Theoretical study of lifetimes and polarizabilities in Ba+

0.17 MB·English

by  E. Iskrenova-Tchoukova

Theoretical study of lifetimes and polarizabilities in Ba+
E. Iskrenova-Tchoukova and M. S. Safronova
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716
(Dated: February 3, 2008)
The6s−np (n=6−9)electric-dipolematrixelementsand6s−nd (n=5−7)electric-quadrupole
j j
matrix elements in Ba+ are calculated using therelativistic all-order method. The resulting values
are used to evaluate ground state dipole and quadrupole polarizabilities. In addition, the electric-
dipole6pj−5dj′ matrix elementsand magnetic-dipole 5d5/2−5d3/2 matrix element arecalculated
using the same method in order to determine the lifetimes of the 6p , 6p , 5d , and 5d
8 1/2 3/2 3/2 5/2
levels. The accuracy of the 6s−5d matrix elements is investigated in detail in order to estimate
0 j
the uncertainties in the quadrupole polarizability and 5d lifetime values. The lifetimes of the 5d
0 j
2 states in Ba+ are extremely long making precise experiments very difficult. Our final results for
dipole and quadrupole ground state polarizabilities are αE1 = 124.15 a30 and αE2 = 4182(34) a50,
n respectively. Theresultinglifetimevaluesareτ =7.83ns,τ =6.27ns,τ =81.5(1.2)s,
a 6p1/2 6p3/2 5d3/2
andτ =30.3(4) s. The extensivecomparison with othertheoretical and experimentalvaluesis
J 5d5/2
carried out for both lifetimes and polarizabilities.
6
2

I. INTRODUCTION lent system allowing for precise theoretical predictions,
]
h and, in some cases, for evaluation of the theoretical un-
p The atomic properties of Ba+ ion are of particular certainties that do not directly rely on the comparison
- with experiment. It is also an excellent testing case for
m interest owing to the prospects of studying the parity
further studies of Ra+ ion, where the correlationcorrec-
nonconservation (PNC) with a single trapped ion [1].
o Progress on the related spectroscopy with a single Ba+ tions are expected to be larger owing to a largercore. A
t
a ion is reported in [2, 3], and precision measurements of project to measure PNC in a single trapped radium ion
s. light shifts in a single trapped Ba+ ion have been re- recentlystartedattheAcceleratorInstitute(KVI)ofthe
c portedin[4]. ThePNCinteractionsgivesrisetonon-zero University of Groningen [16].
i
s amplitudes for transitions that are otherwise forbidden Inthiswork,wecalculate6s−npj (n=6−9),6pj−5dj′
y bytheparityselectionrules,suchas6s−7selectric-dipole electric-dipole matrix elements, 6s − ndj (n = 5 − 9)
h
p transition in Cs. The study of parity nonconservation electric-quadrupole matrix elements, and 5d5/2 −5d3/2
in cesium [5, 6] involving both high-precision measure- magnetic-dipole matrix element in Ba+. This set of ma-
[
ments and several high-precision calculations provided trixelementsisneededforaccuratecalculationofground

1 anatomic-physicstestofthe standardmodelofthe elec- statedipoleandquadrupolepolarizabilitiesandlifetimes
v
troweak interactions and yielded the first measurement of the 6p , 6p , 5d , and 5d levels. We carefully
0 1/2 3/2 3/2 5/2
of the nuclear anapole moment (see [7] for the review investigate the uncertainty in our values of 6s−5d ma-
6 j
0 of study of fundamental symmetries with heavy atoms). trix elements in order to estimate the uncertainties in
