EPJ manuscript No. (will be inserted by the editor) 7 0 0 Multiconfiguration Dirac-Hartree-Fock calculations 2 of transition rates and lifetimes of the eight lowest n a excited levels of radium J 2 2 Jacek Bieron´1, Paul Indelicato2, and Per J¨onsson3 ] 1 InstytutFizykiimienia Mariana Smoluchowskiego h Uniwersytet Jagiellon´ski p Reymonta4, 30-059 Krak´ow, Poland - m e-mail: [email protected] 2 Laboratoire Kastler Brossel, E´cole Normale Sup´erieure o CNRS;Universit´e P.et M. Curie - Paris 6 t a Case 74; 4, place Jussieu, 75252 Paris CEDEX 05, France . e-mail: [email protected] s c 3 Nature, Environment,Society i Malm¨o University s y S-205 06 Malm¨o, Sweden h e-mail: [email protected] p [ 1 Abstract. ThemulticonfigurationDirac-Hartree-Fock(MCDHF)modelhasbeen v employedtocalculatethetransitionratesbetweentheninelowestlevelsofradium. 9 Thedominantrateswerethenusedtoevaluatetheradiativelifetimes. Thedecay 3 of the metastable 7s7p3P0 state through 2-photon E1M1 and hyperfine induced 2 channels is also studied. 1 0 7 1 Introduction 0 / Recentadvancesintrappingandspectroscopyoffree,neutralatomsmakeitpossibletoextend s c the searchfor time reversalviolation effects into the domain of radioactiveelements [1]. In the si last decade several heavy atoms were considered as candidates for experimental searches [2]. y There are at least two ongoing atomic trap experiments (in Kernfysisch Versneller Instituut h in Netherlands [3,4] and in Argonne National Lab in the U.S. [5,6]) whose aim is to detect p the electric dipole moment of radium. The advantage of radium lies in octupole deforma- : v tions of nuclei in several isotopes [7], simple electronic structure (ground state configuration i [Kr]4d104f145s25p65d106s26p67s2 yields the closed-shell singlet state 1S ) as well as in coin- X 0 cidental proximity of two atomic states of opposite parity, 7s7p3P and 7s6d3D , which are 1 2 r separated by a very small energy interval 5 cm−1. a The data on atomic spectrum of radium compiled in the tables of Moore [8] came from the experimental investigations of Rasmussen [9], with subsequent revisions by Russell [10]; both go back to the 1930s. These data cover only 69 classified lines. The isotope shifts and hyperfine structures of radium were measured by the group of Wendt [11,12,13,14,15,16] in the 1980s. They have studied both atomic Ra I and ionic Ra II spectra, and obtained the isotope shifts, magnetic dipole hyperfine constants A and electric quadrupole constants B of the 7s7p1P , 7s7p3P , 7s7p3P , and 7s7d3D levels of neutral radium, as well as the magnetic 1 1 2 3 dipole hyperfine constants A of the 7s2S , 7p2P , and 7p2P levels of Ra+ ion, together 1/2 1/2 3/2 with the electric quadrupole constant B of the 7p2P state of Ra+. Hyperfine structures of 3/2 singly-ionisedradiumhavebeenalsothesubjectofseveraltheoreticalpapers[17,18,19,20,21,22]. 2 Will be inserted by theeditor The excitation energies of several states of neutral and singly-ionised radium (and barium) were later calculated in the framework the relativistic coupled-cluster theory by the group of Kaldor [23,24]. More recently there have been three papers from our group in which we calculatedthelifetimesofthe7s7p1P and7s6d3D states[25];thehyperfinestructureconstants 1 2 of all levels belonging to the two lowest excited-state configurations [26]; and the electric field gradientsgeneratedbytheelectroniccloudinthe 7s7p1P ,7s7p3P ,and7s7p3P states,which 1 1 2 in turn (combined with the measured values of the electric quadrupole constants B) yielded thenuclearelectricquadrupolemomentofradium-223isotope[27].Theexcitationenergiesand lifetimes of several states of radium (and barium) were calculated recently in the framework of the combined configuration-interaction and many-body perturbation theory by Dzuba and Ginges [28]. In the present paper we calculated the transition probabilities between the states arising fromthe threelowestconfigurationsofradium:7s2 1S ,7s7p3P ,7s7p1P ,7s6d3D ,and 0 0,1,2 1 1,2,3 7s6d 1D , as well as the lifetimes of these states. The purpose of the present paper is fourfold. 