Excitation energies, polarizabilities, multipole transition rates, and lifetimes of ions along the francium isoelectronic sequence
Relativistic many-body perturbation theory is applied to study properties of ions of the francium isoelectronic sequence. Specifically, energies of the , , , and states of Fr-like ions with nuclear charges are calculated through third order; reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for , , and electric-dipole transitions; and , , and multipole matrix elements are evaluated to obtain the lifetimes of low-lying excited states. Moreover, for the ions calculations are also carried out using the relativistic all-order single-double method, in which single and double excitations of Dirac-Fock wave functions are included to all orders in perturbation theory. With the aid of the SD wave functions, we obtain accurate values of energies, transition rates, oscillator strengths, and the lifetimes of these six ions. Ground state scalar polarizabilities in Fr I, Ra II, Ac III, and Th IV are calculated using relativistic third-order and all-order methods. Ground state scalar polarizabilities for other Fr-like ions are calculated using a relativistic second-order method. These calculations provide a theoretical benchmark for comparison with experiment and theory.
pacs:31.15.Ar, 31.15.Md, 32.10.Fn, 32.70.Cs
A detailed investigation of radiative parameters for electric dipole (E1) transitions in Fr-like ions with = 89–92 was presented recently by Biémont et al. (2004). The electronic structure of Fr-like ions consists of a single electron outside of a core with completely filled =1, 2, 3, 4 shells and , , , , and subshells. In Fig. 1, we plot one-electron DF energies of valence , , and states as functions of . We find that the valence orbital is more tightly bound than the and orbitals at low stages of ionization ( = 87–89), while the and orbitals are more tightly bound for highly ionized cases ( 90). Competition between the , , and orbitals leads to problems for calculations, making it difficult to obtain very accurate excitation energies and line strengths for the transitions between the low-lying , , and states.
Relativistic Hartree-Fock and Dirac-Fock atomic structure codes were used in Ref. Biémont et al. (2004) to perform calculations of radiative transition rates and oscillator strengths for a limited number of transitions using the energies given by Blaise and Wyart (), where experimental values were given for the , , (with =10–33), and (with =12–31) levels of neutral Fr, for 24 levels for Fr-like Th, and for seven levels of Fr-like Ac and U. Adopted energy level values in the , , and ( 30) series of neutral francium were presented by Biémont et al. (1998). Experimental measurements of energy levels of the and states in Fr I were reported recently in Refs. Simsarian et al. (1999); Grossman et al. (2000a). The experimental energies of 13 levels of Fr-like Ra were reported in the NIST compilation web ().
Lifetime measurements for neutral francium were presented in Refs. Zhao et al. (1997); Simsarian et al. (1998); Grossman et al. (2000b); Aubin et al. (2003); Gomez et al. (2005) for the , , , and levels. In those papers, experimental measurements were compared with ab-initio calculations performed by Johnson et al. (1996), Dzuba et al. (1995, 2001), Safronova et al. (1999), and Safronova and Johnson (2000). Third-order many-body perturbation theory was used in Ref. Johnson et al. (1996) to obtain the E1 transition amplitude for neutral alkali-metal atoms. The correlation potential method and the Feynman diagram technique were used in Refs. Dzuba et al. (1995, 2001) to calculate E1 matrix elements in neutral francium and Fr-like radium. Atomic properties of Th IV ion were studied by Safronova et al. (2006) using a relativistic all-order method.
The present third-order calculations of excitation energies of the , , , and states in Fr-like ions with nuclear charges start from a closed-shell Dirac-Fock potential for the 86 electron radon-like core. We note that Th IV is the first ion in the francium isoelectronic sequence with a [Rn] ground state instead of the [Rn] ground state as for Fr I, Ra II, and Ac III. Correlation corrections become very large for such systems as was demonstrated by Savukov et al. (2003), where the ratio of the second-order and DF removal energies for the [Xe] ground state in Ce IV and Pr V were 18% and 11%, respectively.
