Portal for High-Precision Atomic Data and Computation

We use a linearized coupled-cluster method to calculate the atomic properties of atoms and ions with one valence electron (monovalent systems). This method is also referred to as all-order in the literature as it involves summing series of dominant many-body perturbation terms to all orders. In the single-double (SD) all-order approach, single and double excitations of the Dirac-Fock orbitals are included. The SDpT all-order approach also includes classes of the triple excitations. Omitted higher excitations are estimated by the scaling procedure which can be started from either SD or SDpT approximations. We carry out four all-order computations for each of the electric-dipole matrix elements, ab initio SD and SDpT and scaled SD and SDpT. Either SD or SD scaled data are taken as final values based on the comparison of different contributions to the matrix elements. An algorithm is used to determine the uncertainties of the electric-dipole matrix elements based on the spread of the four results, size of the correlation correction, and comparison of different contributions to the matrix elements.

A detailed description of the SD and SDpT all-order approaches with all formulas is given in this pdf document, edited from edited from M. S. Safronova, High-precision calculations of atomic properties and parity nonconservation in systems with one valence electron, Ph. D. Thesis, University of Notre Dame (2001).

References:
[1] All-Order Methods for Relativistic Atomic Structure Calculations (review paper), M.S. Safronova and W.R. Johnson, Advances in Atomic Molecular and Optical Physics series, volume 55, 194 (2007). Link
[2] Relativistic many-body calculation of energies, lifetimes, hyperfine constants, and polarizabilities in ⁷Li W. R. Johnson, U. I. Safronova, A. Derevianko, and M. S. Safronova, Phys. Rev. A 77, 022510 (2008). Link
[3] Relativistic many-body calculation of energies, lifetimes, polarizabilities, and hyperpolarizabilities in Li-like Be+, U. I. Safronova and M. S. Safronova, Phys. Rev. A 87, 032502 (2013). Link
[4] High-accuracy calculation of energies, lifetimes, hyperfine constants, multipole polarizabilities, and blackbody radiation shift in ³⁹K, U.I. Safronova and M. S. Safronova, Phys. Rev. A 78, 052504 (2008). Link
[5] Blackbody radiation shift, multipole polarizabilities, oscillator strengths, lifetimes, hyperfine constants, and excitation energies in Ca⁺, M.S. Safronova and U.I. Safronova, Phys. Rev. A 83, 012503 (2011). Link
[6] Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in ⁸⁷Rb, M. S. Safronova and U. I. Safronova, Phys. Rev. A 83, 052508 (2011) . Link
[7] Magic wavelengths, matrix elements, polarizabilities, and lifetimes of Cs, M. S. Safronova, U. I. Safronova, and Charles W. Clark Phys. Rev. A 94, 012505 (2016). Link
[8] Calculation of parity-nonconserving amplitude and other properties of Ra⁺, Rupsi Pal, Dansha Jiang, M.S. Safronova, and U.I. Safronova, Phys. Rev. A 79, 062505 (2009). Link

Matrix element pages contain the absolute values of the reduced electric-dipole (E1) matrix elements in atomic units, \(ea_0\), where \(e\) is elementary charge and \(a_0\) is the Bohr radius. Uncertainties are given in parenthesis. See below for the method description. Matrix elements of irreducible tensor operators \(T_q^K\) (\(K=1\) for the E1 matrix element) are evaluated using the Wigner-Eckart theorem: \[ \langle j_1,m_1|T_q^k|j_2,m_2 \rangle = (-1)^{j_1-m_1} \left\{ \begin{array}{ccc} J_1 & k & J_2\\ -m_1 & q & m_2 \end{array} \right\} \langle j_1||T_q^k||j_2 \rangle. \] The convention for reduced matrix element \( \langle j_1||T_q^k||j_2 \rangle \) used in this work [1,2,3] is related to another convention for reduced matrix elements \( ( j_1||T_q^k||j_2 )\) used in Refs. [4,5] . We note that Refs. [4,5] use another convention for reduced matrix elements that differs from out by a factor of \( \frac{1}{\sqrt{2j_1+1}}\). Experimental values are listed with references instead of the theoretical results in cases where high-precision experimental lifetimes of matrix elements are available. Click on the Ref button to see the original reference.
For convenience, we also provide the vacuum wavelength for all transitions in nm. Since most of the relevant energies for these systems have been measured, we use the energy data from the NIST ASD database [6], with a corresponding reference. Note that we use Level rather than Lines ASD data to compute the wavelength, with ASD quoted energy uncertainties. These data are only given for reference and more accurate wavelength may be available in the NIST ASD lines database. A few of the energies are not listed in ASD. In these cases, we use theoretical recommended values for the energies to compute the wavelengths. The ab initio data are adjusted based on experimental results for other levels to improve accuracy. Click on ``Energies'' to see these theoretical values. Theoretical uncertainties are given in these cases.

