Finite Nuclear Mass Effects Research Articles

A numerical code for the analysis of magnetic white-dwarf spectra that includes field effects on the chemical equilibrium

We present a new magnetic-atmosphere model code for obtaining synthetic spectral fluxes of hydrogen-rich magnetic white dwarfs. To date, observed spectra have been analyzed with models that neglect the magnetic field's effects on the atomic populations. In this work, we incorporate state-of-art theory in the evaluation of numerical densities of atoms, free electrons, and ions in local thermodynamical equilibrium under the action of a magnetic field. The energy distribution of atoms is rigorously evaluated for arbitrary field strength. This energy pattern includes going from tightly bound states to metastable or truly bound, highly excited states embedded in the continuum, that is, over the first Landau level. Finite nuclear mass effects and the coupling between the internal atomic structure and the motion of the atom across the magnetic field are also considered. Synthetic fluxes are generated with integrations of numerical solutions of polarized radiative transfer over the visible stellar disk using a spherical $t$-design method. The atmosphere code is tested with observations from the Sloan Digital Sky Survey for a group of known magnetic white dwarfs. Physical stellar parameters are obtained from least-squares fits to the observed energy distribution and compared with results of previous works. We show that the use of zerofield ionization equilibrium in spectral analyses can lead to underestimated effective temperatures for highly magnetic white dwarfs.

Abinitio determination of the polarizability of neon

The static electric-dipole polarizability $\ensuremath{\alpha}$ of the neon atom was determined with a relative uncertainty of only about 0.003% using state-of-the-art ab initio approaches. The new value, $\ensuremath{\alpha}=2.661\phantom{\rule{0.16em}{0ex}}067(77)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, is almost five times more accurate than the previous ab initio estimate, $\ensuremath{\alpha}=2.660\phantom{\rule{0.16em}{0ex}}80(36)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, by Lesiuk et al. [Phys. Rev. A 102, 052816 (2020)]. Similar to their work, we calculated $\ensuremath{\alpha}$ using ab initio methods up to full configuration interaction and added corrections for finite nuclear mass and size, relativistic, and quantum electrodynamics (QED) effects. The uncertainty reduction of this work was achieved in particular by employing extremely large basis sets, including newly developed ones of 11Z, 12Z, and 13Z quality. Moreover, the finite nuclear mass effects and most of the relativistic contributions were calculated at much higher levels of theory than in the work of Lesiuk et al. However, we adopted their values for the orbit-orbit part of the relativistic correction and for the Bethe logarithm needed to compute the QED correction. The uncertainty of our final value is still an order of magnitude larger than that of the experimental value recently measured by Gaiser and Fellmuth [Phys. Rev. Lett. 120, 123203 (2018)] with an uncertainty of only a few parts per million using dielectric-constant gas thermometry. Yet, our ab initio value agrees with their value, ${\ensuremath{\alpha}}_{\mathrm{exp}}=2.661\phantom{\rule{0.16em}{0ex}}057(7)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, almost within the experimental uncertainty. This could indicate that the higher-order relativistic corrections and QED contributions, which dominate our uncertainty budget, are more accurate than expected considering the uncontrolled approximations involved in their calculation.

Hyperfine structure of the 2P3 state in Be9 and the nuclear quadrupole moment

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.

Magic wavelengths for the helium 23S1→21P1 forbidden transition

A series of magic wavelengths for the forbidden transition of $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ in helium are determined by performing large-scale full relativistic configuration interaction calculations based on Dirac-Coulomb-Breit Hamiltonian with the mass shift operator included. QED corrections to magic wavelengths are taken into account by using the perturbation theory in nonrelativistic configuration interaction calculations. The 1335.55(2)-nm magic wavelength for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}(M=\ifmmode\pm\else\textpm\fi{}1)\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}(M=0)$ transition is found to be sensitive to the finite nuclear mass, relativistic, and QED effects, and its measurement might provide a test of atomic structure theory.

