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Project titleTheoretical investigation of the properties and structure of the nuclei using atomic and molecular systems

AffiliationPetersburg Nuclear Physics Institute named by B.P. Konstantinov of National Research Centre "Kurchatov Institute",

Keywordsmagnetic and electric moments, nuclear magnetic resonance, electronic structure, hyperfine structure, relativistic effects

Annotation
Due to the breakthrough development of experimental techniques for manipulating atoms, ions and molecules, such systems are increasingly becoming "laboratories" for studying various physical phenomena, they are used to build atomic clocks, act as elements in the construction of quantum computers, used for precision measurement of fundamental physical constants, testing theories of fundamental interactions [Rev. Mod. Phys. 90, 025008 (2018); Nature Communications 8, 15484 (2017)]. However, in many cases, when considering these tasks, problems arise due to the inaccuracy of information about the structure and properties of atomic nuclei. This is what becomes the main source of errors in the prediction of a number of atomic and molecular properties. A series of works performed by us in recent years [Phys. Rev. Research 2, 013368 (2020); Phys. Rev. Lett., 120, 093001 (2018); Phys. Rev. C 104, 034316 (2021)], shows that in many cases the uncertainties indicated in the reference tables for various nuclear properties are considerably underestimated, which leads to various “paradoxes” and “puzzles” when using these data [https://physicsworld. com/a/has-the-hypefine-puzzle-been-solved/]. To date, the reliability of direct calculations of the properties of the nucleus is limited. This is especially true for heavy nuclei. Therefore, these properties are studied experimentally. To date, a wide amount of experimental data has been obtained on isotopic shifts of the energies of electronic transitions in atoms, which can make it possible to study entire chains of isotope nuclei, follow the evolution of the shell structure, the appearance of magic numbers, the formation of exotic forms of nuclei, etc. Analysis of data on the properties of short-lived nuclei, with an extremely large or small number of neutrons, can provide information on the parameters for the equation of state of nuclear matter. However, for the interpretation of experiments, it is required to know the so-called. electronic factors for isotopic shifts. In the project we’ll interpret data on various isotope chains of such atoms as Tl, Si, Al, Au, etc., at a new, very high theoretical level, for which either there are incomplete theoretical data, obtained within the simplified models, or such data are not available at all. Now a series of molecular experiments is planned to measure a practically unexplored property of the nucleus - its anapole moment using the Stark interference technique. The anapole moment is a manifestation of the forces inside the nucleus which violate spatial parity, and was previously measured only once and, with a large error, for the 133Cs nucleus [C. S. Wood et al., Science 275, 1759 (1997)]. Planned experiments with molecules are expected to make it possible for the first time to measure the anapole moment of a nucleus with high accuracy. The project will develop approaches that will allow extracting the values of the anapole moment from them at the highest level of accuracy. Moreover, the proposed project will investigate an idea of a new type of experiment to search for anapole moments of nuclei using molecules. In this project, we plan to develop original approaches to extract the magnetic moments of nuclei. To do this, it is proposed to consider various schemes that involve combining different data, for example, on hyperfine splittings in molecules and atoms, or data on nuclear magnetic resonance and hyperfine structure. Such approaches are expected for the first time to be able to circumvent the limitations associated with the lack of reliable information on the distribution of magnetization in the nuclei of heavy elements. A number of problems proposed for solution in the project are due to the direct interest/request to us from the experimenters. To solve the problems, the approaches of the 2019 project will be adapted and developed. In the project 2019 these methods have already made it possible to solve problems of interpreting experiments for other properties of nuclei at a record level of accuracy.

