Physical Review A

Hyperfine interaction in excited states of rubidium and cesium: S states

Charles F. Mulks, Taesul Lee, T. P. Das, S. D. Mahanti

Published Mar 1, 1976• 4 citations

Abstract

The one-electron contributions to the hyperfine constants for the excited $S$ states of rubidium and cesium are calculated and compared with recent experimental results. From this comparison, the expected many-body contributions to the hyperfine constants are derived and shown to systematically decrease for higher excited states, in agreement with the situation found in earlier work on sodium. The physical origin of this behavior of the correlation effect is discussed.

DOI: 10.1103/physreva.13.1271

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Relativistic many-body investigation of hyperfine interactions in excited S states of alkali metals: Francium and potassium

Alfred Owusu, R. W. Dougherty, G. Gowri +2 more

Physical Review AJul 1, 199710 citations

Investigating Role of Electron Correlation Effects via Triple Excitations for Precise Evaluation of Energies and Hyperfine Structure Constants in $^{23}$Na

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physics.atom-phCross-listedarXiv:2607.05012Jul 7, 2026

Hyperfine Interaction, Zeeman and Stark Effects for Excited States in Rubidium

G Belin, Lennarth Holmgren, Sune R. Svanberg

Physica ScriptaJun 1, 197635 citations

Hyperfine-structure measurements in the Rydberg S and P states of rubidium and cesium

John W. Farley, Pericles G. Tsekeris, R. Gupta

Physical Review AApr 1, 197728 citations

Ab initio studies of electron correlation effects in magnetic dipolar hyperfine interaction of Cs

Yong-Bo Tang, Bing-Qiong Lou, Ting-Yun Shi

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Comments on the hyperfine structure of the 4D2 state of rubidium

Ingvar Lindgren, Johannes Lindgren, Ann-Marie Pendrill

Physical Review AMay 1, 197731 citations

Physical Review A

Hyperfine interaction in excited states of rubidium and cesium: S states

Charles F. Mulks, Taesul Lee, T. P. Das, S. D. Mahanti

Published Mar 1, 1976• 4 citations

Publisher

Abstract

The one-electron contributions to the hyperfine constants for the excited

S

states of rubidium and cesium are calculated and compared with recent experimental results. From this comparison, the expected many-body contributions to the hyperfine constants are derived and shown to systematically decrease for higher excited states, in agreement with the situation found in earlier work on sodium. The physical origin of this behavior of the correlation effect is discussed.

DOI: 10.1103/physreva.13.1271

Related papers

Relativistic many-body investigation of hyperfine interactions in excited S states of alkali metals: Francium and potassium

