Hyperfine and Zeeman interactions of the a(1) [ 3 Σ + 1 ] state of PbO
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Hyperfine and Zeeman interactions of the a (1)[ Σ +1 ] state of PbO A.N. Petrov ∗ Petersburg Nuclear Physics Institute, Gatchina, Leningrad district 188300, Russia andInstitute of Physics, Saint Petersburg State University, Saint Petersburg, Petrodvoretz 198904, Russia
The role of the interaction with the nearest electronic state Σ +0 − on the hyperfine structure andmagnetic properties of the a (1)[ Σ +1 ] state of PbO is assessed. The accounting for this contributionleads to difference between g -factors of the J = 1 Ω-doublet levels, ∆ g = 37 × − , that is ina good agreement with the experimental datum ∆ g = 30(8) × − . The contribution of thisinteraction rapidly grows with J . For J = 30 the difference of g -factors of Ω-doublet states reaches100%; for hyperfine constants it is 18%. These differences also depend on the electric field andfor E = 11 V/cm for PbO the difference in g -factors turn to zero. The latter is important forsuppressing systematic effects in the electron electric dipole moment search experiment. The use of a (1) excited state of PbO molecule has beenproposed to search for electric dipole moment (EDM) ofthe electron [1]. This experiment is a serious test of the“new physics” beyond the Standard Model including dif-ferent supersymmetric models [2–5]. Because of that themolecule was intensively investigated both theoretically[6–10] and experimentally [11–13].In the adiabatic approximation rotational levels of the a (1) state of PbO are determined by the effective spin-rotational Hamiltonian H sr = B ′ J + A k ( J e · n )( I · n ) + µ B G k ( J e · n )( B · n ) − D n · E (1)Here B ′ is the rotational constant, J , J e , I are theelectron-rotational, electron and nuclear angular momen-tum operators, respectively (in this paper we will mea-sure angular momentum in units of ¯ h ), E and B are exter-nal electric and magnetic fields, D is the molecular-framedipole moment, n is a unit vector along the molecularaxis, ζ , directed from Pb to O, µ B is Bohr magneton.The hyperfine constant A k and g -factor G k are deter-mined by the expressions [14] A k = 1Ω µ Pb I h Ψ e Σ + ± | X i (cid:18) α i × r i r i (cid:19) ζ | Ψ e Σ + ± i , (2) G k = 1Ω h Ψ e Σ + ± | J eζ + S eζ | Ψ e Σ + ± i , (3)where S e is the electron spin operator, µ Pb is the mag-netic moment of Pb, α i are the Dirac matrices for the i -th electron, r i is its radius-vector in the coordinate sys-tem centered on the Pb atom, Ω = h Ψ Σ + ± | J eζ | Ψ Σ + ± i = ±
1. From naturally abundant isotopes only
Pb ( I =1 /
2) has nonzero µ Pb , for Pb and Pb I = 0 and,therefore, µ Pb = 0.The parameters B ′ , A k , G k and D can be obtainedboth theoretically from calculation of the electronic wave-function Ψ ea (1) and by fitting the experimentally observed transitions to the parameters of the spin-rotationalHamiltonian (1). Comparison of theoretical and experi-mental values gives us information about accuracy of thecalculated wavefunction Ψ ea (1) and, therefore, also givesinformation about accuracy of the calculated effectiveelectric field, W d , seen by an unpaired electron [7, 8].Note, that W d can not be measured independently, butit is required for extracting d e from the EDM experiment.The experimentally observed parameters A k , G k also canbe used for a semiemperical evaluation of W d [6].Previous investigations of PbO were based on the spin-rotational Hamiltonian (1). The main goal of the presentwork is to account for the interaction with the nearestelectronic state Σ +0 − , which modifies the form of thisHamiltonian. To the best of our knowledge this is thefirst investigation of such kind for open shell diatomics.In the present paper the hyperfine structure of rota-tional levels was obtained by numerical diagonalizationof the Hamiltonian in the basis set of electronic rotationalwavefunctionsΨ e Σ + ± θ JM, ± ( α, β ) U M I , Ψ e Σ +0 − θ JM, ( α, β ) U M I , (4)where θ JM, Ω ( α, β ) = p (2 J + 1) / πD JM, Ω ( α, β, γ = 0)and U M I are rotational and nuclear spin wavefunctions, M and M I = ± / J and I , on the laboratory axis z . When elec-tronic matrix elements are known then matrix elementson the basis set (4) can be calculated with the help of theangular momentum algebra [15]. Required diagonal elec-tronic matrix elements, being, in fact, the parameters ofthe spin-rotational Hamiltonian (1), are known from ex-periments. For the fifth vibrational level of the a (1) stateof PbO they are B ′ = 0 . − , A k = − . G k = 1 . D = 1 .
