Calculation of parity nonconserving amplitude and other properties of Ra+

0.22 MB·English

by  Rupsi Pal

Calculation of parity nonconserving amplitude and other properties of Ra+
Rupsi Pal, Dansha Jiang and M. S. Safronova
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA
U. I. Safronova
University of Nevada, Reno, NV 89557-0042, USA
We have calculated parity nonconserving 7s 6d amplitude E in 223Ra+ using high-
− 3/2 PNC
precision relativistic all-order method where all single and double excitations of the Dirac-Fock
9 wave functions are included to all orders of perturbation theory. Detailed study of the uncertainty
0 of the parity nonconserving (PNC) amplitude is carried out; additional calculations are performed
0 toestimatesomeofthemissingcorrelation corrections. Asystematicstudyoftheparityconserving
2
atomic properties, including the calculation of the energies, transition matrix elements, lifetimes,

n hyperfineconstants,quadrupolemomentsofthe6dstates, aswellasdipoleandquadrupoleground
a state polarizabilities, is carried out. The results are compared with other theoretical calculations
J and available experimental values.

7
2 PACSnumbers: 31.15.ac, 11.30.Er,31.15.ap,31.15.ag

]
h I. INTRODUCTION state of Fr [5] and an isotopic chain experiment in Yb
p
[6] are underway. We note that when an experimental
-
m study is conducted in a single isotope, both theoretical
There are two separate reasons for parity violation
andexperimentaldeterminationsofPNCamplitudesare
o studiesinanatom: tosearchfornewphysicsbeyondthe
required while the experiments conducted with isotopic
t
a standardmodel of the electroweakinteractionby precise chainsshouldallowtoremovethedependenceonthethe-
. evaluation of the weak charge Q , and to probe parity
s w ory. However,accuratetheoreticalvaluesforanumberof
c violationinthenucleusbyevaluatingthenuclearanapole atomic properties are useful for this type of experiments
i moment. Theatomic-physicstestsofthestandardmodel
s as well.
y thatarecompletedtodatewerecarriedoutbycomparing
Thepresentworkismotivatedbytheprojectthatwas
h experimental weak charges of atoms Q , which depend
W recentlystartedattheAcceleratorInstitute(KVI)ofthe
p on input from atomic theory, with predictions from the
University of Groningen [7] to measure PNC amplitude
[
standardmodel[1]. Themostpreciseexperimentalstudy in a single trapped radium ion. Ra+ is a particularly

1 to date, a 0.35% measurement in Cs was carried out by good candidate for the PNC study owing to high value
v the Boulder group [2] using a Stark interference scheme
of the nuclear charge Z and, correspondingly, large ex-
5 formeasuringthe ratioofthe PNC amplitudeEPNC and pected PNC effects. The 7s 6d transition in Ra+
9 thevectorpartoftheStark-inducedamplitudeβfortran- − 3/2
1 is of special interest owing to the long life of the 6d3/2
sitions between states of the same nominal parity. The
4 state and its sensitivity to both spin independent PNC
valueoftheweakchargeinCswasultimatelyfoundtobe
. and spin dependent PNC [8]. The 7s 6d transition
1 consistentwiththetheoriesofthestandardmodel. How- − 3/2
0 in Ra+ is also being considered for the development of
ever,suchcomparisonsprovideimportantconstraintson
9 optical frequency standards at the same laboratory [9].
its possible extensions. A recent analysis [3] of parity-
0 The parity violation experiments are also accompanied
violating electron-nucleus scattering measurements com-
: byanumberofmeasurementsofparity-conservingquan-
v bined with atomic PNC measurements placed tight con-
i tities; as a result we have included a systematical study
X straints on the weak neutral-current lepton-quark inter-
of such properties in this work.
actions at low energy, improving the lowerbound on the
r In summary, we have calculated the PNC amplitude
a scale of relevant new physics to 1 TeV.
∼ for the 7s 6d3/2 transitionin 223Ra+ together with the
−
Experimental measurements of the spin-dependent lifetimesofthe 7pand6dstates,energylevelsforns,np,
contribution to the PNC 6s 7s transition in 133Cs led nd, andnf states, transitionmatrix elements for anum-
→
toavalueofthecesiumanapolemomentthatisaccurate ber of the E1 and E2 transitions, quadrupole moments
toabout14%[2]. Theanalysisofthis experiment,which of the 6d states, groundstate dipole and quadrupole po-
requiredacalculationofthenuclearspin-dependentPNC larizabilities, and magnetic-dipole hyperfine constants A
amplitude, led to constraints on weak nucleon-nucleon for the 7s,7p, and 6d states using the relativistic all-
couplingconstantsthat areinconsistentwith constraints order method. The all-order method has proved to be
from deep inelastic scattering and other nuclear experi- veryreliableforcalculatingthepropertiesofalkali-metal
ments,aspointedoutin[4]. Therefore,newexperiments atomsandsinglyionizedmonovalentions(see,forexam-
(and associated theoretical analysis) are needed to re- ple, Refs. [10, 11, 12, 13, 14, 15, 16]). The effect of Breit
solve the issue. Currently, a microwave experiment to interactiononthePNCamplitudeisalsoevaluated. The
measure the spin-dependent PNC amplitude in the 7s sensitivity of the PNC amplitude to the nuclear radius
2
and varying neutron distribution has been studied. Our operatorsrepresented in the second quantization as Z =
results are compared with other theoretical values and z a†a :
ij ij i j
available experimental data.