Confined variational calculation of positronium-hydrogen scattering below the positronium excitation threshold

The confined variational method is used to study $S$-, $P$-, and $D$-wave positronium-hydrogen scattering with the scattering energy between 0.068 and 3.333 eV. Accurate phase shifts and $S$-wave scattering lengths are calculated and compared with other theoretical methods. Existing discrepancies for the $D$-wave phase shifts are resolved. According to collision momentum, total orbital angular momentum, and spin configuration, the distortion effects of positronium are quantitatively studied. Finally, the finite nuclear mass effects are investigated.

Two-photon decay rates in heliumlike ions: Finite-nuclear-mass effects

The spontaneous two-photon decay rates from the $1s2s\phantom{\rule{0.16em}{0ex}}^{1}S_{0}$ level to the ground state $1{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}$ in helium and its isoelectronic ions through neon ($Z=10$) are calculated, including the effects of finite nuclear mass. In all cases the length and velocity results agree to eight or more figures, demonstrating that the theoretical formulation correctly takes into account the effects of mass scaling, mass polarization, and motion of the nucleus in the center-of-mass frame. Algebraic relationships are derived and tested relating the expansion coefficients in powers of $\ensuremath{\mu}/M$ for mass polarization, where $\ensuremath{\mu}$ is the electron reduced mass and $M$ is the nuclear mass. Muonic, pionic, and antiprotonic helium are included as extreme test cases where mass polarization is large. The results are compared with experiment and previous calculations.

QED calculation of the dipole polarizability of helium atom

The QED contribution to the dipole polarizability of the $^{4}\mathrm{He}$ atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to 0.048 557 2(14) in atomic units, confirms the small magnitude of this quantity found in the only previous calculation [G. \L{}ach, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 92, 233001 (2004)], but differs from it by about 5%. The origin of this difference is explained. The total QED correction of the order of ${\ensuremath{\alpha}}^{3}$ in the fine-structure constant $\ensuremath{\alpha}$ amounts to $30.6671(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, including the $0.1822\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ contribution from the electric-field derivative of the Bethe logarithm and the $0.011\phantom{\rule{0.16em}{0ex}}12(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ correction for the finite nuclear mass, with all values in atomic units. The resulting theoretical value of the molar polarizability of helium-4 is $0.517\phantom{\rule{0.16em}{0ex}}254\phantom{\rule{0.16em}{0ex}}08(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{3}/\text{mol}$ with the error estimate dominated by the uncertainty of the QED corrections of order ${\ensuremath{\alpha}}^{4}$ and higher. Our value is in agreement with but an order of magnitude more accurate than the result $0.517\phantom{\rule{0.16em}{0ex}}254\phantom{\rule{0.16em}{0ex}}4(10)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{3}/\text{mol}$ of the most recent experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)].

Calculations of Relativistic, Correlation, Nuclear, and Quantum-Electrodynamics Corrections to Energy and Ionization Potential of the Ground State of Helium-Like Ions

Nonrelativistic and relativistic variational calculations of energies and ionization potentials of the ground state of helium-like ions with nuclear number Z = 2−20 have been performed. All leading corrections to the total energy have been calculated, including the contribution of electron correlations, relativistic and quantum-electrodynamics corrections, and contributions due to the finite nuclear size (field shift) and finite nuclear mass (nuclear recoil) effects. Relativistic wave functions were calculated using the Dirac–Coulomb–Breit Hamiltonian.

Few-electron atomic ions in non-relativistic QED: The ground state

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z≤20 the finite nuclear mass effects do not change 4–5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3–4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for Z≤12, and not less than 12 s.d. for Z≤50 for helium-like as well as for Z≤20 for lithium-like ions by a compact meromorphic function in variable λ=Z−ZB (here ZB is the 2nd critical charge (Turbiner et al., 2016)), P9(λ)∕Q5(λ). It is found that both the Majorana formula – a second degree polynomial in Z with two free parameters – and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z≤20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that ≳99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z≤20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z for Z≤20.

