arXiv:2106.16161 · Physical Review A

Fine and hyperfine splitting of the low-lying states of Be9

Mariusz Puchalski, Jacek Komasa, Krzysztof Pachucki

Published Aug 31, 2021• 12 citations

Abstract

We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest $^{1,3}P_{J}, ^{3}S_{1}, ^{3}P_{J}^{e}$, and $^{1,3}D_{J}$ excited states of the $^{9}\mathrm{Be}$ atom using explicitly correlated Gaussian functions, and we report on the breakdown of the standard hyperfine structure theory. Because of the strong hyperfine mixing, which prevents the use of common hyperfine constants, we formulate a description of the fine and hyperfine structure that is valid for an arbitrary coupling strength and may have wide applications in many other atomic systems.

DOI: 10.1103/physreva.104.022824

Related papers

Hyperfine structure of the $2^3S-$state in the ${}^9$Be atom

Alexei M. Frolov, María Belén Ruiz

arXiv:1301.6402Jan 27, 20130 citations

Model for the hyperfine structure of electronically excited KCs molecules

Andrea Orbán, Romain Véxiau, O. Krieglsteiner +4 more

arXiv:1507.06519Physical Review ASep 21, 201519 citations

Fine and hyperfine structure of the Li− spectrum

K. T. Cheng, Jonathan E. Hardis, E. J. Dehm +1 more

Physical Review AAug 1, 198414 citations

Introduction

Howard R. Reiss

Optics ExpressMar 30, 199810 citations

Hyperfine structure of the 2pP122 state in Be+9

J. J. Bollinger, Joseph S. Wells, David J. Wineland +1 more

Physical Review AApr 1, 198555 citations

Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved ( 2S−2S ) systems

Ulrich D. Jentschura, Vincent Debierre, Chandra Mani Adhikari +2 more

arXiv:1701.08813Physical Review AFeb 2, 201717 citations

arXiv:2106.16161 · Physical Review A

Fine and hyperfine splitting of the low-lying states of Be9

Mariusz Puchalski, Jacek Komasa, Krzysztof Pachucki

Published Aug 31, 2021• 12 citations

arXiv HTML PDF

Abstract

We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest

^{1,3}P_{J}, ^{3}S_{1}, ^{3}P_{J}^{e}

, and

^{1,3}D_{J}

excited states of the

^{9}\mathrm{Be}

atom using explicitly correlated Gaussian functions, and we report on the breakdown of the standard hyperfine structure theory. Because of the strong hyperfine mixing, which prevents the use of common hyperfine constants, we formulate a description of the fine and hyperfine structure that is valid for an arbitrary coupling strength and may have wide applications in many other atomic systems.

DOI: 10.1103/physreva.104.022824

Related papers

Hyperfine structure of the $2^3S-$ state in the ${}^9$ Be atomHyperfine structure of the $2^3S-$state in the ${}^9$Be atom

Alexei M. Frolov, María Belén Ruiz

A number of bound states properties and the hyperfine structure of the 23S(L=0)-state in the Be atom are determined from the results of highly accurate computations. The energies of the hyperfine structure levels for this state are ε(F=12) = -13725.635(7) MHz, ε(F=32) = -5490.254(7) MHz and ε(F=52) = 8235.381(7) MHz, respectively. The hyperfine structure splittings are Δ12 = 8235.381(7) MHz and Δ23 = 13725.635(7) MHz, respectively. The spectrum of low-lying (bound) triplet states in the Be atom is briefly investigated.A number of bound states properties and the hyperfine structure of the 23S(L=0)-state in the Be atom are determined from the results of highly accurate computations. The energies of the hyperfine structure levels for this state are ε(F=12) = -13725.635(7) MHz, ε(F=32) = -5490.254(7) MHz and ε(F=52) = 8235.381(7) MHz, respectively. The hyperfine structure splittings are Δ12 = 8235.381(7) MHz and Δ23 = 13725.635(7) MHz, respectively. The spectrum of low-lying (bound) triplet states in the Be atom is briefly investigated.

arXiv:1301.6402Jan 27, 20130 citations

Model for the hyperfine structure of electronically excited KCs molecules

Andrea Orbán, Romain Véxiau, O. Krieglsteiner +4 more

A model for determining the hyperfine structure of the excited electronic states of diatomic bialkali heteronuclear molecules is formulated from the atomic hyperfine interactions and is applied to the case of bosonic $^{39}\mathrm{KCs}$ and fermionic $^{40}\mathrm{KCs}$ molecules. The hyperfine structure of the potential-energy curves of the states correlated to the $\mathrm{K}(4s\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{S}_{1/2})+\mathrm{Cs}(6p\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{P}_{1/2,3/2})$ dissociation limits is described in terms of different coupling schemes depending on the internuclear distance $R$ . These results provide a step in the calculation of the hyperfine structure of rovibrational levels of these excited molecular states in the perspective of the identification of efficient paths for creating ultracold ground-state KCs molecules.A model for determining the hyperfine structure of the excited electronic states of diatomic bialkali heteronuclear molecules is formulated from the atomic hyperfine interactions and is applied to the case of bosonic $^{39}\mathrm{KCs}$ and fermionic $^{40}\mathrm{KCs}$ molecules. The hyperfine structure of the potential-energy curves of the states correlated to the $\mathrm{K}(4s\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{S}_{1/2})+\mathrm{Cs}(6p\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{P}_{1/2,3/2})$ dissociation limits is described in terms of different coupling schemes depending on the internuclear distance $R$. These results provide a step in the calculation of the hyperfine structure of rovibrational levels of these excited molecular states in the perspective of the identification of efficient paths for creating ultracold ground-state KCs molecules.

