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Bulletin of the American Physical Society

Indirect spin-orbit K splittings in strontium

Jirakan Nunkaew, E. S. Shuman, Thomas F. Gallagher

Published May 21, 2009• 1 citation
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Abstract

We use delayed field ionization to observe the microwave resonance transitions of Sr from $5snf$ to $5sng$, $5snh$, and $5sni$. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}g3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the $K$ splittings. From the $K$ splittings we extract the ionic dipole and quadrupole matrix elements, $⟨5s|r|5p⟩=3.65(25){a}_{0}$ and $⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}$. From the dipole matrix element we obtain the dipole polarizability of ${\text{Sr}}^{+}$. With the dipole polarizability and the quantum defect differences of adjacent $l$ states, we can determine the ${\text{Sr}}^{+}$ quadrupole polarizability. The dipole and quadrupole polarizabilities of ${\text{Sr}}^{+}$ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.

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Indirect spin-orbit K splittings in strontium

Jirakan Nunkaew, E. S. Shuman, Thomas F. Gallagher

We use delayed field ionization to observe the microwave resonance transitions of Sr from $5snf$ to $5sng$, $5snh$, and $5sni$. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}>3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the $K$ splittings. From the $K$ splittings we extract the ionic dipole and quadrupole matrix elements, $⟨5s|r|5p⟩=3.65(25){a}_{0}$ and $⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}$. From the dipole matrix element we obtain the dipole polarizability of ${\text{Sr}}^{+}$. With the dipole polarizability and the quantum defect differences of adjacent $l$ states, we can determine the ${\text{Sr}}^{+}$ quadrupole polarizability. The dipole and quadrupole polarizabilities of ${\text{Sr}}^{+}$ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.

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Studies of high-$n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160$, strontium $5sng, 5snh$, and $5sni$ Rydberg states are reported. Measurements using microwave spectroscopy provide the quantum defects for these states and furnish a valuable test of quantum defects reported for lower-$\ensuremath{\ell}$ states in the high-$n$ limit. With increasing $\ensuremath{\ell}$ the relativistic spin-orbit interaction becomes comparable to the electron-electron interaction resulting in the breakdown of the standard $LS$-coupling scheme. Heavier alkaline-earth quasi-two-electron systems differ significantly from a true two-electron system such as helium because indirect spin-orbit effects are dramatically enhanced. We explore the high-$n$ limit of this transition between $LS$ and $jj$ coupling. For high-$\ensuremath{\ell}$ states the quantum defects are determined predominantly by the polarizability of the inner electrons.

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Indirect spin-orbit interaction in high- L Rydberg states with 2S1/2 cores

Erica L. Snow, R. A. Komara, M. A. Gearba +1 more

In high-$L$ Rydberg states, where exchange interactions are of negligible size, spin-orbit interactions are expected to determine the energy differences between states of different spin orientations. This is well studied in the case of helium, where it results in a fourfold splitting of each high-$L$ term, coming from the two electrons' spin-orbit interactions. A similar structure might be expected in heavier Rydberg states, whose core ion is a ${}^{2}{S}_{1/2}$ state. However, observations in barium Rydberg states and recent observations in Rydberg states of ${\mathrm{Si}}^{2+}$ reveal significant deviations from this expectation. Both can be explained as an indirect effect of spin-orbit interactions in virtual excitations of the core ion. A perturbative treatment predicts an indirect spin-orbit interaction for the Rydberg state proportional to the inverse sixth power of the Rydberg electron's radial coordinate. The significance of this term, relative to the direct magnetic interactions, varies by more than six orders of magnitude between the three systems studied, helium, ${\mathrm{Si}}^{2+},$ and barium.

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Determination of dipole and quadrupole polarizabilities of Ba+ by measurement of the fine structure of high- L n=9 and 10 Rydberg states of barium

Erica L. Snow, M. A. Gearba, R. A. Komara +2 more

The fine structure of high-angular-momentum $n=9$ and 10 Rydberg states of barium has been measured precisely, using the resonant excitation Stark ionization spectroscopy method. Optical transitions corresponding to $(n,{n}^{\ensuremath{'}})=(10,30)$, (9,17), and (9,20) were induced with a Doppler-tuned ${\mathrm{CO}}_{2}$ laser, determining the fine-structure energies corresponding to all $n=9$ and 10 levels with $L\ensuremath{\geqslant}6$. The pattern of these fine-structure energies conforms closely with an effective potential model, by comparison with which the dipole and quadrupole polarizabilities of ${\mathrm{Ba}}^{+}$ can be determined. Combining our data with earlier measurements made it possible to deduce, in addition, the portion of ${\ensuremath{\alpha}}_{2}$ due to the lowest excited $D$ state of ${\mathrm{Ba}}^{+}$. Our best estimates of these three properties are ${\ensuremath{\alpha}}_{1}=124.30(16){a}_{0}^{3}$, ${\ensuremath{\alpha}}_{2}=2462(361){a}_{0}^{5}$, and ${\ensuremath{\alpha}}_{2}^{0}=1828(88){a}_{0}^{5}$.

