Hyperfine structure of the 2\,^3\!P state in ^9Be and the nuclear quadrupole moment
HHyperfine structure of the P state in Be and the nuclear quadrupole moment
Mariusz Puchalski and Jacek Komasa
Faculty of Chemistry, Adam Mickiewicz University, Uniwersytetu Pozna´nskiego 8, 61-614 Pozna´n, Poland
Krzysztof Pachucki
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: February 3, 2021)We have performed accurate calculations of the hyperfine structure of the P state in the Be atom withthe help of highly optimized, explicitly correlated functions, accounting for the leading finite nuclear mass,radiative, nuclear structure and relativistic effects. By comparison with measurements, we have determined the Be nuclear quadrupole moment to be Q N = 0 . barns, which is not only the most accurate result,but also disagrees with previous determinations. The electric quadrupole moment Q N of nuclei is a mea-sure of the deformation of the nuclear charge distribution withrespect to the spherical symmetry. This deformation comesfrom a strong dependence of nuclear forces on the orientationof the nucleon spin and is present in atomic nuclei with a spinequal to or greater than one, as it is for Be with I = 3 / . Inprinciple, Q N could be determined from the nuclear structuretheory, but such calculations with controlled accuracy havenot yet been feasible. Only very recently and only for such asimple nucleus as the deuteron, theoretical predictions basedon the chiral effective field theory have reached the uncer-tainty of 1% [1], and this result agreed with the more precisevalue from the rotational spectroscopy of the HD molecule[2].Originally, Q N for many nuclei was determined by the elec-tron scattering of the nuclei, for example for Li [3]. However,for Be the available experimental data are not only limited inaccuracy but also differ noticeably from each other, indicat-ing that the scattering results are not always conclusive [4, 5].Nowadays, reliable Q N values can be derived from a combi-nation of theoretical and atomic or molecular spectroscopicdata. This has been realized for LiH, LiF, and LiCl molecules[6], leading to a Q N ( Li) consistent with the nuclear scatter-ing value [3]. Although appropriate theoretical calculationswere performed for BeH + [7, 8] as well as for the , Be − anion [9] years ago, there have been no corresponding mea-surements reported so far. Therefore, the method of choicefor finding Q N ( Be) is atomic spectroscopy.In the accurate determination of Q N from atomic spec-troscopy, it is important to understand the electron-nucleusinteraction at the fundamental level. Recent advances in mea-surements of electronic [10] and muonic atoms [11], togetherwith progress in the theoretical description of atomic spectra[12], indicate the importance of the nuclear structure effects.Therefore, accurate knowledge of the nuclear quadrupole mo-ment will give access to details of the electron-nucleus inter-actions which, so far, have not been visible, like for examplethe nuclear quadrupole polarizability [13].The ground state S of Be is fully symmetric; thus, thenuclear quadrupole moment does not lead here to any ob-servable effect. In contrast, in the lowest excited P level,which is metastable, the hyperfine interactions cause level splitting, and this splitting can be accurately measured [14].So, it is atomic structure theory that must interpret the hyper-fine structure in terms of the magnetic dipole and the electricquadrupole moments of the nucleus. Still, the same hyper-fine interactions that lead to the hyperfine splitting also causethe hyperfine mixing of different P J levels. This mixinghad been accounted for in Ref. 14 but in a very simplifiedway. Although the accuracy of these approximations has beenquestioned [7], consecutive atomic structure calculations, us-ing modern and sophisticated approaches, have relied on theseoriginal inaccurate calculations of the hyperfine mixing. Anexception is the work by Beloy et al. [15], in which this mix-ing was calculated in the second order of perturbation theoryusing the relativistic CI+MBPT approach, but its numericalaccuracy turned out to be insufficient to obtain Q N with acompetitive accuracy.In this work, we employ well-optimized explicitly cor-related Gaussian wave functions to have good control overthe numerical accuracy. With these wave functions we per-form calculations of the Be hyperfine structure, with a com-plete treatment of the hyperfine mixing of different P J levels. Moreover, we fully account for the leading radia-tive (QED), finite nuclear mass, and nuclear structure effects,while dealing with relativistic and higher order corrections ap-proximately. Our result for the electric quadrupole moment Q N ( Be) show that all of its previous determinations were notas accurate as claimed, due to a very approximate treatment ofthe hyperfine mixing and neglect of the radiative (QED) cor-rections.
