aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Hyperfine structure of S states in Li and Be + V. A. Yerokhin
Center for Advanced Studies, St. Petersburg State Polytechnical University,Polytekhnicheskaya 29, St. Petersburg 195251, Russia
A large-scale configuration-interaction (CI) calculation is reported for the hyperfine splitting of the2 S and 3 S states of Li and Be + . The CI calculation based on the Dirac-Coulomb-Breit Hamil-tonian is supplemented with a separate treatment of the QED, nuclear-size, nuclear-magnetizationdistribution, and recoil corrections. The nonrelativistic limit of the CI results is in excellent agree-ment with variational calculations. The theoretical values obtained for the hyperfine splitting arecomplete to the relative order of α and improve upon results of previous studies. PACS numbers: 31.15.Ar, 32.10.Fn, 31.30.Gs
The hyperfine structure (hfs) of few-electron atomshas been an attractive subject of theoretical studies fordecades, one of the reasons being a few ppm accuracyachieved in experiments on Li and Be + [1, 2]. Inter-est in this topic was enhanced even further recently, inview of prospects of using hfs data to get an access tothe neutron halo structure, the proton charge distribu-tion, and the nuclear vector polarizability, particularlyfor isotopes of Be + [3, 4]. Despite the considerable at-tention received, a high-precision theoretical descriptionof hfs in few-electron atoms remains a difficult task. Themain problem lies in the high singularity of the hfs inter-action and, as a consequence, in the dependence of thecalculated results on the quality of the correlated wavefunction near the nucleus.Among numerous theoretical investigations performedpreviously for Li and Be + , two apparently most accurateones are the multiconfigurational Dirac-Fock (MCDF)calculation [5] and the Hylleraas-type variational calcu-lation [6]. Both studies report good agreement with theexperiment, but they are not entirely consistent with eachother in treatment of individual corrections. The MCDFcalculation does not include the binding QED effects and,in the case of Li, the nuclear magnetization distributioneffect. The variational calculation yields accurate resultsfor the nonrelativistic Fermi contact term but treats therelativistic effects in an effective way only, by rescalingthe hydrogenic result. This indicates that neither of thesestudies is complete at the relative order of α ( α is thefine-structure constant). The aim of the present inves-tigation is to perform a high-precision calculation of thehfs splitting in Li and Be + , with a complete treatmentof all corrections ∼ α .A possible way to accomplish this task would be to sup-plement the nonrelativistic calculation [6] with a rigorousevaluation of the relativistic correction, whose expressionwas recently derived by Pachucki [7]. Such a calculationhas not been performed so far. In the present work, therelativistic correction will be accounted for by means ofthe Dirac-Coulomb-Breit Hamiltonian.The magnetic dipole hfs splitting of an energy levelof an nS state is conveniently represented in terms of a dimensionless function G n ( Z ) defined as [8]∆ E n = 43 α ( Zα ) n mm p µµ N I + 12 I mc (1 + m/M ) G n ( Z ) , (1)where µ is the nuclear magnetic moment, µ N = | e | / (2 m p ) is the nuclear magneton; m , m p , and M arethe masses of the electron, the proton, and the nucleus,respectively; I is the nuclear spin quantum number, and Z is the nuclear charge number. The function G definedin this way is unity for a non-relativistic point-nucleusH-like atom.Within the leading relativistic approximation, the elec-tron correlation can be described by the Dirac-Coulomb-Breit equation, which is solved by the configuration-interaction (CI) Dirac-Fock (DF) method in the presentwork. The many-electron wave function Ψ( P JM ) withthe parity P , the momentum quantum number J , andthe momentum projection M is represented as a sum ofconfiguration-state functions (CSFs),Ψ( P JM ) = X r c r Φ( γ r P JM ) . (2)The CSFs are obtained as linear combinations of theSlater determinants constructed from the positive-energysolutions of the Dirac equation with the frozen-core DFpotential. The mixing coefficients c r are determined bydiagonalizing the Hamiltonian matrix. The hfs splittingis obtained as the expectation value of the hfs operatoron the many-electron wave function (2). The correspond-ing formulas are well-known, see, e.g. , [5]. To performa CI calculation, we devised a code, incorporating andadapting a number of existing packages [9] for settingup the CSFs, calculating angular-momentum coefficients,and diagonalizing the Hamiltonian matrix. The largestnumber of CSFs simultaneously handled was about a halfof a million, with the number of nonzero elements in theHamiltonian matrix of about 5 billions. A thorough opti-mization of the code was carried out, in order to keep thetime and memory consumption of the calculation withinreasonable limits.The dominant part of the hfs splitting in light atomsis delivered by the Dirac-Coulomb Hamiltonian. This TABLE I: The Dirac-Coulomb-Breit part of the hfs splitting,in terms of G ( Z ). l max Li 2 S Be + S Coulomb 1 0 .