P
Ψ Z Ψ
w v
Z = h | | i . (3)
wv
Ψ Ψ Ψ Ψ
II. THEORY h v| vih w| wi
p
Substituting theexpressionforthewavefunctionfrom
In this section, we briefly discuss the all-ordermethod
Eq.(1) in the above equation and simplifying, we get
which has been used to calculate the wavefunctions and
the matrix elements necessary to evaluate the observed
z +Z(a)+ +Z(t)
wv
properties. The all-order method relies on including all Z = ··· , (4)
wv
(1+N )(1+N )
singleanddoubleexcitationsofthecoreandvalenceelec- v w
trons from the lowest-order wave function:
p
where z is the lowest-order DF matrix element and
wv
|Ψvi=[1+ ρmaa†maa+ 12 ρmnaba†ma†nabaa Zqu(aa)d,r·a·t·ic,Zfu(tn)catinodnsnoorfmthaleizsaitnigolnetaenrdmsdoNuibalereexlicnietaatrioonr
ma mnab
X X coefficients [10, 19]. The expression in Eq. (4) does not
+ ρ a† a + ρ a† a†a a ]Φ . (1)
mv m v mnva m n a v | vi dependonthe natureofthe operatorZ,only onits rank
mX6=v mXna and parity. Therefore, all matrix elements calculated in
Here, Φ isthelowest-orderatomicwavefunctiontaken this work (E1, M1, E2, hyperfine, and PNC matrix ele-
v
tobet|heifrozen-coreDFwavefunctionofastatev;a†,a ments) are calculated using the same general code.
i j Corrections to the all-order equations from the
are single-particle creation and annihilation operators,
dominant class of triple excitation terms are also
ρ and ρ are the single core and valence excitation
ma mv
evaluated where needed by including the term
coefficients, and ρ and ρ are double core and
mnab mnva 1ρ a† a†a†a a a Φ into the SD wave func-
valence excitation coefficients, respectively. Indices at 6 mnrvab m n r v b a| vi
tion (1) and considering its effect on the energy and
thebeginningofthealphabet,a,b, ,refertooccupied
··· single valence excitation coefficient equations pertur-
corestates,thoseinthemiddleofthealphabetm,n, ,
··· batively (SDpT approach). Other classes of triple and
refertoexcitedstates,andindexvdesignatesthevalence
higher excitations are included where needed using
orbital.
the scaling procedure by multiplying single excitation
To derive equations for the excitation coefficients, the
coefficients ρ by the ratio of the “experimental” and
all-order wave function (1) is substituted into the many- mv
corresponding (SD or SDpT) correlation energies [10].
body Schr¨odinger equation H Ψ = E Ψ , and terms
v v
| i | i The “experimental” correlation energies are determined
on the left- and right-hand sides are matched, based on
as the difference of the total experimental energy and
the number and type of operators they contain. Hamil-
the DF lowest-order values. The calculation of the
tonianH =H +V istakentobetherelativisticno-pair
0 I
matrix elements is then repeated with the modified
Hamiltonian:
excitation coefficients. We refer the reader to the review
N
[16] and references therein for the detailed description
H = ε :a†a :,
0 i i i of the all-order method and its extensions. The various
Xi=1 atomic properties calculated using the all-order method
1
V = g :a†a†a a :, (2) described above are discussed in detail in the following
I 2 ijlk i j l k sections.
ijkl
X
whereε arethesingle-particleenergies,:: designatenor-
i
malorderingoftheoperatorswithrespecttoclosedcore,
III. PROPERTIES OF RA+
and g are the two-body Coulomb matrix elements.
ijkl
The all-order equations are solved numerically using a
A. Energies
finite basis set of single-particle wave functions which
are linear combinations of B-splines. We have used 70
basis set B-spline orbitals of order 8 defined on a non- Results of our calculations of energies for a number
linear grid with 500 points within a spherical cavity of of Ra+ levels are summarized in Table I. The first
radius 80 a.u. A large spherical cavity is needed to ac- six columns of Table I give the lowest-order DF ener-
commodate all the valence orbitals required for our cal- gies E(0), the all-order SD energies ESD, the part of
culation. A sufficiently large number of grid points were the third-order energies omitted in the SD calculation
enclosed within the nucleus to accommodate the influ- E(3) , first-order Breit contribution B(1), second-order
extra
ence of the nucleus on certain atomic properties such as Coulomb-Breit B(2) corrections, and Lamb shift contri-
parity-violatingmatrixelementsandhyperfineconstants. bution, E (see Ref. [22] for detail). We take the sum
LS
Theresultingsingle-double(SD)excitationcoefficients ofthesesixcontributionstobeourfinalall-orderresults,
areusedtocalculatematrixelementsofvariousone-body ESD listed in the seventh column of Table I.
tot
3
TABLE I: Contributions to the energies of Ra II: lowest-order (DF) E(0), single-double Coulomb all-order correlation energy
ESD,third-ordertermsnotincludedintheSDvalueE(3) ,first-orderBreitandsecond-orderCoulomb-BreitcorrectionsB(n),
extra
andLambshiftE . ThetotalenergiesESD arecomparedwithexperimentalenergiesE [17,18],δE =ESD -E . Our
LS tot expt tot expt
predicted energy values are listed for the 9p and 10p energy levels in separate rows. Units: cm−1.
1/2 j
nlj E(0) ESD E(3) B(1) B(2) E ESD E δESD
extra LS tot expt
7s -75898 -6692 1152 147 -250 33 -81508 -81842 334
1/2
6d -62356 -8042 1152 155 -398 0 -69488 -69758 270
3/2
6d -61592 -7034 926 114 -360 0 -67947 -68099 152
5/2
7p -56878 -4027 587 102 -109 0 -60326 -60491 165
1/2
7p -52906 -3020 433 63 -90 0 -55519 -55633 114
3/2
8s -36860 -1745 316 46 -74 7 -38311 -38437 126
1/2
7d -31575 -1590 245 39 -92 0 -32973 -33098 125
3/2
7d -31204 -1456 204 29 -84 0 -32509 -32602 93
5/2
5f -28660 -4438 371 11 -63 0 -32780 -32854 74
5/2
5f -28705 -4159 353 8 -61 0 -32564 -32570 6
7/2
8p -30053 -1298 201 39 -42 0 -31152 -31236 84
1/2
8p -28502 -1034 156 25 -36 0 -29391 -29450 59
3/2
9s -22004 -741 136 21 -33 2 -22618 -22677 59
1/2
9p -18748 -605 96 20 -21 0 -19259
1/2
-19305a
9p -17975 -495 76 13 -18 0 -18399 -18432 33
3/2
8d -19451 -683 105 18 -40 0 -20051 -20107 56
3/2
8d -19261 -634 90 13 -37 0 -19829 -19868 39
5/2
10s -14651 -388 72 11 -18 1 -14972 -15004 32
1/2
10p -12838 -335 53 11 -11 0 -13120
1/2
-13144a
10p -12397 -278 43 7 -10 0 -12635
3/2
-12653a
9d -13226 -366 56 10 -22 0 -13548 -13578 30
3/2
9d -13118 -342 49 7 -20 0 -13424 -13447 23
5/2
10d -9587 -221 34 6 -13 0 -9780
3/2
10d -9519 -207 30 4 -12 0 -9704
5/2
a Ourpredicted values.
