Relativistic Many-body Methods Research Articles

High-accuracy optical clocks with sensitivity to the fine-structure constant variation based on Sm[formula omitted]

We identify two metastable excited states in Sm10+ highly charged ion as candidates for high accuracy optical clocks. Several atomic properties relevant to optical clock development are calculated using relativistic many-body methods. This includes energy levels, transition amplitudes, lifetimes, scalar polarizabilities, black body radiation shift, and the sensitivity to the fine structure constant variation. We found that the clock transitions are not sensitive to perturbation, e.g., relative black body radiation shifts are ∼10−19. The enhancement factor for the α variation is ∼ 0.8 for one clock transition and ∼ 16 for another.

Optical lines of Xe9+ and Xe10+ ions measured at a low-energy electron beam ion trap

This work reported the investigation of optical spectra of Rh-like Xe9+ and Ru-like Xe10+ ions at a low-energy electron beam ion trap. The line from 4 d9 2D3/2–2D5/2 M1 transition of Xe9+ ions was remeasured, the wavelength of which was determined to be 598.365(5) nm, and the uncertainty was significantly improved. In the region of 200–700 nm, four spectral lines emitted from Xe10+ ions were directly observed for the first time. The relativistic many-body perturbation method was adopted for the investigation of fine structure in Xe10+ ions. The theoretical and experimental results were in good agreement within 2%, and the four observed lines of Xe10+ ions were identified.

Highly charged ion (HCI) clocks: Frontier candidates for testing variation of fine-structure constant

Attempts are made to unify gravity with the other three fundamental forces of nature. As suggested by higher dimensional models, this unification may require space and time variation of some of the dimensionless fundamental constants. In this scenario, probing temporal variation of the electromagnetic fine structure constant (α=e2ℏc) in a low energy regime at the cosmological time scale is of immense interest. Atomic clocks are ideal candidates for probing α variation because their transition frequencies are measured with ultra-high precision. Since atomic transition frequency is a function of α, measurement of a clock frequency at different temporal and spatial locations can yield signatures to ascertain variation of α. Highly charged ions (HCIs) are very sensitive to variation of α and are least affected by external perturbations, making them excellent platforms for searching for temporal variation of α. In this work, we overview HCIs suitable for building atomic clocks because of their spectroscopic features and sensitivity to variation of α. The selection of HCI clock candidates is outlined based on two general rules—by analyzing trends in fine structure splitting and level-crossing patterns along a series of isoelectronic atomic systems of the periodic table. Two variants of relativistic many-body methods in the configuration interaction and coupled-cluster theory frameworks are employed to determine the properties of HCIs proposed for atomic clocks. These methods treat electronic correlation and relativistic effects under the frame of the Dirac-Coulomb Hamiltonian rigorously, while other higher-order relativistic effects are included approximately. Typical systematic effects of the HCI clock frequency measurements are discussed using the calculated atomic properties. This review will help understand limits and potentials of the proposed HCIs as the prospective atomic clock candidates and guide future HCI clock experiments.

HYPERFINE STRUCTURE PARAMETERS FOR COMPLEX ATOMS WITHIN RELATIVISTIC MANY-BODY PERTURBATION THEORY

The calculational results for the hyperfine structure (HFS) parameters for the Mn atom (levels of the configuration 3d64s) and the results of advanced calculating the HFS constants and nuclear quadrupole moment for the radium isotope are obtained on the basis of computing within the relativistic many-body perturbation theory formalism with a correct and effective taking into account the exchange-correlation, relativistic, nuclear and radiative corrections. Analysis of the data shows that an account of the interelectron correlation effects is crucial in the calculation of the hyperfine structure parameters. The fundamental reason of physically reasonable agreement between theory and experiment is connected with the correct taking into account the inter-electron correlation effects, nuclear (due to the finite size of a nucleus), relativistic and radiative corrections. The key difference between the results of the relativistic Hartree-Fock Dirac-Fock and many-body perturbation theory methods calculations is explained by using the different schemes of taking into account the inter-electron correlations as well as nuclear and radiative ones.

