The ground and a few excited states of the beryllium atom in external uniform magnetic fields are calculated by means of our 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35*10^9T. With changing field strength the ground state of the Be atom undergoes three transitions involving four different electronic configurations which belong to three groups with different spin projections S_z=0,-1,-2. For weak fields the ground state configuration arises from the 1s^2 2s^2, S_z=0 configuration. With increasing field strength the ground state evolves into the two S_z=-1 configurations 1s^22s 2p_{-1} and 1s^2 2p_{-1}3d_{-2}, followed by the fully spin polarised S_z=-2 configuration 1s2p_{-1}3d_{-2}4f_{-3}. The latter configuration forms the ground state of the beryllium atom in the high field regime \\gamma>4.567. The analogous calculations for the Be^+ ion provide the sequence of the three following ground state configurations: 1s^22s and 1s^22p_{-1} (S_z=-1/2) and 1s2p_{-1}3d_{-2} (S_z=-3/2).The ground and a few excited states of the beryllium atom in external uniform magnetic fields are calculated by means of our 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35*10^9T. With changing field strength the ground state of the Be atom undergoes three transitions involving four different electronic configurations which belong to three groups with different spin projections S_z=0,-1,-2. For weak fields the ground state configuration arises from the 1s^2 2s^2, S_z=0 configuration. With increasing field strength the ground state evolves into the two S_z=-1 configurations 1s^22s 2p_{-1} and 1s^2 2p_{-1}3d_{-2}, followed by the fully spin polarised S_z=-2 configuration 1s2p_{-1}3d_{-2}4f_{-3}. The latter configuration forms the ground state of the beryllium atom in the high field regime \\gamma>4.567. The analogous calculations for the Be^+ ion provide the sequence of the three following ground state configurations: 1s^22s and 1s^22p_{-1} (S_z=-1/2) and 1s2p_{-1}3d_{-2} (S_z=-3/2).

Sep 1, 200011 citations

Bound state spectrum of the triplet states in the Be atom

Alexei M. Frolov, María Belén Ruiz

arXiv:1310.2657Chemical Physics LettersFeb 12, 201415 citations

MF -dependent hyperfine-induced 5s5pP0o3–5s2S01 clock transition rates in an external magnetic field for Sr87

Xiaotong Lu, Feng Guo, Yebing Wang +4 more

In this paper, we theoretically calculate the natural lifetime of the $5s5p\phantom{\rule{4pt}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state of the $^{87}\mathrm{Sr}$ atom and evaluate the effect of an external magnetic field on the lifetimes of Zeeman sublevels. Using the multiconfiguration Dirac-Hartree-Fock method, the hyperfine-induced $5s5p^{3}\mathrm{P}_{0}^{o}--5{s}^{2}\phantom{\rule{0.28em}{0ex}}^{1}\mathrm{S}_{0}$ clock transition rate of $^{87}\mathrm{Sr}$ is estimated as $6.39(37)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, corresponding to a lifetime of 156(9) s. Our result closely agrees with previous measurements after correcting the decay rate due to the black-body radiation effect. Additionally, the transition rates of Zeeman sublevels are calculated as a function of the magnetic-field strength. Our theoretical results reveal that transition rates between different excited Zeeman states are discrepant, and the lifetime of magnetic sublevel $+{M}_{F}^{e}\phantom{\rule{4pt}{0ex}}(\ensuremath{-}{M}_{F}^{e}$; the superscript $e$ refers to the excited state, i.e., the $5s5p\phantom{\rule{0.28em}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state) will increase (decrease) with the magnetic-field strength. Moreover, we measure the coupling strength between the clock laser and the $^{87}\mathrm{Sr}$ atoms, and our findings align well with theoretical estimations regarding changes in normalized coupling strength with external magnetic-field strength.In this paper, we theoretically calculate the natural lifetime of the $5s5p\phantom{\rule{4pt}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state of the $^{87}\mathrm{Sr}$ atom and evaluate the effect of an external magnetic field on the lifetimes of Zeeman sublevels. Using the multiconfiguration Dirac-Hartree-Fock method, the hyperfine-induced $5s5p^{3}\mathrm{P}_{0}^{o}--5{s}^{2}\phantom{\rule{0.28em}{0ex}}^{1}\mathrm{S}_{0}$ clock transition rate of $^{87}\mathrm{Sr}$ is estimated as $6.39(37)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, corresponding to a lifetime of 156(9) s. Our result closely agrees with previous measurements after correcting the decay rate due to the black-body radiation effect. Additionally, the transition rates of Zeeman sublevels are calculated as a function of the magnetic-field strength. Our theoretical results reveal that transition rates between different excited Zeeman states are discrepant, and the lifetime of magnetic sublevel $+{M}_{F}^{e}\phantom{\rule{4pt}{0ex}}(\ensuremath{-}{M}_{F}^{e}$; the superscript $e$ refers to the excited state, i.e., the $5s5p\phantom{\rule{0.28em}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state) will increase (decrease) with the magnetic-field strength. Moreover, we measure the coupling strength between the clock laser and the $^{87}\mathrm{Sr}$ atoms, and our findings align well with theoretical estimations regarding changes in normalized coupling strength with external magnetic-field strength.