Fine and hyperfine structure Hamiltonian.—
The most ac-curate approach for light atomic systems solves at first thenonrelativistic Hamiltonian H = (cid:126)p m N + (cid:88) a (cid:126)p a − (cid:88) a Zr a + (cid:88) a
1) 3 ( L i L j ) (2) L (2 L − , (12)where the coefficients a , a , a , b, c , c are independent of J , but are specific to the particular state. These coefficientscan be obtained, for example, from the matrix elements of H fs and H hfs in the decoupled | M L , M S (cid:105) or | J, M J (cid:105) basis. Oncethese coefficients are known, the above Hamiltonian can be di-agonalized yielding the hyperfine levels. Alternatively, the hy-perfine structure can be represented in terms of J -dependent A J and B J coefficients H hfs , eff = A J (cid:126)I · (cid:126)J + B J I i I j ) (2) I (2 I −
1) 3 ( J i J j ) (2) J (2 J − , (13) which are conventionally used to represent the measured val-ues, while the fine structure is given in terms of differences inthe centroid energies.If the atomic levels had a definite value J , the fine structurewould be given by the expectation value of H eff , namely E fs ( J ) = c + c / for J = 2 − c − c / for J = 1 − c + 5 c / for J = 0 , (14)and the hyperfine structure by A J = (cid:26) a / a / a / for J = 2 a / a / − a / for J = 1 ,B J = (cid:26) b for J = 2 − b/ for J = 1 . (15)Because, in general, one cannot assume that the levels have adefinite value of J , the effective hfs Hamiltonian in Eq. (12)has to be diagonalized numerically. Nonetheless, the hyper-fine structure can still be represented in terms of A J and B J coefficients, and we use them for comparison of theoreticalpredictions with experimental results and for the determina-tion of the nuclear quadrupole moment Q N . Relativistic, radiative, and finite nuclear size corrections.—
The hyperfine Hamiltonian in Eq. (5) represents the leadinghyperfine interactions, but there are also many small correc-tions which are often overlooked in literature. These correc-tions contain terms with higher powers of the fine structureconstant α . Because most of them are proportional to theFermi contact interaction, we account for them in terms ofthe following factor ˜ a = a (1 + (cid:15) ) . (16)Below, we briefly describe contributions included in the (cid:15) term.The O ( α ) correction is analogous to that in hydrogenic sys-tems [17] and consists of two parts. The first part is due to thenuclear recoil H (5)rec = (cid:20) − Z απ mm N ln (cid:16) m N m (cid:17)(cid:21) (cid:126)I · (cid:88) a Z α g N m m N (cid:126)σ a π δ ( r a ) (17)and numerically is very small, almost negligible. The recoilcontribution to (cid:15) amounts to (cid:15) rec = − .
000 011 . The secondpart of the O ( α ) correction is due to the finite nuclear size andthe nuclear polarizability, and is given by [17, 18] H (5)fs = (cid:2) − Z α m r Z (cid:3) (cid:126)I · (cid:88) a Z α g N m m N (cid:126)σ a π δ ( r a ) , (18)where r Z is a kind of effective nuclear radius called theZemach radius. Disregarding the inelastic effects, this radiuscan be written down in terms of the electric charge ρ E andmagnetic-moment ρ M densities as r Z = (cid:90) d r d r (cid:48) ρ E ( r ) ρ M ( r (cid:48) ) | (cid:126)r − (cid:126)r (cid:48) | . (19)Nevertheless, the inelastic, i.e. polarizability, corrections canbe significant, but because they are very difficult to calculate,they are usually neglected. In this work we account for pos-sible inelastic effects by employing r Z from a comparison ofvery accurate calculations of hfs in Be + with the experimen-tal value, namely r Z = 4 . [18]. Because this correc-tion is also proportional to the contact Fermi interaction, werepresent it in terms of (cid:15) fs = − .
000 615 .Next, there are radiative and relativistic corrections of therelative order O ( α ) . The radiative correction, beyond thatincluded by the free electron g-factor, is [17] H (6)rad = Z α (cid:18) ln 2 − (cid:19) (cid:126)I · (cid:88) a Z α g N m m N (cid:126)σ a π δ ( r a ) (20)and the corresponding (cid:15) factor is (cid:15) rad = − .
000 384 . The O ( α ) relativistic and higher order corrections are much morecomplicated. They have been calculated for the ground stateof Be + [18]. Here we take this result and assume that itis proportional to the Fermi contact interaction, and obtain (cid:15) rel = 0 .