ment [18] in Table II. The calculations in both Ref. [20]
TABLE II: Comparison of the excitation energies important
and Ref. [21] use high-precision all-order methods, but
tothecalculationofthe7s 6d PNCamplitude. Allresults
are in cm−1. − 3/2 represent very different approaches. The calculations
in Ref. [20] are performed using the correlation poten-
Transition Present Ref. [20] Ref. [21] Expt. tial method. The results of Ref. [21] are obtained us-
7s−7p1/2 21182 21279 21509 21351 ing coupled-cluster method including single, double, and
7s 7p 25989 26226 26440 26209
− 3/2 partial triple excitations. The results of Ref. [20] are in
6d 7p 9162 9468 9734 9267
3/2− 1/2 better agreementwith experimentfor the 7s 7p transi-
6d3/2−7p3/2 13969 14415 14665 14125 tions andthe results from the present worka−re in better
agreementwithexperimentforthe6d 7ptransitions.
3/2
−
Large discrepancies of the coupled-cluster results from
The column labeled δESD in Table I gives differences Ref. [21] for the 6d 7p transitions with experiment are
−
between our ab initio results and the experimental val- somewhat surprising and may indicate insufficient num-
ues [17, 18]. The SD results are in good agreement with ber of higher partial wave functions in the basis set. In
the experimental values taking into account very large our calculations, all partial wave up to lmax = 6 are
sizeofthe high-ordercorrelationcorrections. Wepredict explicitly included in all calculations and extrapolation
the energies of the 9p , 10p , and 10p levels using for higher number of partial waves is carried out for the
1/2 1/2 3/2
our theoretical results and differences between our and dominant second-order correlation energy contribution.
experimental values for the known np levels. The pre-
dictedvalues arelistedinTable Iandareexpectedto be
accurate to a few cm−1. B. Electric-dipole matrix elements
We compareour results forthe excitationenergiesim-
portant to the calculation of the 7s 6d PNC am- Wecalculateallallowedreducedelectric-dipolematrix
3/2
−
plitude with other theoretical calculations and experi- elementsbetweenns,np,andn dstateswheren=7 10
1
−
4
sults by the appropriate angular factors for the purpose
TABLEIII:Comparisonofthepresentresultsfortheabsolute
of comparison. The calculations of the Refs. [9, 21] are
valuesoftheelectric-dipolereducedmatrixelementsinRaII
carried out using the coupled-cluster method.
with other theoretical calculations. All results are in atomic
We have also listed the lowest-order DF values in the
units. Thelowest-orderDFvaluesarelistedinthecolumnla-
first column of the table to illustrate the size of the cor-
beled“DF”toillustratethesizeofthecorrelation correction.
NegativesignoftheDFvalueforthe8p 7s transition relationcorrectionsforvarioustransitions. Negativesign
1/2− 1/2
indicates that the lowest-order value is of the opposite sign ofthe DF value for the 8p 7s transitionindicates
1/2 1/2
−
with thefinal result. that the lowest-order values is of the opposite sign with
the final result. The correlation corrections for the pri-
Transition DF Present Ref.[20] Ref.[9] Ref.[21]
mary 7s 7p and 7p 6d transitions are quite large,
7p 7s 3.877 3.254 3.224 3.28 3.31
7p1/2−8s1/2 2.637 2.517 2.534 18-25%. −The correlati−on corrections for the remaining
7p1/2−9s1/2 0.716 0.702 0.708 strongtransitionsaregenerallysmaller,2-10%. Alltheo-
1/2− 1/2
7p 6d 4.446 3.566 3.550 3.64 3.68 reticalvaluesareingoodagreementforthesetransitions.
1/2− 3/2
77pp11//22−−78dd33//22 41..552874 41..249405 41..345382 mOuenrtvwaliuthesrfeosrul7tss−of7Rpefa.n[d207]pth−an6dthaorseeinofbReetftse.r[9a,gr2e1e]-.
7p 7s 5.339 4.511 4.477 4.54 4.58 Theagreementisgenerallypoorerforthetransitionswith
3/2− 1/2
7p 8s 4.810 4.644 4.663 smallvaluesofthe matrixelementsasexpectedowingto
3/2− 1/2
7p 9s 1.078 1.035 1.036 verylargesizeofthecorrelationcorrections. Sincediffer-
3/2− 1/2
7p3/2−6d3/2 1.881 1.512 1.504 1.54 1.56 ent methods omit or include somewhat different classes
7p3/2−7d3/2 2.488 2.384 2.407 of the high-order corrections, discrepancies are expected
7p 8d 0.733 0.652 0.641
3/2− 3/2 when such corrections are large. The issue of the very
7p 6d 5.862 4.823 4.816 4.92
3/2− 5/2 small matrix elements, such as 8p 7s, is also discussed
7p3/2−7d5/2 7.249 6.921 6.995 in Ref. [20]. −
7p 8d 2.227 2.011 1.954
3/2− 5/2
8p 7s -0.125 0.047 0.088 0.04
1/2− 1/2
8p1/2−8s1/2 7.371 6.949 6.959 C. Polarizabilities
8p 9s 5.227 5.012 5.035
1/2− 1/2
8p 6d 0.105 0.049 0.013 0.07
1/2− 3/2 We calculate the static dipole and quadrupole polar-
8p 7d 10.21 9.553 9.540
8p1/2−8d3/2 7.184 7.010 7.104 izabilities of the Ra+ ion in its ground 7s state. The
1/2− 3/2 static polarizability is calculated as the sum of three
8p 7s 0.625 0.395 0.339 0.50
8p3/2−8s1/2 9.880 9.294 9.320 terms representing contributions from the ionic core αc,
3/2− 1/2 asmallcounteractingtermtocompensateforthe excita-
8p 9s 9.244 9.022 9.036
3/2− 1/2 tions from the core states to the valence state α , and
8p 6d 0.168 0.144 0.127 0.15 vc
8p33//22−−7d33//22 4.331 4.035 4.028 valence polarizability αv:
8p 8d 4.047 4.002 4.034
8p33//22−−6d35//22 0.462 0.378 0.347 0.40 α=αc+αvc+αv. (5)
8p 7d 13.37 12.55 12.53
3/2− 5/2
8p 8d 11.68 11.49 11.58
3/2− 5/2
1. Dipole polarizability
The valence polarizability contributes over 90% of the
totalvalue ofthe electric-dipolepolarizabilityandis cal-
and n =6 10 using the method described above. The
1 − culated using sum-over-statesapproach:
subsetofthesematrixelementsiscomparedwiththecor-
relation potential calculations of Ref. [20] and coupled-
1 7s D np 2 7s D np 2
clustercalculationsofRefs.[9,21]inTableIII. Absolute α (E1)= |h || || 1/2i| + |h || || 3/2i| .
