A. Y. Potekhin, A. V. Turbiner
The quadrupole moment of a hydrogen atom in a magnetic field for field strengths from 0 to 4.414e13 G is calculated by two different methods. The first method is variational, and based on a single trial function. The second method deals with a solution of the Schroedinger equation in the form of a linear combination of Landau orbitals.
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Relativistic formulas for the electric quadrupole moment of a hydrogenlike atom, induced by the hyperfine interaction, are derived for $ns_{1/2}$ and $np_{1/2}$ states. Both the magnetic dipole and electric quadrupole hyperfine interactions are taken into account. The formulas are valid for ions with arbitrary nuclear charge and spin. The induced quadrupole moment is compared with the nuclear quadrupole moment for a wide range of hydrogenlike ions.
Molecular hydrogen ions are of metrological relevance due to the possibility of precise theoretical evaluation of their spectrum and of external-field-induced shifts. We report the results of the calculations of the rate of laser-induced electric quadrupole transitions between a large set of ro-vibrational states of ${\rm H_2^+}$. The hyperfine and Zeeman structure of the E2 transition spectrum and the effects of the laser polarization are treated in detail. We also present the nuclear spin-electron spin coupling constants, computed with a precision 10 times higher than previously.
The systematic shifts of the transition frequencies in the molecular hydrogen ions are of relevance to ultra-high-resolution radio-frequency, microwave and optical spectroscopy of these systems, performed in ion traps. We develop the ab-initio description of the interaction of the electric quadrupole moment of this class of molecules with the static electric field gradients present in ion traps. In good approximation, it is described in terms of an effective perturbation hamiltonian. An approximate treatment is then performed in the Born-Oppenheimer approximation. We give an expression of the …
We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earli…
In this paper we present tabulated data for magnetic-dipole-to-electric-quadrupole cross-susceptibilities ($χ_{\textrm{M}1 \to \textrm{E}2}$) for Dirac one-electron atoms with a pointlike, spinless and motionless nucleus of charge $Ze$. Numerical values of this susceptibility for the hydrogen atom ($Z=1$) and for hydrogenic ions with $2 \leqslant Z \leqslant 137$ are computed from the general analytical formula, recently derived by us [P. Stefa{ń}ska, Phys. Rev. A 93 (2016) 022504], valid for an arbitrary discrete energy eigenstate. In this work we provide 30 tables with the values of $χ_{\tex…
We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength B of the external field, the only electric multipole moments, which are induced by the perturbation in the atom, are those of an even order. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); 30, 2747(E) (1997)], we derive a closed-form expression for the electric quadrupole moment induced in the atom in an arbitrary discrete energy eigenstate. The result, which has the form of a double fin…
The three-body Schrödinger equation of the H$_2^+$ hydrogen molecular ion with Coulomb potentials is solved in perimetric coordinates using the Lagrange-mesh method. The Lagrange-mesh method is an approximate variational calculation with variational accuracy and the simplicity of a calculation on a mesh. Energies and wave functions of up to four of the lowest vibrational bound or quasibound states for total orbital momenta from 0 to 40 are calculated. The obtained energies have an accuracy varying from about 13 digits for the lowest vibrational state to at least 9 digits for the third vibratio…
Spherically-symmetric ground states of alkali-metal atoms do not posses electric quadrupole moments. However, the hyperfine interaction between nuclear moments and atomic electrons distorts the spherical symmetry of electronic clouds and leads to non-vanishing atomic quadrupole moments. We evaluate these hyperfine-induced quadrupole moments using techniques of relativistic many-body theory and compile results for Li, Na, K, Rb, and Cs atoms. For heavy atoms we find that the hyperfine-induced quadrupole moments are strongly (two orders of magnitude) enhanced by correlation effects. We further a…
We consider the application of a basic principle of quantum theory, the tracelessness of a certain class of hamiltonians, to the precision spectroscopy of the molecular hydrogen ions. We show that it is possible to obtain the spin-averaged transition frequencies between states from a simple weighted sum of experimentally accessible spin-dependent transition frequencies. We discuss the cases ${\rm H}_{2}^{+}$ and ${\rm HD}^{+}$, which are distinct in the multipole character of their rovibrational transitions. Inclusion of additional frequencies permits canceling also the electric quadrupole shi…