In the present paper, dipole matrix elements are calculated using both relativistic many-body perturbation theory, complete through third-order, and the relativistic all-order method restricted to single and double (SD) excitations. Such calculations permit one to investigate the convergence of perturbation theory and estimate the theoretical error in predicted data. To obtain lifetime predictions, multipole matrix elements for , and transitions are also evaluated. Additionally, scalar polarizabilities for the ground state in Fr I, Ra II, and Ac III are calculated using relativistic third-order and SD methods. Finally, scalar polarizabilities of the ground state of Fr-like ions with nuclear charge = 91–100 are calculated in second-order MBPT.
|Fr I, = 87|
|Ra II, = 88|
|Ac III, = 89|
|Th IV, = 90|
|Pa V, = 91|
|U VI, = 92|
|Fr I, = 87|
|Ra II, = 88|
|Ac III, = 89|
|Th IV, = 90|
|U VI, = 92|
Ii Third-order and all-order MBPT calculations of energies
We start from the “no-pair” Hamiltonian Sucher (1980)
where and can be written in a second-quantized form as
Negative-energy (positron) states are excluded from the sums. The quantities are eigenvalues of the one-electron Dirac-Fock equations with a frozen core and is a two-particle Coulomb matrix element. Our calculations start from a DF potential for a closed-subshell radon-like ion.
The all-order single-double (SD) method was discussed previously in Refs. Blundell et al. (1989); Liu (); Blundell et al. (1991); Safronova et al. (1998, 1999); Safronova and Johnson (2004). Briefly, we represent the wave function of an atom with one valence electron atom as with
where is the lowest-order atomic wave function, which is taken to be the frozen-core DF wave function of a state . Substituting the wave function into the many-body Schrödinger equation, with Hamiltonian given by the Eqs. (1–3), one obtains the coupled equations for the single- and double-excitation coefficients , , , and . The coupled equations for the excitation coefficients are solved iteratively. We use the resulting excitation coefficients to evaluate multipole matrix elements and hyperfine constants. This method includes contribution of important classes of MBPT corrections to all orders.
The SD valence energy does not include a all third-order MBPT corrections. The missing part of the third-order contribution, , is written out in Ref. Safronova et al. (1998) and must be calculated separately. We use our third-order energy code to separate out and add it to the . For notational simplicity, we drop the index in the designations in the text and tables below.
Results of our energy calculations for low-lying states of Fr I–U VI are summarized in Table 1. Columns 2–7 of Table 1 give the lowest-order DF energies , second- and third-order Coulomb correlation energies, and , first-order Breit contribution , second-order Coulomb-Breit corrections, and the Lamb shift contribution, . The sum of these six contributions is our final third-order MBPT result listed in the eighth column. First-order Breit energies (column of Table 1) include retardation, whereas the second-order Coulomb-Breit energies (column of Table 1) are evaluated using the unretarded Breit operator. We list all-order SD energies in the column labelled and the part of the third-order energies omitted in the SD calculation in column . We note that includes , part of , and dominant higher-order corrections. The sum of the six terms , , , , , and gives the final all-order results listed in the eleventh column of the table.
As expected, the largest correlation contribution to the valence energy comes from the second-order term, . This term is simple to calculate in comparison with and terms. Thus, we calculate the term with higher numerical accuracy than and . The second-order energy includes partial waves up to and is extrapolated to account for contributions from higher partial waves (see, for example, Refs. Safronova et al. (1996, 1997)). As an example of the convergence of with the number of partial waves , we consider the state in U VI. Calculations of with = 6 and 8 yield = -33598 and -347559cm, respectively. Extrapolation of these calculations yields -35163 and -35227 cm, respectively. Therefore, we estimate the numerical uncertainty of to be 64 cm. This is the largest contribution from the higher partial waves, since the numerical uncertainty of is equal to 34 cm, and the numerical uncertainty of is equal to 1 cm. The numerical uncertainty of the second-order energy calculation for all other states ranges from 1 cm to 5 cm. We use = 6 in our third-order and all-order calculations, owing to the complexity of these calculations. Therefore, we use our high-precision calculation of described above to account for the contributions of the higher partial waves, i.e. we replace [ = 6] with the final high-precision second-order value . The contribution given in Table 1 accounts for that part of the third-order MBPT correction not included in the SD energy. The values of are quite large and including this term is important.
We find that the correlation corrections to energies are especially large for states. For example, is about 15% of and is about 36% of for states. Despite the evident slow convergence of the perturbation theory expansion, the energy from the third-order MBPT calculation is within 0.9% of the measured energy. The correlation corrections are so large for the states that inclusion of correlation leads to the different ordering of states using the DF and energies. If we consider only DF energies the appears to be the ground state for Th IV, but the full correlation shows that the is the ground state. The correlation corrections are much smaller for all other states; the ratios of and are equal to 6%, 5%, and 2% for the , , and states, respectively.