Investigating properties of Cl− and Au− ions using relativistic many-body methods

We investigate the ground state properties of singly charged chlorine (Cl−) and gold (Au−) negative ions by employing four-component relativistic many-body methods. In our approach, we attach an electron to the respective outer orbitals of chlorine (Cl) and gold (Au) atoms to determine the Dirac–Fock (DF) wave functions of the ground state configurations of Cl− and Au−, respectively. As a result, all the single-particle orbitals see the correlation effects due to the appended electron of the negative ion. After obtaining the DF wave functions, lower-order many-body perturbation methods, random-phase approximation, and coupled-cluster (CC) theory in the single and double approximation are applied to obtain the ground state wave functions of both Cl− and Au− ions. Then, we adopt two different approaches to the CC theory—a perturbative approach due to the dipole operator to determine electric dipole polarizability and an electron detachment approach in the Fock-space framework to estimate ionization potential. Our calculations are compared with the available experimental and other theoretical results.

Tune-out and magic wavelengths, and electric quadrupole transition properties of the singly charged alkaline-earth metal ions

In this paper, we present the tune-out and magic wavelengths of the Mg+, Ca+, Sr+ and Ba+ alkaline earth-metal ions by determining dynamic E1 polarizabilities. Furthermore, we have evaluated the electric quadrupole (E2) matrix elements of a large number of forbidden transitions using an all-order relativistic many-body method and compare them with the previously reported values for a few selective transitions. Compilation of both the E1 and E2 transition matrix elements, will now provide a more complete knowledge about the transition properties of the considered singly charged alkaline earth-metal ions. Similarly, the listed precise values of tune-out and magic wavelengths due to the dominant E1 polarizabilities can be helpful to conduct experiments using the above ions with reduced systematics. Therefore, all these data will be immensely useful for various applications for carrying out the high-precision experiments and laboratory simulations in atomic physics, and interpreting transition lines in the astrophysical observations. This work is a continuation of our earlier reported data on the electric dipole (E1) matrix elements and lifetimes of the metastable states of the alkaline earth ions in Kaur et al. (2021).

Radiative transition properties of singly charged magnesium, calcium, strontium and barium ions

Accurate values of electric dipole (E1) amplitudes along with their uncertainties for a number of transitions among low-lying states of Mg+, Ca+, Sr+, and Ba+ are listed by carrying out calculations using a relativistic all-order many-body method. By combining experimental wavelengths with these amplitudes, we quote transition probabilities, oscillator strengths and lifetimes of many short-lived excited states of the above ions. The uncertainties in these radiative properties are also quoted. We also give electric quadrupole (E2) and magnetic dipole (M1) amplitudes of the metastable states of the Ca+, Sr+, and Ba+ ions by performing similar calculations. Using these calculated E1, E2 and M1 matrix elements, we have estimated the transition probabilities, oscillator strengths and lifetimes of a number of allowed and metastable states. These quantities are further compared with the values available from the other theoretical studies and experimental data in the literature. The present data will be useful for the astrophysical observations, laboratory analysis and simulations of spectral properties in the above considered alkaline-earth metal ions.

Determination of the dipole polarizability of the alkali-metal negative ions

We present electric dipole polarizabilities ($\alpha_d$) of the alkali-metal negative ions, from H$^-$ to Fr$^-$, by employing four-component relativistic many-body methods. Differences in the results are shown by considering Dirac-Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DC Hamiltonian with the lower-order quantum electrodynamics interactions. At first, these interactions are included self-consistently in the Dirac-Hartree-Fock (DHF) method, and then electron correlation effects are incorporated over the DHF wave functions in the second-order many-body perturbation theory, random phase approximation and coupled-cluster (CC) theory. Roles of electron correlation effects and relativistic corrections are analyzed using the above many-body methods with size of the ions. We finally quote precise values of $\alpha_d$ of the above negative ions by estimating uncertainties to the CC results, and compare them with other calculations wherever available.