28 a . u . [11–13]. For purposesof the present study, it is not required to account forthe small difference between the rotational constants of , , PbO molecules. The differences in propertiesdiscussed below are relevant only to the fact that theisotope
Pb has hyperfine structure. The off-diagonal electronic matrix elements were calculated in the presentstudy by the configuration interaction method with thegeneralized relativistic effective core potential [16, 17].The scheme of the calculation is the same as that in thepaper [8]. The calculated matrix elements are∆2 = B ′ h Ψ e Σ +1 | J e + | Ψ e Σ +0 i = 0 .
17 cm − , (5) µ Pb I h Ψ e Σ +1 | X i (cid:18) α i × r i r i (cid:19) + | Ψ e Σ +0 i = − . , (6) G ⊥ = h Ψ e Σ +1 | J e + + S e + | Ψ e Σ +0 i = 1 . . (7)It is known that Hamiltonian (1) leads to two-fold de-generacy of levels with different signs of Ω. This degen-eracy is in fact only approximate. When the interaction(5) is taken into account each rotational level splits ontwo sublevels, called Ω-doublet levels. One of them iseven ( p = 1) and the other one is odd ( p = −
1) withrespect to changing the sign of electrons and nuclear co-ordinates. In order to reproduce experimental value ofthe Ω-doubling, 5 . J ( J + 1) MHz [12], the matrix el-ement (5) has to be equal to 0.15 cm − . We considerthis a good agreement, but will use experimental valuehereafter. The states with p = ( − J are denoted as e and with p = ( − J +1 as f states. Note that the wave-functions Ψ e Σ +0 − θ JM, ( α, β ) U M I are f states, and they donot interact (see below) with e states of the a (1), unlessparity is not conserved, due to weak interactions.Interactions (6) and (7) lead to different hyperfinestructure and magnetic properties of the e and f lev-els. One can estimate from the second order perturbationtheory that contribution from the terms |h Ψ e Σ + ± θ JM, ± U M I | ˆ H hfs(mag) | Ψ e Σ +0 − θ JM, U M I i| / (cid:16) E Σ +1 − E Σ +0 (cid:17) (8)is small. Here ˆ H hfs and ˆ H mag are Hamiltonians of the hy-perfine interaction and the interaction with the externalmagnetic field, respectively. However, the terms2Re( h Ψ e Σ + ± θ JM, ± U M I | B ′ JJ e | Ψ e Σ +0 − θ JM, U M I i ×h Ψ e Σ +0 − θ JM, U M I | ˆ H hfs(mag) | Ψ e Σ + ± θ JM, ± U M I i ) / (cid:16) E Σ +1 − E Σ +0 (cid:17) (9)are much larger and their influence on the spectrum ofthe a (1) state is observable.In Table I we give calculated g -factors for f states withdifferent quantum numbers J of , PbO molecule.For e states calculated g e = 1 . J . We define g -factors so that the Zeeman splitting TABLE I: The g -factors for f -states of , PbO as a func-tion of J . For e states g e = 1 . J . J g f J g f J for f - and e -levels of PbO. Results of Ref.[11] are given in thelast column (see Eqs.(1) and (2) therein). These results arefor both f - and e -levels.J this work Ref. [11]f e1 3188 3187 31952 1905 1903 19133 1356 1353 13585 863 858 85910 458 449 44915 317 304 30420 248 230 23030 181 154 154 is equal to g e ( f ) µ B B z M/J ( J + 1). Our calculations weredone using the finite field method.The obtained difference g f − g e = 37 × − for J = 1 isin good agreement with the experimental result g f − g e =30(8) × − [12]. As it is seen from Table I the differenceis rapidly increasing with J , and for J = 30 g f is abouttwo times larger than g e . Another point to note is thatmatrix elements (5) and (7) do not contribute to g e andit remains J -independent and unchanged. This is due tothe mentioned above parity selection rule. Limiting bythe terms (9) we obtain g f − g e = ∆ · G ⊥ · J ( J + 1) (cid:16) E Σ +1 − E Σ +0 (cid:17) = 1 . × − J ( J + 1)that is in a