Relativistic coupled-cluster calculations of nuclear spin-dependent parity non-conservation in Cs, Ba + and Ra +
RRelativistic coupled-cluster calculations of nuclear spin-dependent paritynon-conservation in Cs, Ba + and Ra + B. K. Mani and D. Angom
Physical Research Laboratory, Navarangpura-380009, Gujarat, India
We have developed a relativistic coupled-cluster theory to incorporate nuclear spin-dependentinteraction Hamiltonians perturbatively. This theory is ideal to calculate parity violating nuclearspin-dependent electric dipole transition amplitudes, E NSDPNC , of heavy atoms. Experimental ob-servation of which is a clear signature of nuclear anapole moment, the dominant source of nuclearspin-dependent parity violation in atoms and ions. We apply the theory to calculate E NSDPNC of Cs,which to date has provided the best atomic parity violation measurements. We also calculate E NSDPNC of Ba + and Ra + , candidates of ongoing and proposed experiments. PACS numbers: 31.15.bw, 11.30.Er, 31.15.am
The effects of parity nonconservation (PNC) in atomsoccur in two forms, nuclear spin-independent (NSI) andnuclear spin-dependent (NSD). The former is well stud-ied and experimentally observed in several atoms. Thesignature of the later (NSD) has been observed only inone experiment with Cs [1] and the same experiment hasprovided the most accurate results on NSI atomic PNCas well. In an atom or ion the most dominant sourceof NSD-PNC is the nuclear anapole moment (NAM), aparity odd nuclear electromagnetic moment. It was firstsuggested by Zeldovich [2] and arises from parity violat-ing phenomena within the nucleus.One major hurdle to a clear and unambiguous ob-servation of NAM is the large NSI signal, which over-whelms the NSD signal. However, proposed experi-ments with single Ba + ion [3] could probe PNC in the s / − d / transition, where the NSI component is zero.This could then provide an unambiguous observation ofNSD-PNC and NAM in particular. The ongoing exper-iments with atomic Ytterbium [4] is another possibility,the 6 s S − s d D transition, to observe NSD-PNCwith minimal mixture from the NSI component. One cru-cial input, which is also the source of large uncertainty,to extract the value of NAM is the input from atomictheory calculations. Considering this, it is important toemploy reliable and accurate many-body theory in theatomic theory calculations.The coupled-cluster (CC) theory[5, 6] is one of themost reliable many-body theory to incorporate electroncorrelation in atomic calculations. It has been used withgreat success in nuclear [7], atomic [8–10], molecular [11]and condensed matter [12] physics. In atomic physics, therelativistic coupled-cluster (RCC) theory has been usedextensively in atomic properties calculations, for exam-ple, hyperfine structure constants [10, 13] and electro-magnetic transition properties [14, 15]. In atomic PNCcalculations too, RCC is the preferred theory and severalgroups have used it to calculate NSI-PNC of atoms [16–18]. However, the calculations in Ref. [16] are entirelybased on RCC with a variation we refer to as perturbedRCC (PRCC), where as the calculations in Ref. [17, 18]are based on sum over states with CC wave functions. To date, the use of PRCC in atomic PNC is limitedto NSI-PNC. In this letter we report the PRCC theoryto calculate NSD-PNC in atoms. Such a development istimely as the recent experimental proposals on Ba + andRa + [19] and observation of large enhancement in atomicYb [4] shall require precision atomic theory to examinethe systematics and interpret the results. It must perhapsbe mentioned that, in an earlier work we had developedand calculated electric dipole moment of atomic Hg [20]using PRCC theory. RCC theory .