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Relativistic calculations of C6 and C8 coefficients for strontium dimers

S. G. Porsev, Marianna S. Safronova, Charles W. Clark

The electric-dipole and quadrupole polarizabilities of the $5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}$ state and the ${C}_{6}$ and ${C}_{8}$ coefficients for the ${}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ and ${}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}$ dimers of strontium are calculated using a high-precision relativistic approach that combines configuration interaction and linearized coupled-cluster methods. Our recommended values of the long-range dispersion coefficients for the ${0}_{u}$ and ${1}_{u}$ energy levels are ${C}_{6}({0}_{u})=3771(32)$ a.u. and ${C}_{6}({1}_{u})=4001(33)$ a.u., respectively. They are in good agreement with recent results from experimental photoassociation data. We also calculate ${C}_{8}$ coefficients for Sr dimers, which are needed for precise determination of long-range interaction potential. We confirm the experimental value for the magic wavelength, where the Stark shift on the ${}^{1}{S}_{0}\text{\ensuremath{-}}{}^{3}{P}_{1}^{o}$ transition vanishes. The accuracy of calculations is analyzed and uncertainties are assigned to all quantities reported in this work.

arXiv:1409.5985•Physical Review A•Nov 18, 2014•13 citations

Effective core polarizabilities in Ba high-l Nsnldouble Rydberg atoms

Pierre Camus, S. Cohen

We present experimental energy-level positions for Ba high-l Nsnl double Rydberg states with N=8 (n=13--15, l\ensuremath{\geqslant}7) and N=9 (n=13--15, l\ensuremath{\geqslant}7). These data as well as previously reported measurements for N=7 (n=12--15, l\ensuremath{\geqslant}5) and N=10 (n=13--15, l\ensuremath{\geqslant}10) are parametrized in order to yield values for the ${\mathrm{Ba}}^{+}$ Ns effective dipole ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ and quadrupole ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ polarizabilities. The obtained ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ values are 1.25(5)\ifmmode\times\else\texttimes\fi{}${10}^{3}$, 7.05(4)\ifmmode\times\else\texttimes\fi{}${10}^{3}$, 2.95(4)\ifmmode\times\else\texttimes\fi{}${10}^{4}$, 9.39(15)\ifmmode\times\else\texttimes\fi{}${10}^{4}$ in ${\mathrm{a}}_{0}^{3}$ units and the ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ values are 1.7(8)\ifmmode\times\else\texttimes\fi{}${10}^{4}$, 1.8(2)\ifmmode\times\else\texttimes\fi{}${10}^{5}$, -1.0(2)\ifmmode\times\else\texttimes\fi{}${10}^{6}$, -1.0(2)\ifmmode\times\else\texttimes\fi{}${10}^{7}$ in ${\mathrm{a}}_{0}^{5}$ units for N=7, 8, 9, and 10, respectively. The ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ evolution with N is found to follow a ${\mathrm{N}}^{\mathrm{*}6.9(1)}$ power law, while calculated static polarizabilities present somewhat lower power dependence. From this small difference it is not possible to appreciate the importance of nonadiabatic effects between the two asymmetrically excited electrons. On the other hand, the larger discrepancies appearing between the calculated static and fitted quadrupole polarizabilities show clear evidence of nonadiabatic effects. Particularly the negative ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ sign for N\ensuremath{\geqslant}9 is attributed to the increasing strength of dynamical effects on the Ns core electron caused by the motion of the outer high-l electron. It gives evidence of the increasing importance of nonadiabatic radial correlation between two planetary electrons when the Ns inner electron is sufficiently radially extended to be characterized as an inner Rydberg electron without, nevertheless, any appreciable overlapping with the outer electron.