RELATIVISTIC CALCULATION OF THE HYPERFINE STRUCTURE PARAMETERS FOR COMPLEX ATOMS WITHIN MANY-BODY PERTURBATION THEORY

The hyperfine structure parameters and electric quadrupole moment of the 201Hg mercury isotope the Mn atom are estimated within the relativistic many-body perturbation theory formalism with a correct and effective taking into account the exchange-correlation, relativistic, nuclear and radiative corrections. Analysis of the data shows that an account of the interelectron correlation effects is crucial in the calculation of the hyperfine structure parameters. The fundamental reason of physically reasonable agreement between theory and experiment is connected with the correct taking into account the inter-electron correlation effects, nuclear (due to the finite size of a nucleus), relativistic and radiative corrections. The key difference between the results of the relativistic Hartree-Fock Dirac-Fock and many-body perturbation theory methods calculations is explained by using the different schemes of taking into account the inter-electron correlations as well as nuclear and radiative ones.

The role of relativistic many-body theory in probing new physics beyond the standard model via the electric dipole moments of diamagnetic atoms

The observation of electric dipole moments (EDMs) in atomic systems due to parity and time-reversal violating (P,T-odd) interactions can probe new physics beyond the standard model and also provide insights into the matter-antimatter asymmetry in the Universe. The EDMs of open-shell atomic systems are sensitive to the electron EDM and the P,T-odd scalar-pseudoscalar (S-PS) semi-leptonic interaction, but the dominant contributions to the EDMs of diamagnetic atoms come from the hadronic and tensor-pseudotensor (T-PT) semi-leptonic interactions. Several diamagnetic atoms like 129Xe, 171Yb, 199Hg, 223Rn, and 225Ra are candidates for the experimental search for the possible existence of EDMs, and among these 199Hg has yielded the lowest limit till date. The T or CP violating coupling constants of the aforementioned interactions can be extracted by combining the EDM measurements from with atomic and nuclear calculations. In this work, we report the calculations of the EDMs of the above atoms by including both the electromagnetic and P,T-odd violating interactions simultaneously. These calculations are performed by employing relativistic many-body methods based on the random phase approximation (RPA) and the singles and doubles coupled-cluster (CCSD) method starting with the Dirac-Hartree-Fock (DHF) wave function in both cases. The differences in the results from both the methods shed light on the importance of the non-core-polarization electron correlation effects that are accounted for by the CCSD method. We also determine THE electric dipole polarizabilities of these atoms, which have computational similarities with EDMs and compare them with the available experimental and other theoretical results to assess the accuracy of our calculations.

Conforming the measured lifetimes of the5d2D3/2,5/2states in Cs with theory

We find very good agreement between our theoretically evaluated lifetimes of the $5d \ ^2D_{3/2}$ and $5d \ ^2D_{5/2}$ states of Cs with the experimental values reported in [Phys. Rev. A {\bf 57}, 4204 (1998)], which were earlier evinced to be disagreeing with an earlier rigorous theoretical study [Phys. Rev. A {\bf 69}, 040501(R) (2004)] and with another precise measurement [Opt. Lett. {\bf 21}, 74 (1996)]. In this work, we have carried out calculations of the radiative transition matrix elements using many variants of relativistic many-body methods, mainly in the coupled-cluster theory framework, and analyze propagation of the electron correlation effects to elucidate their roles for accurate evaluations of the matrix elements. We also demonstrate contributions explicitly from the Dirac-Coulomb interactions, frequency independent Breit interaction and lower order quantum electrodynamics (QED) effects. Uncertainties to these matrix elements due to different possible sources of errors are estimated. By combining our calculated radiative matrix elements with the experimental values of the transition wavelengths, we obtain the transition probabilities due to both the allowed and lower order forbidden channels. Adding these quantities together, the lifetimes of the above two states are determined precisely and plausible reasons for the reported inconsistencies between the earlier theoretical calculations and the experimental results have been pointed out.

Relativistic Many-body Calculations for Electric Dipole Moments in \(^{129}\)Xe, \(^{199}\)Hg, \(^{223}\)Rn, \(^{225}\)Ra and \(^{171}\)Yb Atom