v
valuesofthereducedmatrixelementsinatomicunitsare 3 n Enp1/2 −E7s Enp3/2 −E7s !
X
listed in the table. All present values with the exception (6)
of the 7p 8s, 7p 8s, 8p 7s and 8p 7s The sum over n in Eq. (6) converges extremely fast. In
1/2 3/2 1/2 3/2
− − − −
transitionsareab initio SDvalues. Forthesefourtransi- fact, the first term with n = 7 contributes 99.8% of the
tions, we used scaling procedure described above to pro- total value. As a result, we calculate the first few terms
vide recommended values as we expect the scaled values (with n = 7 10) using our all-order matrix elements
−
tobemoreaccuratebasedonCs“bestset”dataRef.[23]. from Table III and experimental energies [17, 18] where
The calculations of Ref. [20] are carried out using fitted available. Theremainderαtail iscalculatedintheDFap-
v
Bruecknerorbitals(i.e. includesemi-empiricalcorrection proximationwithoutlossofaccuracy. Theioniccorecon-
to the correlation operator) and include core polariza- tribution α and term α are calculated in the random-
c vc
tion, structure radiation, and normalization corrections. phase approximation (RPA). The RPA core value is ex-
WenotethatRef.[20]quotesradialintegralsratherthan pected to be accurate to better than 5% (see Ref. [24]
reduced matrix elements, so we have multiplied their re- and references therein). All contributions to the dipole
5
TABLE IV: Contributions to the ground state dipole polar- TABLEV:Contributionstothegroundstatequadrupolepo-
izability of Ra+. Thecontributionsfrom the(7 10)p states larizabilityandtheE2reducedmatrixelementsofRa+ina.u.
−
aregivenseparately. Ourresultiscomparedwith calculation The comparison of our result with other theoretical calcula-
from Ref. [9]. All results are in a.u. tion [9] is also presented.
Contribution α Contribution E2 α
E1 E2
7p 7s 36.29 6d 7s 14.74(15) 789(13)
1/2− 3/2−
7p 7s 56.79 6d 7s 18.86(17) 1136(16)
3/2− 5/2−
8p 7s 0.00 7d 7s 14.21(30) 182(3)
1/2− 3/2−
8p 7s 0.23 7d 7s 16.49(38) 243(4)
3/2− 5/2−
(9 10)p 7s 0.04 8d 7s 5.63(4) 22.6(2)
− − 3/2−
8d 7s 6.79(6) 32.6(2)
αmain 93.35 5/2−
v 9d 7s 3.30(3) 7.0(1)
α 13.74 3/2−
c 9d 7s 4.03(3) 10.4(1)
αtail 0.11 5/2−
v 10d 7s 2.27(3) 3.1
α -0.98 3/2−
vc 10d 7s 2.79(3) 4.7
Total 106.22 5/2−
Theory [9] 106.12 αmain 2430(21)
v
αtail 35(10)
v
α 68(12)
c
Total 2533(26)
polarizability are listed in Table IV. The contributions
Theory [9] 2547.5
from n=7 10 are given together as αmain.
− v
The value of the ground state Ba+ polarizability cal-
culated by the same approach [24] is in near perfect
sults by 0.7 to 2.3% depending on the transition. We
agreementwiththeexperiment[25](to0.2%). Moreover,
the theoretical SD 6p lifetimes in Ba+ are also in excel- have also carried out the ab initio all-order calculation
withinclusionofthe triple valence excitationcoefficients
lent agreement with experimental values [24]. We note
as described in the Section II (SDpT approach). The
that lifetime experiments are conducted entirely differ-
scaling procedure was repeated starting from the SDpT
ently fromthe polarizability measurementof[25]. There
are two differences betweenthe Ba+ and Ra+ dipole po- approximationforthedominant7s 6d3/2 and7s 6d5/2
− −
transitions. These additional calculations allow us to di-
larizabilitycalculations: increasedioniccorecontribution
rectly evaluate the uncertainty in our calculations since
and increased size of the correlation corrections. The
core contribution increases from 8% in Ba+ to 13% in they produce different evaluations of the omitted corre-
Ra+ and the correlation correction contribution to the lation correction. We take the uncertainty in the calcu-
lation of the 7s 6d and 7s 6d to be the maxi-
7s 7p matrix elements increases by about 3% (from 3/2 5/2
− −
− mum of the difference of out final SD scaled results with
16.6% to 19.1% for the 7s 7p transition). Neither
− 1/2 ab initio and scaled SDpT data. We note that SD ap-
of these changes is expected to significantly decrease the
accuracy of the Ra+ ground state dipole polarizability proach generally underestimates the correlation energy
in comparison with the Ba+ one. Therefore, we expect and SDpT approachgenerally overestimatesthe correla-
tion energy used in the scaling procedure. The scaled
our value to be accurate to better than 1%. Our result
SD and SDpT results are rather close, further confirm-
is in agreement with the coupled-cluster calculation of
ing the validity of this procedure and of our uncertainty
Ref. [9].
estimate. Therefore, we take the uncertainty of the re-
maining transitions to be the difference of the final SD
scaledandabinitio SDpTvalues. Theresultingfinalma-
2. Quadrupole polarizability
trixelementsandtheiruncertaintiesarelistedinTableV
in column labeled “E2”. The relative uncertainty of the
The valence part of the quadrupole polarizability is
corresponding polarizability values is twice the relative
calculated using the sum-over-states approach as:
uncertainty of the matrix elements since we assume the
experimental energies be accurate to all figures quoted.