214 470 3 0 .
390 159 92 0 .
215 167 8 0 .
390 798 63 0 .
215 304 4 0 .
390 938 74 0 .
215 346 2 0 .
390 984 75 0 .
215 362 9 0 .
391 003 86 0 .
215 371 9 0 .
391 014 57 0 .
215 376 5 0 .
391 020 2 ∞ .
215 384 8(49) 0 .
391 030 4(61)Breit 0 .
000 015 9 0 .
000 038 6Total 0 .
215 400 7(49) 0 .
391 069 0(61)MCDF [5] 0 .
215 287 0 .
390 984Hylleraas a [6] 0 .
215 379(13) 0 .
391 023(34) a the sum of the nonrelativistic, the relativistic, and thenuclear-charge distribution terms. was the most demanding part of the calculation since ahigh relative precision was required. The one-electronorbitals for constructing CFSs were obtained by thedual-kinetic-balance (DKB) B-spline basis set method[10] for the Dirac equation. For a given number of B-splines n a , all eigenstates were taken with the energy0 < ε ≤ mc (1 + Zα E max ) and the orbital quantumnumber l ≤ l max , where E max was varied between 0 . l max , between 1 and 7. Three main sets of one-electron orbitals were employed in the present work: (A)20 s p d f g h with n a = 44 and E max = 3, (B)14 s p d f g h i k with n a = 34 and E max =0 .
5, and (C) 25 s p d with n a = 54 and E max = 6.Here, the notation, e.g. , 20 p means 20 p / p / . Cal-culational results were first obtained with the set (A)and then corrected for contributions of the higher partialwaves with the set (B) and for a more complete represen-tation of the Dirac spectrum with the set (C). The set ofCSFs used in the calculation was obtained by taking allsingle, double, and triple excitations from the referenceconfiguration with at least one electron orbital with l ≤ Breit interaction into the Dirac-CoulombHamiltonian yields only a small correction in the caseof Li and Be + . Since the effect is small, it is sufficientto use a much shorter basis set for its evaluation, whichsimplifies the computation greatly.The results of our CI calculation of the Dirac-Coulomb-Breit part of the ground-state hfs in Li and Be + arepresented in Table I. The Fermi model was employedfor the nuclear-charge distribution, with the nuclear-charge radii [11] < r > / = 2 .
9, 1, and 1 . α , the finite nuclear-charge distribution (FNC) correctionwas evaluated separately and subtracted from the CI val-ues. The point-nucleus results thus obtained were fittedto a polynomial in α , assuming the absence of the linearterm. In this way, the CI results with the physical valueof α were separated into three parts: the nonrelativisticpoint-nucleus contribution, the relativistic point-nucleuscorrection, and the FNC correction. The numerical re-sults for them are listed in Table II.The FNC correction was evaluated for both the hy-drogenic wave functions and the CI many-electron wavefunctions. In the latter case, a series of the CI calcula-tions with different values of the nuclear-charge radius R was performed and the FNC correction was extracted bya fit, using the analytical form of the R dependence [8].It was found that, with an accuracy of ∼ QED effects induce the largest correction to beadded to the Dirac-Coulomb-Breit hfs value. For nS states of few-electron atoms, the QED correction can bewritten in the same form as for hydrogen [15], δG n ( Z ) = απ G NR n ( Z ) (cid:26)
12 +
Zα π (cid:18) ln 2 − (cid:19) +( Zα ) (cid:20) −
83 ln ( Zα ) + a ln( Zα ) + a (cid:21)(cid:27) , (3)where G NR n is the nonrelativistic hfs value. The firstthree coefficients in the Zα expansion (3) are the sameas for hydrogen. The higher-order terms a and a are different and not known at present. One can, how-ever, estimate them with their hydrogenic values [16, 17]: a (2 s ) = − . a (2 s ) = 11 . a (3 s ) = − . a (3 s ) = 9 . ∼
40% due to the neglect ofthe binding QED effects [i.e., the terms in (3) beyond thefirst one].The nuclear structure effects have significant influenceon hfs and should be taken into account. Their rigor-ous description is a demanding problem. The way forits solution was paved in recent studies [4, 18]. Practi-cal realizations of this approach, however, are so far re-stricted by two- and three-nucleon systems [18] and their
TABLE II: Individual contributions to the hfs splitting, in terms of G ( Z ). The experimental values for the function G for Liwere inferred from the original references by using the nuclear magnetic moment µ/µ N = 3 .
