery of the Lamb shift in hydrogen (S=22S
1/2 − 22P
1/2), led to the development of modern quantum electrodynamics with its renormalization procedure. The theory enables us to calculate these effects precisely as finite radiative corrections. By now the experimental value of g
e
-2has been measured to about 4 ppb, and the theoretical value, which is expressed as a power series in the fine-structure constant a, has been evaluated to better than 1 ppb, assuming the value of α is known.
The hyperfine structure of the one-electron ion ââ²°â¹Bi{sup 82+} is evaluated within the framework of a dynamical model in which the electron is assumed to interact with the valence proton through the exchange of a photon.
We describe the status of the problem of the electron structure of superheavy atoms with nuclear charge Z > Zc ; here Zcapprox170 is the critical value of the nuclear charge, at which the energy of the ground state of the 1S1/2 electron reaches the limit of the lower continuum of the solutions of the Dirac equation (ε = - mec2) . We discuss the dependence of Zc on the nuclear radius R and on the character of the distribution of the electric charge inside the nucleus, and also the form of the wave functions at Z close to Zc . Owing to the Coulomb barrier , the state of the electron remains localized at Z > Zc , in spite of the fact that its energy approaches the continuum. An analysis of the polarization of the vacuum in a strong Coulomb field shows that a bare nucleus with supercritical charge Z > Zc produces spontaneously two positrons and, in addition a charge density with a total of two units of negative charge in the vacuum. The distribution of this density is localized in a region of dimension r ~ h/mec at the nucleus. The possibility of experimentally observing the effect of quasistatic production of positrons in the collision of two bare uranium nuclei (i.e., without electrons) is discussed. A brief review is presented of work on the motion of levels with increasing depth of the potential well in other relativistic equations (Kelin-Gordon, Proca, etc.).
The spin-exchange optical pumping technique has been used to measure the gyromagnetic ratios of hydrogen, tritium, and free electrons in terms of the electronic gyromagnetic ratio of Rb85. A direct measurement was also made of the ratio of the electronic g factors of tritium and hydrogen. The 60-G magnetic field in which these measurements were made was produced by a precision-wound solenoid which was surrounded by three concentric magnetic shields. The magnetic field varied by 5 parts in 106 over the region occupied by a typical absorption flask. The free electrons were produced by ionization resulting from the beta decay of the radioactive tritium atoms. Measurements were made in various sizes of absorption flasks and in flasks containing different pressures of helium, argon, and neon buffer gases. The measured g-factor ratio did not depend upon the nature or pressure of the buffer gas. The results of these measurements were as follows: gJ(Rb)g(e)=1+(6.3+/-1.0)×10- 6 gJ(Rb)gJ(H)=1+(23.74+/-0.1)×10-6 gJ(T)gJ(H)=1-(0.11+/-0.3)×10- 6 gJ(H)g(e)=1-(17.4+/-1.0)×10- 6.
DOI:https://doi.org/10.1103/PhysRev.72.241
For an atom or monatomic ion in a magnetic field H there will be an induced shielding field H'(0) at the the nucleus given by H'(0)=(eH3mc2)v(0) where v(0) is the electrostatic potential produced at the nucleus by the atomic electrons. Using the Thomas-Fermi model, Lamb put this expression into a calculable form. However, in modern nuclear induction and resonance absorption experiments it is important to have a more precise knowledge of the magnitude of this shielding field. In this paper computed values of v(0) are given for all atoms and singly charged ions which have been treated by the Hartree or Hartree-Fock approximations to the self-consistent field method. By interpolation a list of H'(0)H values for all neutral atoms is given. Although it is impossible to check the accuracy of these values experimentally it is estimated from other evidence that they can be trusted to within five percent. An exception must be made, however, for the heaviest atoms where the relativity effect becomes appreciable, amounting to an estimated six percent correction to H'(0)H for Z=92. Finally, the usefulness of accurate values of the atomic shielding field in analyzing the total shielding field in molecules is discussed.
The self-energy contribution to the hyperfine splitting of the ground state of Bi82+ is calculated for a point nucleus. It is found that the theoretical value of the wavelength of the ground-state hyperfine-splitting transition of 209Bi82+ is λ=244(1) nm and is in good agreement with experiment.