1 7s Q nd 2 7s Q nd 2
α (E2)= |h || || 3/2i| + |h || || 5/2i| . The sum over n converges far slower than in the case of
v
5 n End3/2 −E7s End5/2 −E7s ! the dipole polarizability so calculating a first few terms
X
(7) to high precision is essential to obtain an accurate final
All contributions to the quadrupole polarizability are value. The tail contribution, while small, is significant
listed in Table V. The correlation correction to the E2 and has to be treated with care. We estimated that DF
matrix elements is dominated by a single term among value for the main (n = 6 10) term is larger than our
−
twenty terms in the numerator of Eq. (4). As described final all-order result by 22%. Therefore, we decrease the
in detail in Ref. [24], additional omitted correlation cor- DF tail of 45 a.u. by 22% and take the difference of the
rectiontothistermmaybeestimatedbythescalingpro- DFtailandthefinaladjustedvaluetobeitsuncertainty.
cedure described above. The scaling modifies the SD re- The core contribution is calculated in the RPA approxi-
6
TABLE VI: Contributions to the lifetimes of the 7p1/2 and TABLE VII: Lifetimes of the 6d3/2 and 6d5/2 states of Ra+
7p states. ThetransitionsratesAaregivenin106 s−1 and in seconds. Comparison of our results with other theoretical
3/2
thelifetimes are given in ns. calculations is presented.
7p 7p Term τ(6d ) τ(6d )
1/2 3/2 3/2 5/2
A(7p 7s) 104.4 A(7p 7s) 185.5 Present 0.638(10) 0.303(4)
1/2− 3/2−
A(7p 6d ) 10.3 A(7p 6d ) 3.3 Theory [9] 0.627(4) 0.297(4)
1/2− 3/2 3/2− 3/2
PA 114.7 A(7p 6d ) 22.8 Theory [20] 0.641 0.302
3/2− 5/2
τ(7p ) 8.72 ns PA 211.6
1/2
τ(7p ) 4.73 ns
3/2
M16d 6d matrixelementis1.55a.u. TheE2and
5/2 3/2
−
M1transitionratescontributingtothe6d lifetime are
mation;wetakethedifferencebetweenDFandRPAcore 5/2
3.255 s−1 and 0.049 s−1. We verified that the contribu-
valuestobetheuncertaintyofthecorecontribution. Our
tion of the 6d 6d E2 transition is negligible.
final value is in agreement with the result of Ref. [9]. 5/2− 3/2
Our results for the 6d and 6d lifetimes are pre-
3/2 5/2
sented in Table VII together with other theoretical val-
ues. Our values for the lifetimes of the 6d states are in
D. Lifetimes of the 7p and 6d states
better agreement with those published by Dzuba et al.
[20] than with the results of Sahoo et al. [9]; however,
The lifetimes τ of the 7p and 6dstates in Ra+ arecal-
the discrepancies with Ref. [9] are small. We also list
culatedastheinverseofthe sumofthetransitionproba-
theuncertaintiesofourvaluesinTableVII. Therelative
bilities A. The 7p states decay via strong electric-dipole
uncertainties in our values of the 6d lifetimes are twice
transitions. TotaloffiveE1transitionscontributeto the
the relative uncertainties in the values of the E2 matrix
lifetimes of these two states: 7p 7s, 7p 6d ,
1/2 − 1/2 − 3/2 elements listed in Table V. We note that the estimated
7p 7s,7p 6d ,and7p 6d . Theelectric-
3/2− 3/2− 3/2 3/2− 5/2 uncertainties quoted in Ref. [9] are obtained by carrying
dipole transition rates are calculated using formula
outcalculationswithdifferentbases;i.e. theyarenumer-
ical uncertainties resulting from the particular choice of
2.02613 1018 i D f 2 the basis set and do not include estimation of the miss-
AE1 = × |h || || i| s−1, (8)
if λ3 2j +1 ing correlation effects. In our calculations, the basis set
i
iscomplete(70splinesforeachpartialwave)andincreas-
where λ is the wavelength of the transition in ˚A and ing its size does notchangethe result. Our uncertainties
i D f is the electric-dipole reduced matrix element include estimation of the terms beyond triple contribu-
h || || i
in atomic units. We use the experimental wavelength tions as described above as well as uncertainty owing to
[17, 18] and our all-order matrix elements listed in Ta- truncation of the partial waves above l > 6. Therefore,
ble III when evaluating the transition rates. The results while our uncertainty is higher for 6d state than the
3/2
aresummarizedinTableVI. Wefindthatwhilethecon- one quoted in Ref. [9], it represents an attempt to pro-
tributions of the 7s 7p transitions to the 7p lifetimes vide an actual boundary for the recommended value of
−
aredominant,thecontributionsofthe7p 6dtransitions this lifetime.
−
are significant (over 10%).
Onlyonetransition,6d 7s,hastobeconsideredfor
3/2
−
the calculation of the 6d3/2 lifetime. The corresponding E. Quadrupole moments of the 6d states
transition rate is calculated as
1.11995 1018 i Q f 2 We also calculated the values of the quadrupole mo-
AE2 = × |h || || i| s−1, (9)
if λ5 2ji+1 ments of the 6d3/2 and 6d5/2 states since these proper-
ties are of interest to the investigation of possible use of
where λ is the wavelength of the transition in ˚A and Ra+ for the development of optical frequency standard
i Q f is the electric-quadrupole reduced matrix ele- [9]. ThequadrupolemomentΘ(γJ)canbeexpressedvia
h || || i
ment in atomic units. the reduced matrix element of the quadrupole operator
Two transitions have to be considered in the calcula- Q as
tion of the 6d lifetime: E2 6d 7s transition and
5/2 5/2
M1 6d 6d transition. The M−1 transition rate is (2J)!