256 426 8(17) [12]. Li 2 S Li 3 S Be + S Be + S Nonrelativistic 0 .
215 251 a .
168 340 a .
390 544 a .
335 066 a Ref. [6] 0 .
215 254 (4) 0 .
168 351 (13) 0 .
390 549 (9)Relativistic 0 .
000 205 a .
000 159 a .
000 664 a .
000 564 a Finite nuclear charge − .
000 055 a − .
000 043 a − .
000 139 a − .
000 119 a Dirac-Coulomb-Breit 0 .
215 401 (5) 0 .
168 456 (9) 0 .
391 069 (6) 0 .
335 510 (9)QED 0 .
000 182 (4) 0 .
000 143 (4) 0 .
000 289 (12) 0 .
000 250 (12)Bohr-Weisskopf − .
000 024 (3) − .
000 019 (2) − .
000 062 (17) − .
000 053 (14)Specific mass shift 0 .
000 002 0 .
000 002 0 .
000 002 0 .
000 002Negative-continuum 0 .
000 002 0 .
000 002 0 .
000 005 0 .
000 005Total theory 0 .
215 563 (7) 0 .
168 584 (10) 0 .
391 304 (22) 0 .
335 714 (21)Ref. [6] 0 .
215 54 (2) 0 .
168 58 (2) 0 .
391 27 (4)Experiment 0 .
215 561 1 (1) b .
168 60 (2) c .
391 260 (1) e ∗ .
171 5 (4) d .
391 240 (6) e † a These three entries are inferred from the corresponding Dirac-Coulomb-Breit values; their sum is expected to be moreaccurate than each of the entries separately; b Ref. [1]; c Ref. [13]; d Ref. [14]; e ∗ Ref. [2] with µ/µ N = − .
177 432(3) [12]; e † Ref. [2] with µ/µ N = − .
177 49(2) [12]. extension for more complex nuclei like Li and Be looksproblematic.The most widely used approach up to now is to accountfor the nuclear magnetization distribution [the Bohr-Weisskopf (BW) effect] by means of the Zemach formula[19], which is simple and apparently model independent.Such approach ignores inelastic effects, which can yielda large contribution [18], and it is not clear what uncer-tainty should be ascribed to such results. In the presentstudy, we calculate the BW correction within the single-particle (SP) nuclear model [20, 21], in which the nuclearmagnetic moment is assumed to be induced by the oddnucleon. This model is expected to be reasonably ade-quate for Li since it reproduces well the observable nu-clear magnetic moment basing on just the free-nucleon g factors, the difference being only 15%. For Be, the devi-ation is four times larger and the SP approach is expectedto yield worse results.Within the SP model, the BW effect can be accountedfor by adding a multiplicative magnetization-distributionfunction to the standard point-dipole hfs interaction [21].The distribution function is induced by the wave func-tion of the odd nucleon and is obtained by solving theSchr¨odinger equation with the Woods-Saxon potentialand an empirical spin-orbit interaction included. Theparameters of the potential were taken from [22]. TheBW correction was calculated for both the one-electronwave functions and the CI many-electron wave functions.It was found that, with a very good accuracy ( < . ∼
10% in the case of Liand by ∼
30% in the case of Be. The Zemach-formularesult of [5] for Be is larger than the one of [6] by a factorof four, which is due, we believe, to a misinterpretation ofthe Zemach formula in [5]. Our computational results forthe BW correction are presented in Table II. The error bars specified were obtained as the difference of the SPand the Zemach values and should be regarded as order-of-magnitude estimations of the error. We checked thatsimilar evaluations of the nuclear effect on hfs in He + agree well with a much more elaborate calculation of [18].The leading recoil contribution is given by the massscaling factor (1+ m/M ) − included into the definition ofthe function G in (1). The remaining correction (withinthe nonrelativistic approach) is due to the specific massshift (SMS) and is very small for the S states. We calcu-late it by introducing the SMS term ( m/M ) P i