001 664 . This is the only approximation we assumein this work, and as a consequence we neglect the mixing ofthe P state with the nearby lying P . Exactly for thisreason we will use only the hyperfine splitting of the P state, which does not mix with P for the determination of Q N . The resulting total correction is (cid:15) = (cid:15) rec + (cid:15) fs + (cid:15) rad + (cid:15) rel = 0 . · − . (21)Some previous works present these multiplicative correctionfor all individual hyperfine contributions, but in our opinionthis cannot be fully correct because higher order relativisticcorrections may include additional terms, beyond that in H eff in Eq. (12). These corrections are expected to be much smallerthan the experimental uncertainty for the B J coefficient, andtherefore are neglected. ECG wave function and expectation values.—
Let us nowmove to the calculations of the a i , b , and c i coefficients. Toobtain sufficiently high accuracy for these parameters, we ex-press the four-electron atomic wave function as a linear com-bination of properly symmetrized explicitly correlated Gaus-sian (ECG) functions, Ψ ( { (cid:126)r a } ) = K (cid:88) n =1 t n A (cid:2) φ n ( { (cid:126)r a } ) χ { a } (cid:3) , (22)where t n are linear coefficients, A is the antisymmetrizationoperator over electronic indices, and { a } and { (cid:126)r a } denotethe sequence of electron indices and coordinates, respectively.The electronic P symmetry of the states was enforced usingthe following spatial functions φ i ( { (cid:126)r a } ) = r ia exp (cid:2) − (cid:88) b w b r b − (cid:88) c The hyperfine transition fre-quencies ν J ( F ; F +1) can be expressed in terms of the A J and B J parameters and vice versa. For this, one writes (cid:104) H hfs , eff (cid:105) F = A J A IJF + B J B IJF + C J C IJF (33)where A IJF = 12 K (34) B IJF = 3 / K ( K + 1) − I ( I + 1) J ( J + 1)2 I (2 I − J (2 J − (35)with K = F ( F + 1) − I ( I + 1) − J ( J + 1) and the totalangular momentum (cid:126)F = (cid:126)S + (cid:126)L + (cid:126)I . The octupole term C IJF is given e.g. by Schwartz [22] and Jaccarino [23].For J = 1 we arrive at A = − ν (cid:18) 12 ; 32 (cid:19) − ν (cid:18) 32 ; 52 (cid:19) (36) B = 13 ν (cid:18) 12 ; 32 (cid:19) − ν (cid:18) 32 ; 52 (cid:19) , (37)and for J = 2 A = − ν (cid:18) 12 ; 32 (cid:19) − ν (cid:18) 32 ; 52 (cid:19) − ν (cid:18) 52 ; 72 (cid:19) (38) TABLE I. Convergence of the nonrelativistic energy and theoretical fine and hyperfine structure parameters for the P state of Be (in MHz).Mass m N = 9 . 012 183 07(8) u [20] and magnetic moment µ/µ N = − . 177 432(3) [21] were used for the Be nucleus. K E/ a . u . c c a a a b/Q N (MHz / barn) − . 566 340 608 5 32 986 . 385 5 399 . − . 116 70 − . 698 781 14 . 788 494 27 . 150 476 − . 566 341 144 7 32 987 . 849 5 399 . − . 126 66 − . 698 923 14 . 788 389 27 . 149 896 − . 566 341 380 0 32 989 . 401 5 399 . − . 130 99 − . 698 986 14 . 788 467 27 . 149 112 − . 566 341 441 5 32 989 . 854 5 399 . − . 129 61 − . 699 124 14 . 788 579 27 . 148 954 − . 566 341 466 0 32 990 . 027 5 399 . − . 128 99 − . 699 186 14 . 788 630 27 . 148 898 ∞ − . 566 341 474(8) 32 990 . . − . 128 4(6) − . 699 24(5) 14 . 788 68(5) 27 . 148 87(3) B = 25 ν (cid:18) 12 ; 32 (cid:19) + 25 ν (cid:18) 32 ; 52 (cid:19) − ν (cid:18) 52 ; 72 (cid:19) (39) C = − ν (cid:18) 12 ; 32 (cid:19) + 150 ν (cid:18) 32 ; 52 (cid:19) − ν (cid:18) 52 ; 72 (cid:19) . (40)The above listed parameters (36)-(40) were evaluated us- TABLE II. A J , B J , and C J parameters determined using Eqs. (36)-(40) from experimental [14] and theoretical hfs frequencies ν J (inMHz). The experimental values for A and B parameters inferredin this work are more accurate than those presented in the originalwork [14] because we included the C parameter in their determina-tions. Experimental Theoretical Difference A − . − . − . B − . − . − . A − . − . − . B . . . C − . − . . ing both experimental and theoretical transition frequencies ν J ( F ; F + 1) , and their numerical values are presented in Ta-ble II. The experimental values of ν J (in MHz) were takenfrom Ref. 14, while theoretical ones were found by diagonal-ization of the effective Hamiltonian of Eq. (12). Most impor-tantly, the parameter b was fixed by matching theoretical andexperimental B , which holds for b = 1 . , (41)where the uncertainty originates from that of the experimentalone in B .The differences between experimental and theoretical val-ues, shown in Tab. II, are not discrepancies but result fromneglected higher order relativistic and QED effects, whichare beyond those