Hyperfine structure of S states in Li and Be^+

0.12 MB·English

by  V. A. Yerokhin

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
Alarge-scaleconfiguration-interaction(CI)calculationisreportedforthehyperfinesplittingofthe
22S and32S statesof7Liand9Be+. TheCIcalculation basedontheDirac-Coulomb-BreitHamil-
tonian is supplemented with a separate treatment of the QED, nuclear-size, nuclear-magnetization
distribution, and recoil corrections. The nonrelativistic limit of the CI results is in excellent agree-
8 ment with variational calculations. The theoretical values obtained for the hyperfine splitting are
0 complete to therelative order of α2 and improve upon results of previousstudies.
0
2 PACSnumbers: 31.15.Ar,32.10.Fn,31.30.Gs

n
a
The hyperfine structure (hfs) of few-electron atoms dimensionless function G (Z) defined as [8]
J n
has been an attractive subject of theoretical studies for
8 decades, one of the reasons being a few ppm accuracy 4 α(Zα)3 m µ 2I+1 mc2
2 ∆E = G (Z),
achieved in experiments on Li and Be+ [1, 2]. Inter- n 3 n3 mp µN 2I (1+m/M)3 n
] est in this topic was enhanced even further recently, in (1)
h
view of prospects of using hfs data to get an access to where µ is the nuclear magnetic moment, µ =
p N
the neutron halo structure, the proton charge distribu- |e|/(2m ) is the nuclear magneton; m, m , and M are
- p p
m tion, and the nuclear vector polarizability, particularly the masses of the electron, the proton, and the nucleus,
o for isotopes of Be+ [3, 4]. Despite the considerable at- respectively; I is the nuclear spin quantum number, and
t tention received, a high-precision theoretical description Z is the nuclear charge number. The function G defined
a
ofhfs infew-electronatomsremainsadifficulttask. The in this way is unity for a non-relativistic point-nucleus
.
s mainproblemliesinthehighsingularityofthehfs inter- H-like atom.
c
i action and, as a consequence, in the dependence of the Withintheleadingrelativisticapproximation,theelec-
s calculated results on the quality of the correlated wave troncorrelationcanbe describedbythe Dirac-Coulomb-
y
function near the nucleus. Breit equation, which is solved by the configuration-
h
p interaction (CI) Dirac-Fock (DF) method in the present
Among numerous theoreticalinvestigations performed
[ work. The many-electron wave function Ψ(PJM) with
previouslyforLiandBe+,twoapparentlymostaccurate
the parity P, the momentum quantum number J, and
1 ones are the multiconfigurational Dirac-Fock (MCDF)
the momentum projection M is represented as a sum of
v calculation [5] and the Hylleraas-type variational calcu-
9 configuration-state functions (CSFs),
lation [6]. Both studies report good agreement with the
5
experiment,buttheyarenotentirelyconsistentwitheach
3 Ψ(PJM)= c Φ(γ PJM). (2)
otherintreatmentofindividualcorrections. The MCDF r r
4
. calculationdoesnotincludethebindingQEDeffectsand, Xr
1
in the case of Li, the nuclear magnetization distribution
0 The CSFs are obtained as linear combinations of the
effect. The variationalcalculationyieldsaccurateresults
8 Slaterdeterminantsconstructedfromthepositive-energy
0 for the nonrelativistic Fermi contact term but treats the solutions of the Dirac equation with the frozen-core DF
: relativistic effects in an effective way only, by rescaling potential. The mixing coefficients c are determined by
v r
thehydrogenicresult. Thisindicatesthatneitherofthese
i diagonalizing the Hamiltonian matrix. The hfs splitting
X studies is complete at the relative order of α2 (α is the
is obtained as the expectation value of the hfs operator
r fine-structure constant). The aim of the present inves- onthemany-electronwavefunction(2). Thecorrespond-
a tigation is to perform a high-precision calculation of the
ing formulas are well-known, see, e.g., [5]. To perform
hfs splitting in Li and Be+, with a complete treatment
a CI calculation, we devised a code, incorporating and
of all corrections ∼α2.
adapting a number of existing packages [9] for setting
uptheCSFs,calculatingangular-momentumcoefficients,
Apossiblewaytoaccomplishthistaskwouldbetosup-
and diagonalizing the Hamiltonian matrix. The largest
plementthenonrelativisticcalculation[6]witharigorous
numberofCSFssimultaneouslyhandledwasaboutahalf
evaluationoftherelativisticcorrection,whoseexpression
of a million, with the number of nonzero elements in the
was recently derived by Pachucki [7]. Such a calculation
Hamiltonianmatrixofabout5billions. Athoroughopti-
has not been performed so far. In the present work, the
mizationofthecodewascarriedout,inordertokeepthe
relativistic correction will be accounted for by means of
time and memory consumption of the calculation within
the Dirac-Coulomb-BreitHamiltonian.
reasonable limits.