2 Firstly,weintendedtoextendthetransitionratecalculationsonalllevelsarisingfromthethree lowest configurations (i.e. 7s2, 7s7p, and 7s6d). We included (1) transitions which contribute appreciablytolifetimes;(2)transitionsinvolving7s6d3D and7s7p3P levelsbecausethey are 2 1 ofinterestinEDMexperiments(themixinginducedbyT-oddinteractionsisstrongestbetween thesetwolevels);(3)transitionswhichmaybeimportantfortrapping;(4)transitionswhichare stronger than 1/s (a somewhat arbitrary threshold). Secondly, new comparison of our results became available when the two most recent papers [28,5] appeared in print. These in turn permitted further tests ofthe newly developed[29] parallelversionof the GRASP package[30] and calibration the theoretical model for the calculations of the spectroscopic properties of radium. Finally, we present a summary of the available theoretical data on the spectroscopic properties of eight lowest excited levels of radium, with the hope that they may be of help for the experimental groups, that are currently in the process of setting up the atomic traps for the searchofpermanentelectricdipole moments.The Argonnegrouphasalreadymeasured[5] the frequency andthe rateofthe3P − 1S transitioninthe 225Ra isotope,anddeterminedthe 1 0 88 lifetime of the 3P level. The lifetime of this level have been previously calculated by Hafner 1 and Schwarz [31] and by Bruneau [32]. 2 Theory The modified version [29] of the GRASP implementation [30] of the multiconfiguration Dirac- Hartree-Fock method [33] was used in the present paper. The starting point is the Dirac- Coulomb Hamiltonian HDC =Xcαi·pi+(βi−1)c2+ViN +X1/rij, (1) i i>j whereVN isthemonopolepartoftheelectron-nucleusCoulombinteraction.Thewavefunction for a particular atomic state (Ψ) is obtained as the self-consistent solution of the Dirac-Fock equation [30] in a basis of symmetry adapted configuration state functions (Φ) NCF Ψ(ΓPJM)= X ciΦ(γiPJM). (2) i The basis NCF was systematically enlarged [34,35] to yield increasingly accurate approxima- tions to the exact wavefunction. All calculations were done with the nucleus modeled as a variable-density sphere, where a two-parameter Fermi function [36] was employed to approxi- matethechargedistribution.TheBreitandQEDcorrectionswereestimatedwiththestep-wise proceduredescribedin[34].Theywereappliedonlytotwotransitionrates,asexplainedinsec- tion 4.3 below. Will be inserted by theeditor 3 3 Method The wavefunctions were obtained with the active space method in which configuration state functions of a particular parity and symmetry are generated by substitutions from a reference configurationto an active set of orbitals.The active set and the multiconfigurationexpansions are increased systematically. The whole process is governed by convergence of the expectation values. The calculations were divided into two stages. Each stage was further divided into several consecutive steps. In the first stage, the spectroscopic and virtual spinorbitals were generatedinrelativelysmallmulticonfigurationexpansions.Thespectroscopicorbitalsrequired to form a reference wavefunction were obtained with a minimal configuration expansion, with full relaxation. Then virtual orbitals were generated in five consecutive steps. At each step the virtual set has been extended by one layer of virtual orbitals. A layer is defined as a set of virtual orbitals with different angular symmetries. In the present paper five layers of virtual orbitals of each of the s, p, d, f, g, h symmetries were generated. At each step the configurationexpansions were limited to single and double substitutions from valence shells to all new orbitals and to all previously generated virtual