4 The analysis of the Cs experiment, which requireda cal- thequadrupolepolarizabilityandthe5dj lifetimevalues.
. culation of the nuclear spin-dependent PNC amplitude, It is particularly important to independently determine
1
ledtoconstraintsonweaknucleon-nucleoncouplingcon- these uncertainties because of significant inconsistencies
0
8 stants that are inconsistent with constraints from deep between different measurements of the 5d3/2 and 5d5/2
0 inelastic scattering and other nuclear experiments [8]. lifetimes [15, 17, 18, 19, 20, 21, 22]. There are also
: More PNC experiments in other atomic systems, such largediscrepanciesbetweenexperimentaldeterminations
v
as Ba+, are needed to resolve this issue. The prospects ofthe5d−6squadrupolematrixelementfromthelifetime
i
X formeasuringparityviolationinBa+ havebeenrecently experimentsandstudiesoftheRydbergstatesofbarium
r discussed in [3]. [12,13, 14]. The experimentalvaluesofthe groundstate
a
Ba+ is also of particular interest for developing op- quadrupole polarizability from Refs. [12, 14, 23] differ
tical frequency standard [9] and quantum information by a factor of two; our value of the quadrupole polar-
processing [10, 11] owing to the extremely long lifetimes izability is in agreement with Ref. [14]. We note that
of 5d states. The accuracy of optical frequency stan- there are no inconsistencies between the experimental
dards is limited by the frequency shift in the clock tran- lifetimes [24, 25, 26] of the 6pj levels and experimen-
sitions caused by the interaction of the ion with exter- taldeterminationsofthe electric-dipolegroundstate po-
nal fields. Therefore, knowledge of atomic properties is larizability [12, 14, 23]. The experimental values of the
needed for the analysis of the ultimate performance of electric-dipolepolarizabilityofthe Ba+ ioninits ground
such frequency standard. state [12, 14, 23] are also in agreement with each other
AnothermotivationforstudyofBa+isanexcellentop- and our theoreticalvalue. Our lifetimes of the 6p1/2 and
portunityfortestsoftheoreticalandexperimentalmeth- 6p3/2 levels are in agreement with experimental values
ods,inparticularinlightofrecentmeasurementsofBa+ [24, 25, 26] within expected accuracy (1%).
atomicproperties[2,3,12,13,14,15]. Ba+ isamonova- The paper is organized as follows. In Section II, we
2
give a short description of the method used for the cal- 9inasphericalcavityofradius80a.u. Suchcavitysizeis
culation of the matrix elements. In Section III, we dis- chosen to accurately represent all orbitals of interest to
cuss the calculation of the electric-dipole polarizability the present study. The resulting excitation coefficients
and conduct comparative analysis of the correlationcor- ρ , ρ , ρ , and ρ are used to calculate the
ma mv mnab mnva
rections to the ns−np matrix elements in Ba+, Cs, and one-body E1, M1, and E2 matrix elements.
Ca+. The 6s−5d quadrupole matrix elements and the The SD all-order method yielded results for the pri-
ground state quadrupole polarizability are discussed in mary ns−np E1 matrix elements of alkali-metal atoms
j
Section IV, and the lifetimes are discussed in Section V. thatareinagreementwithexperimentto0.1%-0.5%[28].
A consistency study of the 5d lifetime and groundstate Wenotethatwhiletheall-orderexpressionforthematrix
j
quadrupole polarizability measurements in Ba+ is pre- elements contains 20 terms that are linear or quadratic
sented in Section IV. functions of the excitation coefficients, only two terms
are dominant for all matrix elements considered in this
work:
II. METHOD
Z(a) = (z ρ˜ +z ρ˜∗ ) (3)
am wmva ma vmwa
We calculate the reduced multipole matrix elements Xma
using the relativistic all-order method [27, 28, 29] which
and
is a linearized coupled-cluster method where all single
and double excitations of the Dirac-Fock wave function Z(c) = (z ρ +z ρ∗ ), (4)
wm mv mv mw
are included to all orders of perturbation theory. The
m
presentimplementation of the method is suitable for the X
calculationofmatrixelementsofanyone-bodyoperator, where ρ˜mnab = ρmnab−ρnmab and zwv are lowest-order
i.e., the calculations of the E1, E2, and M1 matrix el- matrix elements of the corresponding operator. In the
ements are carried out in the same way. We refer the caseoftheelectric-quadrupoletransitionsstudiedinthis
reader to the review [29] and references therein for the work,the second term Z(c) is overwhelmingly(by an or-
detailed description of the all-order method. der of magnitude) larger than any other term. In such
Briefly, our starting point is the relativistic no-pair cases,it wasfoundnecessaryto include atleastpartially
Hamiltonian [30] expressed in second quantization as triple excitations into the wave function
H = ǫi :a†iai :+12 gijkl :a†ia†jalak :, (1) |ΨSvDpTi=|ΨSvDi+ 61 ρmnrvaba†ma†na†rabaaav|Φvi
Xi Xijkl mXnrab (5)
where a†,a are single-particle creation and annihilation and to correct single excitation coefficient ρmv equation
i j fortheeffectoftripleexcitations[28,32,33,34]. Wehave
operators,respectively, ǫ is the Dirac-Fock (DF) energy
i
conductedsuchacalculationforthe6s−5d ,6s−6d ,and
forthe state i,g arethe two-bodyCoulombintegrals, j j
ijkl
6s−7d electric-quadrupolematrixelementsandreferto
and : : indicates normal order of the operators with re- j
the corresponding results as SDpT values (i.e. including
specttotheclosedcore. Thesingle-double(SD)all-order
all single, double, and partial triple excitations).