Expected results
The purpose of the ongoing project is to develop original approaches for the solution of the problem of the reliable extraction of the fundamental properties of the nucleus from high-precision atomic and molecular experiments, as well as to explore the possibilities of conduction of new experiments to study the nucleus. The most important property of the nucleus that characterizes its structure is its root-mean-square radius. To obtain information about it, precise experiments can be performed to measure isotopic shifts in the energies of electronic transitions in atoms. If there is a measurement of the isotopic shift for one such transition, it requires the involvement of theory to interpret the experiment in terms of the difference in the squares of the radii of the isotopes under consideration. At the moment, “raw” experimental data obtained for the Si isotope chain are available; in addition, experiments for Al isotopes are being conducted. An important aspect of such studies is the possibility of studying a pair of the so-called mirror nuclei: one nucleus from such a pair comprise Z protons and N neutrons, and the other one comprise N protons and Z neutrons. If we neglect the dependence of the interactions of nucleons on their charges, studying such a mirror pair and knowing the difference in the squares of the nuclear radii (due to the distribution of positively charged protons), we can obtain information about the difference in the squares of the radii of the neutron distribution in the nucleus. The information obtained can then be used to find the parameters of nuclear forces and to test the theory of the nuclear structure. Measurements of the radius of the 32Si nucleus can complete the data on the radii in a pair of 32Ar-32Si mirror nuclei, since the radius of the 32Ar nucleus is already known [Nuclear Physics A, 607, 1 (1996)]. In addition, one of the nuclei of the silicon isotopes under consideration, 34Si, presumably has a “bubble" structure with a reduced central density, which is also of significant fundamental interest [Nature Physics 13, 152 (2017)]. The result of our work will be a reliable interpretation of these measurements in terms of the differences of the squares of the radii of the isotopes from the chain under consideration. A feature of our consideration will be the solution of a non-trivial problem of taking into account the effects of high-order electronic correlation, including one that exceeds the current world “standards” in this field of research. In the 2019 project, we showed this possibility when considering other properties of the nucleus, for example, in [Phys. Rev. C 104, 034316 (2021)], we have shown that the approach of the coupled cluster method in the formulation for the Fock space can significantly outperform approaches like the multiconfigurational Dirac-Fock method [Phys. Rev. Lett. 87, 133003 (2001)] in describing the states of the neutral bismuth atom. Such approach is very often used to calculate the electronic factors of isotopic shifts [J. Phys. G: Nucl. Part. Phys. 38 025104 (2011)]. Theoretical studies of isotope shifts are also carried out by a group of theorists from Australia and Germany (see [Phys. Rev. Lett. 128, 073001 (2022); Phys. Rev. A 103, 032811 (2021)] and references therein). But they are mainly interested in the task of searching for new physics using such experiments. In their work, they take into account various contributions to isotopic shifts such as nuclear deformation. In our approach, these effects can also be taken into account. Note that recently in [New J. Phys. 22, 012001 (2020)] (by Indian researchers) the coupled cluster method was applied to the calculation of the electronic factors of isotope shifts in the indium atom. However, these calculations took into account only single and double cluster amplitudes with a perturbative correction for triples for some valence electrons. In our approach [Phys. Rev. C 104, 034316 (2021); Symmetry, 12, 1101 (2020)] it is possible to fully take into account three-fold cluster amplitudes, and not perturbatively, but iteratively. This makes it possible to qualitatively improve the calculation results. In addition, it is not obvious whether the approach [New J. Phys. 22, 012001 (2020)] is developed to describe two or three particles. As far as we know, for the latter case, the approach taking into account iterative triple cluster amplitudes is implemented only in the code, the author of which is one of the main executors of this project. In this regard, we expect to obtain the most accurate results for electronic factors of Al and Si isotopic shifts compared to other approaches developed by other groups to date. We also plan to interpret the data of measurements of isotopic shifts performed on chains of thallium isotopes at our Institute (NRC KI-PNPI) at the IRIS facility [A.E. Barzakh et al, Phys. Rev. C 88, 024315 (2013)]. Similar studies will be performed for gold isotopes. Combination of the experimental and theoretical data will help us to find radii for nuclei of different isotopes, including short-lived ones with an extremely large or small number of neutrons. Such studies will help to better understand the structure of nuclear forces that underlie our knowledge about the properties of the nucleus and nuclear matter (neutron stars), nuclear transformations, etc. It should be noted that groups of experimentalists have contacted us and asked for the interpretation of these measurements of isotopic shifts. In this project, we plan to develop original approaches for extracting magnetic moments of nuclei. In one of them, it is planned to combine the results of measurements of nuclear magnetic resonance and the hyperfine structure of atoms with the results of precision calculations of properties determined by the electronic structure. For example, we plan to clarify the value of the magnetic moment of the 197Au nucleus. Now, the errorbar of the value is mainly determined by the lack of reliable information about the nuclear magnetization distribution function. It is expected that the combination of experimental data on the observation of nuclear magnetic resonance and hyperfine