—In the RCC method, the atomic stateis expressed in terms of T and S , the closed-shell andone-valence cluster operators respectively, as | Ψ v (cid:105) = e T (0) (cid:104) S (0) (cid:105) | Φ v (cid:105) , (1)where | Φ v (cid:105) is the one-valence Dirac-Fock reference state.It is obtained by adding an electron to the closed-shellreference state, | Φ v (cid:105) = a † v | Φ (cid:105) . In the coupled-clustersingles doubles (CCSD) approximation T (0) = T (0)1 + T (0)2 and S (0) = S (0)1 + S (0)2 . The open-shell cluster operatorsare solutions of the nonlinear equations [21] (cid:104) Φ pv | ¯ H N + { ¯ H N S (0) }| Φ v (cid:105) = E att v (cid:104) Φ pv | S (0)1 | Φ v (cid:105) , (2a) (cid:104) Φ pqva | ¯ H N + { ¯ H N S (0) }| Φ v (cid:105) = E att v (cid:104) Φ pqva | S (0)2 | Φ v (cid:105) , (2b)where ¯ H N = e − T (0) H N e T (0) is the similarity transformedHamiltonian and the normal ordered atomic Hamilto-nian H N = H − (cid:104) Φ | H | Φ (cid:105) . And, E att v = E v − E , isthe attachment energy of the valence electron. The T (0) are solutions of a similar set of equations, however, with S (0) = 0. A similar set of equations may be derived in thecase of two-valence systems and use it in the wave func-tion and properties calculations of atoms like Yb [22]. Perturbed RCC theory .—The perturbed RCC method[23, 24], unlike the standard time-independent perturba-tion theory, implicitly accounts for all the possible inter-mediate states in properties calculations. Consider theNSD-PNC interaction Hamiltonian H NSDPNC = G F µ (cid:48) W √ (cid:88) i α i · I ρ N ( r ) , (3) a r X i v : . [ phy s i c s . a t o m - ph ] J un as the perturbation. Here, µ (cid:48) W is the weak nuclear mo-ment of the nucleus and ρ N ( r ) is the nuclear density. Thetotal atomic Hamiltonian is H A = H DC + λH NSDPNC , (4)where λ is the perturbation parameter. Mixed parity hy-perfine states | (cid:101) Ψ v (cid:105) are then the eigen states of H A . Tocalculate | (cid:101) Ψ v (cid:105) from RCC, we define a new set of clus-ter operators T (1) , which unlike T (0) connects the ref-erence state to opposite parity states. This is the re-sult of incorporating one order of H NSDPNC and for this rea-son we refer to T (1) as the perturbed cluster operators.Although hyperfine states are natural to H NSDPNC , clus-ter operator T (1) is defined to operate only in the elec-tronic space and is a rank one operator. For this define H NSDelec = ( G F µ (cid:48) W ) / ( √ (cid:80) i α i ρ N ( r ), which operates onlyin the electronic space, so that H NSDPNC = H NSDelec · I . Theclosed-shell exponential operator in PRCC is e T (0) + λ T (1) · I and the atomic state is | (cid:101) Ψ (cid:105) = e T (0) (cid:104) λ T (1) · I (cid:105) | Φ (cid:105) . (5)Similarly, the mixed parity state from one-valence PRCCtheory is | (cid:101) Ψ v (cid:105) = e T (0) (cid:104) λ T (1) · I (cid:105) (cid:104) S (0) + λ S (1) · I (cid:105) | Φ v (cid:105) . (6)As T (1)1 is one particle and rank one operator, in termsof c-tensors T (1)1 = (cid:88) ap τ pa C (ˆ r ) , (7)where C i are c-tensor operators. Similarly, the tensorstructure of T (1)2 is T (1)2 = (cid:88) abpq (cid:88) l ,l τ pqab ( l , l ) { C l (ˆ r ) C l (ˆ r ) } , (8)where {· · · } indicates the two c-tensor operators cou-ple to a rank one tensor operator. Based on the tensorstructures, the perturbed cluster operators are diagram-matically represented as shown in Fig. 1. For the doubles T (1)2 , to indicate the multipole structure, an additionalline is added to the interaction line. The cluster opera-tors are solutions of the equations (cid:104) Φ pv |{ ¯ H N S (1) } + { ¯ H N T (1) } + { H N T (1) S (0) } + ¯ H NSDelec + { ¯ H NSDelec S (0) }| Φ v (cid:105) = ∆ E v (cid:104) Φ pv | S (1)1 | Φ v (cid:105) , (9a) (cid:104) Φ pqvb |{ ¯ H N S (1) } + { ¯ H N T (1) } + { H N T (1) S (0) } + ¯ H NSDelec + { ¯ H NSDelec S (0) }| Φ v (cid:105) = ∆ E v (cid:104) Φ pqvb | S (1)2 | Φ v (cid:105) , (9b) T (1)1 T (1)2 S (1)1 S (1)2 FIG. 1. Diagrammatic representation of single and doubleexcitation perturbed cluster operators. The short line on theinteraction line of T (1)2 and S (1)2 is to indicate the multipolestructure of these operators. Where we have used the relations (cid:104) Φ pv | T (1) | Φ v (cid:105) = 0,and (cid:104) Φ pv | T (1) S | Φ v (cid:105) = 0, as the bra state is valence ex-cited. An approximate form of Eq. (9), but which con-tains all the important many-body effects, are the lin-earized cluster equations. This is obtained by consider-ing ¯ H N T (1) ≈ H N T (1) , and ¯ H NSDelec ≈ H NSDelec + H NSDelec T (0) .We refer to this as the linear approximation and use itextensively to check the results. E NSDPNC calculations .—If | Ψ v (cid:105) and | Ψ w (cid:105) are atomic statesof same parity, then the H NSDPNC induced electric dipoletransition amplitude E NSDPNC = (cid:104) (cid:101) Ψ w || D || (cid:101) Ψ v (cid:105) , where D isthe dipole operator. Similarly, the transition amplitudewithin the electronic sector is E NSDelec = (cid:104) Φ w || ¯ D (cid:104) T (1) + S (1) + T (1) S (cid:105) + (cid:104) T (1) + S (1) + T (1) S (cid:105) † ¯ D + S † ¯ D (cid:104) T (1) + S (1) + T (1) S (cid:105) + (cid:104) T (1) + S (1) + T (1) S (cid:105) † ¯ D S || Φ v (cid:105) , (10)where ¯ D = e T † D e T , is the dressed electric dipole op-erator. It is evident that ¯ D is a non-terminating seriesof the closed-shell cluster operators. It is non-trivial toincorporate T to all orders in numerical computations.For this reason ¯ D approximated as ¯ D ≈ D + D T (0) + T (0) † D + T (0) † D T (0) . This captures all the importantcontributions arising from the core-polarization and pair-correlation effects. Terms not included in this approxi-mation are third and higher order in T (0) . The expressionused in our calculations is then E NSDelec ≈ (cid:104) Φ w || DT (1) + T (0) † DT (1) + T (1) † D T (0) + T (1) † D + DT (1) S (0) + T (1) † S (0) † D + S (0) † DT (1) + T (1) † D S (0) + DS (1) + S (1) † D + S (0) † DS (1) + S (1) † D S (0) || Φ v (cid:105) . (11)From our previous study of properties calculations [21],we conclude that the contributions from the higher orderare negligible. Coupling with nuclear spin .