The third-order and all-order results are compared with experimental values in Table 2. The energies are given relative to the ground state to facilitate comparison with experiment. Experimental energies for Fr I, Ac III, Th IV, and U VI are taken from Blaise and Wyart () and energies of Fr-like Ra are taken from the NIST compilation web (). Differences of our third-order and all-order calculations with experimental data, and , respectively, are given in the two final columns of Table 2. In general, the SD results agree better with the experimental values than the third-order MBPT values. Exceptions are the cases where the third-order fortuitously give results that are close to experimental values. Comparison of results from two last columns of Table 2 shows that the ratio of and is about three for the states. As expected, including correlation to all orders led to significant improvement of the results. Better agreement of all-order energies with experiment demonstrates the importance of the higher-order correlation contributions.
Below, we describe a few numerical details of the calculation. We use the B-spline method described Johnson et al. (1988a) to generate a complete set of basis DF wave functions for use in the evaluation of the MBPT expressions. We use 50 splines of order for each angular momentum. The basis orbitals are constrained to a spherical cavity of radius = 90–30 a.u for Fr I–U VI. The cavity radius is chosen large enough to accommodate all orbitals considered here and small enough that 50 splines can approximate inner-shell DF wave functions with good precision. We use 40 out 50 basis orbitals for each partial wave in our third-order and all-order energy calculations, since contributions from the highest-energy orbitals are negligible.
ii.1 dependence of energies in Fr-like ions
In Fig. 2, we illustrate the -dependence of the second and third-order energy corrections for the valence , , , and states of Fr-like ions. The second-order energy is a smooth function of for the and states, but exhibits a few sharp features for the and states. These very strong irregularities occur for the state =96, 99 and state for =99 and are explained by accidentally small energy denominators in the MBPT expressions for the correlation corrections to the energy. The third-order energy is a smooth function of for the and states, but exhibits a similar sharp features for the state. Most of the sharp features for the and states occur at very high values of , 97. Comparison of the and corrections for the and states is illustrated by Fig. 3. This figure shows also the smooth dependence of second- and third-order corrections to the and fine-structure intervals.
The total energies divided by in Fr-like ions are shown in Fig. 4. We plot the five energy levels for the , , and states. The -dependence of is smooth up to very high . Irregularities observed in Fig. 4 for high are explained by vanishing energy denominators in MBPT expressions for correlation corrections.
|Fr I, = 87||Th IV, = 90|
|Ra II, = 88||Pa V, = 91|
|Ac III, = 89||U VI, = 92|
|Ac III, = 89|
|Th IV, = 90|
|U VI, = 92|
|Fr I, = 87|
|Ra II, = 88|
Iii Electric-dipole matrix elements, oscillator strengths, transition rates, and lifetimes in Fr-like ions
iii.1 Electric-dipole matrix elements
The matrix element of a one-particle operator is given by Blundell et al. (1989)
where is the exact wave function for the many-body “no-pair” Hamiltonian . In MBPT, we expand the many-electron wave function in powers of as
where is the corresponding one-particle matrix element Johnson et al. (1995). Since is a DF function we designate by below.
The second-order Coulomb correction to the transition matrix element in the DF case with potential is given by Johnson et al. (1996)
where is the lowest-order (DF) matrix element given by Eq. (7), and the terms , are linear or quadratic functions of the excitation coefficients introduced in Eq. (4). The normalization terms are quadratic functions of the excitation coefficients. As a result, certain sets of many-body perturbation theory terms are summed to all orders. In contrast to the energy, all-order SD matrix elements contain the entire third-order MBPT contribution.
The calculation of the transition matrix elements provide another test of the quality of atomic-structure calculations and another measure of the size of correlation corrections. Reduced electric-dipole matrix elements between low-lying states of Fr-like systems with = 87–92 calculated in various approximations are presented in Table 3.
Our calculations of the reduced matrix elements in the lowest, second, and third orders are carried out following the method described above. The lowest order DF value is obtained from Eq. (7). The values are obtained as the sum of the second-order correlation correction given by Eq. (8) and the DF matrix elements . The second-order Breit corrections are rather small in comparison with the second-order Coulomb correction (the ratio of to is about 0.2%–2%).
The third-order matrix elements include the DF values, the second-order results, and the third-order correlation correction. includes random-phase-approximation terms (RPA) iterated to all orders Johnson et al. (1996).
We find correlation corrections to be very large, 10-25%, for many cases. All results given in Table 3 are obtained using length form of the matrix elements. Length-form and velocity-form matrix elements differ typically by 5–20% for the DF matrix elements and 2–5 % for the second-order matrix elements in these calculations.