The nuclear size and mass effects on muonic hydrogen-like atoms embedded in Debye plasma

Effects of finite nuclear size and finite nuclear mass are investigated for muonic atoms and muonic ions embedded in the Debye plasma. Both nuclear charge radii and nuclear masses are taken into account with experimentally determined values. In particular, isotope shifts of bound state energies, radial probability densities, transition energies, and binding energies for several atoms are studied as functions of Debye length. The theoretical model based on semianalytical calculations, the Sturmian expansion method, and the perturbative approach has been constructed, in the nonrelativistic frame. For some limiting cases, the comparison with previous most accurate literature results has been made.

Tune-out wavelength around 413 nm for the helium23S1state including relativistic and finite-nuclear-mass corrections

The tune-out wavelength at 413 nm for the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state of helium is expected to be sensitive to finite nuclear mass, relativistic, and quantum electrodynamic (QED) corrections, which provides a scheme for testing atomic structure theory [J. Mitroy and L.-Y. Tang, Phys. Rev. A 88, 052515 (2013)]. In the present work, a large-scale full-configuration-interaction calculation based on both the Dirac-Coulomb-Breit Hamiltonian and the nonrelativistic Hamiltonian is performed for the dynamic dipole polarizabilities of helium in the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state. The tune-out wavelengths for the magnetic sublevels ${M}_{J}=0$ and ${M}_{J}=\ifmmode\pm\else\textpm\fi{}1$ are determined to be 413.0801(4) nm and 413.0859(4) nm, respectively, at sub-ppm accuracy, including finite nuclear mass and relativistic corrections. Our value for the ${M}_{J}=1$ sublevel agrees with the measured value of 413.0938(20)(9) nm [B. M. Henson et al., Phys. Rev. Lett. 115, 043004 (2015)] at the level of 19 ppm. The discrepancy between these two values is mainly due to the uncalculated QED contribution. Our current value confirms quantitatively the prediction of Mitroy and Tang. Also, for the state of $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ we find that the corrections due to finite nuclear mass and relativistic effects to the static dipole polarizability of 315.7227(4)${a}_{0}^{3}$ are about 600 ppm and 310 ppm, respectively, which are about 1.4 and 5.4 times larger than those for the ground state. A measurement at the level of 10 ppm for the static dipole polarizability of helium in $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ can be used to determine the transition matrix element between $2{\phantom{\rule{0.28em}{0ex}}}^{3}S$ and $2{\phantom{\rule{0.28em}{0ex}}}^{3}P$ at the level of ${10}^{\ensuremath{-}5}$.

Deuteron and triton magnetic moments from NMR spectra of the hydrogen molecule

We present a theory and calculations of the nuclear magnetic shielding with finite nuclear mass effects and determine magnetic moments of deuteron and triton using the known NMR spectra of HD and HT molecules. The results $\mu_d = 0.857\,438\,234\,6(53)\;\mu_N$ and $\mu_t = 2.978\,962\,471(10)\;\mu_N$ are more accurate and in a good agreement with the currently accepted values.

Oscillator strengths and radiative decay rates for spin-changing S–P transitions in helium: finite nuclear mass effects

We have calculated the electric dipole (E1) and magnetic quadrupole (M2) oscillator strengths and spontaneous decay rates for 24 spin-changing transitions of atomic helium. We included the effects of the finite nuclear mass and the anomalous magnetic moment of the electron augmented by the recently derived Pachucki term. The specific transitions for 4He are and with and for For the E1 calculations we used the Breit approximation and pseudostate expansions to perform the perturbation sums over intermediate states in both the length and velocity gauge as a check on both numerical accuracy and validity of the transition operators. The corrections for the nuclear mass and the electron anomaly tend to cancel, indicating that if one is included, then so should be the other. The tables give mass- and anomaly-dependent coefficients permitting the easy generation of results for the other isotopes of helium.

Intercombination transitions of the carbon-like isoelectronic sequence

Energy levels, wavelengths, transition rates, oscillator strengths, and lifetimes between the 2s22p2 3P1, 2s22p2 3P2, and 2s2p3 5S2 levels of ions in the carbon-like (C-like) isoelectronic sequence (nuclear charges Z = 7–92) are calculated in the valence and core-valence limits using the multiconfiguration Dirac—Fock method. The Breit interaction, quantum electrodynamics (QED), and finite nuclear mass effects are taken into account in subsequent relativistic configuration-interaction calculations. The calculated energies and transition rates are compared with the critically evaluated experimental values and other recent calculated results. Our calculated data are in good agreement with these data.

