Fine and Hyperfine Splitting of the Low-Lying States of $^ 9$ Be

Fine and hyperfine splitting of the low-lying states of 9Be Mariusz Puchalski,1 Jacek Komasa,1 and Krzysztof Pachucki2 1Faculty of Chemistry, Adam Mickiewicz University, Uniwersytetu Poznanskiego´ 8, 61-614 Poznan,´ Poland 2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: August 13, 2021) 1;3 We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest PJ , 3 3 e 1;3 9 S1, PJ , and DJ excited states of the Be atom using explicitly correlated Gaussian functions and report on the breakdown of the standard hyperfine structure theory. Because of the strong hyperfine mixing, which prevents the use of common hyperfine constants, we formulate a description of the fine and hyperfine structure that is valid for an arbitrary coupling strength and may have wide applications in many other atomic systems. PACS numbers: 31.15.ac, 31.30.J- I. INTRODUCTION The controlled accuracy is achieved by means of a full vari- ational optimization of the wave function and by transforma- tion of singular operators to an equivalent but more regular The main drawback of atomic structure methods based form. The price paid for using the explicitly correlated func- on the nonrelativistic wave function represented as a lin- tions is the rapid increase in the complexity of calculations ear combination of determinants of spin-orbitals (Hartree- with each additional electron; therefore, application of these Fock, configuration interaction, multi-configurational self- functions has so far been limited to few-electron systems only. consistent field, etc.) is the difficulty in providing results with reliably estimated uncertainties. While the accuracy of Before passing to the main topic, which is the 4-electron nonrelativistic energy can be assessed by increasing the space beryllium (Be) atom, let us briefly describe recent advances of electronic configurations, the irregular convergence of main the calculation of 1-, 2-, and 3-electron atomic systems. trix elements, especially involving singular operators, such Hydrogenic systems are the only ones in which theoretical as those for relativistic or quantum electrodynamics (QED) predictions including QED effects are sufficiently accurate to corrections, often does not allow for presenting any uncer- determine the nuclear (proton, deuteron) charge radius from tainties. Therefore, this deficiency limits the use of these the measured transition frequencies [10]. Being apparently methods in the high-accuracy-demanding applications, e.g. simple, hydrogenic systems are a cornerstone for the imple- testing quantum electrodynamics [1–3], determination of the mentation of QED in bound states, which relies on the ex- nuclear charge radii [4–7], the nuclear electromagnetic mo- pansion of binding energy in powers of the fine structure con- ments [8, 9], or physical constants [10], and searching for new stant α 1 137. Similarly, for two- and more-electron sys- physics [11]. tems, one also performs expansion in α, as long as the nu- On the other hand, an alternative approach based on the clear charge∼ ~ Z is not too large. This allows description of Dirac-Coulomb Hamiltonian, with the wave function repre- an atomic system in terms of the successively smaller effects, 2 4 sented as a determinant of four-component spin-orbitals of i.e. nonrelativistic energy α , relativistic correction α , lead- 5 positive energy, can reach reasonably convergence on rela- ing QED of order α , and so on. For the helium atom all 6 tivistic energies and matrix elements [12]. But so far there is these expansion terms are calculated up to the order α [15], 7 no formulation of QED theory on the top of Dirac-Coulomb with some states up to the order α [16]. Such high order Hamiltonian with projection on positive one-electron ener- calculations are feasible with explicitly correlated exponen- gies. Therefore numerical convergence does not say much tial basis functions, for which analytic integrals are known. about uncertainties due to omitted QED effects, including Atomic systems with three electrons present greater difficulty those related to negative energy orbitals. Another problems for the accurate calculation of their energy levels despite ob- arise when the hyperfine effects are not negligible compared taining very precise wave functions with explicitly correlated to the fine structure splitting. In the previous works on this Hylleraas or ECG functions. Nevertheless, several highly ac- + topic (e.g. [13, 14]), it has been demonstrated the hyperfine curate results have been obtained for Li and Be , including mixing of fine structure levels can be satisfactorily accounted