wave function is written as
We note that such approachworkspoorly when terms
Z(a) and Z(c) are of similar order of magnitude (such as
1
|ΨSvDi= 1+ ρmaa†maa+ 2 ρmnaba†ma†nabaa all E1 transition considered here) owing most likely to
ma mnab cancellation of high-order corrections to terms Z(a) and
X X
Z(c). The term Z(a) is not directly corrected for triple
+ ρ a† a + ρ a† a†a a |Φ i (2) excitationsinthe SDpT extensionofthe methodleading
mv m v mnva m n a v v to consistent treatment of the higher-order correlations
m6=v mna
X X
only when the second term is overwhelminglydominant.

where |Φ i is the lowest-orderwavefunction takento be WereferthereadertoRef.[35]forthedetaileddiscussion
v
the frozen-coreDF wavefunction ofa state v. Indices at of triple excitations. The results of the matrix element
the beginning ofthe alphabet, a,b, ···, referto occupied calculation are discussed in the following sections.
corestates,thoseinthemiddleofthealphabetm,n,···,
refertoexcitedstates,andindexvdesignatesthevalence
orbital. The all-order equations for the excitation coeffi- III. BA+ GROUND STATE DIPOLE
cients ρ , ρ , ρ , and ρ are solved iteratively POLARIZABILITY
ma mv mnab mnva
with a finite basis set, and the correlationenergy is used
as a convergence parameter. The basis set is defined The ground state dipole or quadrupole polarizability
in a spherical cavity on non-linear grid and consists of can be represented as a sum of the valence polarizabil-
single-particlebasisstates which arelinear combinations ity α and the polarizability of the ionic core α [28].
v core
ofB-splines[31]. Weuseabasissetof50splinesoforder Thecalculationofthecorepolarizabilityassumesallowed
3
excitations to any excited state including the valence
TABLEI:Contributions to theground state6s scalar dipole
shell,whichrequirestheintroductionofthesmallcounter
polarizability α in Ba+ in units of a3. Comparison with
terms α to subtract out 1/2 of the contribution corre- E1 0
vc experiment and other calculations. The absolute values of
sponding to the 6s shell excitation [28]. The core polar-
corresponding SD all-order reduced electric-dipole matrix el-
izabilities havebeen calculatedinrandom-phaseapprox- ementsd (in a.u.) are also given.
imation (RPA) in Ref. [36]. The accuracy of the RPA
values is expected to be on the order of 5%[34]. We cal- Contribution d αE1
culated the αvc term the in the RPA approximation for 6s−6p1/2 3.3357 40.18
consistencywithαcore value. The valencedipole polariz- 6s−6p3/2 4.7065 73.82
ability for the 6s state of Ba+ is calculated as sum-over- 6s−7p 0.0621 0.06
1/2
states 6s−7p 0.0868 0.01
3/2
1 |h6s||d||np i|2 |h6s||d||np i|2 α 0.03
1/2 3/2 tail
α = + .(6)
v,E1 3 n Enp1/2 −E6s Enp3/2 −E6s ! αcore 10.61
X α -0.51
vc
ThesumovertheprincipalquantumnumberninEq.(6)
Total 124.15
converges very rapidly and very few first terms have to
Expt. [14] 123.88(5)
be calculated to high precision. In this work, we use SD
Expt. [23] 125.5(10)
all-order matrix elements and experimental energies for
terms with n = 6−9 and evaluate the remainder α Theory [41] 123.07
tail
in the Dirac-Fock approximation. The contributions to Theory [42] 126.2
the dipole polarizability are summarized in Table I. We Theory [43] 124.7
alsolisttheabsolutevaluesofcorrespondingSDall-order
reducedelectric-dipolematrix elements d. The contribu-
tionofthetermswithn=6isoverwhelminglydominant.