References:
[1] G. Racah, Phys. Rev. A 62, 438 (1942). DOI: https://doi.org/10.1103/PhysRev.62.438
[2] A. R. Edmonds., Angular Momentum in Quantum Mechanics. Princeton University Press, Princeton, New Jersey, 1974.
[3] Atomic Structure Theory: Lectures on Atomic Physics, Walter R. Johnson, Springer Verlag, Berlin (ISBN 978-3-540-68010-9). A textbook on atomic structure for students with a background in quantum mechanics.
[4] Angular Momentum, by D.M. Brink and G.R. Satchler, Oxford library of the physical sciences, Clarendon Press (1962), https://books.google.co.in/books?id=i1TNswEACAAJ
[5] Daniel A. Steck, ‘‘Rubidium 85 D Line Data,’’ available online at http://steck.us/alkalidata (revision 2.2.3, 9 July 2021).
[6] Kramida, A., Ralchenko, Yu., Reader, J. and NIST ASD Team (2020). NIST Atomic Spectra Database (version 5.8), [Online]. Available: https://physics.nist.gov/asd [Tue Dec 29 2020]. National Institute of Standards and Technology, Gaithersburg, MD. DOI: https://doi.org/10.18434/T4W30F

Transition rate pages contain transition probabilities \(A_{wv}\) (in s\(^{-1}\)) from the upper (initial) level \(w\) to lower (final) energy level \(v\). These are used to compute radiative lifetimes and corresponding branching ratios. Matrix elements and wavelengths used to compute transition rates are listed for convenience; see matrix element page info. Most of the transition probabilities are for the electric-dipole transitons, electric-quadrupole and magnetic-dipole transition rates and correpsonding matrix elements are listed for transitions from metastable states. The buttons for the metastable states are of different color for easy identification. Other multipole transition probabilities are listed for highly-charged ions.

Multipole transition probabilities \(A_{wv}\) for the electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), magnetic quadrupole (M2), and electric octupole (E3) and magnetic octupole (M3) are calculated from the wavelength and matrix elements using the formulas: \begin{eqnarray} A_{wv}(E1) &=& \frac{2.02613\times 10^{18}} {(2J_w+1)\lambda ^{3}} \, \, S(E1), \nonumber \\ A_{wv}(M1) &=& \frac {2.69735\times 10^{13}}{(2J_w+1)\lambda ^{3}} \, \, S(M1),\nonumber \\ A_{wv}(E2)& =& \frac{1.11995\times 10^{18}}{(2J_w+1)\lambda ^{5}} \, \, S(E2),\nonumber \\ A_{wv}(M2)& =& \frac{1.49097 \times 10^{13}}{(2J_w+1)\lambda ^{5}} \, \, S(M2),\nonumber \\ A_{wv}(E3)& = &\frac{3.14441 \times 10^{17}}{(2J_w+1)\lambda ^{7}} \, \, S(E3),\nonumber \\ A_{wv}(M3) &= &\frac{4.18610 \times 10^{12}}{(2J_w+1)\lambda ^{7}} \, \, S(M3),\nonumber \end{eqnarray} \(\require{mediawiki-texvc}\) where the wavelength \(\lambda\) is in \(\AA\)(= \(10^{-10}\)m) and the line strength \(S\) are determined as the squares of the corresponding reduced matrix elements. The matrix elements are in atomic units for \(E_k\) matrix elements and in \(\mu_B\) for \(M_k\) matrix elements (k=1,2,3). The uncertainties in the transition rate values are determined from the uncertainties in the matrix elements and the wavelengths.