arXiv:1507.06519Physical Review ASep 21, 201519 citations

Fine and hyperfine structure of the Li− spectrum

K. T. Cheng, Jonathan E. Hardis, E. J. Dehm +1 more

The fine and hyperfine structures of the doubly excited, quintet states of the $^{7}\mathrm{Li}^{\ensuremath{-}}$ ion are of comparable magnitude. This leads to strong second-order hyperfine corrections and to significant distortion of the Land\'e interval rule. We discuss the results of our many-body calculations in this system, and also in the analogous lithium quartet systems for which experimental data exist. We find that the hyperfine matrix elements of these few-electron systems may be adequately computed without detailed consideration of the electron correlations. Fine structure is significantly affected by such correlations. Experimental verification of the ${\mathrm{Li}}^{\ensuremath{-}}$ structure is underway.The fine and hyperfine structures of the doubly excited, quintet states of the $^{7}\mathrm{Li}^{\ensuremath{-}}$ ion are of comparable magnitude. This leads to strong second-order hyperfine corrections and to significant distortion of the Land\'e interval rule. We discuss the results of our many-body calculations in this system, and also in the analogous lithium quartet systems for which experimental data exist. We find that the hyperfine matrix elements of these few-electron systems may be adequately computed without detailed consideration of the electron correlations. Fine structure is significantly affected by such correlations. Experimental verification of the ${\mathrm{Li}}^{\ensuremath{-}}$ structure is underway.

Physical Review AAug 1, 198414 citations

Introduction

Howard R. Reiss

Most of atomic physics is concerned with the behavior of valence electrons, where energies are of the order of the square of the fine structure constant (alpha approximately 1/137) times the rest energy of the electron, mc(2). Well-known relativistic corrections, such as fine splitting of energy levels, are smaller than usual atomic transition energies by two extra powers of the fine structure constant, and are even smaller for hyperfine splitting. In valence-electron atomic physics, therefore, relativistic effects are normally treated as small corrections.Most of atomic physics is concerned with the behavior of valence electrons, where energies are of the order of the square of the fine structure constant (alpha approximately 1/137) times the rest energy of the electron, mc(2). Well-known relativistic corrections, such as fine splitting of energy levels, are smaller than usual atomic transition energies by two extra powers of the fine structure constant, and are even smaller for hyperfine splitting. In valence-electron atomic physics, therefore, relativistic effects are normally treated as small corrections.

Optics ExpressMar 30, 199810 citations

Hyperfine structure of the 2pP122 state in Be+9

J. J. Bollinger, Joseph S. Wells, David J. Wineland +1 more

An optical-optical double resonance technique has been used on beryllium ions stored in a Penning trap to measure the magnetic dipole hyperfine interaction constant ${A}_{\frac{1}{2}}$ of the $2p^{2}P_{\frac{1}{2}}$ level in $^{9}\mathrm{Be}^{+}$ . The measured value of ${A}_{\frac{1}{2}}=\ensuremath{-}118.6(3.6)$ MHz is in good agreement with theoretical calculations. The $2p^{2}P$ fine-structure splitting and the $2s^{2}S_{\frac{1}{2}}\ensuremath{\rightarrow}2p^{2}P_{\frac{3}{2},\frac{1}{2}}$ optical transition frequencies have also been measured.An optical-optical double resonance technique has been used on beryllium ions stored in a Penning trap to measure the magnetic dipole hyperfine interaction constant ${A}_{\frac{1}{2}}$ of the $2p^{2}P_{\frac{1}{2}}$ level in $^{9}\mathrm{Be}^{+}$. The measured value of ${A}_{\frac{1}{2}}=\ensuremath{-}118.6(3.6)$ MHz is in good agreement with theoretical calculations. The $2p^{2}P$ fine-structure splitting and the $2s^{2}S_{\frac{1}{2}}\ensuremath{\rightarrow}2p^{2}P_{\frac{3}{2},\frac{1}{2}}$ optical transition frequencies have also been measured.