The magnetic dipole hfs splitting of an energy level The dominant part of the hfs splitting in light atoms
of an nS state is conveniently represented in terms of a is delivered by the Dirac-Coulomb Hamiltonian. This
2
cantly from the MCDF values [5]. The comparisonleads
TABLEI:TheDirac-Coulomb-Breit partofthehfssplitting,
us to a conclusion that the dominant part of the rel-
in terms of G(Z).
ativistic correction can indeed be accounted for by an
lmax Li 22S Be+ 22S effective scaling of the hydrogenic results, as was argued
Coulomb 1 0.2144703 0.3901599 in [6]. A complete evaluation of the relativistic correc-
2 0.2151678 0.3907986 tion within the Zα-expansion approach, however,has to
3 0.2153044 0.3909387 be performed along the way paved in [7], which has not
4 0.2153462 0.3909847 been done yet.
5 0.2153629 0.3910038 Comparison with the results of [6] would become pos-
6 0.2153719 0.3910145
sible on a much higher level of accuracy if we identified
7 0.2153765 0.3910202
thenonrelativisticpartofourCIresults. Suchanidenti-
∞ 0.2153848(49) 0.3910304(61)
fication was carriedout by repeating our CI calculations
Breit 0.0000159 0.0000386
for different values of α (namely, three values with ra-
Total 0.2154007(49) 0.3910690(61)
′
tios α/α=0.9, 1, and 1.1 were used). For each value of
MCDF [5] 0.215287 0.390984
Hylleraasa [6] 0.215379(13) 0.391023(34) α,thefinitenuclear-chargedistribution(FNC)correction
a wasevaluatedseparatelyandsubtractedfromtheCIval-
thesum of the nonrelativistic, therelativistic, and the
ues. The point-nucleus results thus obtained were fitted
nuclear-charge distribution terms.
to a polynomial in α, assuming the absence of the linear
term. In this way, the CI results with the physical value
of α were separated into three parts: the nonrelativistic
was the most demanding part of the calculation since a
point-nucleus contribution, the relativistic point-nucleus
high relative precision was required. The one-electron
correction, and the FNC correction. The numerical re-
orbitals for constructing CFSs were obtained by the
sults for them are listed in Table II.
dual-kinetic-balance (DKB) B-spline basis set method
The FNC correction was evaluated for both the hy-
[10] for the Dirac equation. For a given number of B-
drogenic wave functions and the CI many-electron wave
splines n , all eigenstates were taken with the energy
0 < ε ≤amc2(1 + ZαEmax) and the orbital quantum functions. In the latter case, a series of the CI calcula-
tionswithdifferentvalues ofthe nuclear-chargeradiusR
number l ≤ lmax, where Emax was varied between 0.5
wasperformedandtheFNC correctionwasextractedby
and6andlmax,between1and7. Threemainsetsofone-
a fit, using the analytical form of the R dependence [8].
electronorbitalswereemployedinthe presentwork: (A)
Itwasfoundthat,withanaccuracyof∼0.5%,therewas
20s20p19d19f18g18h with na =44 and Emax =3, (B) noscreeningeffectontherelativevalueofthiscorrection.