Effect of the electric quadrupole moment, $Q$, is studied for positron-atom bound systems. It is demonstrated that for $Q >50$ a.u. the electric quadrupole potential is sufficiently strong to bind positron (or electron) even in the absence of the dipole polarization potential. Such large values of $Q$ are not known for atomic ground states, however, they exist in molecules and excited atoms. In the state $2s2p~^3P^o_2$ of beryllium, the quadrupole contribution makes difference between stable bound state and decay to Be$^+$ ion and positronium. In a majority of atoms the quadrupole contribution…
Atomic quadrupole moments and hyperfine constants of the metastable $^2 D_{3/2, 5/2}$ states of Ca$^+$, Sr$^+$, Ba$^+$, Yb$^+$, and Hg$^+$ are calculated by the multiconfiguration Dirac-Hartree-Fock and relativistic configuration-interaction methods. For Hg$^+$, the configuration is $5d^9 6s^2$. For the other ions, the configuration consists of a single $d$-electron outside a set of closed shells. Current interest in the quadrupole moments of these states is due to the fact that optical transitions of these ions may be useful as references for frequency standards. Energy shifts of the metastab…
Collective effects in deformed atomic nuclei present possible avenues of study on the non-spherical distribution of neutrons and the violation of the local Lorentz invariance. We introduce the weak quadrupole moment of nuclei, related to the quadrupole distribution of the weak charge in the nucleus. The weak quadrupole moment produces tensor weak interaction between the nucleus and electrons and can be observed in atomic and molecular experiments measuring parity nonconservation. The dominating contribution to the weak quadrupole is given by the quadrupole moment of the neutron distribution, t…
A method for highly accurate calculations of atomic electric quadrupole moments (EQM) is presented, using relativistic general-excitation-rank configuration interaction wavefunctions based on Dirac spinors. Application to the clock transition states of the thulium atom employing up to full Quadruple excitations for the atomic wavefunction yields a final value of $Q_{zz}({^2F}_{7/2}) = 0.07 \pm 0.07$ a.u., establishing that the thulium electronic ground state has an exceptionally small EQM. A detailed analysis of this result is presented which has implications for EQMs of other atoms with unpai…
The electric quadrupole moment for the $5d^{2}D_{3/2}$ state of $\mathrm{^{171}Yb^{+}}$, has been calculated using the relativistic coupled-cluster method. Earlier a similar calculation was performed for the 4d $^2D_{5/2}$ state of $\mathrm{^{88}Sr^{+}}$ which is the most accurate determination to date {[}PRL, \textbf{96}, 193001 (2006)]. The present calculation of the electric quadrupole moment of $\mathrm{^{171}Yb^{+}}$ yielded a value $2.157 ea_{0}^{2}$ where the experimental value is $2.08(11)ea_{0}^{2}$; $a_{0}$ is the Bohr radius and $e$ the elementary charge. We discuss in this paper ou…
The relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) and the non-relativistic multiconfiguration Hartree-Fock (MCHF) methods have been employed to calculate the magnetic dipole and electric quadrupole hyperfine structure constants of zinc. The calculated electric field gradients for the $ 4s 4p \,\, ^3 \! P^o_{1} $ and $ 4s 4p \,\, ^3 \! P^o_{2} $ states, together with experimental values of the electric quadrupole hyperfine structure constants, made it possible to extract a nuclear electric quadrupole moment $ Q(^{67}{\rm Zn}) = 0.122(10) $ b. The error bar has been evaluated in a q…
The electric quadrupole moment for the $4d ^2D_{5/2}$ state of $\mathrm{^{88}Sr^+}$, one of the most important candidates for an optical clock, has been calculated using the relativistic coupled-cluster theory. The result of the calculation is presented and the important many-body contributions are highlighted. The calculated electric quadrupole moment is $(2.94\pm0.07)ea_{0}^{2}$, where $a_{0}$ is the Bohr radius and $e$ the electronic charge while the measured value is $(2.6\pm0.3)ea_{0}^{2}$. This is so far the most accurate determination of the electric quadrupole moment for the above ment…
Nuclear magnetic quadrupole moments (MQMs), like intrinsic electric dipole moments of elementary particles, violate both parity and time-reversal symmetry and therefore probe physics beyond the Standard Model of particle physics. We report on accurate relativistic coupled cluster calculations of the nuclear MQM interaction constants in BaF, YbF, BaOH, and YbOH. We elaborate on estimates of the uncertainty of our results. The implications of experiments searching for nonzero nuclear MQMs are discussed.