Alfred Owusu, R. W. Dougherty, G. Gowri +2 more

To enhance the current understanding of mechanisms contributing to magnetic hyperfine interactions in excited states of atomic systems, in particular, alkali-metal atom systems, the hyperfine fields in the excited $5{}^{2}{S}_{1/2}--8{}^{2}{S}_{1/2}$ states of potassium and $8{}^{2}{S}_{1/2}--12{}^{2}{S}_{1/2}$ states of francium atoms have been studied using the relativistic linked-cluster many-body perturbation procedure. The net theoretical values of the hyperfine fields for the excited states studied are in excellent agreement with available experimental data for both atoms. There is a significant decrease in importance of the correlation contribution in going from the ground state to the excited states, the correlation contributions as ratios of the direct contribution decreasing rapidly as one moves to the higher excited states. However, the contribution from the exchange core polarization (ECP) effect is nearly a constant fraction of the direct effect for all the excited states considered. Physical explanations are offered for the observed trends in the contributions from the different mechanisms. A comparison is made of the different contributing effects to the hyperfine fields in potassium and francium to those in the related system, rubidium, studied earlier. Extrapolating from our results to the highly excited states of alkali-metal atoms, referred to as the Rydberg states, it is concluded that in addition to the direct contribution from the excited valence electron to the hyperfine fields, a significant contribution is expected from the ECP effect arising from the influence of exchange interactions between electrons in the valence and core states.To enhance the current understanding of mechanisms contributing to magnetic hyperfine interactions in excited states of atomic systems, in particular, alkali-metal atom systems, the hyperfine fields in the excited $5{}^{2}{S}_{1/2}--8{}^{2}{S}_{1/2}$ states of potassium and $8{}^{2}{S}_{1/2}--12{}^{2}{S}_{1/2}$ states of francium atoms have been studied using the relativistic linked-cluster many-body perturbation procedure. The net theoretical values of the hyperfine fields for the excited states studied are in excellent agreement with available experimental data for both atoms. There is a significant decrease in importance of the correlation contribution in going from the ground state to the excited states, the correlation contributions as ratios of the direct contribution decreasing rapidly as one moves to the higher excited states. However, the contribution from the exchange core polarization (ECP) effect is nearly a constant fraction of the direct effect for all the excited states considered. Physical explanations are offered for the observed trends in the contributions from the different mechanisms. A comparison is made of the different contributing effects to the hyperfine fields in potassium and francium to those in the related system, rubidium, studied earlier. Extrapolating from our results to the highly excited states of alkali-metal atoms, referred to as the Rydberg states, it is concluded that in addition to the direct contribution from the excited valence electron to the hyperfine fields, a significant contribution is expected from the ECP effect arising from the influence of exchange interactions between electrons in the valence and core states.

Physical Review AJul 1, 199710 citations

Investigating Role of Electron Correlation Effects via Triple Excitations for Precise Evaluation of Energies and Hyperfine Structure Constants in $^{23}$ NaInvestigating Role of Electron Correlation Effects via Triple Excitations for Precise Evaluation of Energies and Hyperfine Structure Constants in $^{23}$Na

Vaibhav Katyal, B. K. Sahoo

Accurate determination of hyperfine structure constants in atomic systems provides important insight into the interplay of electron correlation and relativistic effects in the nuclear region. Although sodium (Na) is a relatively light atom, previous all-order relativistic many-body calculations of the magnetic dipole hyperfine constants for the low-lying states of $^{23}$ Na show noticeable discrepancies with experiment. To address this, we calculate the ionization potentials and hyperfine structure constants of $^{23}$ Na using relativistic coupled-cluster theory with explicit inclusion of triple excitations. We further incorporate corrections from the Breit interaction, quantum electrodynamics, and the Bohr-Weisskopf (BW) effect. Results from lower-order methods are also presented to assess the importance of different physical contributions across states. Our calculations demonstrate that contributions from the lower-order relativistic and BW effects play almost similar roles with the electron correlation effects, including triple excitations, and are essential for reconciling theoretical predictions with experimental observations. This study can also serve as a useful guide for understanding the role of triples in heavier alkali systems.Accurate determination of hyperfine structure constants in atomic systems provides important insight into the interplay of electron correlation and relativistic effects in the nuclear region. Although sodium (Na) is a relatively light atom, previous all-order relativistic many-body calculations of the magnetic dipole hyperfine constants for the low-lying states of $^{23}$Na show noticeable discrepancies with experiment. To address this, we calculate the ionization potentials and hyperfine structure constants of $^{23}$Na using relativistic coupled-cluster theory with explicit inclusion of triple excitations. We further incorporate corrections from the Breit interaction, quantum electrodynamics, and the Bohr-Weisskopf (BW) effect. Results from lower-order methods are also presented to assess the importance of different physical contributions across states. Our calculations demonstrate that contributions from the lower-order relativistic and BW effects play almost similar roles with the electron correlation effects, including triple excitations, and are essential for reconciling theoretical predictions with experimental observations. This study can also serve as a useful guide for understanding the role of triples in heavier alkali systems.