layers.These were augmented by small subsets of dominant single and double substitutions from core and valence shells, with further restriction, that at most one electron may be promoted from core shells (which means, that in the case of a double substitution the second electron must be promoted from a valence shell). All configurationsfromearliersteps were retained,with allpreviously generatedorbitalsfixed, and all new orbitals made orthogonal to others of the same symmetry. The initial shapes of radial orbitals were obtained in Thomas-Fermi potential, and then driven to convergence with theself-consistencythresholdsetto10−8.Allradialorbitalswereseparatelyoptimizedforeach of the nine atomic states of interest. The Optimal Level form of the variationalexpression[36] was applied in all variational calculations. In the second stage, the configuration-interaction calculations (i.e., with no changes to the radial wavefunctions) were performed, with multiconfiguration expansions tailored in such a way, as to capture the dominant electron correlation contributions to the expectation values. The valence andcore-valenceeffects constitute the dominantelectroncorrelationcontributions in the oscillator strength calculations [37], therefore all single and double substitutions were allowed from several core shells and both valence shells (i.e., 7s2, 7s7p, or 7s6d, depending on the state) to all virtual shells, with the same restriction as above, i.e. that at most one electron may be promoted from core shells. The virtual set was systematically increased from one to five layers, until the convergence of transition rates was obtained. In a similar manner, severalcore shells were systematically opened for electronsubstitutions — from the outermost 6p to 5s5p5d6s6p shells. The effects of substitutions from 4s4p4d4f shells were neglected. We estimatedthemseparatelyforthreestatesanddiscoveredthattheychangethecalculatedvalues oftransitionratesbynomorethanafractionofapercent.The transitionrateswerecalculated with the biorthonormal technique [38,39], which permits the application of standard Racah algebra, while retaining the advantage of wavefunctions separately optimized for each state. Experimental values of transition energies from Moore’s tables [8] were used in calculations of transition rates. 4 Results 4.1 The metastable 7s7p3P state 0 In principle there are three possible decay channels of the 7s7p3P state. It can decay to (the 0 onlylowerlying)groundstatethrough(1)ablackbodyradiationinduceddecay,(2)a2-photon E1M1transition,orthrough(3)ahyperfineinducedtransition.Thefirstisbeyondthescopeof the present paper, since it depends on the ambient temperature. Of the other two, the former canbe estimated throughan order-of-magnitudecomparisonwith the E1M11s2p3P – 1s21S 0 0 two-photontransitioninhelium-likeheavyions.Westartedfromarecentevaluation[40],which gives 3.14×109 s−1 for Ra. In order to obtain a good dependence on transition energy and radialmatrix elements,weevaluatedthe E1M1matrixelements (followingEq.(2)ofRef. [40]) 4 Will be inserted by theeditor for He-like Ra (using only 1s2p3P as an intermediate state) and for neutral Ra (using only 1 the 7s7p3P ). Then we integrated over photon energies with the five-point Gauss-Legendre 1 formula.Finally,weevaluatedtheratioofthesetwovaluestoscaletheHe-likeE1M1rate.This gives an order of magnitude estimate of 9.6×10−3 s−1. Such lifetime (≈100 s) of the 7s7p3P 0 state would be comparable to the lifetimes of nuclei of several radium isotopes (it would be significantly shorter only in comparison with the nuclei of the most stable radium isotopes spanning the mass range 223-229).This would be the case of spin-zero isotopes of radium. For the isotopes of radium with a nonzero value of nuclear spin, the hyperfine-induced transition must also be considered. We estimated the 7s7p3P – 7s21S transition rate