Theories that extend the Standard Model of particle physics often introduce new interactions that violate charge-parity (CP) symmetry. CP-violating effects within an atomic nucleus can be probed by measuring its nuclear magnetic quadrupole moment (MQM). The sensitivity of such a measurement is enhanced when using a heavy polar molecule containing a nucleus with quadrupole deformation. We determine how the energy levels of a molecule are shifted by the MQM and how those shifts can be measured. The measurement scheme requires molecules in a superposition of magnetic sub-levels that differ by man…
CP-violating interaction of the nuclear magnetic quadrupole moment (MQM) with electrons in the ytterbium mono-hydroxide molecule, $^{173}$YbOH, is considered. Both the magnetic quadrupole moment (MQM) of the $^{173}$Yb nucleus and the molecular interaction constant WM are estimated. Electron correlation effects are taken into account within the relativistic Fock-space coupled cluster method. Results are interpreted in terms of the strength constants of CP-violating nuclear forces, neutron EDM, QCD vacuum angle $θ$, quark EDM and chromo-EDM.
The high spectral resolution R about 45,000 provided by IGRINS (Immersion Grating INfrared Spectrometer) at MacDonald Observatory and R 100,000 achieved by CRIRES (CRyogenic high-resolution InfraRed Echelle Spectrograph) at VLT (Very Large Telescope) challenges the present knowledge of infrared spectra. aims We aim to predict the full infrared spectrum of molecular hydrogen at a comparable accuracy. methods We take advantage of the recent theoretical ab initio studies on molecular hydrogen to compute both the electric quadrupole and magnetic dipole transitions taking place within the ground el…
Intensities of all fine-structure resolved rovibrational electric quadrupole absorption lines in $^{16}$O$_2$($X^{3}Σ^{-}_{g}$) calculated with a new $\textit{ab initio}$ quadrupole moment curve
针对¹⁶O₂（X³Σ⁻₉）振转电四极吸收线强度计算问题，本文采用新的从头算四极矩曲线和中间耦合方法，计算了v≤35、J≤40的所有谱线强度，其结果与HITRAN数据库一致，为2024版数据库更新提供了振动泛频和热带强度数据。
Nuclei with spin $I \ge 1$ have a weak quadrupole moment which leads to tensor contribution to the parity non-conserving interaction between nuclei and electrons. We calculate this contribution for Yb$^+$, Fr and Ra$^+$ and found it to be small. In contrast, in many lanthanides (e.g., Nd, Gd, Dy, Ho, Er, Pr, Sm) and Ra close levels of opposite parity lead to strong enhancement of the effect making it sufficiently large to be measured. Another possibility is to measure the PNC transitions between the hyperfine components of the ground state of Bi. Since nuclear weak charge is dominated by neutr…
We have performed accurate calculations of the hyperfine structure of the $2\,^3\!P$ state in the $^9$Be atom with the help of highly optimized, explicitly correlated functions, accounting for the leading finite nuclear mass, radiative, nuclear structure and relativistic effects. By comparison with measurements, we have determined the $^9$Be nuclear quadrupole moment to be $Q_{\rm N} = 0.05350(14)$ barns, which is not only the most accurate result, but also disagrees with previous determinations.
We investigate electric quadrupole hyperfine structure constant ($B$) results in $^{135}$Ba$^+$ and $^{137}$Ba$^+$ and compare their ratios with the reported measurements of the ratio between the nuclear quadrupole moment ($Q$) values of these isotopes. We carry out confidence tests for the reported experimental $B$ values and calculations of $B/Q$ from the present work. Inconsistencies in the experimental $B$ values are observed in both the isotopes from different experiments performed using the same techniques. The present calculations are carried out using an all order relativistic many-bod…
We report the value of the electric quadrupole moment of $^{209}$Bi extracted from the atomic data. For this, we performed electronic structure calculations for the ground $^4S^o_{3/2}$ and excited $^2P^o_{3/2}$ states of atomic Bi using the Dirac-Coulomb-Breit Hamiltonian and the Fock space coupled cluster method with single, double, and full triple amplitudes for the three-particle Fock space sector. The value of the quadrupole moment of $^{209}$Bi, $Q(^{209}$Bi$)=-418(6)$~mb, derived from the resulting electric field gradient values and available atomic hyperfine splittings is in excellent …
The influence of oscillating quadrupole fields on atomic energy levels is examined theoretically and general expressions for the quadrupole matrix elements are given. The results are relevant to any ion-based clock in which one of the clock states supports a quadrupole moment. Clock shifts are estimated for $^{176}$Lu$^+$ and indicate that coupling to the quadrupole field would not be a limitation to clock accuracy at the $\lesssim10^{-19}$ level. Nevertheless, a method is suggested that would allow this shift to be calibrated. This method utilises a resonant quadrupole coupling that enables t…
Precision measurements of the Lu$^+$ $^3D_1$ and $^3D_2$ quadrupole moments have been carried out giving $Θ(^3D_1)=0.63862(74)\,e a_0^2$ and $Θ(^3D_2)=0.8602(14)\,e a_0^2$, respectively. The measurements utilize the differential shift between ions in a multi-ion crystal so that effects of external field gradients do not contribute leaving only the well defined Coulomb interaction. At this level of precision, hyperfine-mediated corrections will likely be important.