5/2 − 3/2 Θ(γJ)= Ψ(γJ) Q Ψ(γJ) .
calculated as (2J 2)!(2J +3)!h k k i
−
2.69735 1013 i M1 f 2 (11)
AM1 = × |h || || i| s−1. (10) The calculatpion follows that of the E2 matrix elements.
if λ3 2j +1
i AsinthecaseoftheE27s ndmatrixelements,asingle
−
We use the experimental wavelengths [17, 18] and our correlation correction term is dominant, and the omit-
all-order matrix elements listed in Table V when evalu- ted correlation contributions may be estimated via the
ating the E2 transition rates. Our result for the reduced scaling procedure. We have conducted four different cal-
7
rounded off value µ = 0.271(2) µ from [29] was used
I N
TABLE VIII: Quadrupole moments of the 6d and 6d
3/2 5/2 in Ref. [9]. The values for A/g were quoted in Ref. [21],
states in Ra+ in a.u. I
so we multiplied their values by 0.18067 for comparison.
State SD SDpT SDsc SDpTsc Final Ref. [9] Thedifferencesbetweenourresultsandexperimentalval-
6d3/2 2.814 2.868 2.839 2.829 2.84(3) 2.90(2) ues are 1.3%, 0.7%, and 4% for 7s, 7p1/2, and 7p3/2
6d5/2 4.311 4.380 4.342 4.329 4.34(4) 4.45(9) states, respectively. We note that the uncertainty in the
value of the nuclear magnetic moment is 0.7%. Larger
difference of the A(7p ) SDpT value with the experi-
3/2
TABLEIX:Magnetic-dipolehyperfineconstantsA(MHz)for mentissimilartothatoneinCs[11],wherethedifference
the 7s, 7p1/2, 7p3/2, 6d3/2, and 6d5/2 states in 223Ra+ calcu- of the SDpT value for the 6p3/2 magnetic-dipole hyper-
latedusingSDandSDpTall-orderapproaches. Lowest-order fine constant with experiment is 3.5%. Interestingly, the
(DF)valuesarealsolistedtoillustratethesizeofthecorrela- Cs SDpT values are below the experimental ones while
tioncorrections. Thepresentvaluesarecomparedwithother the Ra+ SDpT results are above the experimental val-
theoretical[9,21]andexperimentalvaluesfromRefs.[27,28]. ues. This can be explained by the uncertainty in the
treatmentof the finite size correction,uncertainty in the
State DF SD SDpT Ref. [21] Ref. [9] Expt. valueofRanuclearmagneticmoment,andthedifference
7s 2614 3577 3450 3557 3567 3404(2) in the size and distribution ofthe correlationcorrections
6d3/2 52.92 81.51 79.56 79.80 77.08 in Cs and Ra+.
6d 19.24 -23.98 -24.08 -23.90
5/2
7p 444.5 699.5 671.5 671.0 666.9 667(2)
1/2
7p 33.91 56.62 54.40 56.53 56.75 56.5(8)
3/2 IV. PARITY NONCONSERVATION
Nuclear-spin independent PNC effects in atoms are
culations: ab initio SD and SDpT, and scaled SD and caused by the exchange of a virtual Z boson between
0
SDpT ones to evaluate the uncertainty in the final val- an electron of the atom and a quark in the nucleus,
ues. The results are summarized in Table VIII. The or between two atomic electrons [30]. The second ef-
correlation correction to the quadrupole moments is on fect is extremely small and will not be consideredin this
theorderof20%. Ourvaluesarecomparedwithcoupled- work. ThedominantPNCinteractionbetweenanatomic
cluster calculationofRef. [9]. Ourresults are lowerthan electron and the nucleus is described by a Hamiltonian
that of Ref. [9]. This issue has been discussed in detail A V , which is the product of axial-vector electron cur-
e N
in Ref. [26] where we have demonstrated that CCSD(T) rent A and vector nucleon current V . The PNC inter-
e N
method mayoverestimatequadrupolemoments by a few action leads to a non-zero amplitude for transitions oth-
percentowingtothecancellationofvariousterms. Omis- erwise forbidden by the parity selectionrule, suchas the
sion of orbitals with l > 4 from the basis set may also 6d 7stransitioninsinglyionizedradium. Combining
3/2
−
lead to higher values. experimental measurements and theoretical calculations
of the PNC amplitude permits one to infer the value of
the weak charge Q for precise atomic-physics tests of
W
F. Magnetic-dipole hyperfine constants the standard model.
The7s 6d PNCamplitudeinRa+canbeevaluated
3/2
−
as a sum over states:
Ourresultsforthemagnetic-dipolehyperfineconstants
A(MHz) in223Ra+ arecomparedwith theory[9,21] and ∞ 6d D np np H 7s
3/2 1/2 1/2 PNC
experiment [27, 28] in Table IX. The gyromagneticratio E = h | | ih | | i
PNC
E E
gI for 223Ra is taken to be gI = 0.1803 and corresponds nX=2 7s− np1/2
to the value µI =0.2705(19)µN fromRef. [29]. We note ∞ 6d3/2 HPNC np3/2 np3/2 D 7s
that the magnetic moment of 223Ra have not been di- + h | | ih | | i,(12)
E E
rectly measured but recalculated from measurements of n=2 6d3/2 − np3/2
X
213Ra and 225Ra nuclear magnetic moments in Ref. [29].
where D is the dipole transition operator. The values of
The magnetization distribution is modeled by a Fermi
m are customary taken to be m = 1/2 for all states.
distributionwith the same parametersasourchargedis- j j
The PNC Hamiltonian H is given by
tribution (c= 6.862 fm and 10%-90% thickness param- PNC
eter is taken to be t=2.3 fm). The lowest-order values G
F
H = Q γ ρ(r), (13)
are also listed to demonstrate the size of the correlation PNC W 5
2√2
corrections for various states. The triple contributions
are important for the hyperfine constants and are par- where G is the universal Fermi coupling constant, Q
F W
tially included as described in Section II. These values is the weak charge and γ is the Dirac matrix associated
5
are listed in column labeled “SDpT”. The SD values are with pseudoscalars. The quantity ρ(r) is a nuclear den-
also listed for comparison in column labeled “SD”. sityfunction,whichisapproximatelytheneutrondensity.