14s14p14d13f13g13h12i12kwithna =34andEmax = The QED effects induce the largest correction to be
0.5, and (C) 25s25p24d with na = 54 and Emax = 6. added to the Dirac-Coulomb-Breit hfs value. For nS
Here, the notation, e.g., 20p means 20p 20p . Cal-
1/2 3/2 states of few-electron atoms, the QED correction can be
culational results were first obtained with the set (A)
written in the same form as for hydrogen [15],
andthencorrectedforcontributionsofthehigherpartial
waveswiththeset(B)andforamorecompleterepresen- α 1 5
δG (Z) = GNR(Z) +Zαπ ln2−
tationofthe Dirac spectrumwith the set(C).The setof n π n (cid:26)2 (cid:18) 2(cid:19)
CSFs used in the calculation was obtained by taking all
8
single, double, and triple excitations from the reference +(Zα)2 − ln2(Zα)+a21ln(Zα)+a20 , (3)
(cid:20) 3 (cid:21)(cid:27)
configurationwithatleastoneelectronorbitalwithl≤1
present. The triple excitations that were left out in this where GNR is the nonrelativistic hfs value. The first
n
way were found to yield a negligible contribution. In- three coefficients in the Zα expansion (3) are the same
clusion of the Breit interaction into the Dirac-Coulomb as for hydrogen. The higher-order terms a21 and a20
Hamiltonian yields only a small correction in the case are different and not known at present. One can, how-
of Li and Be+. Since the effect is small, it is sufficient ever,estimatethemwiththeirhydrogenicvalues[16,17]:
to use a much shorter basis set for its evaluation, which a21(2s) = −1.1675, a20(2s) = 11.3522, a21(3s) =
simplifies the computation greatly. −2.3754, and a20(3s) = 9.7474. A 100% uncertainty
TheresultsofourCIcalculationoftheDirac-Coulomb- is ascribed to this approximation. Essentially the same
Breit part of the ground-state hfs in 7Li and 9Be+ are treatmentofthe QEDcorrectionwasreportedin[6]; the
presented in Table I. The Fermi model was employed QED results of [5] differ by ∼40% due to the neglect of
for the nuclear-charge distribution, with the nuclear- thebindingQEDeffects[i.e.,thetermsin(3)beyondthe
charge radii [11] <r2>1/2= 2.431(28) fm for Li and first one].
<r2>1/2= 2.518(11) fm for Be. The uncertainties spec- The nuclear structure effects have significant influence
ified in the table include the estimated error due to the on hfs and should be taken into account. Their rigor-
incompleteness of the basis and due to the finite nuclear ous description is a demanding problem. The way for
size. The error of the Breit part was found to be negli- its solution was paved in recent studies [4, 18]. Practi-
gible. Our results are in reasonable agreement with the cal realizations of this approach, however, are so far re-
nonrelativistic variational results [6] but deviate signifi- strictedbytwo-andthree-nucleonsystems[18]andtheir
3
TABLE II: Individual contributions to the hfs splitting, in terms of G(Z). The experimental values for the function G for Li
were inferred from theoriginal references byusing thenuclear magnetic moment µ/µN =3.2564268(17) [12].
7Li 22S 7Li 32S 9Be+ 22S 9Be+ 32S
Nonrelativistic 0.215251a 0.168340a 0.390544a 0.335066a
Ref. [6] 0.215254(4) 0.168351(13) 0.390549(9)
a a a a
Relativistic 0.000205 0.000159 0.000664 0.000564
Finite nuclear charge −0.000055a −0.000043a −0.000139a −0.000119a
Dirac-Coulomb-Breit 0.215401(5) 0.168456(9) 0.391069(6) 0.335510(9)
QED 0.000182(4) 0.000143(4) 0.000289(12) 0.000250(12)
Bohr-Weisskopf −0.000024(3) −0.000019(2) −0.000062(17) −0.000053(14)
Specificmass shift 0.000002 0.000002 0.000002 0.000002
Negative-continuum 0.000002 0.000002 0.000005 0.000005
Total theory 0.215563(7) 0.168584(10) 0.391304(22) 0.335714(21)
Ref. [6] 0.21554(2) 0.16858(2) 0.39127(4)
Experiment 0.2155611(1)b 0.16860(2)c 0.391260(1)e∗
0.1715(4)d 0.391240(6)e†
a
These threeentries are inferred from thecorresponding Dirac-Coulomb-Breit values; theirsum is expected to be more
accurate than each of theentries separately; b Ref. [1]; c Ref. [13]; d Ref. [14]; e∗ Ref. [2] with µ/µN =−1.177432(3) [12];
e† Ref. [2] with µ/µN =−1.17749(2) [12].
extensionformore complex nucleilike 7Liand9Be looks bars specified were obtained as the difference of the SP
problematic. and the Zemachvalues andshould be regardedas order-
Themostwidelyusedapproachuptonowistoaccount of-magnitude estimations of the error. We checked that
for the nuclear magnetization distribution [the Bohr- similar evaluations of the nuclear effect on hfs in 3He+
Weisskopf(BW) effect] by means of the Zemachformula agreewellwithamuchmoreelaboratecalculationof[18].