isotope shifts for the charge radii determination [4, 17, 18], arXiv:2106.16161v2 [physics.atom-ph] 12 Aug 2021 for by the second-order perturbation theory. However, when and fine [19] and hyperfine [20] splitting. The fine structure 2 6 the hyperfine splitting is of the same order or even larger than splitting of the lithium 2 PJ state with the inclusion of O α the fine structure splitting, the perturbative approach fails, and QED corrections [19, 21] agrees well with even more accurate one can no longer use the standard A and B hyperfine pa- experimental values [22, 23], while current theoretical predic-( ) J J 6~7 rameters. tions for the Li ground state hyperfine splitting are limited by insufficient knowledge of the nuclear structure, and not by It is desirable therefore to develop tools which provide high the atomic structure theory. and controlled accuracy, like those based on nonrelativistic QED (NRQED) theory and representation of the nonrelativis- The experience gained from the above-mentioned systems tic wave function in terms of explicitly correlated basis func- can be exploited to a large extent in four-electron systems, but tions, e.g. exponential, Hylleraas, or Gaussian (ECG) ones. these ones are much more demanding in calculations. The at- 2 tempts employing Hylleraas wave functions [24–26] are lim- tries, we have introduced in this work a Cartesian angular mo- ited to nonrelativistic energy because of the lack of effec- mentum representation, which is tailored for use with many- tive methods of evaluation of ”relativistic” four-electron in- electron explicitly correlated basis functions and which sim- tegrals. Therefore, our choice is the use of the ECG method, plifies evaluation of matrix elements. which performs very well for both nonrelativistic and relativistic contributions [24–46]. Nonetheless, we note that be- E(cm-1) -1 cause Gaussian-type wave functions do not satisfy the Kato 70 000 E(cm ) cusp condition, the complete calculation of the α6 correction K=1/2 62053.81 62053.80 is still unfeasible. This unsolved problem limits the current 1 2s 3d; D2 K=3/2 62053.79 3 capabilities of the ECG method in the application to three- 2s 3d; DJ 62053.78 1 o and more-electron systems. After all, the ECG functions are 60 000 2s 3p; P1 2p2; 3Pe 62053.77 2s 3p; 3Po J J K=5/2 62053.76 so far the best suited for the four-electron systems and such an 2 1 2p ; D2 62053.75 application to the Be atom will be presented here. 1 2s 3s; S0 3 In our previous works on Be we have obtained accurate en- 2s 3s; S1 2 1 1 ergies for the ground 2s S0 and the excited 2s2p P1 states 50 000 59 700 −1 [42], the difference of which at 42 565:441 11 cm , agrees 59 698 −1 J=2 well with the experimental value of 42 565:450 1 13 cm 59 696 1 o J=1 obtained by Cook et al. [47] and with( later) calculations 2s 2p; P1 J=0 59 694 [48]. Last year, two more transitions were measured( ) to a 40 000 2 1 1 59 692 high accuracy—the wavenumber of the 2s S0 2s3d D2 −1 1 line, equal to 64 428:403 21 55 cm , and of the 2s2p P1 1 −1 2s3d D2 line, equal to 21 862:952 9 14 cm , were− reported by Cook et al. [49]. In this( case,) no theoretical results at an− 30 000 adequate level of accuracy have been( calculated) yet. More- 21 984 over, in the beryllium atom, of particular theoretical interest 21 982 3 J=2 is the lowest 2s2p P excited state, because it is metastable. 3 o 2s 2p; PJ 21 980 So far though, its energy has not been measured and calcu- 20 000 J=1 1 lated to such a high accuracy as for the 2s2p P state. An J=0 21 978 old but the most accurate experimental excitation energy from 21 976 3 −1 the ground to the 2s2p P1 level equal to 21 978:92 1 cm [50] is in agreement with the less accurate recent theoretical 10 000 value of 21 978:2 11 cm−1 by Kedziorski et al. [46].( ) Quite recently, the hyperfine splitting of the 2s2p 3P state has been accurately calculated( ) and, with the help of the 50-years old measurements by Blachman [51], has been employed to de- 2 1 termine the most accurate value of the electric quadrupole 0 2s ; S0 moment of 9Be [9], but it is in disagreement with all previ- FIG. 1. The lowest energy levels of theBe atom. The levels for which ous determinations. Moreover, all the other Be energy levels the energy is evaluated in this work are drawn in solid lines and the lying below the ionization threshold of 75 192:64 cm−1 have −1 remaining ones in dashed line. The fine structure of the triplet states large uncertainties, being in the range 0:01 0:2 cm [52], of our interest is also shown, J L S; K I S. and there are no corresponding accurate theoretical results to compare with.

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