with a 0 0 simple three-statemodel,inwhichthe wavefunctionsofthehyperfine componentsofthe upper 3P state are described by a symmetry-adapted configuration-state-function expansion of the 0 form |7s7p 3P IFi =c |7s7p 3P IFi+c |7s7p 1P IFi+c |7s7p 3P IFi (3) 0 HFS 0 0 1 1 2 1 runningovertheappropriatehyperfinecomponentsIF ofthe1P and3P states.Thehyperfine- 1 1 induced transition rate (in s−1) may be approximately expressed as A(7s7p 3Po →7s2 1S )= 0 0 2.02613×1018 2 (cid:12)c h7s7p 1PokQ(1)k7s2 1S i+c h7s7p 3PokQ(1)k7s2 1S (cid:12) (4) 3λ3 (cid:12) 1 1 1 0 2 1 1 0(cid:12) (cid:12) (cid:12) where h7s7p 1PokQ(1)k7s2 1S i and h7s7p 3PokQ(1)k7s2 1S i are reduced matrix elements for 1 1 0 1 1 0 the electric dipole operator, and λ is the transition wavelength (in ˚A). The coefficients c and 1 c are relatedto the off-diagonalmagnetic dipole constants AHFS(1P ,3P ) and AHFS(3P ,3P ) 2 M1 1 0 M1 1 0 AHFS(1P ,3P ) AHFS(3P ,3P ) c1 =pI(I +1)∆ME1(1P −1 3P0), c2 =pI(I +1)∆ME(13P −1 3P0) (5) 1 0 1 0 (see [41] or [42] for full derivation). The sum in the expansion (3) should run over all excited states,butthesuminEq.(4)isusuallydominatedbythosestates,forwhichthetransitionrates tothegroundstatearelargeandatthesametimetheenergydenominatorsinEq.(5)aresmall. Incaseofthe7s7p3P stateofradiumthe7s7p3P and7s7p1P statesdominate.Thecontribu- 0 1 1 tionofthe reducedmatrix elementh7s7p3PokQ(1)k7s2 1S iis 3.7times largerthanthatofthe 1 1 0 h7s7p 1PokQ(1)k7s2 1S i matrix element. The contributions of other states are much smaller, 1 1 0 due to the presence of the energy denominators in the coefficients (5) and to the fact that the correspondingoff-diagonalmatrixelementsaremanyordersofmagnitudesmaller(actuallythey are exactly zero, unless non-orthogonalitybetween fully relaxed wavefunctions and correlation effectsarenotneglected),andtheycanbesafelyignoredatpresentleveloftheoverallaccuracy. Therateofthehyperfine-inducedtransitionisisotope-dependent,i.e.itdependsonthenuclear spinandonthenuclearmagneticmoment.Asinourpreviouspaper[27],the223Raisotopewas 88 chosen to set the nuclear parameters (the transition rate A(3P −1S ) may be readily recalcu- 0 0 latedforotherradiumisotopes,forwhichnuclearspinsandmagneticmomentsareknown).The nuclear spin of 223Ra is I = 3/2, and the nuclear magnetic dipole moment µ = 0.2705(19)µ 88 N was takenfromthe paper of Arnoldet al [13]. The electric dipole transitionrates A(1P − 1S ) 1 0 andA(3P −1S )fromtable1(inBabushkingauge)wereused.Theoff-diagonalmagneticdipole 1 0 constants AHFS(1P ,3P ) = 540 MHz, and AHFS(3P ,3P ) = 1172 MHz, were evaluated with M1 1 0 M1 1 0 the use of the same wavefunctions, and in the same approximation, as described in section (3) above.Togetherthey yield hyperfine-induced transitionrate A(3P − 1S ) = 0.0210s−1 in case 0 0 of constructive interference, and A(3P − 1S ) = 0.0070 s−1 in case of destructive interference. 0 0 The above evaluations of the E1M1 and hyperfine-induced rates were performed indepen- dently. As it turned out, the two contributions are of the same size, therefore they have to be treated simultaneously. To this end we employed the effective Hamiltonian method [43]. It requires also an evaluation of the AHFS(3P ,1P ) matrix element (851 MHz). The method is M1 1 1 valid beyond the limits of the perturbation method exposed above, in particular it does not require that the energy separation between levels is large compared to level widths. It yields Will be inserted by theeditor 5 A 3P − 1S =0.02935s−1 (if the two-photontransitionis switched off the effective Hamilto- (cid:0) 0 0(cid:1) nianmethodyieldsA 3P − 1S =0.0197s−1).Intheprocessofthiscalculation,wefoundout (cid:0) 0 0(cid:1) bycomparisonwiththeresultsofmdfgme code[44]thatnon-orthogonalitybetweenspinorbitals playsveryimportantroleintheevaluationofnon-diagonalhyperfinematrixelements(neglect- ingitmayevenleadtoasignchange).Weemployedanewcode,developedbyoneofus(P.J.)to