structure, as well as precise calculations of the electronic structure will allow one to circumvent this problem. Note that the effects of the finite distribution of magnetization inside the nucleus are studied in the world by several groups (from Russia, Australia, USA), see, for example, [Phys. Rev. A 103, 032824 (2021); Phys. Rev. A 105, 052802 (2022)] (and references). But this effect is considered for the hyperfine constants of atoms. In the problem under consideration, we are talking about taking into account the contribution of this effect to the shielding constant. In [Phys. Rev. Lett. 107, 043004 (2011); Phys. Rev. A 85, 022512 (2012)], such a problem was considered for hydrogen-like ions in the framework of a particular model of the magnetization distribution. In the work of our group [arXiv:2204.13015 [physics.atom-ph] (2022)], we solved such a problem for the many-electron ReO4- molecule. But, if we consider the problem with gold, then it is known that the effects of the distribution of magnetization are poorly described by simple models, for example, this is analyzed at a high level in [Phys. Rev. A 103, 032824 (2021)]. Therefore, instead of a specific simple model, we plan to consider the possibility of using experimental data in combination with precision calculations of the electronic structure in order to take into account the considered contribution. The possibility of such an approach has never been considered before. This will open up new prospects for the accuracy of extracting nuclear magnetic moments from experimental data. At the moment, for many nuclei, the values of magnetic moments are limited by the possibilities of experimental data interpretation, despite the fact that the accuracy of the experiments themselves is very high. The magnetic moment of the nucleus is one of several of its most important fundamental characteristics along with its root-mean-square radius. Knowledge of the magnetic moments of nuclei with high accuracy is required for the problems of probing quantum electrodynamics, calculating nuclear magnetic anomalies, which are necessary for studying the magnetic structure of the nucleus, etc. We also plan to investigate another new possibility of finding the magnetic moment of the nucleus - using data on the hyperfine structure of molecules. As a test system for this approach, it is planned to consider the thorium monoxide molecule, for which the experiments for hyperfine structure measurement are planned. The interest in this system is due to the fact that in some cases it turns out to be easier to interpret the data of molecular experiments than atomic ones (this conclusion was made by us according to the results of the 2019 project) [L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii, D.E. Maison, A.E. Barzakh, Phys. Rev. C 104, 034316 (2021)]. That is, our proposed approach will presumably open up new possibilities for finding magnetic moments of nuclei for which experiments can be performed to measure only the hyperfine structure of atoms or molecules, and NMR spectroscopy for some reason is not possible (for example, due to the instability of nuclei, etc.). This approach will make it possible to clarify/find out the magnetic moments of radioactive nuclei. One of the key ideas here is also to take into account the contribution of the effect of the distribution of magnetization over the nucleus, the original way of considering which in molecules was proposed in [J. Chem. Phys. 153, 114114 (2020)]. We note that today none of the other groups of researchers (from France, the Netherlands, India, Japan) involved in the prediction of the hyperfine structure of molecules considers effects of this type, confining to the point magnetic dipole moment of the nucleus approximation. One of the interesting, but practically unexplored properties of the nucleus is its anapole moment. The anapole moment of the nucleus is a manifestation of intranuclear interactions that violate spatial parity [Zeldovich, Sov. Phys. JETP 6, 1184 (1958);V. V. Flambaum and I. B. Khriplovich, Sov. Phys. JETP 52, 835 (1980); V. V. Flambaum, I. B. Khriplovich, and O. P. Sushkov, Phys. Lett. B146, 367 (1984)]. The anapole moment was measured only once for the caesium 133Cs nucleus and with a large error [C. S. Wood et al., Science 275, 1759 (1997)]. New experiments are being planned to measure the anapole moments of nuclei using molecular cations held in a Penning trap. It is expected that anapole moments can be measured with high accuracy. However, a reliable interpretation of these experiments, i.e., extracting the values of anapole moments, will require high-precision calculations of the molecular constants of the interaction of these moments with electrons. We will perform calculations of these characteristics for the molecular cation SiO+ and/or similar ones planned for the experiment on the highest level of accuracy. Finally, we plan to consider (propose) a scheme of a new type of experiment for measuring nuclear anapole moments. For this purpose, the contribution of the indirect parity nonconserving interaction of nuclei in a diatomic molecule of thallium monofluoride induced by the anapole moment of the thallium nucleus will be studied. In the 2019 project, we have developed a unique approach to predicting the value of the constant of the indirect interaction of the magnetic dipole moments of nuclei in a diatomic molecule, based on the theory of relativistic coupled cluster. We have shown that the accuracy of this approach is much higher than the accuracy of the relativistic density functional theory method, which is at best used in the world to solve this problem for heavy-atom compounds, e.g. [Phys. Chem. Chem. Phys., 19, 8400 (2017)]. Therefore, the generalization of this approach to the problem of the indirect interaction of the anapole moment of one nucleus and the magnetic dipole moment of another nucleus, as expected, will be able to give results at an accuracy level that has no analogues. Measurements of anapole moments will allow one to obtain completely unique information about parameters of nuclear forces parametes that violate spatial parity [Phys. Rev. C 65, 045502 (2002)].