Electric-dipole matrix elements evaluated in the all-order SD approximation are given in columns labeled (Eq. (9)) of Table 3. The SD matrix elements include completely, along with important fourth- and higher-order corrections. The fourth-order corrections omitted from the SD matrix elements were discussed recently by Derevianko and Emmons (2002). The values are smaller than the values and larger than the values for all transitions given in Table 3.
In Fig. 5, we illustrate the -dependences of the line strengths for the , , and transitions. Two sets of line strengths values and are presented for each transition. The values of and are obtained as and , respectively. It should be noted that the values are scaled by to provide better presentation of the line strengths. The difference between and curves increases with increasing ; for the transition, the ratio of the second-order ( - ) and the first-order () contributions is equal to 15% and 37% for =87 and 92, respectively.
iii.2 Form-independent third-order transition amplitudes
We calculate electric-dipole reduced matrix elements using the form-independent third-order perturbation theory developed by Savukov and Johnson in Ref. Savukov and Johnson (2000). Previously, a good precision of this method has been demonstrated for alkali-metal atoms. In this method, form-dependent “bare” amplitudes are replaced with form-independent random-phase approximation (“dressed”) amplitudes to obtain form-independent third-order amplitudes to some degree of accuracy. As in the case of the third-order energy calculation, a limited number of partial waves with is included. This restriction is not very important for considered here ions because third-order correction is quite small, but it gives rise to some loss of gauge invariance. The gauge independence serves as a check that no numerical problems occurred.
Length and velocity-form matrix elements from DF, second-order (RPA), and third-order calculations are given in Table 4 for the limited number of transitions in Fr-like systems with = 87–100. The values differ in and forms by 2–15 % for the transitions. The very large difference (by a factor of 2–3) is observed in the values for transitions as illustrated in Table 4. The second-order RPA contribution removes this difference in values, and the and columns with the headings are almost identical. There are, however, small differences (0.002%–0.2%) in the third-order matrix elements. These remaining small differences can be explained by limitation in the number of partial waves taken into account in the in the third-order matrix element evaluations that we already discussed in the previous section describing the energy calculations.
iii.3 Oscillator strengths, transition rates, and lifetimes
We calculate oscillator strengths and transition probabilities for the eight , , and electric-dipole transitions in Ra II, Ac III, Th IV, U VI and for the seven , , and electric-dipole transitions in Fr I. Wavelengths (Å), weighted transition rates (s), and oscillator strengths in Ac III, Th IV, U VI are given in Table 5 for Ac III, Th IV, U VI ions and in Table 6 for Ra II and Fr I.
The SD data ( and ) are compared with theoretical ( and ) results from Ref. Biémont et al. (2004). The experimental energies were used to calculate the , , , and . Therefore, we really compare the values of the electric-dipole matrix elements. The SD and RHF results for and transitions disagree by about 6–25 % with exception of the transitions where disagreement is 60 %. The largest disagreement (by a factor of 2–5) is observed between the SD and RHF results for the transitions. The correlation corrections are especially large for these transitions as illustrated in Table 3. Therefore, we expect that our values, which include correlation correction in rather complete way, will disagree with RHF calculations which appear not to include any correlation effects for transitions which involve states. Our conclusion is confirmed by comparison of the and and our and results (see, also Ref Safronova et al. (2006)). Our values for transitions rates and oscillator strengths are in reasonable agreement (10–20 %) with RHF data.
The SD data and for Fr I and Ra II given Table 6 are compared with theoretical data (, ) given in Refs. Biémont et al. (1998, 2004) and theoretical ( and ) values from Ref. Dzuba et al. (2001). The wavelengths given in Table 6 are taken from NIST compilation web () for Ra II and from Ref. Blaise and Wyart () ( = 12237.409 cm, = 13923.998 cm), Ref. Simsarian et al. (1999) ( = 19732.523 cm), and Ref. Grossman et al. (2000a) ( = 24244.831 cm and = 24333.298 cm) for Fr I. Comparison of the and results obtained by three different approximations shows better agreement between our SD and MBPT Dzuba et al. (2001) results than between our and HFR results obtained Biémont et al. in Refs. Biémont et al. (2004, 1998), owing to more complete treatment of the correlation in our calculation and in Ref. Dzuba et al. (2001).
Our SD lifetime results are compared in Table 7 with experimental measurements presented in Refs. Simsarian et al. (1998); Grossman et al. (2000b); Gomez et al. (2005) for the , , levels in neutral francium. We find that our SD lifetimes are in excellent agreement with precise measurements provided in Refs. Simsarian et al. (1998); Grossman et al. (2000b); Gomez et al. (2005).
|Ra II, = 88|
|Ac III, = 89|