Physical Review A

Metastable 4Fo state of Sr− bound by core polarization

H. W. van der Hart

Published Apr 1, 1998• 2 citations

Abstract

Energies have been obtained in $\mathrm{LS}$ coupling for low-lying states in ${\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}$ using a model potential approach including core polarization. An indication has been obtained for the existence of a new quasibound state in ${\mathrm{Sr}}^{\mathrm{\ensuremath{-}}},$ $5s4d5p{}^{4}{F}^{o},$ bound by 30.0 meV with respect to the $5s5p{}^{3}{P}^{o}$ state. Core polarization is essential for obtaining binding for this state. The quasibound ${5s5p}^{2}{}^{4}{P}^{e}$ state is estimated to lie 548.4 meV below the $5s5p{}^{3}{P}^{o}$ state of Sr in good agreement with experiment.

DOI: 10.1103/physreva.57.2502

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Physical Review A

Metastable 4Fo state of Sr− bound by core polarization

H. W. van der Hart

Published Apr 1, 1998• 2 citations

Publisher

Abstract

Energies have been obtained in

\mathrm{LS}

coupling for low-lying states in

{\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}

using a model potential approach including core polarization. An indication has been obtained for the existence of a new quasibound state in

{\mathrm{Sr}}^{\mathrm{\ensuremath{-}}},

5s4d5p{}^{4}{F}^{o},

bound by 30.0 meV with respect to the

5s5p{}^{3}{P}^{o}

state. Core polarization is essential for obtaining binding for this state. The quasibound

{5s5p}^{2}{}^{4}{P}^{e}

state is estimated to lie 548.4 meV below the

5s5p{}^{3}{P}^{o}

state of Sr in good agreement with experiment.

DOI: 10.1103/physreva.57.2502

Related papers

Energies of metastable 4So states in the alkaline-earth sequence

H. W. van der Hart

Theoretical binding energies have been obtained for the metastable ${\mathrm{np}}^{3}{}^{4}{S}^{o}$ states in the alkaline-earth sequence using a B-spline configuration-interaction approach. The results agree with available experimental and theoretical data within 5 meV for ${\mathrm{Be}}^{\ensuremath{-}}$ and within 10 meV for ${\mathrm{Mg}}^{\ensuremath{-}}.$ The binding energies of the ${p}^{3}{}^{4}{S}^{o}$ state in ${\mathrm{Mg}}^{\ensuremath{-}},$ ${\mathrm{Ca}}^{\ensuremath{-}},$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ are larger than the one in ${\mathrm{Be}}^{\ensuremath{-}}$ by roughly 250 meV due to the stronger interactions with the ${d}^{2}p$ configurations. For ${\mathrm{Ca}}^{\ensuremath{-}},$ the ${}^{4}{S}^{o}$ binding energy is determined to be 555.5 meV, including a contribution of $\ensuremath{-}70$ meV due to dielectronic core polarization. For ${\mathrm{Sr}}^{\ensuremath{-}}$ the ${}^{4}{S}^{o}$ binding energy is 557.9 meV, including a decrease of only 6 meV due to dielectronic core polarization. This behavior for the ${}^{4}{S}^{o}$ states in ${\mathrm{Ca}}^{\ensuremath{-}}$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ is ascribed to the different neutral-atom states to which the ${}^{4}{S}^{o}$ states are bound.Theoretical binding energies have been obtained for the metastable ${\mathrm{np}}^{3}{}^{4}{S}^{o}$ states in the alkaline-earth sequence using a B-spline configuration-interaction approach. The results agree with available experimental and theoretical data within 5 meV for ${\mathrm{Be}}^{\ensuremath{-}}$ and within 10 meV for ${\mathrm{Mg}}^{\ensuremath{-}}.$ The binding energies of the ${p}^{3}{}^{4}{S}^{o}$ state in ${\mathrm{Mg}}^{\ensuremath{-}},$ ${\mathrm{Ca}}^{\ensuremath{-}},$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ are larger than the one in ${\mathrm{Be}}^{\ensuremath{-}}$ by roughly 250 meV due to the stronger interactions with the ${d}^{2}p$ configurations. For ${\mathrm{Ca}}^{\ensuremath{-}},$ the ${}^{4}{S}^{o}$ binding energy is determined to be 555.5 meV, including a contribution of $\ensuremath{-}70$ meV due to dielectronic core polarization. For ${\mathrm{Sr}}^{\ensuremath{-}}$ the ${}^{4}{S}^{o}$ binding energy is 557.9 meV, including a decrease of only 6 meV due to dielectronic core polarization. This behavior for the ${}^{4}{S}^{o}$ states in ${\mathrm{Ca}}^{\ensuremath{-}}$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ is ascribed to the different neutral-atom states to which the ${}^{4}{S}^{o}$ states are bound.