Data format for the transition probabilities: The uncertainties are listed in parenthesis before the scientific formatting E+0X, so 5.351(32)E+06 means the value of 5.351E+06 s\(^{-1}\) which has an uncertainty of 0.032E+06 s\(^{-1}\).

The radiative lifetime of an upper state \(w\) is calculated as an inverse of the total transition probability \[ \tau_w=\frac{1}{\sum_v A_{wv}},\] where the denominator contains the sum over all possible transition rates. The lifetimes are given in ns. Note that the blackbody contribution is not included and may contribute outside of the quoted uncertainties for higher excited states. The branching ratios are calculated as a ratio of a transition rate to the total transition rate: \[ R_{wv}=\frac{A_{wv}}{\sum_v A_{wv}}.\] This quantity is dimensionless.

The electric quadrupole moments are listed for metastable states. The electric quadrupole moments \( \Theta \) of an atom in an electronic state with angular momentum \(J\) is defined as the diagonal matrix element of the \(q=0\) component of the electric quadrupole operator \(Q\) in a spherical basis [1] \[ \Theta = \langle \Psi_{JM_J} | Q|\Psi_{JM_J}\rangle,\] with the magnetic quantum number \(M_J\) taken to be equal to its maximum value, \(M_J=J\) [1]. Therefore, \[\Theta = \frac{(2J)!}{\sqrt{(2J-2)!(2J+3)!}} \langle \Psi_J || Q|| \Psi_J \rangle,\] where \( \langle \Psi_J || Q|| \Psi_J\rangle \) is a reduced matrix element of the electric quadrupole operator \( Q \). The values are given in atomic units, \( ea_0^2 \).
[1] W. M. Itano, Phys. Rev. A 73, 022510 (2006), DOI

Polarizability pages contain static polarizabilities for a large number of states and dynamic polarizabilities for the low-lying states for the case of linear polarization. The values for the monovalent systems are calculated using the sum-over-states approach described, for example, in the following paper: [Magic wavelengths, matrix elements, polarizabilities, and lifetimes of Cs, M. S. Safronova, U. I. Safronova, and Charles W. Clark Phys. Rev. A 94, 012505 (2016).] The total dynamic polarizability \(\alpha(\omega) \) is computed as \[\alpha(\omega)=\alpha_0(\omega)+\alpha_2(\omega) \, \frac{3m_j^2 -j(j+1)}{j(2j-1)}, \] where \(\alpha_0(\omega) \) and \(\alpha_2(\omega) \) are scalar and tensor dynamic polarizabilities and are written as: \begin{eqnarray} \alpha_{0}(\omega)&=&\frac{2}{3(2j+1)} \sum_{j'} \frac{(E -E')|\langle j ||{\bf D}||j' \rangle|^2} {\omega^2 - (E -E')^2}, \nonumber \end{eqnarray} and \begin{eqnarray} \alpha_{2}(\omega) &=& \sqrt{\frac{40 j (2j-1)}{3(j +1)(2j+3)(2j+1)}} \sum_{j'}(-1)^{j+j'} \nonumber \\ &\times & \left\{ \begin{array}{ccc} j & 2 & j \\ 1 & j' &1 \end{array}\right\} \frac{ (E -E')|\langle j||{\bf D}||j' \rangle|^2} {\omega^2 - (E -E')^2}, \nonumber \end{eqnarray} where \(j'\) are the angular momentum of the intermediate states, \(E\) and \(E'\) are energies of the corresponding states. For \( j=3/2 \) and \(|m_j|=1/2\), \(\alpha(\omega)=\alpha_0(\omega)-\alpha_2(\omega) \). For \( j=3/2 \) and \( |m_j|=3/2 \), \(\alpha(\omega)=\alpha_0(\omega)+\alpha_2(\omega) \). Tensor polarizability is zero for \( j=1/2 \). Static polarizability values can be obtained by substituting \(\omega=0\) in above expressions.