good agreement with Table I.In Table II, the hyperfine splitting (HFS) calculatedbetween F = J − / F = J + 1 / e and f states of PbO. Also theresults obtained by applying Eq. (1) and (2) of ref. [11]are listed. Eqs. (1) and (2) of ref. [11] give HFS in theframework of the Hamiltonian (1). The interaction withthe Σ +0 − is not taken into account in the (1), thereforeEqs. (1) and (2) of ref. [11] give the same HFS for e and f states of the a (1). Similarly to g factors, the hyperfinestructure of e states is not affected when interactions (5)and (6) are taken into account. However, there is a smalldifference between the hyperfine splittings calculated byEqs. (1) and (2) in Ref. [11] and that calculated for e states in this paper. This difference is related with thefact that the mixing between the states with ∆ J = ± J = 1 , M ± e or f levels ismeasured. This Stark effect induced by the interactionwith the electron EDM that violate both parity ( P ) andtime reversal ( T ) invariance, and is not related with the(large) dipole moment D presented in the (1). For detailssee pp. 1–3 in Ref. [3]. In the external electric field thestates J = 1, M = ± P and T are violated. However an external magnetic fieldremove degeneracy between them and can mimic the ex-istence of the EDM. For J = 1 levels the systematics dueto spurious magnetic fields can be suppressed if the dif-ference between g e and g f can be made smaller [12]. Theexternal electric field mixes e and f levels. Therefore, onthe first glance, one can expect that when increasing theelectric field the initial small difference between g e and g f can be made zero. However, it was found in [12] thatthis difference for , PbO is actually increases as theelectric field increases. This fact was explained by M.G.Kozlov (see acknowledgments in [13]) by accounting forthe mixing with J = 2 level. In the present paper wereproduce this result for spinless isotopes of led and alsocalculate g -factors for J = 1 , F = 1 / , / PbO. For
PbO, g -factors was defined so that theZeeman shift is given by g e ( f ) µ B B z F ( F + 1) + J ( J + 1) − / F ( F + 1) J ( J + 1) M F . With this definition they will coincide with g -factors of , PbO in the limit of zero hyperfine interaction. Thecorresponding results are given in Fig. (1). One can seethat difference between g e and g f for J = 1 , F = 3 / E increases. However, for F = 1 / , J = 1 at E ≈
11 V/cm g e and g f becomeequal. The plotted g e and g f for J = 1 , PbO arein agreement with Fig. (5) of ref. [13]. Large deviationof g -factors for J = 1 , F = 3 / PbO from thosefor J = 1 of , PbO is explained by mixing of the J = 1 , F = 3 / J = 2 , F = 3 / PbOthat is induced by the hyperfine interaction.In the EDM experiment the maximum Stark split-ting, 2W d · d e , between F = 1 / , M F = ± / F = 1 / , M F = ± / E = 11 V/cm the obtained splitting isabout 75% of the maximal value.In this work we account for non-adiabatic interactionof a (1)[ Σ +1 ] state only with the state Σ +0 − . There areseveral reasons for this. One can see [18, 19] that the Σ +0 − state is the nearest one to the a (1) state. All other states, except ∆, are more than an order of magnitudefurther away. Accounting for the non-adiabatic interac-tion with the ∆ state (the same Ω = 1 as in a (1)) willlead only to a small modification of the parameters of thespin-rotational Hamiltonian (1). Since we use the exper-imental data, those interactions with the ∆ and otherΩ = 1 states are taken into account. Though the interac-tion with ∆ can not be described in the framework ofthe Hamiltonian (1), it will not lead in the leading orderto the difference in properties of the f and