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Relativistic semiempirical-core-potential calculations of Sr+ using Laguerre and Slater spinors

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A relativistic description of the structure of heavy alkali-metal atoms and alkali-like ions using S-spinors and L-spinors is developed. The core wave function is defined by a Dirac-Fock calculation using an S-spinor basis. The S-spinor basis is then supplemented with a large set of L-spinors for calculation of the valence wave function in a frozen-core model. The numerical stability of the L-spinor approach is demonstrated by computing the energies and decay rates of several low-lying hydrogen eigenstates, along with the polarizabilities of a $Z=60$ hydrogenic ion. The approach is then applied to calculate the dynamic polarizabilities of the $5s, 4d$ , and $5p$ states of ${\mathrm{Sr}}^{+}$ . The magic wavelengths at which the Stark shifts between different pairs of transitions are 0 are computed. Determination of the magic wavelengths for the $5s\ensuremath{\rightarrow}4{d}_{\frac{3}{2}}$ and $5s\ensuremath{\rightarrow}4{d}_{\frac{5}{2}}$ transitions near 417 nm (near the wavelength for the $5s\ensuremath{\rightarrow}5{p}_{j}$ transitions) would allow determination of the oscillator strength ratio for the $5s\ensuremath{\rightarrow}5{p}_{\frac{1}{2}}$ and $5s\ensuremath{\rightarrow}5{p}_{\frac{3}{2}}$ transitions.A relativistic description of the structure of heavy alkali-metal atoms and alkali-like ions using S-spinors and L-spinors is developed. The core wave function is defined by a Dirac-Fock calculation using an S-spinor basis. The S-spinor basis is then supplemented with a large set of L-spinors for calculation of the valence wave function in a frozen-core model. The numerical stability of the L-spinor approach is demonstrated by computing the energies and decay rates of several low-lying hydrogen eigenstates, along with the polarizabilities of a $Z=60$ hydrogenic ion. The approach is then applied to calculate the dynamic polarizabilities of the $5s, 4d$, and $5p$ states of ${\mathrm{Sr}}^{+}$. The magic wavelengths at which the Stark shifts between different pairs of transitions are 0 are computed. Determination of the magic wavelengths for the $5s\ensuremath{\rightarrow}4{d}_{\frac{3}{2}}$ and $5s\ensuremath{\rightarrow}4{d}_{\frac{5}{2}}$ transitions near 417 nm (near the wavelength for the $5s\ensuremath{\rightarrow}5{p}_{j}$ transitions) would allow determination of the oscillator strength ratio for the $5s\ensuremath{\rightarrow}5{p}_{\frac{1}{2}}$ and $5s\ensuremath{\rightarrow}5{p}_{\frac{3}{2}}$ transitions.

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Renormalization of the singular attractive 1∕r4 potential

Mary Alberg, Michel Bawin, Fabian Brau

We study the radial Schr\"odinger equation for a particle of mass $m$ in the field of a singular attractive ${g}^{2}∕{r}^{4}$ potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. [Phys. Rev. A 64, 042103 (2001)], we solve analytically the corresponding ``renormalization-group flow'' equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of $s$ -wave polarization bound electrons in the field of ${\mathrm{C}}_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius $(\ensuremath{\sim}3.37\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.We study the radial Schr\"odinger equation for a particle of mass $m$ in the field of a singular attractive ${g}^{2}∕{r}^{4}$ potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. [Phys. Rev. A 64, 042103 (2001)], we solve analytically the corresponding ``renormalization-group flow'' equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of $s$-wave polarization bound electrons in the field of ${\mathrm{C}}_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius $(\ensuremath{\sim}3.37\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.

arXiv:quant-ph/0412112Physical Review AFeb 17, 20059 citations

Energies have been obtained in

\mathrm{LS}

coupling for low-lying states in

{\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}

using a model potential approach including core polarization. An indication has been obtained for the existence of a new quasibound state in

{\mathrm{Sr}}^{\mathrm{\ensuremath{-}}},

5s4d5p{}^{4}{F}^{o},

bound by 30.0 meV with respect to the

5s5p{}^{3}{P}^{o}

state. Core polarization is essential for obtaining binding for this state. The quasibound

{5s5p}^{2}{}^{4}{P}^{e}

state is estimated to lie 548.4 meV below the

5s5p{}^{3}{P}^{o}

state of Sr in good agreement with experiment.