Physical Review A•Mar 1, 1995•10 citations
We use delayed field ionization to observe the microwave resonance transitions of Sr from 5snf5snf5snf to 5sng5sng5sng, 5snh5snh5snh, and 5sni5sni5sni. The energies of the 5sn\ensuremathℓ5sn\ensuremath{\ell}5sn\ensuremathℓ, \ensuremathℓg3\ensuremath{\ell}g3\ensuremathℓg3 states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the KKK splittings. From the KKK splittings we extract the ionic dipole and quadrupole matrix elements, ⟨5s∣r∣5p⟩=3.65(25)a0⟨5s|r|5p⟩=3.65(25){a}_{0}⟨5s∣r∣5p⟩=3.65(25)a0​ and ⟨5s∣r2∣4d⟩=12(14)a02⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}⟨5s∣r2∣4d⟩=12(14)a02​. From the dipole matrix element we obtain the dipole polarizability of Sr+{\text{Sr}}^{+}Sr+. With the dipole polarizability and the quantum defect differences of adjacent lll states, we can determine the Sr+{\text{Sr}}^{+}Sr+ quadrupole polarizability. The dipole and quadrupole polarizabilities of Sr+{\text{Sr}}^{+}Sr+ are \ensuremathαd=86(11)a03{\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}\ensuremathαd​=86(11)a03​ and \ensuremathαq=1.1\ifmmode×\else\texttimes\fi103(10)a05{\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}\ensuremathαq​=1.1\ifmmode×\else\texttimes\fi103(10)a05​, respectively.
We use delayed field ionization to observe the microwave resonance transitions of Sr from 5snf5snf5snf to 5sng5sng5sng, 5snh5snh5snh, and 5sni5sni5sni. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}>3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the KKK splittings. From the KKK splittings we extract the ionic dipole and quadrupole matrix elements, ⟨5s∣r∣5p⟩=3.65(25)a0⟨5s|r|5p⟩=3.65(25){a}_{0}⟨5s∣r∣5p⟩=3.65(25)a0​ and ⟨5s∣r2∣4d⟩=12(14)a02⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}⟨5s∣r2∣4d⟩=12(14)a02​. From the dipole matrix element we obtain the dipole polarizability of Sr+{\text{Sr}}^{+}Sr+. With the dipole polarizability and the quantum defect differences of adjacent lll states, we can determine the Sr+{\text{Sr}}^{+}Sr+ quadrupole polarizability. The dipole and quadrupole polarizabilities of Sr+{\text{Sr}}^{+}Sr+ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.
Studies of high-$n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160$, strontium 5sng,5snh5sng, 5snh5sng,5snh, and 5sni5sni5sni Rydberg states are reported. Measurements using microwave spectroscopy provide the quantum defects for these states and furnish a valuable test of quantum defects reported for lower-$\ensuremath{\ell}$ states in the high-nnn limit. With increasing $\ensuremath{\ell}$ the relativistic spin-orbit interaction becomes comparable to the electron-electron interaction resulting in the breakdown of the standard LSLSLS-coupling scheme. Heavier alkaline-earth quasi-two-electron systems differ significantly from a true two-electron system such as helium because indirect spin-orbit effects are dramatically enhanced. We explore the high-nnn limit of this transition between LSLSLS and jjjjjj coupling. For high-$\ensuremath{\ell}$ states the quantum defects are determined predominantly by the polarizability of the inner electrons.
In high-LLL Rydberg states, where exchange interactions are of negligible size, spin-orbit interactions are expected to determine the energy differences between states of different spin orientations. This is well studied in the case of helium, where it results in a fourfold splitting of each high-LLL term, coming from the two electrons' spin-orbit interactions. A similar structure might be expected in heavier Rydberg states, whose core ion is a 2S1/2{}^{2}{S}_{1/2}2S1/2​ state. However, observations in barium Rydberg states and recent observations in Rydberg states of Si2+{\mathrm{Si}}^{2+}Si2+ reveal significant deviations from this expectation. Both can be explained as an indirect effect of spin-orbit interactions in virtual excitations of the core ion. A perturbative treatment predicts an indirect spin-orbit interaction for the Rydberg state proportional to the inverse sixth power of the Rydberg electron's radial coordinate. The significance of this term, relative to the direct magnetic interactions, varies by more than six orders of magnitude between the three systems studied, helium, Si2+,{\mathrm{Si}}^{2+},Si2+, and barium.