Therefore,theuncertaintyinourcalculationofthedipole
polarizability is dominated by the uncertainties in the TABLEII:ContributionsofdifferenttermstotheBa+,Ca+,
and Cs ns−np reduced matrix elements in a.u.
6s−6p and 6s−6p matrix elements. 1/2
1/2 3/2
Tostudytheuncertaintyinthesevalues,weinvestigate
Contribution Ba+ Cs [37] Ca+
theimportanceofthecontributionsfromvariouscorrela-
6s−6p 6s−6p 4s−4p
tion correction terms and the overall size of the correla- 1/2 1/2 1/2
tion correction. The contributions to the 6s−6p ma- DF 3.891 5.278 3.201
1/2
trixelementaresummarizedinTableII. Thebreakdown Z(a) -0.387 -0.334 -0.200
of the contributions to the 6s−6p matrix element is Z(c) -0.209 -0.485 -0.120
3/2
essentially the same, and we do not list it here. We also Other 0.041 0.019 0.016
give the breakdown of the correlation correction for the Total 3.336 4.478 2.898
same transition in Cs and 4s−4p transition in Ca+.
1/2 Correlation 16.6% 17.9% 10.5%
Cs values are taken from Ref. [37]. Final Ca+ value has
been published in Ref. [38]. As we noted in Section II,
only two terms give large contributions to the correla-
tioncorrection. Whiletherearesomecancellationsinthe
other terms, all them are at least an order of magnitude and later experiments confirmed the theory values. We
smaller. Unfortunately, there is no straightforward way refer the reader to Ref. [38] for more detailed discussion
to evaluate the uncertaintyin theZ(a) term (aswe show of this issue. It would have been very interesting to see
inthelatersectionitcanbedoneforZ(c)). Therefore,we the 4p lifetimes in Ca+ remeasured to resolve this issue.
cannotmakeanuncertaintyestimatethatisindependent Based on the similar size of the correlation corrections
onexperimentalobservations. However,we note that Cs for Cs and Ba+, we expect similar accuracy of our data
6s−6p transitionsareextremelywellstudiedbyanum- (on the order of 0.5%). Therefore, the resulting accu-
j
ber of different experimental approaches (see, for exam- racy of our dipole polarizability is expected to be on the
ple, [39] and references therein), and all-order SD data orderof1%. We findthatourvalueis inexcellentagree-
are in agreement with Cs experimental values to 0.2%- ment with both experimental values [14, 23] when our
0.4% [28]. The breakdown of terms for Ba+ is slightly estimated uncertainty is taken into account. Our results
different than for Cs but is very similar to Ca+. As ex- areingoodagreementwithothertheoreticalcalculations
pected, the size of the correlations is larger in Ba+ than [41, 42, 43]. We also note that the h6s|d|6pi matrix ele-
in Ca+. Unfortunately, there is only one high-precision menthasbeenrecentlyextractedfromtheK splittingsof
measurement of the 4p Ca+ lifetimes [40] that is in sig- thebound6snlstatesinRef.[13],andtheresultingvalue
j
nificant (2%) disagreement with high-precision theoreti- h6s|d|6pi = 4.03(12) is in excellent agreement with our
cal results. Similar discrepancies existed for the alkali- result h6s|d|6pi = 4.08 (normalized spherical harmonics
metalatommeasurementsdonewiththe sametechnique C is factored out here for comparison).
1
4
TABLE III: Absolute values of electric-quadrupole 6s−5d and 6s−5d reduced matrix elements in Ba+ calculated in
3/2 5/2
different approximations in a.u. Columns labeled “DF” and “III” are lowest-order Dirac-Fock and third-order MBPT values,
respectively. The third-order results calculated with maximum number of partial values l =6 and l =10 are given to
max max
illustrate the contribution of the higher partial waves. Breit correction is given separately. The all-order ab initio results are
given in columns labeled “SD” and “SDpT”, respectively; these results include contributions from higher partial waves and
Breit correction. The corresponding scaled values are listed in columns labeled “SD ” and “SDpT ”. The calculation of the
sc sc
uncertainties of the finalvalues is described in detail in text.