evaluate correlatedoff-diagonalhyperfine matrix elements from GRASP wavefunctions,taking into account the effect of non-orthogonalitybetween spinorbitals. The accuracyis limited by the electric dipole matrix elements h7s7p3PokQ(1)k7s2 1S i and 1 1 0 h7s7p1PokQ(1)k7s21S iaswellasbytheoff-diagonalmagneticdipoleconstantsAHFS(3P ,1P ), 1 1 0 M1 1 1 AHFS(3P ,3P ) and AHFS(1P ,3P ). In case of the matrix element h7s7p 3PokQ(1)k7s2 1S i we M1 1 0 M1 1 0 1 1 0 mayassumethe5%relativeaccuracyforthe 3P − 1S transitionratefromexperiment[5].The 1 0 accuracyoftheA(1P −1S )rateismoredifficulttoestimate(seeref.[25]).Thisisthestrongest, 1 0 ’allowed’ transition in the radium spectrum, but (as mentioned above) the contribution of its matrixelementh7s7p1PokQ(1)k7s21S itothetotalvalueofthecalculatedrateofthehyperfine- 1 1 0 inducedtransition3P −1S is3.7timessmallerthanthatoftheh7s7p3PokQ(1)k7s21S imatrix 0 0 1 1 0 element, so even a relatively large error bar would be quenched. We may very conservatively take the entire difference between the Babushkin and Coulomb gauge final values from table 1 in reference [25] as the error limit, obtaining a relative accuracy 25% for the A(1P − 1S ) 1 0 transition probability. The accuracy of the diagonal magnetic dipole constants was estimated to be 6% (see [26]). Since an off-diagonalconstant depends on both states rather than on one, we, again quite conservatively, doubled the ’diagonal’ error limit estimate and assumed 12% as the contributions of the off-diagonal hyperfine constants to the error bar. Eventually, the above procedure yields the lifetime of the metastable 3P state τ = 34(15)s of 223Ra isotope. 0 88 The lifetime isbasedonboththe 2-photonandhyperfine-inducedchannels.The errorbardoes not include the 2-photon contribution. 4.2 The 7s7p 3P state 1 Figure 1 presents the transition probability A(3P −1S ) as a function of the size of the mul- 1 0 ticonfiguration expansion. The transition rates were calculated in Babushkin and Coulomb gauges,which in the non-relativistic formulationcorrespondto length and velocity form of the transitionintegral,respectively.Bothcurveswereobtainedin the ’core-valence’approximation described above. The resulting Babushkin and Coulomb values are compared with the experi- mentalresultobtainedbytheArgonnegroup[5]andwiththreeavailabletheoreticalvalues(the value A=4·106s−1 obtained by Bruneau [32] did not fit within the vertical scale of figure 1). It is clearly seen in the figure 1 that the core-valence correlation effects are saturated in the framework of the single and restricted double substitutions and five layers of virtual orbitals, as described in section (3) above. The remaining difference between the final Babushkin and Coulomb gauge values may be attributed to the omitted core-core effects. We have made an attempt to estimate the contribution of the core-core correlation to 3D − 1S and 1P − 1S 2 0 1 0 transition rates [25], but for other transitions the gauge differences and comparisonswith data obtained by other authors, where available, are the only indications of the accuracy of our calculatedrates.Althoughthegaugedifferencemustnotbetreatedastheerrorbarperse,itis a usefulindicatorinpartiallysaturatedmulticonfigurationcalculationsoftransitionrates.The values obtained in the Babushkin gauge are weighted toward the outer parts of the electronic wavefunctions, while the Coulomb gauge values weight more inner parts, where the core-core effects arise. Therefore partially saturated expansions often produce Babushkin and Coulomb gauge values converging toward different limits, as in figure 1, with the difference arising from the omitted core-core effects. This is also the reason why Babushkin gauge results are usu- ally treated as more reliable, which seems to be confirmed by the agreement of the Babushkin gaugetransitionrateA(3P −1S )withexperiment(althoughagreementthiscloseisverylikely 1 0 accidental; and good agreement for one level is not enough to justify a more general rule). 