Theoretical binding energies have been obtained for the metastable

{\mathrm{np}}^{3}{}^{4}{S}^{o}

states in the alkaline-earth sequence using a B-spline configuration-interaction approach. The results agree with available experimental and theoretical data within 5 meV for ${\mathrm{Be}}^{\ensuremath{-}}$ and within 10 meV for ${\mathrm{Mg}}^{\ensuremath{-}}.$ The binding energies of the

{p}^{3}{}^{4}{S}^{o}

state in ${\mathrm{Mg}}^{\ensuremath{-}},$ ${\mathrm{Ca}}^{\ensuremath{-}},$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ are larger than the one in ${\mathrm{Be}}^{\ensuremath{-}}$ by roughly 250 meV due to the stronger interactions with the

{d}^{2}p

configurations. For ${\mathrm{Ca}}^{\ensuremath{-}},$ the

{}^{4}{S}^{o}

binding energy is determined to be 555.5 meV, including a contribution of $\ensuremath{-}70$ meV due to dielectronic core polarization. For ${\mathrm{Sr}}^{\ensuremath{-}}$ the

{}^{4}{S}^{o}

binding energy is 557.9 meV, including a decrease of only 6 meV due to dielectronic core polarization. This behavior for the

{}^{4}{S}^{o}

states in ${\mathrm{Ca}}^{\ensuremath{-}}$ and ${\mathrm{Sr}}^{\ensuremath{-}}$ is ascribed to the different neutral-atom states to which the

{}^{4}{S}^{o}

states are bound.

{3p}^{5}4s{}^{3}{P}_{2}^{o}

state of neutral argon. Our calculations, based on many-body perturbation theory, yield a binding energy of 24.9 meV. As a test of the present computational method, we also include the results of a comparative study of ${\mathrm{Ca}}^{\ensuremath{-}}$.

The binding energies of the

{4s}^{2}4p^{2}P

ground state in ${\mathrm{Ca}}^{\ensuremath{-}}$ have been determined by means of state-selective photoionization of Ca atoms produced in s-wave photodetachment to be $24.55\ifmmode\pm\else\textpm\fi{}0.10$ meV for the

J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2

and $19.73\ifmmode\pm\else\textpm\fi{}0.10$ meV for the

J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2

component. The sensitivity of the detection technique applied has also made it possible to disprove previous claims for the observation of a metastable ${4s4p}^{2}^{4}P$ state in ${\mathrm{Ca}}^{\ensuremath{-}}$.

Z=60

hydrogenic ion. The approach is then applied to calculate the dynamic polarizabilities of the

5s, 4d

, and

5p

states of

{\mathrm{Sr}}^{+}

. The magic wavelengths at which the Stark shifts between different pairs of transitions are 0 are computed. Determination of the magic wavelengths for the $5s\ensuremath{\rightarrow}4{d}_{\frac{3}{2}}$ and $5s\ensuremath{\rightarrow}4{d}_{\frac{5}{2}}$ transitions near 417 nm (near the wavelength for the $5s\ensuremath{\rightarrow}5{p}_{j}$ transitions) would allow determination of the oscillator strength ratio for the $5s\ensuremath{\rightarrow}5{p}_{\frac{1}{2}}$ and $5s\ensuremath{\rightarrow}5{p}_{\frac{3}{2}}$ transitions.

We study the radial Schr\"odinger equation for a particle of mass

m

in the field of a singular attractive

{g}^{2}∕{r}^{4}

potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. [Phys. Rev. A 64, 042103 (2001)], we solve analytically the corresponding ``renormalization-group flow'' equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of

s

-wave polarization bound electrons in the field of

{\mathrm{C}}_{60}

molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius $(\ensuremath{\sim}3.37\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.