Testing quantum electrodynamics in the lowest singlet states of the beryllium atom

High-precision results are reported for the energy levels of the $2\phantom{\rule{0.16em}{0ex}}{}^{1}S$ and $2\phantom{\rule{0.16em}{0ex}}{}^{1}P$ states of the beryllium atom. Calculations are performed using fully correlated Gaussian basis sets and taking into account the relativistic, quantum electrodynamics (QED), and finite nuclear mass effects. Theoretical predictions for the ionization potential of the beryllium ground state $75\phantom{\rule{0.16em}{0ex}}192.699(7)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ and the $2\phantom{\rule{0.16em}{0ex}}{}^{1}P\ensuremath{\rightarrow}2\phantom{\rule{0.16em}{0ex}}{}^{1}S$ transition energy $42\phantom{\rule{0.16em}{0ex}}565.441(11)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ are compared to known but less accurate experimental values. The accuracy of the four-electron computations approaches that achieved for the three-electron atoms, which enables determination of the nuclear charge radii and precision tests of QED.

Relativistic, QED, and nuclear mass effects in the magnetic shielding of H3e

The magnetic shielding σ of H3e is studied. The complete relativistic corrections of order O(α2), leading QED corrections of order O(α3 ln α), and finite nuclear mass effects of order O(m/mN) are calculated with high numerical precision. The resulting theoretical predictions for σ=59.967 43(10)×10−6 are the most accurate to date among all elements and support the use of H3e as a NMR standard.

Nuclear recoil effect on the magnetic-dipole decay rates of atomic levels

The effect of finite nuclear mass on the magnetic-dipole transition probabilities between fine-structure levels of the same term is investigated. Based on a rigorous QED approach a nonrelativistic formula for the recoil correction to first order in m_e/M is derived. Numerical results for transitions of experimental interest are presented.

Upper bound to the critical binding nuclear charge for a three-electron atomic system

The three-electron ground state atomic energy in the Z → 2 limit, where Z is the nuclear charge, is studied using high precision Hylleraas-type calculations, with the objective to investigate the stability of the ground state of the helium negative ion. Finite nuclear mass effects and relativistic corrections are incorporated in the computations. The calculations reveal that the critical binding nuclear charge, Zc, the value below which the three-electron system is no longer stable, is bounded above by Zc = 2.000 001.

Accurate long-range coefficients for two excited like isotope He atoms:He(2P1)–He(2P1),He(2P1)–He(2P3), andHe(2P3)–He(2P3)

A general formalism is used to express the long-range potential energies in inverse powers of the separation distance between two like atomic or molecular systems with P symmetries. The long-range molecular interaction coefficients are calculated for the molecular symmetries {delta}, {pi}, and {sigma}, arising from the following interactions: He(2 {sup 1}P)-He(2 {sup 1}P), He(2 {sup 1}P)-He(2 {sup 3}P), and He(2 {sup 3}P)-He(2 {sup 3}P). The electric quadrupole-quadrupole term C{sub 5}, the van der Waals (dispersion) term C{sub 6}, and higher-order terms C{sub 8} and C{sub 10} are calculated ab initio using accurate variational wave functions in Hylleraas coordinates with finite nuclear mass effects. A comparison is made with previously published results where available.

Long-range interactions forHe(nS)–He(n′S)andHe(nS)–He(n′P)

The energetically lowest five states of a helium atom are: He($1^1S$), He($2^3S$), He($2^1S$), He($2^3P$), and He($2^1P$). Long-range interaction coefficients $C_3$, $C_6$, $C_8$, $C_9$, and $C_{10}$ for all $S-S$ and $S-P$ pairs of these states are calculated precisely using correlated wave functions in Hylleraas coordinates. Finite nuclear isotope mass effects are included.