Transition DF III (l =6) III (l =10) Breit SD SDpT SD SDpT Final
max max sc sc
6s−5d 14.76 11.82 11.75 -0.07 12.42 12.66 12.63 12.59 12.63(9)
3/2
6s−5d 18.38 14.86 14.78 -0.09 15.55 15.84 15.80 15.76 15.80(11)
5/2
in Section II. We also carried out semi-empirical scal-
TABLEIV:Contributionstothegroundstate6squadrupole
ing in both approximations by multiplying single exci-
polarizability α in Ba+ and their uncertainties in units of
E2 tation coefficients ρ by the ratio of the “experimen-
a5. The absolute values of corresponding all-order reduced mv
0 tal” and corresponding (SD or SDpT) correlation ener-
electric-quadrupole matrix elements Q (in a.u.) and their
gies [32]. The “experimental” correlation energies are
uncertainties are also given.
determined as the difference of the total experimental
Contribution Q αE2 energy and the DF lowest-order values. The calculation
6s−5d 12.63(9) 1436(20) of the matrix elements is then repeated with the modi-
3/2
6s−6d 16.83(5) 270(2) fied excitation coefficients. The accuracy of such scaling
3/2
6s−7d 5.68(5) 23.7(4) procedure for the similar cases was discussed in detail
3/2
6s−8d 3.09(6) 6.3(3) in Refs. [33, 34, 44]. The reasoning for such a scaling
3/2
procedure in third-order perturbation theory (scaling of
6s−9d 2.07(4) 2.7(1)
3/2 the self-energy operator)has been discussed in Ref. [45].
6s−5d5/2 15.8(1) 1932(27) We list SD, SDpT, and the corresponding scaled results
6s−6d 20.30(6) 392(2) (labeled“SD ”and“SDpT ”)inTableIII. Thelowest-
5/2 sc sc
6s−7d 6.98(6) 35.7(6) orderDFresultsarelistedtoillustratethesizeofthecor-
5/2
6s−8d 3.83(8) 9.6(4) relation corrections. We demonstrate the size of the two
5/2
6s−9d 2.57(5) 4.192) othercorrections,contributionofthehigherpartialwaves
5/2
andBreit correction,in the same table. The firstcorrec-
α 24(6)
tail tionresultsfromthetruncationofthepartialwavesinall
αcore 46(2) sumsinall-ordercalculationatl =6. All-ordercalcu-
max
Total 4182(34) lationwithhighernumberofpartialwavesisunpractical.
Expt.[14] 4420(250) Therefore, we carry out the third-order MBPT calcula-
Expt.[12] 2462(361) tion (following Ref. [45]) including all partial waves up
Expt.[23] 2050(100) to lmax = 6 and lmax = 10 and take the difference of
these two values to be the contribution of the omitted
partial waves that we add to ab initio all-order results.
IV. BA+ GROUND STATE QUADRUPOLE Weverifiedthatthecontributionofthel =9−10partial
POLARIZABILITY waves is very small justifying the omission of contribu-
tions from l > 10. The Breit correction is calculated as
thedifferenceofthethird-orderresultswithtwodifferent
The valence part of the quadrupole polarizability is
basis sets. The second basis set is generated with taking
given in the sum-over-statesapproach by
into account one-body part of the Breit interaction. We
1 |h6s||Q||nd i|2 |h6s||Q||nd i|2 note that scaled values should not be corrected for ei-
3/2 5/2
αv,E2 = + .(7) ther partialwavetruncationerroror Breitinteractionto
5 E −E E −E
n nd3/2 6s nd5/2 6s ! avoid possible double-counting of the same effects. We
X
TheRPAcorevalue[36]is46a5,andtheα termisneg- take SDsc values as our final results. The uncertainty
0 vc
of the final values is calculated as follows: the uncer-
ligible. Thetermscontainingthe6s−5d and6s−5d
3/2 5/2 tainty in the Z(c) term is evaluated as the spread of the
matrix elements give overwhelmingly dominant contri-
most high-precision values (SD , ab initio SDpT, and
bution to the total values. Therefore, we study these sc
SDpT ), the remaining theoretical uncertainty in the
transitions in more detail and evaluate their uncertain- sc
Coulomb correlation correction is taken to be the same
ties. Unlike the case of the E1 transitions considered
earlier, Z(c) term contributes over 90% of the total cor- astheuncertaintyinthedominantZ(c)term. Weassume
100%uncertaintiesinthecontributionsofthehigherpar-
relation correction. Therefore, we carried out the cal-
tial waves and Breit correction. The final uncertainty of
culation using both SD and SDpT approaches described
5
the6s−5d matrixelements(0.7%)isobtainedbyadding
j
thesefouruncertaintiesinquadrature. Wenotethatthis TABLE V: Lifetimes of the 6pj and 5dj states in Ba+; com-
parison with experiment and other theory. The lifetimes of
procedurefortheuncertaintyevaluationdoesnotrelyon
the 6p states are given in ns, and the lifetimes of the 5d
the experimentalvalues with the exception of the exper- j j
states are given in s.