6 Will be inserted by theeditor size of MCDHF space ] 1 [2006] - s 6 0 2.5 1 [1978] [ e t a r 2 [2000] n o i t i s n 1.5 a r t 0 S 1 1 - - 1 P 3 1 20000 40000 60000 Fig. 1. Transition probability A(3P1 −1S0) in Babushkin (upper curve with circles) and Coulomb (lower curve with squares) gauges, as a function of the multiconfiguration expansion (in core-valence approximation — see text for details). The lone dot with error bars (to the right from the end of the Babushkin curve) represents the experimental result from reference [5]. The three stars at the far right represent the theoretical data obtained by other authors and are denoted by publication year in brackets: [1978] — reference [31]; [2000] — reference [45]; [2006] — reference [28]. 4.3 The 7s6d 3D and 7s7p 1P states 2 1 Thesetwolevelsaredistinguishedbecausetheywerethesubjectofaseparatepaper[25].Inthe present paper we duly quote the data obtained in core-valence approximation, and the reader is referred to the above mentioned paper for further discussion. 4.4 The remaining 7s7p 3P , 7s6d 3D , 7s6d 3D , and 7s6d 1D states 2 1 3 2 The remaining 7s7p 3P , 7s6d 3D , 7s6d 3D , and 7s6d 1D states, together with the states 2 1 3 2 discussedinsections4.1, 4.2,and 4.3above,constitutefull setofstatesarisingfromthe three lowest electronic configurations of radium. Someoftheselevelsshouldalsobeconsideredmetastable,notonly7s7p3P .The3D −3D 0 3 2 transition is the strongest ’direct’ decay channel for the 3D state, but it is in fact very weak 3 (comparable to that of the 3P − 1S transition). The rates of other possible ’direct’ decay 0 0 channels would be still smaller (e.g. M2 transition 3D − 3P rate is comparable to that of the 3 1 3D − 3P transition),thereforethe multiphotonorhyperfine-inducedtransitionsmayalsoplay 2 0 a role in the 3D lifetime. Similar considerations may in principle apply to the 3P , 1D , and 3 2 2 3D states (we did not pursue this issue, though). 2 5 Summary and conclusions Table1presentscalculatedtransitionrates,andthelifetimesarepresentedintable2.Coulomb gauge values of electric multipole transitions from the table 1 were not used to obtain the lifetimes. With the exception of the 3P level, the transition rates are given with 4 significant 0 digitsandthelifetimes with3significantdigits,butthatnotnecessarilyreflectstheiraccuracy. Will be inserted by theeditor 7 Table 1. Calculated transition rates between nine lowest levels of radium [s−1]. Transition multipo- larities are denoted by E1, E2, M1, M2; the Babushkin and Coulomb gauge values by (B) and (C), respectively. HFS means hyperfine-induced.Numbersin bracketsrepresent powers of 10. transition This work Ref. [28] Expt.[5] 3P0− 1S0 HFS+E1M1 2.935[-2] 3P1− 1S0 E1(B) 2.374[6] 2.760[6] 2.37(12)[6] (C) 1.873[6] 3P1− 3P0 M1 1.334[-2] 3P1− 3D1 E1(B) 8.794[1] 9.850[1] (C) 4.025[3] 3P1− 3D2 E1(B) 1.775[-3] 1.572[-3] (C) 1.607[2] 3P2− 3P0 E2(B) 1.185[-2] (C) 1.243[-2] 3P2− 3D1 E1(B) 4.310[3] 4.897[3] (C) 9.722[3] 3P2− 3D2 E1(B) 4.602[4] 5.204[4] (C) 1.109[5] 3P2− 3D3 E1(B) 1.044[5] 1.234[5] (C) 4.201[5] 1P1− 1S0 E1(B) 1.793[8] 1.805[8] (C) 1.795[8] 1P1− 3D1 E1(B) 3.282[4] 4.195[4] (C) 5.222[4] 1P1− 3D2 E1(B) 9.793[4] 2.646[4] (C) 1.441[5] 1P1− 1D2 E1(B) 3.241[5] 3.194[5] (C) 5.875[5] 3D1− 3P0 E1(B) 1.390[3] 1.529[3] (C) 7.940[3] 3D2− 1S0 E2(B) 2.524[-1] 3.032[-1] (C) 1.630[-1] 3D2− 3P0 M2 3.021[-13] 3D2− 3D1 M1 5.082[-4] 3D3− 3D2 M1 6.352[-3] 1D2− 1S0 E2(B) 2.710[1] (C) 2.271[1] 1D2− 3P1 E1(B) 6.960[2] 7.722[3] (C) 1.224[3] 1D2− 3P2 E1(B) 5.930[0] 7.973[0] (C) 7.535[0] As discussed in section(4.2) above,the accuracyof our results is difficult to estimate, but it is probably much worse than 3 or 4 digits, particularly for very weak transitions. The results of the presentcalculationsmay