Physical Review AApr 1, 198555 citations

Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved ( 2S−2S ) systems

Ulrich D. Jentschura, Vincent Debierre, Chandra Mani Adhikari +2 more

The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasidegenerate $2{P}_{1/2}$ levels that are removed from the $2S$ states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift, and the hyperfine effects. The van der Waals shift becomes commensurate with the $2S\ensuremath{-}2{P}_{3/2}$ fine-structure splitting only for close approach ( $R<100{a}_{0}$ , where ${a}_{0}$ is the Bohr radius) and one may thus restrict the discussion to the levels with $n=2$ and $J=1/2$ to a good approximation. Because each $S$ or $P$ state splits into an $F=1$ triplet and an $F=0$ hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64. A careful analysis of the symmetries of the the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states that are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to $1/{R}^{3}$ and $1/{R}^{6}$ and are evaluated within the hyperfine manifolds. When both atoms are metastable $2S$ states, we find an interaction energy of order ${E}_{h}\ensuremath{\chi}{({a}_{0}/R)}^{6}$, where ${E}_{h}$ and $\mathcal{L}$ are the Hartree and Lamb shift energies, respectively, and $\ensuremath{\chi}={E}_{h}/\mathcal{L}\ensuremath{\approx}6.22\ifmmode\times\else\texttimes\fi{}{10}^{6}$ is their ratio.The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasidegenerate $2{P}_{1/2}$ levels that are removed from the $2S$ states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift, and the hyperfine effects. The van der Waals shift becomes commensurate with the $2S\ensuremath{-}2{P}_{3/2}$ fine-structure splitting only for close approach ($R<100{a}_{0}$, where ${a}_{0}$ is the Bohr radius) and one may thus restrict the discussion to the levels with $n=2$ and $J=1/2$ to a good approximation. Because each $S$ or $P$ state splits into an $F=1$ triplet and an $F=0$ hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64. A careful analysis of the symmetries of the the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states that are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to $1/{R}^{3}$ and $1/{R}^{6}$ and are evaluated within the hyperfine manifolds. When both atoms are metastable $2S$ states, we find an interaction energy of order ${E}_{h}\ensuremath{\chi}{({a}_{0}/R)}^{6}$, where ${E}_{h}$ and $\mathcal{L}$ are the Hartree and Lamb shift energies, respectively, and $\ensuremath{\chi}={E}_{h}/\mathcal{L}\ensuremath{\approx}6.22\ifmmode\times\else\texttimes\fi{}{10}^{6}$ is their ratio.

arXiv:1701.08813Physical Review AFeb 2, 201717 citations

We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest

^{1,3}P_{J}, ^{3}S_{1}, ^{3}P_{J}^{e}

, and

^{1,3}D_{J}

excited states of the

^{9}\mathrm{Be}

Hyperfine structure of the

2^3S-

state in the

{}^9

Be atom

^{39}\mathrm{KCs}

and fermionic

^{40}\mathrm{KCs}

molecules. The hyperfine structure of the potential-energy curves of the states correlated to the

\mathrm{K}(4s\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{S}_{1/2})+\mathrm{Cs}(6p\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}{}^{2}{P}_{1/2,3/2})

dissociation limits is described in terms of different coupling schemes depending on the internuclear distance

R

. These results provide a step in the calculation of the hyperfine structure of rovibrational levels of these excited molecular states in the perspective of the identification of efficient paths for creating ultracold ground-state KCs molecules.

An optical-optical double resonance technique has been used on beryllium ions stored in a Penning trap to measure the magnetic dipole hyperfine interaction constant

{A}_{\frac{1}{2}}

of the

2p^{2}P_{\frac{1}{2}}

level in

^{9}\mathrm{Be}^{+}

. The measured value of ${A}_{\frac{1}{2}}=\ensuremath{-}118.6(3.6)$ MHz is in good agreement with theoretical calculations. The

2p^{2}P

fine-structure splitting and the $2s^{2}S_{\frac{1}{2}}\ensuremath{\rightarrow}2p^{2}P_{\frac{3}{2},\frac{1}{2}}$ optical transition frequencies have also been measured.

The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasidegenerate

2{P}_{1/2}

levels that are removed from the

2S

states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift, and the hyperfine effects. The van der Waals shift becomes commensurate with the $2S\ensuremath{-}2{P}_{3/2}$ fine-structure splitting only for close approach (

R<100{a}_{0}

, where

{a}_{0}

is the Bohr radius) and one may thus restrict the discussion to the levels with

n=2

and

J=1/2

to a good approximation. Because each

S

P

state splits into an

F=1

triplet and an

F=0

hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64. A careful analysis of the symmetries of the the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states that are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to

1/{R}^{3}

and

1/{R}^{6}

and are evaluated within the hyperfine manifolds. When both atoms are metastable

2S

states, we find an interaction energy of order ${E}_{h}\ensuremath{\chi}{({a}_{0}/R)}^{6}$, where

{E}_{h}

and

\mathcal{L}

are the Hartree and Lamb shift energies, respectively, and $\ensuremath{\chi}={E}_{h}/\mathcal{L}\ensuremath{\approx}6.22\ifmmode\times\else\texttimes\fi{}{10}^{6}$ is their ratio.