Physical Review AJul 28, 20235 citations

Hartree-Fock calculations of atoms and molecular chains in strong magnetic fields

Daniel Neuhauser, Karlheinz Langanke, S. E. Koonin

We calculate the binding energies of atoms and molecular chains in ${10}^{12}$ G magnetic fields using the Hartree-Fock method. For Z>2 (4) at 1\ifmmode\times\else\texttimes\fi{} ${10}^{12}$ (5\ifmmode\times\else\texttimes\fi{} ${10}^{12}$ ), the isolated atom is energetically favored over the molecular chain.We calculate the binding energies of atoms and molecular chains in ${10}^{12}$ G magnetic fields using the Hartree-Fock method. For Z>2 (4) at 1\ifmmode\times\else\texttimes\fi{}${10}^{12}$ (5\ifmmode\times\else\texttimes\fi{}${10}^{12}$), the isolated atom is energetically favored over the molecular chain.

Physical Review AMar 1, 198652 citations

Hartree-Fock variational method for fermions

Luca Salasnich

Springer eBooksJan 1, 20200 citations

In this paper, we theoretically calculate the natural lifetime of the

5s5p\phantom{\rule{4pt}{0ex}}^{3}\mathrm{P}_{0}^{o}

state of the

^{87}\mathrm{Sr}

atom and evaluate the effect of an external magnetic field on the lifetimes of Zeeman sublevels. Using the multiconfiguration Dirac-Hartree-Fock method, the hyperfine-induced

5s5p^{3}\mathrm{P}_{0}^{o}--5{s}^{2}\phantom{\rule{0.28em}{0ex}}^{1}\mathrm{S}_{0}

clock transition rate of

^{87}\mathrm{Sr}

is estimated as $6.39(37)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, corresponding to a lifetime of 156(9) s. Our result closely agrees with previous measurements after correcting the decay rate due to the black-body radiation effect. Additionally, the transition rates of Zeeman sublevels are calculated as a function of the magnetic-field strength. Our theoretical results reveal that transition rates between different excited Zeeman states are discrepant, and the lifetime of magnetic sublevel $+{M}_{F}^{e}\phantom{\rule{4pt}{0ex}}(\ensuremath{-}{M}_{F}^{e}$; the superscript

e

refers to the excited state, i.e., the

5s5p\phantom{\rule{0.28em}{0ex}}^{3}\mathrm{P}_{0}^{o}

state) will increase (decrease) with the magnetic-field strength. Moreover, we measure the coupling strength between the clock laser and the

^{87}\mathrm{Sr}

atoms, and our findings align well with theoretical estimations regarding changes in normalized coupling strength with external magnetic-field strength.

We calculate the binding energies of atoms and molecular chains in

{10}^{12}

G magnetic fields using the Hartree-Fock method. For Z>2 (4) at 1\ifmmode\times\else\texttimes\fi{}

{10}^{12}

(5\ifmmode\times\else\texttimes\fi{}

{10}^{12}

), the isolated atom is energetically favored over the molecular chain.