physics.atom-phCross-listedarXiv:2607.05012Jul 7, 2026

Hyperfine Interaction, Zeeman and Stark Effects for Excited States in Rubidium

G Belin, Lennarth Holmgren, Sune R. Svanberg

Highly excitedPandDstates in87Rb were examined using a two-step excitation procedure. A conventional rf lamp and a CW dye laser were used for the first and second step, respectively. The following results for the magnetic dipole interaction constanta, the electric quadrupole interaction constantb, the LandégJfactor, and the tensor polarizability α2were obtained in level-crossing and optical-double-resonance experiments: 92P3/2:a= 4.05(3) MHz,b=0.55(3) MHz,gJ= 1.3335(15) 102P3/2:a= 2.60(8) MHz,gJ= 1.3332(20) 82D3/2:a= 2.840(15) MHz,b= 0.17(2) MHz 82D5/2:a= -1.20(15) MHz,gJ= 1.1998(15) 92D3/2:a= 1.900(10) MHz,b= 0.11(3) MHz 92D5/2:a= (-)0.80(15) MHz,gJ= 1.1995(15), α2= 180.3(9.0) MHz/(kV/cm)2 Many-body calculations of the hyperfine structure of Rb have been performed. The polarization effect is included to all orders, whereas the correlation effect is omitted. The negativeafactors for the2D5/2states, indicating large perturbations, are explained by the polarization contribution. A review of experimental hyperfine interaction constants, LandégJfactors and tensor polarizabilities for rubidium is given together with theoretical values of theafactors. The quadrupole moment for87Rb is obtained from thebfactors for the2P3/2and2D3/2states, using quadrupole parameters from the many-body calculations. The correlation effect is included in a semi-empirical way.Highly excitedPandDstates in87Rb were examined using a two-step excitation procedure. A conventional rf lamp and a CW dye laser were used for the first and second step, respectively. The following results for the magnetic dipole interaction constanta, the electric quadrupole interaction constantb, the LandégJfactor, and the tensor polarizability α2were obtained in level-crossing and optical-double-resonance experiments: 92P3/2:a= 4.05(3) MHz,b=0.55(3) MHz,gJ= 1.3335(15) 102P3/2:a= 2.60(8) MHz,gJ= 1.3332(20) 82D3/2:a= 2.840(15) MHz,b= 0.17(2) MHz 82D5/2:a= -1.20(15) MHz,gJ= 1.1998(15) 92D3/2:a= 1.900(10) MHz,b= 0.11(3) MHz 92D5/2:a= (-)0.80(15) MHz,gJ= 1.1995(15), α2= 180.3(9.0) MHz/(kV/cm)2 Many-body calculations of the hyperfine structure of Rb have been performed. The polarization effect is included to all orders, whereas the correlation effect is omitted. The negativeafactors for the2D5/2states, indicating large perturbations, are explained by the polarization contribution. A review of experimental hyperfine interaction constants, LandégJfactors and tensor polarizabilities for rubidium is given together with theoretical values of theafactors. The quadrupole moment for87Rb is obtained from thebfactors for the2P3/2and2D3/2states, using quadrupole parameters from the many-body calculations. The correlation effect is included in a semi-empirical way.

Physica ScriptaJun 1, 197635 citations

Hyperfine-structure measurements in the Rydberg S and P states of rubidium and cesium