We formulate a general method for evaluating relativistic and binding energy corrections to matrix elements of the electromagnetic current for multiparticle composite systems. Application is made to the calculation of the g-factor for bound electrons and compared with recent experiments.
Some of the consequences of the positron theory for the special case of impressed electrostatic fields are investigated. By imposing a restriction only on the maximum value of the field intensity, which must always be assumed much smaller than a certain critical value, but with no restrictions on the variation of this intensity, a formula for the charge induced by a charge distribution is obtained. The existence of an induced charge corresponds to a polarization of the vacuum, and as a consequence, to deviations from Coulomb's law for the mutual potential energy of point charges. Consequences of these deviations which are investigated are the departures from the Coulombian scattering law for heavy particles and the displacement in the energy levels for atomic electrons moving in the field of the nucleus.
A determination of the ratio of g factors of the electron bound in the hydrogen atom to that of the free electron has been made. For the first time mass-independent terms in the theory are confirmed through the alpha3 radiative correction. Our result is gJ (1H, 12S12)gs (e)=1-17.709(13)×10-6.
In the positron theory considerable interest attaches to the consideration of processes in which the occurrence of electrons and positrons is transitory only, such as the scattering of light by a Coulomb field (Delbrück), and the scattering of light by light (Euler and Kockel). Calculations of such effects can frequently be simplified on account of cancellations brought about by the distribution's symmetry between electrons and positrons. An abstract proof is here presented for the theorem which predicts the appearance of such cancellations in the general case. Certain modifications are found to be required when interactions other than the usual electric forces are introduced.
The alpha3 term in the ratio of the hyperfine splitting in the 2S state of the one-electron atom to the hyperfine splitting in the 1S state is recalculated, and a new theoretical value for this ratio is obtained which is in agreement with the experimental value, thereby eliminating a previously reported discrepancy. The calculation consists in the evaluation of the low-momentum parts, of order alpha3 hfs, of the expression for the lowest order radiative level shift in the bound interaction representation with external Coulomb and magnetic dipole fields. By rearranging the terms so as to display the gauge invariance of the matrix elements with respect to the external potentials, considerable simplicity is achieved, and the formulas are easily interpreted as a generalization of the expression for the lowest order Lamb shift. The contribution from soft photon intermediate states is obtained by an extension of the method developed by Schwartz and Tiemann for evaluating the Bethe logarithm, and an appendix contains a tabulation of twelve analogous integrals which were integrated numerically, and which may be of use elsewhere. The calculated value of the ratio is 18(1.000 034 5+/-0.000 000 2) which agrees with the experimental values for hydrogen: 18(1.000 034 6+/-0.000 000 3), and deuterium: 18(1.000 034 2+/-0.000 000 6).
The use of exact Dirac-Coulomb propagators allows the evaluation of binding corrections to the Schwinger correction in ground state muonium hyperfine splitting to all orders. The calculational method is described and the results are used firstly to verify recent perturbative calculations of higher-order binding corrections and secondly to evaluate the residual terms of still higher order. Implications for muonium hyperfine splitting are discussed.
Positronium is the quasistable bound system consisting of an electron and its antiparticle, the positron. Its energy levels can be explained to a high degree of accuracy by the electromagnetic interaction, affording an ideal test of the quantum electrodynamic (QED) theory of bound systems. We have measured the 1 3S1-2 3S1 interval in positronium by Doppler-free two-photon spectroscopy to be 1 233 607 216.4+/-3.2 MHz. We employ continous-wave (cw) excitation to eliminate the problems inherent with pulsed laser measurements of nonlinear transitions. Positronium (Ps) atoms generated in vacuum are excited to the 2S state using cw laser light built up to 2.5 kW circulating power in a resonant Fabry-Pérot cavity. The excited-state atoms are photoionized using a pulsed laser at 532 nm, and the liberated positrons counted as the cw laser is tuned relative to a reference line in tellurium (Te2) molecular vapor. The fit of a detailed theoretical model to the measured line shape determines the Ps resonance frequency relative to the Te2 reference line. The Monte Carlo model includes details of the excitation and detection geometry, the Ps velocity distribution, and the dynamic Stark shift, and gives excellent agreement with the measured line shapes. The quoted 2.6 parts per billion (ppb) uncertainty is dominated by the measurement of the Ps line center relative to the Te2 reference line, with a 1.0-ppb contribution from a recent calibration of our Te2 cell relative to the hydrogen 1S-2S transition frequency. The measurement is in excellent agreement with theory and sufficiently accurate to provide a test of the as-yet-uncalculated alpha4R∞ QED correction. Our measurement tests the alpha2R∞ QED contributions to the energy of the 1 3S1 and 2 3S1 states to 3.5 parts in 105.