imental energies used for scaling.
The contributions to the ground state quadrupole po- τ (ns) τ (ns) τ (s) τ (s)
6p1/2 6p3/2 5d3/2 5d5/2
larizability are given in Table IV. While the n=5 term
Present 7.83 6.27 81.5(1.2) 30.3(4)
is dominant, the contributions of the few next terms are
Theory [46, 47] 7.99 6.39 83.7 30.8
substantial. Therefore, we carry out SD, SDpT, and
Theory [48] 7.89 6.30 81.5 30.3
both scaled calculations for the 6s−6d and 6s−7d
j j
Theory [49] 7.92 6.31 81.4 36.5
matrix elements aswelland repeatthe uncertaintyanal-
Theory [50] 80.1(7) 29.9(3)
ysis described above (we omit Breit and higher-partial
wavecorrectionsheresincesuchpreciseevaluationofthe Theory [15] 82.0 31.6
uncertainties is not needed for these transitions). The Expt. [24] 6.312(16)
6s−8dj and 6s−9dj matrix elements are calculated in Expt. [25] 7.92(8)
third-orderMBPT,andtheir accuracyis takento be 2% Expt. [26] 7.90(10) 6.32(10)
basedon the comparisonof the third-orderandall-order
Expt. [17] 17.5(4)
valuesofthe 6s−7d matrixelements. The remainderis
j Expt. [18] 48(6)
evaluated in the DF approximationand reduced by 23%
Expt. [19] 79.8(4.6)
basedon the comparisonof the DF and third-orderdata
Expt. [15] 89.4(15.6) 32.0(4.6)
for 6s−8d and 6s−9d matrix elements. Its accuracy
j j
is correspondingly taken to be 23%. Expt. [20] 47.0(16)
Our recommended value for the ground state Expt. [21] 32.0(5)
quadrupolepolarizabilityis inagreementwithin the cor- Expt. [22] 34.5(3.5)
responding uncertainties with the most recent experi-
mental work [14]. However, our value for the contribu-
tion of the 6s − 5d transitions to the quadrupole po- 1.11995×1018 S
j AE2 = E2 s−1, (9)
larizability [3368(34)] differs by about a factor of 2 from ab λ5 2j +1
a
the experimental values [12, 13, 14] obtained based on
2.69735×1013 S
the nonadiabatic effects on the Rydberg fine-structure AM1 = M1 s−1, (10)
ab λ3 2j +1
intervals. This issue and the discrepancies in the ex- a
perimental values of the quadrupole polarizabilities are
respectively, where λ is the wavelength of the transition
addressed in detail in Ref. [14]. We note that these ex- in ˚A and S is the line strength. In this work, we calcu-
perimental values of the 6s − 5d contributions to the
j lated the lifetimes of the 6p , 6p , 5d , and 5d
1/2 3/2 3/2 5/2
quadrupole polarizabilities (1524(8) [14] and 1562(93)
levelsinBa+. Theresultsarecomparedwithexperimen-
[13] in the two most recent studies) are significantly in-
talandothertheoreticalvaluesinTable V. Sincethe 6p
consistent with all high-precision calculations of the 5d
j levels are above 5d levels in Ba+, we also needed to cal-
lifetimes [15, 46, 47, 48, 49, 50] carried out by differ-
culate the SD all-order reduced matrix elements for the
ent methods as well as with all experimental lifetime
6p−5d E1 transitions, and our results (in atomic units)
measurements (also carried out by different techniques)
are d(6p −5d ) = 3.034, d(6p −5d ) = 1.325,
[15, 17, 18, 19, 20, 21, 22]. For comparison, the value 1/2 3/2 3/2 3/2
and d(6p −5d )=4.080. These values include con-
1562(93)obtainedfrom the h6s|r2|5di=9.76(29)matrix 3/2 5/2
tributionsfromthehigherpartialwaves(0.6%)and0.1%-
element that was extracted from the K splittings of the
0.2% Breit correction. The correlation corrections to
bound6snl statesinRef.[13]correspondstothe lifetime
these transitions are similar to the ones for the 6s−6p
τ = 170(10) s that is a factor of 2 longer than all j
5d3/2 transitions. Therefore, similar (on the order of 0.5%)
other values. We discuss the lifetimes of the 5d and
3/2 accuracy is expected for these matrix elements. The
5d levels in the next section.