John W. Farley, Pericles G. Tsekeris, R. Gupta

We have measured the hyperfine structure in many highly-excited $S$ and $P$ states of $^{87}\mathrm{Rb}$ and $^{133}\mathrm{Cs}$ . The results for the magnetic-dipole coupling constants are, in MHz, for $^{87}\mathrm{Rb}$ : $A(10^{2}S_{\frac{1}{2}})=56.3(2)$ , $A(11^{2}S_{\frac{1}{2}})=37.4(3)$ ; and for $^{133}\mathrm{Cs}$ : $A(9^{2}S_{\frac{1}{2}})=110.1(5)$ , $A(13^{2}S_{\frac{1}{2}})=18.4(1)$ , $A(14^{2}S_{\frac{1}{2}})=13.4(1)$ , $A(15^{2}S_{\frac{1}{2}})=10.1(1)$ , $A(16^{2}S_{\frac{1}{2}})=7.73(5)$ , $A(17^{2}S_{\frac{1}{2}})=6.06(10)$ , $A(10^{2}P_{\frac{1}{2}})=13.9(2)$ .We have measured the hyperfine structure in many highly-excited $S$ and $P$ states of $^{87}\mathrm{Rb}$ and $^{133}\mathrm{Cs}$. The results for the magnetic-dipole coupling constants are, in MHz, for $^{87}\mathrm{Rb}$: $A(10^{2}S_{\frac{1}{2}})=56.3(2)$, $A(11^{2}S_{\frac{1}{2}})=37.4(3)$; and for $^{133}\mathrm{Cs}$: $A(9^{2}S_{\frac{1}{2}})=110.1(5)$, $A(13^{2}S_{\frac{1}{2}})=18.4(1)$, $A(14^{2}S_{\frac{1}{2}})=13.4(1)$, $A(15^{2}S_{\frac{1}{2}})=10.1(1)$, $A(16^{2}S_{\frac{1}{2}})=7.73(5)$, $A(17^{2}S_{\frac{1}{2}})=6.06(10)$, $A(10^{2}P_{\frac{1}{2}})=13.9(2)$.

Physical Review AApr 1, 197728 citations

Ab initio studies of electron correlation effects in magnetic dipolar hyperfine interaction of Cs

Yong-Bo Tang, Bing-Qiong Lou, Ting-Yun Shi

Abstract The hyperfine-structure constant A of the S , P , and D states in Cs atom with the principal quantum number n ≤ 12 are calculated using the Dirac–Fock approximation, the low-order many-body perturbation theory, and relativistic coupled-cluster method in single and double approximations with and without the inclusion of nonlinear terms. The importance of electron correlation effects, especially the nonlinear corrections of the cluster operators, is demonstrated by comparing the results of various approximations with available experimental values. Moreover, the correlation trends for individual electron correlation effects involving direct, core-polarization, and pair-correlation ones in members of Rydberg series are also investigated in the framework of the coupled-cluster theory. Some interesting features are observed.Abstract The hyperfine-structure constant A of the S , P , and D states in Cs atom with the principal quantum number n ≤ 12 are calculated using the Dirac–Fock approximation, the low-order many-body perturbation theory, and relativistic coupled-cluster method in single and double approximations with and without the inclusion of nonlinear terms. The importance of electron correlation effects, especially the nonlinear corrections of the cluster operators, is demonstrated by comparing the results of various approximations with available experimental values. Moreover, the correlation trends for individual electron correlation effects involving direct, core-polarization, and pair-correlation ones in members of Rydberg series are also investigated in the framework of the coupled-cluster theory. Some interesting features are observed.

Journal of Physics B Atomic Molecular and Optical PhysicsJan 23, 201916 citations

Comments on the hyperfine structure of the 4D2 state of rubidium

Ingvar Lindgren, Johannes Lindgren, Ann-Marie Pendrill

In a recent paper Lee, Rodgers, Das, and Sternheimer have reported on the evaluation of the fine and hyperfine structure of the $4^{2}D$ state of the rubidium atom by considering the first-order polarization of the electron core. By means of more extensive calculations we have shown that the first-order results can be quite unreliable for such strongly perturbed systems. By including polarization effects to all orders and the most important correlation effects, we have been able to reproduce the experimental results for this state quite well. The good agreement obtained by Lee et al. for the spin-dipole parameter is in our opinion largely fortuitous.In a recent paper Lee, Rodgers, Das, and Sternheimer have reported on the evaluation of the fine and hyperfine structure of the $4^{2}D$ state of the rubidium atom by considering the first-order polarization of the electron core. By means of more extensive calculations we have shown that the first-order results can be quite unreliable for such strongly perturbed systems. By including polarization effects to all orders and the most important correlation effects, we have been able to reproduce the experimental results for this state quite well. The good agreement obtained by Lee et al. for the spin-dipole parameter is in our opinion largely fortuitous.