e states thatis a topic of this paper. Moreover, our calculation showthat the corresponding matrix element B ′ h Ψ e Σ +1 | J e − | Ψ e ∆ +2 i ≈ × − cm − is small as compared to (5). Ω = 3 states are not mixed inthe leading order due to the selection rule. The validity ofthe above approximation is approved by the fact that thecalculated and the experimentally obtained differencesof the g -factors for e and f J = 1 states are in goodagreement.Finally we have investigated the influence of the in-teraction with the nearest electronic state Σ +0 − onthe hyperfine structure and magnetic properties of the a (1)[ Σ +1 ] state. We have shown that it is required forits accurate description, especially for g-factors. One cansuppose that similar situation takes place also for otherdiatomics in Ω = 1 states. It is found that the differ-ence between g e and g f for PbO is converged to zeroat E ≈
11 V/cm. The latter is important for the sup-pressing systematic effects in the electron EDM searchexperiment.I am grateful to M.G. Kozlov and A.V. Titov forvery useful discussions. This work supported by RFBRGrants No. 09–03–01034 and by the Ministry of Educa-tion and Science of Russian Federation (Program for De-velopment of Scientific Potential of High School) GrantNo. 2.1.1/1136 ∗ Electronic address: [email protected][1] D. DeMille, F. Bay, S. Bickman, D. Kawall, D. Krause,Jr., S. E. Maxwell, and L. R. Hunter, Phys. Rev. A ,052507 (2000).[2] M. Kozlov and L. Labzowsky, J. Phys. B , 1933 (1995).[3] E. D. Commins, Adv. At. Mol. Opt. Phys. , 1 (1998).[4] A. V. Titov, N. S. Mosyagin, A. N. Petrov, T. A. Isaev,and D. P. DeMille, Progr. Theor. Chem. Phys. B 15 , 253(2006).[5] M. Raidal et al., Eur. Phys. J. C , 13 (2008), arXiv:0801.1826.[6] M. G. Kozlov and D. DeMille, Phys. Rev. Lett. , 133001(2002).[7] T. A. Isaev, A. N. Petrov, N. S. Mosyagin, A. V. Titov,E. Eliav, and U. Kaldor, Phys. Rev. A , 030501(R)(2004). [8] A. N. Petrov, A. V. Titov, T. A. Isaev, N. S. Mosyagin,and D. P. DeMille, Phys. Rev. A , 022505 (2005).[9] E. R. Meyer, J. L. Bohn, and M. P. Deskevich, Phys. Rev.A , 062108 (2006).[10] E. R. Meyer and J. L. Bohn, Phys.Rev. A , 010502(R) (2008), URL http://link.aps.org/abstract/PRA/v78/e010502 .[11] L. R. Hunter, S. E. Maxwell, K. A. Ulmer, N. D. Charney,S. K. Peck, D. Krause, S. Ter-Avetisyan, and D. DeMille,Phys. Rev. A , 030501(R) (2002).[12] D. Kawall, F. Bay, S. Bickman, Y. Jiang, and D. DeMille,Phys. Rev. Lett. , 133007 (2004).[13] S. Bickman, P. Hamilton, Y. Jiang, and D. DeMille, Phys.Rev. A , 023418 (2009).[14] . Yu. Yu. Dmitriev, Yu. G. Khait, M. G. Kozlov, L. N.Labzovsky, A. O. Mitrushenkov, A. V. Shtoff, and A. V.Titov, Phys. Lett. A , 280 (1992).[15] L. D. Landau and E. M. Lifshitz, Quantum mechanics (Pergamon, Oxford, 1977), 3rd ed.[16] A. V. Titov and N. S. Mosyagin, Int. J. Quantum Chem. , 359 (1999).[17] N. S. Mosyagin, A. N. Petrov, A. V. Titov, and I. I. Tupit-syn, Progr. Theor. Chem. Phys. B 15 , 229 (2006).[18] K. P. Huber and G. Herzberg,
Constants of DiatomicMolecules (Van Nostrand-Reinhold, New York, 1979).[19] M. L. Polak, M. K. Gilles, R. F. Gunion, and W. C.Lineberger, Chem. Phys. Lett. , 55 (1993).
Electric field (V/cm) (b) g g f g e (a) g e g f g Electric field (V/cm)
FIG. 1: Calculated g -factors for e ( g e ) and f ( g f ) states.(a) Solid lines correspond to J = 1 , F = 1 / PbO, dashed lines correspond to J = 1 ro-tational levels of , PbO. (b) Solid lines correspond to J = 1 , F = 3 / , | M F | = 3 /
2, dashed lines correspond to J = 1 , F = 3 / , | M F | = 1 / PbO (cid:39) E / ( W d d e ) Electric field (V/cm)
FIG. 2: EDM induced Stark splitting between M F = ± / J = 1 , F = 1 /207