The value g = 0.18067 that corresponds to the In our calculations, we model ρ(r) by the charge form
I
8
TABLE X: Contributions to the Emain in 223Ra+ in units of 10−11iea ( Q /N). D and H are dipole and PNC matrix
PNC | | 0 − W PNC
elements, respectively. Reduced electric-dipole matrix elements are listed for consistency with previous tables; they need to
be multiplied by 1/√6 to obtain relevant values of iDj (m = 1/2 for all states). All values are in a.u. Our results are
j
h | | i
compared with calculations of Ref. [20].
n 6d D np np H 7s E E E Ref. [20]
h 3/2k k 1/2i h 1/2| PNC| i 7s− np1/2 PNC
7 3.566 -2.665 -0.0973 39.882 40.69
8 0.049 -1.590 -0.2306 0.137 0.11
9 0.017 -1.124 -0.2849 0.027 0.02
10 0.008 -0.841 -0.3130 0.009
n 6d H np np D 7s E E E Ref. [20]
h 3/2| PNC| 3/2i h 3/2k k i 6d3/2 − np3/2 PNC
7 -0.047 -4.551 -0.0644 -1.348 -2.33
8 -0.040 -0.405 -0.1837 -0.036 -0.05
9 -0.032 -0.140 -0.2339 -0.008 -0.01
10 -0.026 -0.069 -0.2602 -0.003
factor, which is taken to be a Fermi distribution with is multiplied by the neutron number in the present com-
50%radiusc =c =6.8617fm[31]and10%-90% monlyacceptedunitsof10−11iea ( Q /N),thediffer-
PNC charge 0 W
| | −
thickness parameter t=2.3 fm for 223Ra+ , i.e. we take ence between values for the PNC amplitudes for 223Ra+
ρ(r) to be the same distribution as the charge distribu- and226Ra+ is2%justowingto138/135neutronnumber
tionusedourentireall-ordercalculationoftheRa+ wave ratio. Therefore, the difference may be either explained
functions and corresponding properties. We also investi- by the simple isotope rescaling, difference in the choice
gate how the PNC amplitude vary with changes in both ofthe nucleardensityfunctionparameters,ordifferences
c and c . in the treatment of the correlation correction. The only
PNC charge
significant discrepancy between our calculation and that
The sum over n in Eq. (12) converges very fast in our of Ref. [20] is in the other term with n = 7 (-1.35 vs.
case, and only first few terms need to be calculated ac- -2.33). This difference has to result from the differences
curately. Therefore, we divide our calculation of EPNC in the treatment of the correlation correction since this
into three parts: a main term EPmNaiCn that consists of the entire value comes from the correlation effects. Taking
sumoverstateswithn=7−10,atailEPtaNilC whichisthe into account that the DF value for this term is consis-
sumoverstateswithn=11,..., ,andthecontribution tent with zero and random-phase approximation (RPA)
∞
EPauNtCo from autoionizing states given by the terms with result, -4.08, is larger than the all-order value by nearly
n=2 6. Thecalculationofthemaintermisillustrated a factor of 3, such discrepancy is not very surprising.
−
in Table X, where we list the “best set” of the dipole
andPNCmatrixelementsusedinourcalculationaswell To provide some estimate of the uncertainty in the
as relevant energy differences. The final electric-dipole calculation of the main term, we conduct the “scatter”
matrix elements are taken to be ab initio single-double analysis of the data following the calculation of the Cs
all-order results (following the comparison of the simi- PNC amplitude [32]. In such analysis, sets of data for
lar Cs and Ba+ results with experiment [11, 24]). Re- dipole matrixelements, PNCmatrix elements,andener-
duced electric-dipole matrix elements are listed for con- gies are varied (i.e., taken to be SD, SDpT, expt.) and
sistency with previoustables; they need to be multiplied the scatter in the final PNC values is analyzed. The re-
by1/ (6)toobtainrelevantvaluesof iD j (m =1/2 sults are summarized in Table XI. Our final value (cor-
j
h | | i
for all states). The final PNC matrix elements for the responding to data in Table X) is listed in the last row
p
6d 7p and 6d 8p transitions are taken to of Table XI. We note that essentially entire difference in
3/2 1/2 3/2 1/2
− −
be SD all-order scaled values since the contribution that the results comes from the dominant term (first row of
can be accounted for by scaling is the dominant one for Table X), and the variationin all other terms is insignif-
these cases; remaining PNC matrix elements are taken icant. Therefore, the possible uncertainty in the next
to be ab initio SD values. Experimental energies are term (-1.35), which is bound to be substantial, can not
used where they are available, our predicted energy val- be evaluated by this approach. While we have included
uesfromTableIareusedforthe9p ,10p ,and10p the values with SDpT dipole matrix elements, there is
1/2 1/2 3/2
levels. Our results are comparedwith results of Ref. [20] no reasonto expect these data to be more accuratethan
calculated using the correlation potential method. The SDvalues. This conclusionisbasedonthe breakdownof
mainpartofthePNCamplitudeisoverwhelminglydom- the correlationcorrectioncontributionsandcomparisons
inated by a single termlisted inthe firstrowof Table X. of the similar calculations in other monovalent systems
Our result for this term slightly differs from the calcula- that demonstrate cancellation of some missing effects in
tion in Ref. [20] (by 2.2% ). However, the Ref. [20] does SD approximation but not in SDpT one. As a result,
not list the Ra+ isotope for which the calculation has we conclude that the uncertainty in the dominant term
been conducted. Since the value of the PNC amplitude owing to the Coulomb correlation correction is probably
9
TABLE XI: ‘Scatter” analysis of the main part of the PNC TABLE XIII: Dependence of the lowest-order Ra+ PNC
amplitude (n = 7 10) in 223Ra+. Lowest-order DF and amplitude on the parameters of the nuclear distributions
−
random-phase RPA values are listed for reference. SD la- c (fm) and c (fm). The parameter c is used in
charge PNC charge
bels single-double all-order values, SDpT values include par- thechargedistributionintheall-orderwavefunctioncalcula-
tial triple contributions. tions. Theparameterc isusedinthemodelingthenuclear
PNC
density function in the PNC Hamiltonian. The variation of
Energies iDj iH j Emain
h | | i h | PNC| i PNC thegivenparameterislisted in% forconvenience. Theunits
DF DF DF 38.95 for the PNCamplitude is 10−11iea0( QW/N).