—To couple E NSDelec withnuclear spin I and obtain E NSDPNC , consider the exchangediagram in Fig. 2(a). It arises from the term T (0)2 † D S (1)2 in the PRCC expression of E NSDelec .To demonstrate the non-trivial angular integration, inhyperfine atomic states, the angular momentum diagram a p rq vw (a) k − + j a l − l + k − + j p j r j q k − j w j v IIF w − F v +(b) FIG. 2. Examples of E NSDPNC diagrams (a) one of the exchangediagrams in electronic sector and (b) angular momentum di-agram in terms of hyperfine states and the portion within thedash lines is the electronic component. of the same diagram is shown in Fig. 2(b). Conventionsof phase and angular momentum lines of Lindgren andMorrison [25] are used while drawing the diagram. Theportion of the diagram within the rectangle in dashed-lineis the angular momentum part of the electronic sector.The evaluation of the angular integral of the electronicsector, following Wigner-Eckert theorem, is equivalent to (cid:104) j w m w | (cid:88) l (cid:26) T (0)2 † DS (1)1 (cid:27) l | j v m v (cid:105) = ( − j w − m w × (cid:88) l (cid:18) j w l j v − m w q m v (cid:19) (cid:104) j w || (cid:26) T (0)2 † DS (1)1 (cid:27) l || j v (cid:105) , (12)where { . . . } l represents coupling of rank one tensor oper-ators D and S (1) to an operator of rank l . This couplingis a structure common to any PRCC term of E NSDelec .From the triangular condition, l = 0 , , l contribute de-pends on j v and j w . For example, l = 0 , S / → S / transition of atomic Cs [1],where as only l = 2 contributes to the proposed PNC6 S / → D / transition in Ba + [3].The angular momentum diagram in Fig. 2(b), afterevaluation, reduces to a 9 j -symbol and free line part.Algebraically, the matrix element in the hyperfine statesis (cid:88) l (cid:104) F w m w | (cid:40)(cid:20) T (0)2 † DS (1)1 (cid:21) l I (cid:41) | F v m v (cid:105) = ( − F w − m w × (cid:18) F w F v − m w q m v (cid:19) (cid:104) F w || D eff || F v (cid:105) , (13)where D eff = (cid:80) l { [ T (0)2 † DS (1)1 ] l I } , is the effective dipoleoperator in the hyperfine states. As seen from the an-gular momentum diagram, coupling of angular momentain proper sequence is essential to obtain correct angularfactors. However, the sequence is not manifest in thealgebraic expression. TABLE I. Reduced matrix element, E NSDPNC , of the 6 S / → S / , 6 S / → D / and 7 S / → D / transi-tions between different hyperfine states in Cs, Ba + and Ra + respectively. The values listed are in units of iea × − µ (cid:48) W .Atom Transition This work Other works F f F i DF MBPT PRCC
Cs 3 3 2 .
011 2 .
060 2 .
274 2 .
249 [26]4 4 2 .
289 2 .
338 2 .
589 2 .
560 [26]4 3 5 .
000 4 .
819 5 .
446 6 .
432 [26],7 .
057 [27]3 4 5 .
774 5 .
662 6 .
313 7 .
299 [26],7 .
948 [27] Ba + − . − . − . − .
915 [19], − .
565 [28]2 1 2 .
707 2 .
834 1 .
607 2 .
682 [19],2 .
430 [28] Ba + − . − . − . − .
250 [19] − .
510 [28]2 3 6 .
888 7 .
386 6 .
096 7 .
389 [19],6 .
510 [28] Ra + − . − . − . − .
918 [19], − .
90 [28] Ra + − . − . − . − .
204 [19], − .
65 [28]2 1 30 .
307 32 .
286 15 .
683 30 .
525 [19],24 .
15 [28] Ra + − . − .
788 1 . − .
297 [19]3 2 47 .
336 50 .
917 20 . − .
50 [28]2 2 − . − . − . − .
387 [19], − .
00 [28]TABLE II. Component wise contribution from the coupled-cluster terms for 6 S / → S / transition in Cs,6 S / → D / transition in Ba + , and 7 S / → D / transition in Ra + .Atom Transition DS (1)1 S (1)1 † D DT (1)1 S (0) † DS (1)1 F f F i + c.c. + c.c. Cs 3 3 − .
278 4 . − . − . − .
317 4 . − . − . .
764 6 . − . − . .
657 8 . − . − . Ba + − .
676 0 . − . − . . − .
723 0 . − . Ba + − .
808 1 . − .
620 0 . . − .