The fine structure of high-angular-momentum n=9n=9n=9 and 10 Rydberg states of barium has been measured precisely, using the resonant excitation Stark ionization spectroscopy method. Optical transitions corresponding to $(n,{n}^{\ensuremath{'}})=(10,30)$, (9,17), and (9,20) were induced with a Doppler-tuned CO2{\mathrm{CO}}_{2}CO2​ laser, determining the fine-structure energies corresponding to all n=9n=9n=9 and 10 levels with $L\ensuremath{\geqslant}6$. The pattern of these fine-structure energies conforms closely with an effective potential model, by comparison with which the dipole and quadrupole polarizabilities of Ba+{\mathrm{Ba}}^{+}Ba+ can be determined. Combining our data with earlier measurements made it possible to deduce, in addition, the portion of ${\ensuremath{\alpha}}_{2}$ due to the lowest excited DDD state of Ba+{\mathrm{Ba}}^{+}Ba+. Our best estimates of these three properties are ${\ensuremath{\alpha}}_{1}=124.30(16){a}_{0}^{3}$, ${\ensuremath{\alpha}}_{2}=2462(361){a}_{0}^{5}$, and ${\ensuremath{\alpha}}_{2}^{0}=1828(88){a}_{0}^{5}$.
The electric-dipole and quadrupole polarizabilities of the 5s5p3P1o5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}5s5p3P1o​ state and the C6{C}_{6}C6​ and C8{C}_{8}C8​ coefficients for the 1S0+1S0{}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}1S0​+1S0​ and 1S0+3P1o{}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}1S0​+3P1o​ dimers of strontium are calculated using a high-precision relativistic approach that combines configuration interaction and linearized coupled-cluster methods. Our recommended values of the long-range dispersion coefficients for the 0u{0}_{u}0u​ and 1u{1}_{u}1u​ energy levels are C6(0u)=3771(32){C}_{6}({0}_{u})=3771(32)C6​(0u​)=3771(32) a.u. and C6(1u)=4001(33){C}_{6}({1}_{u})=4001(33)C6​(1u​)=4001(33) a.u., respectively. They are in good agreement with recent results from experimental photoassociation data. We also calculate C8{C}_{8}C8​ coefficients for Sr dimers, which are needed for precise determination of long-range interaction potential. We confirm the experimental value for the magic wavelength, where the Stark shift on the ${}^{1}{S}_{0}\text{\ensuremath{-}}{}^{3}{P}_{1}^{o}$ transition vanishes. The accuracy of calculations is analyzed and uncertainties are assigned to all quantities reported in this work.
We present experimental energy-level positions for Ba high-l Nsnl double Rydberg states with N=8 (n=13--15, l\ensuremath{\geqslant}7) and N=9 (n=13--15, l\ensuremath{\geqslant}7). These data as well as previously reported measurements for N=7 (n=12--15, l\ensuremath{\geqslant}5) and N=10 (n=13--15, l\ensuremath{\geqslant}10) are parametrized in order to yield values for the Ba+{\mathrm{Ba}}^{+}Ba+ Ns effective dipole ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ and quadrupole ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ polarizabilities. The obtained ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ values are 1.25(5)\ifmmode\times\else\texttimes\fi{}103{10}^{3}103, 7.05(4)\ifmmode\times\else\texttimes\fi{}103{10}^{3}103, 2.95(4)\ifmmode\times\else\texttimes\fi{}104{10}^{4}104, 9.39(15)\ifmmode\times\else\texttimes\fi{}104{10}^{4}104 in a03{\mathrm{a}}_{0}^{3}a03​ units and the ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ values are 1.7(8)\ifmmode\times\else\texttimes\fi{}104{10}^{4}104, 1.8(2)\ifmmode\times\else\texttimes\fi{}105{10}^{5}105, -1.0(2)\ifmmode\times\else\texttimes\fi{}106{10}^{6}106, -1.0(2)\ifmmode\times\else\texttimes\fi{}107{10}^{7}107 in a05{\mathrm{a}}_{0}^{5}a05​ units for N=7, 8, 9, and 10, respectively. The ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{d}}^{\ensuremath{'}}$ evolution with N is found to follow a N∗6.9(1){\mathrm{N}}^{\mathrm{*}6.9(1)}N∗6.9(1) power law, while calculated static polarizabilities present somewhat lower power dependence. From this small difference it is not possible to appreciate the importance of nonadiabatic effects between the two asymmetrically excited electrons. On the other hand, the larger discrepancies appearing between the calculated static and fitted quadrupole polarizabilities show clear evidence of nonadiabatic effects. Particularly the negative ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{q}}^{\ensuremath{'}}$ sign for N\ensuremath{\geqslant}9 is attributed to the increasing strength of dynamical effects on the Ns core electron caused by the motion of the outer high-l electron. It gives evidence of the increasing importance of nonadiabatic radial correlation between two planetary electrons when the Ns inner electron is sufficiently radially extended to be characterized as an inner Rydberg electron without, nevertheless, any appreciable overlapping with the outer electron.