We present and compare the results of permanent electric dipole moments (EDMs) of various closed-shell atoms due to the nuclear Schiff moment (NSM) and the tensor-pseudotensor (T-PT) interactions between the atomic nuclei and electrons. In order to highlight the role of electron-correlation effects in obtaining accurate EDM results, we employ a number of relativistic many-body methods including coupled-cluster theory at different degrees of approximation. On combining our results obtained from the relativistic coupled-cluster (RCC) at the levels of singles and doubles excitations (CCSD method) with the available EDM measurements we obtain accurate bounds on the couplings \(S\) and \(C_T\) associated with the respective NSM and T-PT interactions. The most precise EDM measurement on \(^{199}\)Hg in combination with our CC results yield limits on the above couplings as \(S<1.45 \times 10^{-12}|e|\)fm\(^3\) and \(C_T < 2.09 \times 10^{-9}\) respectively. Further combining these bounds with the latest nuclear structure and quantum chromodynamics calculations we infer limits on the strong CP-violating parameter and for the combined up- and down- quark chromo-EDMs as \(|\bar{\theta}| < 1.1 \times 10^{-9}\) and \(|\widetilde{d}_u - \widetilde{d}_d| < 2.8 \times 10^{-26} |e|\)cm, respectively.

Dispersion coefficients for the interactions of the alkali-metal and alkaline-earth-metal ions and inert-gas atoms with a graphene layer

Largely motivated by a number of applications, the van der Waals dispersion coefficients ($C_3$s) of the alkali ions (Li$^+$, Na$^+$, K$^+$ and Rb$^+$), the alkaline-earth ions (Ca$^+$, Sr$^+$, Ba$^+$ and Ra$^+$) and the inert gas atoms (He, Ne, Ar and Kr) with a graphene layer are determined precisely within the framework of Dirac model. For these calculations, we have evaluated the dynamic polarizabilities of the above atomic systems very accurately by evaluating the transition matrix elements employing relativistic many-body methods and using the experimental values of the excitation energies. The dispersion coefficients are, finally, given as functions of the separation distance of an atomic system from the graphene layer and the ambiance temperature during the interactions. For easy extraction of these coefficients, we give a logistic fit to the functional forms of the dispersion coefficients in terms of the separation distances at the room temperature.

Rigorous limits on the hadronic and semileptonicCP-violating coupling constants from the electric dipole moment ofHg199

Relativistic many-body methods at different levels of approximations are employed to gain insights into the passage of the electron correlations from lower to higher levels in the accurate determination of the electric dipole polarizability ($\alpha_d$) and electric dipole moment (EDM) due to the electron-nucleus tensor-pseudotensor (T-PT) and nuclear Schiff moment (NSM) interactions of the $^{199}$Hg ground state. Moreover, plausible reasons for the differences in the previous calculations are pointed out. Comparison between the calculated and experimental results of $\alpha_d$ indicates that our EDM calculations are about 3\% accurate which in combination with the measured value of $^{199}$Hg EDM yield limits for the T-PT coupling constant $C_T < 2.31 \times 10^{-9}$ and the NSM $S<1.55 \times 10^{-12}|e|fm^3$. From this limit of $S$ and the latest nuclear structure calculation, we get improved limits for the strong CP violating parameter $\theta_{QCD} < 2.12 \times 10^{-10}$ and the combined up and down quark chromo-EDMs $(-0.004 \tilde{d}_u + 0.020 \tilde{d}_d) < 1.15 \times 10^{-27} |e| cm$.

Correlation trends in the polarizabilities of atoms and ions in the boron, carbon, and zinc homologous sequences of elements

We investigate the role of electron-correlation effects in calculations of the electric-dipole polarizabilities $\ensuremath{\alpha}$ of elements belonging to three different groups (12--14) of the periodic table. To understand the propagation of the electron-correlation effects at different levels of approximations, we employ the relativistic many-body methods developed, based on first principles, at mean-field Dirac--Fock (DF), third-order many-body perturbation theory [MBPT(3)], random-phase approximation (RPA), and singly and doubly approximated coupled-cluster methods at the linearized (LCCSD) and nonlinearized (CCSD) levels. We observe a variance in the trends of the contributions of the correlation effects in a particular group of elements through the many-body method used; however, they resemble a similar tendency among the iso-electronic systems. Our CCSD results are within sub-1% agreement with the experimental values which are further ameliorated by including the contributions from the important triple excitations (${\mathrm{CCSD}}_{p}\mathrm{T}$ method).