Annotation of the results obtained in 2022
One of the objectives of this project is to develop original approaches to solving the problem of reliable extraction of fundamental properties of nuclei from high-precision experiments with atoms and molecules. The most important property of the nucleus, which characterizes its structure, is its root-mean-square radius. To obtain information about it, one can perform high-precision experiments to measure the isotopic shifts of electronic transition energies in atoms. If measured data on the isotopic shift for one such transition are available, the theoretical consideration is necessary to interpret the experiment in terms of the difference of mean square charge radii of the nuclei of the considered isotopes. In this case, there are two effects that have to be described. The first effect is the field shift associated directly with the difference in charge radii. The second effect is the mass shift (recoil effect). In turn, the latter can be divided into two parts, the normal and specific mass shifts. One of the most important challenges for the theory is the calculation of the electronic factors of these effects. Within the project we have developed a new approach that has allowed us to calculate these atomic constants. For this purpose a new program calculating corresponding matrix elements of the field shift and recoil effect has been developed. Note that relativistic effects play an important role in the recoil effect in neutral heavy atoms. Therefore, we have implemented the completely relativistic approach, which is the most accurate in the framework of the relativistic four-component approximation (QED effects were not considered here). It should be noted that such a completely relativistic approach was not systematically used in previously published papers on neutral many-electron atoms.. Using the developed method we have calculated the field and mass shifts for three transitions in the thallium atom, 6p3/2 - 7s1/2 (535 nm), 6p1/2 - 6d3/2 (277 nm) and 6p1/ 2 - 7s1/2 (378 nm), at a fundamentally new level of accuracy. Note that before our study, in some papers, only the order of magnitude of the specific mass shift was estimated at best, or this contribution was completely ignored in comparison with the normal mass shift (based on the assumption of its smallness). Within this project the latter assumption was proved to be wrong. The use of modern relativistic coupled cluster methods to solve the electronic problem made it possible to reduce the error in the theoretical calculation of electronic factors for transitions in the thallium atom by about an order of magnitude compared to previous studies (from ~30% to ~3%). The completely new level of accuracy achieved also made it possible to extract values of root-mean-square radii of thallium isotopes from experimental data obtained at facilities at the NRC KI - PNPI, CERN, etc. We have analyzed all previously reported experiments and obtained new radii for the following Tl isotopes: 208g, 207g, 205g, 204g, 203g, 202g, 201g, 200g, 199g, 198g, 198m, 197g, 197m, 196g, 196m, 195g, 195m, 194g, 194m, 193g, 193m, 192g, 192m, 191g, 191m, 190g, 190m, 189m, 188m, 187m, 186m1, 186m2, 185g, 185m, 184m1, 184m2, 184m3, 183g, 183m, 182m1, 182m2, 181, 180, 179Tl. In addition to the thallium atom studies, we have performed similar calculations of electronic factors for transitions in Au and Si atoms. One of the most poorly studied characteristics of atomic nuclei is the nuclear anapole moment. Despite the fact that the term “anapole moment” itself was introduced in the 1950s by Zeldovich, to date, the value of the nuclear anapole moment has been experimentally measured only for the cesium nucleus, while the measurement error was quite large. In [Phys. Rev. Research 5, 013191 (2023)], we proposed a new method aimed at experimental measurement of nuclear anapole moments in molecular experiments, and the TlF molecule in the ground electronic state was considered as an object for such a research. The most important problem is the interpretation of such an experiment. This requires the constant of the indirect (through the electronic subsystem) coupling of the Tl nuclear anopole moment with the magnetic moment of the fluorine nucleus to be calculated at a very high level of accuracy. We have developed a theoretical method for such a calculation and applied it to the TlF molecule. An interesting result obtained within the project is that the main contribution to the molecular interaction constant under consideration arises from the negative spectrum of the Dirac Hamiltonian. Using the calculated value of this constant, we estimated the expected value of the signal that can be detected in the proposed experiment. The results obtained seem to be encouraging. According to our estimates, the value of the signal-to-noise ratio equal to 100 can be achieved in just a few hours of the experiment. In addition to the completed studies which were planned, we have also considered other related problems for which the theoretical methods developed within the present project can be used. In particular, we have calculated the hyperfine structure constants of the molecular lutetium monohydroxide cation LuOH+. For a number of potassium isotopes, their nuclear magnetic moments were refined. We have studied the effect of the exchange of a hypothetical axion-like particle between electrons and the nucleus in the special case of the hafnium monofluoride molecular cation.