be consideredas fully convergedincore-valence approximation(convergencehasindeedbeenobservedforalltransitions,similarlytothatshown in figure 1), with core-core effects omitted. With the exception of the 1P − 3D and 1D − 3P transitions (we cannot offer any plau- 1 2 2 1 sible explanation for these discrepancies), our Babushkin gauge values are in reasonably good agreementwiththeresultsofDzubaandGinges[28].Aninterestingfeatureisthelargediscrep- ancybetweentheresultsobtainedinBabushkinandCoulombgaugesfortransitionsconnecting closely-lyingtriplet P and triplet D states.The B/C ratios (i.e. the ratiosof Babushkinversus Coulomb gauge results) turned out to be much closer to unity for those transitions when the experimental energies in the transition operator were replaced by theoretical ones. This obser- vation,togetherwithinconsistentresultsofthecalculationsofenergiesofexcitedstates,hadled 8 Will be inserted by theeditor Table 2. Calculated lifetimes ofeightlowest excitedstatesofradium,comparedwith datafromother authors. state This work Ref. [28] Ref. [45] Ref. [31] Ref. [32] Expt. [5] 3P0 34(15) s a 3P1 421 ns 362 ns 505 ns 420 ns 250 ns 422(20) ns 3P2 6.46 µs 5.55 µs 5.2 µs 1P1 5.56 ns 5.53 ns 5.5 ns 3D1 719 µs 654 µs 617 µs 3D2 3.95 s 3.3 s 15 s 3D3 157 s b 1D2 1.37 ms 0.129 ms c 38 ms (a) based on E1M1 and hfs-induced decay channelsin the22838Raisotope (b) based on 3D3− 3D2 magnetic dipole transition only; other possible decay channels neglected (c) calculated from transition probabilities quoted after Table VIin Ref. [28] us[25]tosuggestanexperimentalverificationofradiumdatainMoore’stables[8].Atthattime we were not awareof the papers by Eliav et al [23] and Landau et al [24], where the excitation energies of radium were calculated in the coupled-cluster approximation. More recently these calculationswereindependently confirmedwithinthe frameworkofthe CI+MBPTtheory[28]. In both cases good agreement with experiment had been achieved. There are two principal differences between the methods andapproximationsused in the abovementionedthree papers withrespecttothemethodsandapproximationsusedinthepresentpaper.Thetransitionrates were the primary targets of the present calculations, not the transition energies. Therefore we optimised the electronic wavefunctions separately for each of the nine states of interest. The core-valencecorrelationeffectswerefullyaccountedfor.Thecore-corecorrelationeffects,which arelessimportantincalculationsoftransitionratesandhyperfinestructures,weretreatedina very crude approximationin the cases of 1S , 3D and 1P states, and were neglected for other 0 2 1 states.The calculations ofenergy leveldifferences requirewell balancedorbitalsets and exten- sivemulticonfigurationexpansions.Theresultsareusuallyinbetteragreementwithexperiment if a common set of orbitals is used for both states. If the orbital sets are separately optimised, the transition energy is obtained as a pure difference of the total energies of the two states of interest. They are both several orders of magnitude (five orders in case of radium) larger than the transition energy itself, therefore our calculated transition energy values may be less accurate than the calculated transition rates and hyperfine structures. Any further refinement of the present calculations would require computer resources,which are currently unavailable. 6 Acknowledgments This workwassupportedby the PolishMinistry ofScience andHigherEducation(MNiSW) in the framework of the scientific grant No. 1 P03B 110 30 awardedfor the years 2006-2009. ◦ Laboratoire Kastler Brossel is Unit´e Mixte de Recherche du CNRS n C8552. P.J. acknowledges the support from the Swedish Research Council (VR). References 1. J.A.Behr,A.Gorelov,D.Melconian,M.Trinczek,W.P.Alford,D.Ashery,P.G.Bricault,L.Cour- neyea,J.M. D’Auria, J. Deutsch et al., Eur. 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