Calculations on polarization properties of alkali metal atoms using Dirac—Fock plus core polarization method

A semi-empirical atomic structure model method is developed in the framework of a relativistic case. This method starts from Dirac—Fock calculations using B-spline basis set. The core-valence electron correction is then treated in a semi-empirical core polarization potential. As an application, the polarization properties of alkali metal atoms, including the static polarizabilities and long-range two-body dispersion coefficients, have been calculated. Our results are in good agreement with the results obtained from ab initio relativistic many-body perturbation method and the available experimental measurements.

Ab initiodetermination of theP- andT-violating coupling constants in atomic Xe by the relativistic-coupled-cluster method

The parity (P) and time-reversal (T) odd coupling constant associated with the tensor-pseudotensor (T-PT) electron-nucleus interaction and the nuclear Schiff moment (NSM) have been determined by combining the result of the measurement of the electric dipole moment (EDM) of a $^{129}\mathrm{Xe}$ atom and our calculations based on the relativistic-coupled-cluster (RCC) theory. Calculations using various relativistic many-body methods have been performed at different levels of approximation. The accuracies of our results are estimated by comparing our dipole polarizability calculations of the ground state of Xe with its most precise available experimental data, and taking into consideration the difference of the results of our RCC single- and double-excitation calculations with and without the important triple excitations as well as the size of our basis set. The nonlinear terms that arise in the RCC theory were found to be crucial for achieving high accuracy in the calculations. We obtain the upper limits for the T-PT electron-nucleus coupling coefficient and NSM as $1.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}e\phantom{\rule{0.28em}{0ex}}{\text{fm}}^{3}$, respectively, by combining our calculations with the available measurement. Our results, in combination with future EDM measurements in atomic Xe, could improve these limits further.

Transition properties of a potassium atom

We report here oscillator strengths, transition rates, branching ratios, and lifetimes due to the allowed transitions in the potassium (K) atom. We evaluate electric dipole ($E1$) amplitudes using an all-order relativistic many-body perturbation method. The obtained results are compared with previously available experimental and theoretical studies. Using the $E1$ matrix elements mentioned above and estimated from the lifetimes of the $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2,3/2}$ states, we determine precise values of static and dynamic polarizabilities for the first five low-lying states in the considered atom. The static polarizabilities of the ground and $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2,3/2}$ states in the present work are more precise than the available measurements in these states. Only the present work employs relativistic theory to evaluate the polarizabilities in the $3d{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{D}_{3/2,5/2}$ states for which no experimental results are known with which to compare. We also reexamine ``magic wavelengths'' for the $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2}\ensuremath{\rightarrow}4s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}$ and $4p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{3/2}\ensuremath{\rightarrow}4s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}$ transitions due to the linearly polarized light, which are useful to perform the state-insensitive trapping of K atoms.

Hyperfine-mediated static polarizabilities of monovalent atoms and ions

We apply relativistic many-body methods to compute static differential polarizabilities for transitions inside the ground-state hyperfine manifolds of monovalent atoms and ions. Knowledge of this transition polarizability is required in a number of high-precision experiments, such as microwave atomic clocks and searches for $\mathit{CP}$-violating permanent electric dipole moments. While the traditional polarizability arises in the second order of interaction with the externally applied electric field, the differential polarizability involves an additional contribution from the hyperfine interaction of atomic electrons with nuclear moments. We derive formulas for the scalar and tensor polarizabilities including contributions from magnetic dipole and electric quadrupole hyperfine interactions. Numerical results are presented for Al, Rb, Cs, ${\mathrm{Yb}}^{+}$, ${\mathrm{Hg}}^{+}$, and Fr.

Hyperfine structure of the metastableP32state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments

Measuring the hyperfine structure (HFS) of long-lived ${^{3}P}_{2}$ states of divalent atoms may offer the opportunity of extracting relatively unexplored nuclear magnetic octupole and electric hexadecapole moments. Here, using relativistic many-body methods of atomic structure and the nuclear shell model, we evaluate the effect of these higher nuclear moments on the hyperfine structure. We find that the sensitivity of HFS interval measurements in $^{87}\text{S}\text{r}$ needed to reveal the perturbation caused by the nuclear octupole moment is on the order of kHz. The results of similar analyses for $^{9}\text{B}\text{e}$, $^{25}\text{M}\text{g}$, and $^{43}\text{C}\text{a}$ are also reported.