Based on the single particle approximation [V. F. Dmitriev {\it et al}, Phys. Rev. C {\bf50}, 2358 (1994), C.-C. Chen, Phys. Rev. A {\bf51}, 2611 (1995)], the Landau quantization associated with an atom with a magnetic quadrupole moment is introduced, and then, rotating effects on this analogue of the Landau quantization is investigated. It is shown that rotating effects can modify the cyclotron frequency and breaks the degeneracy of the analogue of the Landau levels.
By combining our theoretical calculation and recently measured electric quadrupole hyperfine structure constant of the $3d ^2D_{5/2}$ state in the singly ionized $^{43}$Ca, we determine its nuclear quadrupole moment to one percent accuracy. The obtained result, $-0.0444(6)b$, is about ten percent improvement over the considered standard value. We have employed the relativistic coupled-cluster theory at single and double excitations level to calculate the electronic wave functions. The accuracy of these wave functions are estimated by comparing our calculated magnetic dipole hyperfine constants…
Nuclear forces violating parity and time reversal invariance P,T-odd produce P,T-odd nuclear moments, for example, the nuclear Schiff moment. In turn, this moment can induce the electric dipole moment (EDM) in the atom. The contribution to the Schiff moment from the soft collective quadrupole and octupole vibrations in spherical nuclei is calculated in the framework of the quasiparticle random phase approximation with separable quadrupole and octupole forces. The values of nuclear Schiff moments predicted for odd 217-221Ra and 217-221Ra isotopes indicate a possibility of enhancement by a facto…
Recent experiments involving the interaction of optical vortices with atoms in quadrupole transitions have been shown to be accompanied by the exchange of orbital angular momentum (OAM) between the electronic states of the atom and the optical vortex field. Earlier work by both theory and experiment had ruled out the transfer of a vortex OAM to the electronic degrees of freedom in an electric dipole atomic transition and it has been confirmed that the lowest multipolar order involving an OAM transfer to the electronic motion is indeed the electric quadrupole. Hitherto, the quadrupole transitio…
The CP-violating interaction of the nuclear magnetic quadrupole moment (MQM) of the $^{175}$Lu nucleus with electrons in the molecular cation LuOH$^+$ is studied. The resulting effect is expressed in terms of CP-odd parameters, such as quantum chromodynamics angle $\barθ$, quark electric dipole moment (EDM) and chromo-EDM. For this we have performed a calculation of the nuclear MQM as well as the molecular constant that characterises the interaction of this MQM of $^{175}$Lu with electrons. Additionally, we predict the hyperfine structure constants for the ground electronic state of LuOH$^+$. …
Atomic determination of the nuclear quadrupole moment $\mathrm{Q}(^{209}{\rm Bi})$ using the multi-configuration Dirac-Hartree-Fock method
利用 Grasp2018 中的多组态 Dirac-Hartree-Fock 方法计算铋原子电场梯度，结合实验超精细常数提取 $^{209}$Bi 核四极矩为 -422(22) mb，并参与构建世界平均值 -420(17) mb。
The existence of permanent electric dipole moments (EDMs) and magnetic quadrupole moments (MQMs) violate both time reversal invariance ($T$) and parity ($P$). Following the $CPT$ theorem they also violate combined $CP$ symmetry. Nuclear EDM are completely screened in atoms and molecules while interaction between electrons and MQM creates atomic and molecular EDM which can be measured and used to test CP-violation theories. Nuclear MQM are produced by the nucleon-nucleon $T,P$-odd interaction and by nucleon EDM. In this work we study the effect of enhancement of the nuclear MQM due to the nucle…
Deformed nuclei possess enhanced moments violating time reversal invariance ($T$) and parity ($P$). Collective magnetic quadrupole moments (MQM) appear in nuclei with a quadrupole deformation (which have ordinary $T$,$P$-conserving collective electric quadrupole moments). Nuclei with an octupole deformation have a collective electric octupole moment, electric dipole moment (EDM), Schiff moment and MQM in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by $T$ and $P$ conservation, meaning their ex…
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