496 1 . − . Results .—For the calculations reported in the letter,we use Gaussian type orbitals generated with V N − cen-tral potential. The E NSDPNC of Cs, Ba + and Ra + betweenvarious hyperfine states are given in Table. I. There is aclose match between our MBPT results and results fromsimilar works.There are changes when the transition amplitudes arecalculated with PRCC. This can be attributed to theinclusion of higher order correlation effects. However,it require a systematic series of calculations to examinethe nature of the correlation effects from the higher or-der terms which are subsumed in the PRCC calculations.The results of Ra + is a cause for concern, there is alarge cancellation in the F i = 3 → F f = 2 transitionamplitude. However, for the other two transitions of thesame ion, the transition amplitudes are higher than Ba + .In particular, the F i = 2 → F f = 2 transition amplitudeof Ra + is the largest among all the values and thisis in agreement with the previous results. For the neu-tral atom Cs, the PRCC results are larger than MBPT.This indicates, higher order correlation effects enahances E NSDPNC . It is opposite in Ba + and Ra + , the PRCC resultsare lower than MBPT and indicates higher correlation ef-fects have suppression effect.To examine the impact of electron correlation in bet-ter detail, consider the leading order (LO) and next toleading order (NLO) terms as listed in Table. II. In thePRCC calculations, as given in Eq. (11), for Cs these are S (1)1 † D and D S (1)1 , respectively. Here, the former repre-sents H NSDPNC perturbed 7 S / and has larger oppositeparity mixing as it is energetically closer to odd paritystates like 6 P / . The same is not true of 6 S / , whichis represented by D S (1)1 .In the case of Ba + the LO and NLO are D S (1)1 and S (1)1 † D , respectively. Although, not shown in Table. II asimilar pattern is observed in Ra + . The sequence is op-posite to Cs. Reason is, the transitions in these ions areof n S / → n (cid:48) D / type and matrix elements of H NSDPNC involving n (cid:48) D / are negligible. Dominant contributionarises from the sp matrix elements, which are large. So,the term D S (1)1 , which represents H NSDPNC perturbation of n S / is the LO term of these ions. The contributionfrom S (1)1 † D is, however, non-zero as n (cid:48) D / acquiresopposite parity mixing through electron correlation ef-fects. It must be mentioned that, the Dirac-Fock contri-bution is the most dominant, however, in PRCC it is sub-sumed in the LO and NLO terms. The terms which aresecond order in cluster operators, in Eq. (11), are non- zero but small. For comparison, the two dominant con-tributions from the second order term, S (0) † DS (1)1 andit’s hermitian conjugate, is given in the Table. II.We have also calculated the E NSDPNC of the 6 S / → D / and 7 S / → D / transitions in Ba + andRa + , respectively, and the results are given in Table. III.The results from the PRCC are much larger than theMBPT results and this shows, without any ambiguity,electron correlation is the key to get meaningful results.This is on account of d / in the atomic n D / states,which are diffused and leads to larger electron correla-tions. Conclusions .— The PRCC theory we have developedincorporates electron correlation effects arising from aclass of diagrams to all order with a nuclear spin-dependent interaction as a perturbation. It is a suitable
TABLE III. Reduced matrix element, E NSDPNC , of the6 S / → D / and 7 S / → D / transitions be-tween different hyperfine states Ba + and Ra + respectively.The values listed are in units of iea × − µ (cid:48) W .Atom Transition This work Other works F f F i DF MBPT PRCC Ba + .
003 0 .
098 0 .
227 0 .
041 [19]2 1 0 .
002 0 .
048 0 . − Ba + .
003 0 .
125 0 .
235 0 .
043 [19] Ra + − . − .
605 1 . − .
526 [19] Ra + − . − .
324 0 . − .
256 [19] theory for precision calculations of atomic PNC arisingfrom H NSDPNC . With this method, it is possible to incorpo-rate electron correlation effects within the entire config-uration space obtained from a set of spin-orbitals.
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