proportional to the Fermi contact interac-tion. For the A coefficient these effects are of relative order . · − , while for A they are . · − . They are largerfor A due to hyperfine mixing of P with the nearby P state, which we have not taken into account in our calcula-tion. Moreover, the relative difference for the B coefficientis δB /B = 0 . · − , which most probably is also due tothis mixing. For this reason, we have chosen B and not B for the determination of the b coefficient. The centroids of the calculated hyperfine energy levels areshown in Tab. III in the column ‘fs levels’ for all J values ofthe P state. This table contains also experimental [14] andcalculated fine-structure transition frequencies as well as theirdifference. This difference originates again from the higher TABLE III. Theoretical fine structure (fs) energy levels of the P state of Be, and the comparison of experimental [14] and theoreticalfs transition frequencies (in GHz). J fs levels ν exp ( J ; J + 1) ν the ( J ; J + 1) Diff.2 . − . 489 71 . . 379 0 . − . 987 19 . . − . order relativistic and QED effects. Due to a large experimentaluncertainty we cannot conclude much about the magnitude ofthese effects. Determination of Q N ( Be) .— Having calculated the valueof b/Q N = 27 . 148 87(3) MHz/barn, see Tab. I, and fixed b by matching theoretical and experimental values of B , weobtain the electric nuclear quadrupole moment Q N of Be Q N = 0 . 053 50(14) barn . (42)The uncertainty in Q N comes from the experimental error inthe ν hyperfine frequencies. In this context, the uncertaintiesfrom the neglected higher order relativistic and QED effectsin the b parameter, as well as the numerical uncertainties, aremuch smaller and are not shown.Among the literature results listed in Tab. IV and depictedin Fig. 1, only the Q N reported by Beloy et al. [15], thanksto its large uncertainty, agrees with our quadrupole moment.The lack of agreement with the other previous studies indi-cates that it is very difficult to estimate theoretical uncertain-ties using well-established, atomic structure methods such asMCHF or CI+MBPT. Conclusions.— We have determined the electric quadrupolemoment Q N ( Be) with significantly higher accuracy andin disagreement with previous values. The improvementachieved in this work has several independent sources. Thefirst one is the recalculation of the A , B , and C coeffi-cients from experimental hyperfine splitting. More precisely,including C in the effective Hamiltonian H hfs , eff in Eq. (13) TABLE IV. Comparison of the electric quadrupole moment (in barns)of the Be nucleus obtained using atomic structure calculations.Source Q N ( Be) Blachman and Lurio, 1967 [14] . Ray et al. , 1973 [24] . 052 5(3) Sinanoˇglu and Beck, 1973 [25] . 054 94 Beck and Nicolaides, 1984 [26] . 055 45 Sundholm and Olsen, 1991 [27] . 052 88(38) J¨onsson and Fisher, 1993 [28] . 052 56 Nemouchi et al. , 2003 [9] . 052 77 Beloy et al. , 2008 [15] . This work . 053 50(14) enabled us to decrease the experimental uncertainties by a fac-tor of 2. The second improvement is due to the inclusion ofthe hyperfine mixing by exact diagonalization of the effectivefine- and hyperfine-structure Hamiltonian of Eq. (12). Thethird improvement comes from the very accurate calculationof the expectation values of the fine- and hyperfine-structureoperators using explicitly correlated functions allowing for thecomplete electron correlations. The fourth one is due to theexact inclusion of the finite nuclear mass in the fine and hy-perfine interaction in Eqs. (2) and (5). Finally, the fifth sourceof improvement is due to accounting for the relativistic, radia-tive and nuclear structure corrections by appropriate rescalingof the a parameter.Regarding our theoretical uncertainties, they come exclu-sively from neglected higher order relativistic and QED con-tributions, in particular those due to hyperfine mixing of the P and P states. The calculation of the complete O ( α ) correction is possible but technically difficult. It has been per-formed for Be + in [29] due to the possibility to use the ex-ponentially correlated basis functions there. Nevertheless, ifa complete O ( α ) correction is known, one can use both B and B parameters to obtain an even more accurate nuclearquadrupole moment of Be. Q Be ( barn ) Blachman and Lurio, 1967Ray et al. , 1973Sinanoglu and Beck, 1973Beck and Nicolaides, 1984Sundholm and Olsen, 1991Jonsson and Fisher, 1993Nemouchi et al. , 2003Beloy et al. , 2008This work FIG. 1. 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