| | −
DF RPA RPA 37.10
c δc c δc EDF δEDF
SD SD SD 39.05 charge charge PNC PNC PNC PNC
6.8617 0% 6.8617 44.913
Expt. SD SD 39.65
6.8960 0.5% 6.8617 44.853 -0.13%
Expt. SDpT SD 40.22
6.9303 1% 6.8617 44.792 -0.27%
Expt. SD SDpT 38.09
6.9989 2% 6.8617 44.671 -0.54%
Expt. SDpT SDpT 38.65
Expt. SD SDa 38.66 7.2048 5% 6.8617 44.310 -1.34%
sc
a Scaled values are used for the7p1/2−7s and 8p1/2−7s 6.8617 6.8960 0.5% 44.875 -0.08%
matrix elements only, remaining data are taken to beSD. 6.8617 6.9303 1% 44.837 -0.17%
6.8617 6.9989 2% 44.761 -0.34%
6.8617 7.2048 5% 44.531 -0.85%
TABLE XII:Contribution to the E in 223Ra+ and com-
PNC
parison with other theory. Our value for 226Ra+ is obtained 6.8960 0.5% 6.8960 0.5% 44.815 -0.22%
by reducing our 223 value by 0.2% owing to the correc- 6.9303 1% 6.9303 1% 44.717 -0.44%
tion for the different nuclear parameters and multiplying by 6.9989 2% 6.9989 2% 44.523 -0.87%
138/135 neutron number ratio. All results are in units of 7.2048 5% 7.2048 5% 43.954 -2.14%
10−11iea ( Q /N).
0 W
| | −
Isotope Term Value
223 Emain 38.66
PNC
223 Etail -0.02
PNC
223 Eauto 6.83 WealsoinvestigatedthedependenceofthePNCampli-
PNC
223 Breit -0.29 tudeonthevaluesofthenucleardistributionparameters
223 Total 45.18 c andc . Aswedescribedinthebeginningofthis
charge PNC
section,theparameterc isusedinthechargedistri-
226 Total 46.09 charge
bution in the all-order wave function calculations. The
Mixed states [20] 42.9 parameter c is used in the modeling of the nuclear
PNC
Sum over states [20] 45.9
density function ρ(r) in the PNC Hamiltonian given by
226 CCSD [21] 46.1
Eq. (13). Both are modeled by the Fermi distributions;
226 CCSD(T) [21] 46.4
the all-ordercalculationiscarriedwithbothhalf-density
parameters being equal to 6.8617 fm [31]. Since the DF
result is rather close to the final value owing to various
on the order of 2%. We note that completely ab initio cancellations, it is sufficient to carry out this study us-
SDvalueisingoodagreementwithourfinalvalue. Mea- ing DF data. The results are summarized in Table XIII,
surement of the 6d 7p oscillator strength would where we list EDF calculated with varying values or ei-
3/2 − 1/2 PNC
help to reduce this uncertainty. ther one or both parameters. The variation of the given
WecalculateremainingtermsEtail andEauto inboth parameter is listed in % for convenience. The results
PNC PNC
DF and RPA approximations. The RPA results are show that possible uncertainty in the PNC amplitude
listed in Table XII together with our total value for owing to the uncertainty in the value of the charge ra-
the PNC amplitude. The corresponding DF results are dius(thatisunlikelytobe large)isnegligibleincompar-
Eauto = 4.8 and Etail = 1.2. The correction due to ison with the uncertainty in the correlation correction.
PNC PNC
Breit interaction is obtained in the DF approximation. For example, difference in the rms radii for A=223 and
Our finalvalue is comparedwith other calculations from A=226isotopescorrespondstothechangeinc that
charge
Refs. [20, 21]. Our result for the terms with n < 7 and is on the order of 0.5% resulting in only 0.2% change
n > 9 (6.8) is in reasonably good agreement with the in the PNC amplitude. Possible variance in the density
value from Ref. [20] (7.5). The notable feature of Ta- ρ(r)inEq.(13)whichisapproximatelyneutrondensityis
ble XII is an excellent agreement of all rather different higher, but even 5% change in c with the fixed value
PNC
high-precision calculations despite relatively large pos- of the c leads to 0.85% change in the PNC ampli-
charge
sible uncertainties in various terms, with the exception tude value. Table XIII may be used to recalculate the
of the mixed-states result [20]. Further calculations as values of the PNC amplitude between different isotopes
well as experimental measurements will be necessary to since the changein E with the nuclear parametersis
PNC
achieve 1% accuracy in the PNC amplitude. essentially linear.
10
V. CONCLUSION be 45.2 10−11iea ( Q /N). The dependence of the
0 W
× | | −
PNC amplitude on the choice of nuclear parameters is
We have calculated the energies, transition matrix el- studied. This work provides a number of recommended
ements, lifetimes, hyperfine constants, quadrupole mo- values for yet unmeasured properties of Ra+.
ments of the 6d states, as well as dipole and quadrupole
ground state polarizabilities, and PNC amplitude in
223Ra+ using high-precision all-order method. The en-
Acknowledgments
ergies of the 9p , 10p , and 10p levels are pre-
1/2 1/2 3/2
dicted. The results for atomic properties are compared
with available theoretical and experimental data. The This work was supported in part by National Science
PNC amplitude for the 7s 6d transition is found to Foundation Grant No. PHY-07-58088.
3/2
−
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