[19], which is simple and apparently model independent. The leading recoil contribution is given by the mass
Such approach ignores inelastic effects, which can yield scalingfactor(1+m/M)−3includedintothedefinitionof
a large contribution [18], and it is not clear what uncer- the function G in (1). The remaining correction (within
tainty should be ascribed to such results. In the present the nonrelativistic approach) is due to the specific mass
study, we calculate the BW correctionwithin the single- shift(SMS) andis verysmallfor theS states. We calcu-
particle(SP)nuclearmodel[20,21],inwhichthenuclear late it by introducing the SMS term (m/M) i<jpi·pj
magnetic moment is assumed to be induced by the odd into the Dirac-Coulomb Hamiltonian and taPking the in-
nucleon. This model is expected to be reasonably ade- crement of the CI results with and without SMS (p is
quate for 7Li since it reproduces well the observable nu- the momentum operator). Ourresults agreewith the es-
clearmagnetic moment basingon just the free-nucleong timates obtained in [6] but are more accurate. For the
factors,thedifferencebeingonly15%. For9Be,thedevi- 22S and 32S states of Li, we obtain δG=2.0(2)×10−6
ationisfourtimeslargerandtheSPapproachisexpected and 1.9(2)× 10−6, respectively, which should be com-
to yield worse results. paredwith2(5)×10−6and2(20)×10−6from[6],respec-
tively.
Withinthe SPmodel,the BWeffectcanbeaccounted
forbyaddingamultiplicativemagnetization-distribution Thenegative-continuumcontributionmightbeofsome
functiontothestandardpoint-dipolehfsinteraction[21]. importance in calculations involving the operators that
The distribution function is induced by the wave func- mix the upper and the lower components of the Dirac
tion of the odd nucleon and is obtained by solving the wavefunction. Thehfsoperatorisofthiskind,sowehave
Schr¨odinger equation with the Woods-Saxon potential to obtain an estimation for this correction. We calculate
and an empirical spin-orbit interaction included. The the negative-continuum contribution by employing the
parameters of the potential were taken from [22]. The many-body perturbation theory to the first order. The
BW correction was calculated for both the one-electron same one-electron DKB basis set was used as in the CI
wavefunctionsandtheCImany-electronwavefunctions. calculations, with the only difference that all negative-
It was found that, with a very good accuracy (< 0.5%), energy eigenstates were taken.
there was no screeningeffect onthe relativevalue ofthis Our total theoretical values for the hfs splitting pre-
correction. Our calculational results are larger than the sented in Table II agree with the results by Yan et al.
Zemach-formula values of [6] by ∼10% in the case of Li [6] but are more accurate. The present theory agrees
and by ∼30% in the case of Be. The Zemach-formula very well with experiments on Li, the only exception be-
resultof[5]forBeislargerthantheoneof[6]byafactor ing the experimental result [14], which contradicts both
offour,whichisdue,webelieve,toamisinterpretationof the theory and the result of a more recent measurement
theZemachformulain[5]. Ourcomputationalresultsfor [13]. The comparisonoftheoreticalcalculationswiththe
the BW correction are presented in Table II. The error high-precision experiment for the ground state of Be+
4
[2] is complicated by the existence of two different val- reported nonrelativistic values but are more accurate
ues for the nuclear magnetic moment [12]. The smaller due to a rigorous treatment of the relativistic correc-
value yields a better agreement with our theoretical re- tion. The QED, nuclear magnetization distribution, re-
sult, but there is still a 2σ deviation present. Having in coil, and negative-continuum corrections were evaluated
mind that the experimentalresults for the magnetic mo- separately and added to the Dirac-Coulomb-Breitvalue.
mentareinsignificantdisagreementwitheachother,one Detailed comparison with the earlier calculations were
can surmise the presence of underestimated systematic made and some inconsistencies in their previous treat-
effects in one or both of these measurements. We thus ment of individual corrections were revealed. The cal-
employ the comparison presented in Table II to infer an culational results for Li are in good agreement with the
independent value of the magnetic moment, which reads experimental data. For Be+, the theoretical prediction
µ(9Be)/µ =−1.17730(6)andissomewhatsmallerthan deviatesfromtheexperimentalvalueby2or3σ,depend-
N
the both values from [12]. ing on the value of the nuclear magnetic moment used.
In summary, we have performed a large-scale CI cal- KrzysztofPachuckiisgratefullyacknowledgedforsug-
culation of the hfs splitting of the 22S and 32S states gesting the topic of this investigation and valuable dis-
of Li and Be+. The results obtained from the Dirac- cussions. The work was supported by the RFBR grant
Coulomb-Breit Hamiltonian agree with the previously no. 06-02-04007and by the foundation “Dynasty”.
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