5/2 6s−6p transitions contribute about 73%to the respec-
j
tive A totals for the 6p lifetimes. Based on our
b≤a ab j
evaluation of the uncertainty in these matrix elements
P
discussed in Section III, we expect present 6p lifetime
V. LIFETIMES
values to be accurate to about 1%. Our results are in
excellentagreementwithotherrecenttheoretical[48,49]
The lifetime of a state a is calculated as τa = and experimental [24, 25, 26] values. The calculation
( b≤aAab)−1. The E1,E2,andM1 transitionratesAab of Refs. [46, 47] is a third-order MBPT calculation that
are given by [51]: omits higher-order corrections included in the present
P
calculation, slightly different values are expected.
2.02613×1018 S Only one transition contributes to the 5d lifetime:
AE1 = E1 s−1, (8) 3/2
ab λ3 2ja+1 6s−5p3/2E2transition(thecontributionofthe6s−5d3/2
6
M1 transition is negligible). In the case of the 5d life- elements; and 5d −5d magnetic-dipole matrix el-
5/2 5/2 3/2
time, M1 5d −5d transition has to be included as ement. These values areused to evaluate lifetimes ofthe
5/2 3/2
pointed out in [47, 48, 50]. Our SD all-order value for 6p , 6p , 5d ,and5d levelsaswellasdipole and
1/2 3/2 3/2 5/2
this transition (in a.u.) is 1.5493. The correlation cor- quadrupolegroundstatepolarizabilities. Extensivecom-
rection contribution is very small, and the lowest order parison with other theoretical and experimental values
gives essentially the same value, 1.5489. The M1 transi- is carried out. The present values of the dipole polariz-
tioncontributes17%to the A totalforthe 5d ability and 6p lifetimes are in excellent agreement with
b≤a ab 5/2 j
level. experimentalvalues. Weestimatedtheuncertaintyofour
P
We compare our final results for the 5d and 5d theoretical values for these properties to be on the order
3/2 5/2
lifetimes with experimental [15, 17, 18, 19, 20, 21, 22] of1%. Ourrecommendedvalueofthequadrupoleground
andothertheoretical[15,46,47,48,49,50]valuesinTa- state polarizability α = 4182(34)a5 is in agreement
E2 0
bleV. Wenotethatcalculation[49]omitted5d −5d with the most recent experimental work [14]. Our rec-
5/2 3/2
M1 contributionto the 5d lifetime leading to a higher ommendedvaluesforthe5d lifetimesτ =81.5(1.2)s
5/2 j 5d3/2
value, as noted in later work [50]. Our results are in and τ = 30.3(4) s are in agreement with other the-
5d5/2
agreement with other theoretical calculations, most re- oretical calculations, most recent values from [15] mea-
cent values from [15] measured in a beam-laser experi- sured in a beam-laser experiment performed at the ion
ment performed at the ion storage ring CRYRING, as storage ring CRYRING, as well as experimental values
well as experimental values from [19, 21, 22]. from [19, 21, 22].
VI. CONCLUSION
Acknowledgements
In conclusion, we carried out the relativistic all-order
calculations of Ba+ 6s−np (n =6−9), 6p −5d ,
j 1/2 3/2
6p −5d ,and6p −5d electric-dipolematrixel- WethankSteveLundeenandEricaSnowformanyuse-
3/2 5/2 3/2 5/2
ements; 6s−5d , 6s−5d , 6s−6d , 6s−6d , ful discussions. This work was supported in part by the
3/2 5/2 3/2 5/2
6s − 7d , and 6s − 7d electric-quadrupole matrix National Science Foundation Grant No. PHY-04-57078.
3/2 5/2
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