Note that the formulas we use assume that the frequency is at least several linewidths off resonance with the corresponding transitions. Linear polarization parallel to the quantization axis is assumed in all calculations. We use the conventional system of atomic units, a.u., in which \(e\), the electron mass \( m_e \), and the reduced Planck constant have the numerical value 1, and the electric constant \( \epsilon_0 \) has the numerical value \( 1/(4\pi) \). Dipole polarizabilities \( \alpha \) in a.u. have the dimension of volume, and their numerical values presented here are expressed in units of \(a_0^3\). The atomic units for \( \alpha \) can be converted to SI units via \(\alpha/h\, [\rm{Hz/(V/m)}^2] = 2.48832 \times 10^{-8} \alpha \, \rm{[a.u.]} \), where the conversion coefficient is \(\ 4\pi \epsilon_0 a^3_0/h \) and the Planck constant \(\ h \) is factored out.

A few of the energies for some atoms and ions are provided in cm\(^{-1}\) along with theoretical uncertainties. These are theoretical energies for the states not listed in NIST ASD. We use these theoretical recommended values for the energies to compute the wavelengths required for calculation of transition rates, lifetimes and polarizabilities.

The properties of isotopes with a half-life exceeding 10 minutes are provided. All data are taken from available nuclear data sources [1-4]. In the case of Fr, properties of isotope with a half-life exceeding 3 minutes are provided due to their use in parity-violating studies. All data are given for the nuclear ground state. The following data are provided:

1. Nuclear spins [1]
2. Natural abundances [2]
3. Half-life for unstable isotopes [1]
4. Root-mean-square (rms) radii in fermi [3]
5. Recommended values of the magnetic dipole moment. Units: \(\mu\)N (nuclear magneton) [1]
6. Recommended values of the electric quadrupole moment. Units: b (barn, 1b= 10\(^{-28}\) m\(^2\)) [4]

References:
[1] N.J. Stone, Table of recommended nuclear magnetic dipole moments: Part I, long-lived states, INDC(NDS)-079. Available online, Link
[2] Abundance for the naturally occurring isotopes is taken from Juris Meija et al., Isotopic compositions of the elements 2013 (IUPAC Technical Report), Pure Appl. Chem. 88, 293 (2016), DOI
[3] I. Angeli, K.P. Marinova, Table of experimental nuclear ground state charge radii: An update, Atomic Data and Nuclear Data Tables 99, 69 (2013), DOI.Available online: Link
[4] N.J. Stone, Table of nuclear electric quadrupole moments, At. Data Nucl. Data Tables 111-112, 1 (2016), DOI

The values of magnetic-dipole hyperfine constants A are listed in MHz. In Version 2.0 of the portal, the values are taken from published sources, distinguishing theoretical and experimental values. All references are provided with DOI links. More theoretical values will be added from new computations in the next releases of the portal, along with hyperfine constants B.

The following properties are provided as "Atomic data" for highly charged ions, where available: absolute values of the reduced matrix elements for dominant electric dipole (E1), electric quadrupole (E2), electric octupole (E3), magnetic dipole (M1) and magnetic quadrupole (M2), magnetic octupole (M3) transitions with corresponding transition rates (in s\(^{-1}\)) and transition energies, lifetimes, quadrupole moments (in \(|e|a_0^2)\), polarizability (in \(a_0^3)\), \(\alpha\)-variation sensitivity q, enhancement factor K. Electric matrix elements are given in atomic units. Magnetic matrix elements are given in Bohr magnetons, \(\mu_B\). The values are taken from published sources. All references are provided with DOI links.