Physical Review AMay 1, 197731 citations

The one-electron contributions to the hyperfine constants for the excited

S

5{}^{2}{S}_{1/2}--8{}^{2}{S}_{1/2}

states of potassium and

8{}^{2}{S}_{1/2}--12{}^{2}{S}_{1/2}

states of francium atoms have been studied using the relativistic linked-cluster many-body perturbation procedure. The net theoretical values of the hyperfine fields for the excited states studied are in excellent agreement with available experimental data for both atoms. There is a significant decrease in importance of the correlation contribution in going from the ground state to the excited states, the correlation contributions as ratios of the direct contribution decreasing rapidly as one moves to the higher excited states. However, the contribution from the exchange core polarization (ECP) effect is nearly a constant fraction of the direct effect for all the excited states considered. Physical explanations are offered for the observed trends in the contributions from the different mechanisms. A comparison is made of the different contributing effects to the hyperfine fields in potassium and francium to those in the related system, rubidium, studied earlier. Extrapolating from our results to the highly excited states of alkali-metal atoms, referred to as the Rydberg states, it is concluded that in addition to the direct contribution from the excited valence electron to the hyperfine fields, a significant contribution is expected from the ECP effect arising from the influence of exchange interactions between electrons in the valence and core states.

Investigating Role of Electron Correlation Effects via Triple Excitations for Precise Evaluation of Energies and Hyperfine Structure Constants in

^{23}

^{23}

Na show noticeable discrepancies with experiment. To address this, we calculate the ionization potentials and hyperfine structure constants of

^{23}

Na using relativistic coupled-cluster theory with explicit inclusion of triple excitations. We further incorporate corrections from the Breit interaction, quantum electrodynamics, and the Bohr-Weisskopf (BW) effect. Results from lower-order methods are also presented to assess the importance of different physical contributions across states. Our calculations demonstrate that contributions from the lower-order relativistic and BW effects play almost similar roles with the electron correlation effects, including triple excitations, and are essential for reconciling theoretical predictions with experimental observations. This study can also serve as a useful guide for understanding the role of triples in heavier alkali systems.

We have measured the hyperfine structure in many highly-excited

S

and

P

states of

^{87}\mathrm{Rb}

and

^{133}\mathrm{Cs}

. The results for the magnetic-dipole coupling constants are, in MHz, for

^{87}\mathrm{Rb}

A(10^{2}S_{\frac{1}{2}})=56.3(2)

A(11^{2}S_{\frac{1}{2}})=37.4(3)

; and for

^{133}\mathrm{Cs}

A(9^{2}S_{\frac{1}{2}})=110.1(5)

A(13^{2}S_{\frac{1}{2}})=18.4(1)

A(14^{2}S_{\frac{1}{2}})=13.4(1)

A(15^{2}S_{\frac{1}{2}})=10.1(1)

A(16^{2}S_{\frac{1}{2}})=7.73(5)

A(17^{2}S_{\frac{1}{2}})=6.06(10)

A(10^{2}P_{\frac{1}{2}})=13.9(2)

In a recent paper Lee, Rodgers, Das, and Sternheimer have reported on the evaluation of the fine and hyperfine structure of the

4^{2}D

state of the rubidium atom by considering the first-order polarization of the electron core. By means of more extensive calculations we have shown that the first-order results can be quite unreliable for such strongly perturbed systems. By including polarization effects to all orders and the most important correlation effects, we have been able to reproduce the experimental results for this state quite well. The good agreement obtained by Lee et al. for the spin-dipole parameter is in our opinion largely fortuitous.