The interferometer pattern due to a group of spectral lines can be expressed as a Fourier series, and the coefficients of this series can be regarded as the quantities to be measured. These coefficients can also be computed in terms of the positions, intensities, and the parameters describing the shapes of the lines. Although the equations cannot be explicitly solved for the latter quantities in terms of the Fourier coefficients, numerical means can be used to find the parameters which best reproduce the observed values. This method has been applied to two plates of Halpha and Dalpha. The results indicate that to fit the observations with the four strongest lines given by the theory it is necessary to make the separations of the two most intense components some 2 percent less than the theoretical value.
This paper presents the results of an analytic calculation of the corrections of relative order alpha2(memmu) to the muonium ground-state hyperfine splitting due to exchanged photons. Theory and experiment are compared, with these corrections and some radiative-recoil contributions described in an accompanying paper taken into account.
Optical pumping in the 3P0 ground state of Pb207 is studied in detail. Pumping and nutation transient signals are used to investigate relaxation phenomena and to place upper limits on buffer-gas disorientation cross sections. Hg199 has been optically pumped in the same cell yielding |muI(Pb207)muI(Hg199)|=1.14960 (4) and |muI(Pb207)|uncorrected=0.57235 (2)muN. The large (2%) difference between the accepted NMR value and the previously reported optical-pumping value is thus confirmed.
The one-loop self-energy correction to the hyperfine splitting of the 1 s and 2 s states of hydrogenlike ions is calculated for both point and finite nuclei. The results of the calculation are combined with other corrections to find the ground state hyperfine splitting in lithiumlike 209Bi80+ and 165Ho64+.
The beta-spectrum of P32 has been examined with a thin-lens spectrometer. The maximum energy of the 14.3-day beta-activity, as determined from several Kurie plots, was found to be 1.704+/-0.008 Mev. The Kurie plots gave excellent straight lines from the maximum beta-energy to about 0.26 Mev. In phosphorus samples prepared from neutron irradiated sulfur an additional beta-activity was observed having a maximum energy of 0.26+/-0.02 Mev and a half-life of 24.8+/-0.5 days. This low energy beta-group was also observed in phosphorus samples prepared from sulfur and lithium chloride irradiated with x-rays having a maximum energy of 68 Mev. The low energy beta-group was not observed in phosphorus samples prepared from sulfur irradiated with deuterons or phosphorus irradiated with neutrons. The low energy beta-group is ascribed to P33.
An expression is developed for the magnetic field at a nucleus resulting from the application of an external magnetic field to a polyatomic molecule which has no resultant electron orbital or spin angular momenta in the absence of the external field. The field at the nucleus is not the same as the externally applied field because of the field arising from the motion of the electrons in the molecule. The expression for the electron contribution to the magnetic field is shown to consist of two parts. The first is a simple term that is similar to the diamagnetic correction developed by Lamb for atoms. The second is a complicated one arising from second-order paramagnetism and is analogous to the term dependent on the high frequency matrix elements in the theory of molecular diamagnetism. Under certain circumstances the second-order paramagnetic term can become quite large. Since both of these terms are altered when the same nucleus is in different molecules, they at least partially and perhaps completely explain the chemical effect that has been reported by various observers in measurements of nuclear moments. For linear molecules, the second-order paramagnetic term is shown to be directly related to the experimentally measurable spin-rotational magnetic interaction constant of the molecule. This relation is particularly valuable in the important case of molecular hydrogen where it is shown that the correction for second-order paramagnetism is -0.56×10-5. When this is added to the Lamb-type term as calculated by Anderson, the total magnetic shielding constant for molecular H2 becomes 2.68×10-5.
Using a self-consistent relativistic molecular Dirac-Fock-Slater code we have determined the charge density and the diamagnetic shielding at the 209Bi nucleus in the molecule Bi(NO3)3, which was used in the experiment for the g-factor. Our final value for the nuclear moment of 209Bi is 7 I=4.1103ǂ.0005 7 N.