Publications

1. D.E. Maison, L.V. Skripnikov, G. Penyazkov, M. Grau, A.N. Petrov T ,P-odd effects in the LuOH+ cation Physical Review A, Phys. Rev. A 106, 062827 (2022) (year - 2022) https://doi.org/10.1103/PhysRevA.106.062827

2. G. Penyazkov, S.D. Prosnyak, A.E. Barzakh, L.V. Skripnikov Refined theoretical values of field and mass isotope shifts in thallium to extract charge radii of Tl isotopes The Journal of Chemical Physics, J. Chem. Phys. 158, 114110 (2023) (year - 2023) https://doi.org/10.1063/5.0142202

3. J.W. Blanchard, D. Budker, D. DeMille, M.G. Kozlov, L.V. Skripnikov Using parity-nonconserving spin-spin coupling to measure the Tl nuclear anapole moment in a TlF molecular beam Physical Review Research, Phys. Rev. Research 5, 013191 (2023) (year - 2023) https://doi.org/10.1103/PhysRevResearch.5.013191

4. S.D. Prosnyak, D.E. Maison, L.V. Skripnikov Updated Constraints on T ,P -Violating Axionlike-Particle-Mediated Electron–Electron and Electron–Nucleus Interactions from HfF+ Experiment Symmetry, Symmetry 2023, 15(5), 1043 (year - 2023) https://doi.org/10.3390/sym15051043

5. Yu.A. Demidov, M.G. Kozlov, A.E. Barzakh, V.A. Yerokhin Bohr-Weisskopf effect in the potassium isotopes PHYSICAL REVIEW C, Phys. Rev. C 107, 024307 (2023) (year - 2023) https://doi.org/10.1103/PhysRevC.107.024307

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