Electric-field-dependent g factor for the ground state of lead monofluoride, PbF
V.V. Baturo, P. M. Rupasinghe, T. J. Sears, R. J. Mawhorter, J.-U. Grabow, A.N. Petrov
TThe electric field dependent g − factor for the lead monofluoride, PbF, ground state V.V. Baturo,
1, 2
P. M. Rupasinghe, T. J. Sears, R. J. Mawhorter, J.-U. Grabow, and A.N. Petrov
1, 2, ∗ Saint Petersburg State University, St. Petersburg, 199034, Russia Federal state budgetary institution “Petersburg Nuclear Physics Institute”, Gatchina, Leningrad district 188300, Russia Department of Physics, State University of New York at Oswego, Oswego, New York 13126, USA Department of Chemistry, Stony Brook University, Stony Brook, New York 11794-3400, USA Department of Physics and Astronomy, Pomona College, Claremont, California 91711, USA Gottfried-Wilhelm-Leibniz-Universit¨at, Institut f¨ur Physikalische Chemieand Elektrochemie, Lehrgebiet A, Hannover, D-30167 Germany
The electric field dependent g − factor and the electron electric dipole moment (eEDM)-inducedStark splittings for the lowest rotational levels of , PbF are calculated. Observed and calculatedZeeman shifts for
PbF are found to be in very good agreement. It is shown that the
PbFhyperfine sublevels provide a promising system for the eEDM search and related experiments.
Spectroscopic and theoretical work on the PbFmolecule over more than three decades including[1–11] has been reported. Based on data at opticalresolution,[2], Shafer-Ray et al.[4] predicted that theelectric-field-dependent g − factor of the ground state of PbF could cross zero at an electric field of 68 kV/cm.This led to the conclusion that PbF might provide auniquely sensitive probe of the electric dipole momentof the electron, eEDM, d e .Working to verify this, subsequent spectroscopy athigher resolution by McRaven et al.[5] and theoreticalanalysis in Ref.[6] revealed a mis-assignment of theparity of the lowest rotational levels and confusionconcerning the sign of the large Pb Frosch-Foleyd (= - A ⊥ ) hyperfine parameter in the optical workon PbF.[2, 7] The reanalysis performed by Yang etal.[9] with corrected spectroscopic constants showed the g − factor of the ground-state PbF unfortunately doesnot vanish. Nevertheless the very small g − factor inthe Π / ground state of PbF reduces the sensitivityto stray magnetic fields by about a factor of 20 withregard to comparable Σ molecules. This is a significantadvantage in parity non-conservation studies.Analytical expressions for electric-field-dependent g − factor were obtained[4, 9] for PbF under theassumption that mixing of different rotational levels byelectric field is not important. In this paper we take themixing into account by numerical inclusion of a largenumber of rotational states and consider both odd andeven mass Pb isotopologues of PbF.
Pb is the most abundant lead I = 0 isotope with52% natural abundance, while Pb has a nuclear spin I = 1 / PbF has a surprisinglystrong effect on the Zeeman splittings in low-lying fineand hyperfine split levels that has major implications forexperimental e EDM searches. It was shown by Alpheiet al.[8] that a coincidental near-degeneracy of levels of opposite parity in the ground rotational state J = 1 / PbF[12] takes place, caused by near cancellationof energy shifts due to omega-type doubling and the Pb F magnetic hyperfine interactions. Thus
PbFhas also been proposed as a promising candidate forboth anapole moment [8, 13] and temporal variation ofthe fundamental constants experiments [14].The knowledge of g − factors helps to control and sup-press important systematic effects due to stray magneticfields [15–17]. However, neither theoretical nor experi-mental data for g − factors of Pb F for the field-freecase or in an external electric field have been reported todate. The main purpose of the article is to fill this gap.
EXPERIMENTAL DETAILS
As described in detail in our earlier study[7],the rotational Zeeman spectra were taken at theGottfried-Wilhelm-Leibniz-Universit¨at Hannover usinga Fourier-transform microwave (FTMW) spectrometerthat exploits a coaxial arrangement of the supersonicjet and resonator axes (COBRA)[18]. The resultingsensitivity coupled with laser ablation[19] of elementalPb in a neon carrier gas augmented with a few percentof SF enabled the observation of strong and robustsignals, which were essential to measuring the Zeemaneffect data for both PbF and
PbF. As alreadymentioned, due to a cancellation of spin and orbitalcontributions inherent in the Π ground state of PbFthe observed Zeeman splittings are small. Even so, theexcellent signal to noise with long emission decay timesallows frequency measurements for unblended lines atan accuracy of 0.5 kHz and the resolution of transitionsseparated by more than 6 kHz. Figure 1 shows represen-tative Zeeman spectra for the 22541.912 MHz transitionin
PbF. Note that the resonance signals are dou-bled due to velocity structure in the experimental design.Given the high resolution of the jet spectra, the a r X i v : . [ phy s i c s . a t o m - ph ] F e b TABLE I. Observed (∆
U/B ) exp (MHz/G) and calculated (∆ U/B ) th (MHz/G) Zeeman shifts of the J = 1 / → J = 3 / L and U refer to the upper and lower energy level of the transition, respectively.Unsplit line (MHz) F L F U MF L MF U (∆ /B ) exp (∆ /B ) th (∆ /B ) exp − (∆ /B ) th .
501 3/2 5/2 3 / / − . − . . / − / − . − . . − / − / − . − . − . / / − . − . . − / − / − . − . . / / − . − . . − / − / . . − . / / . . − . − / − / . . . / / . . . − / / . . − . .
912 3/2 5/2 − / − / − . − . − . − / / − . − . . / / − . − . . − / − / − . − . − . / / − . − . . − / − / − . − . − . / / . . − . − / − / . . − . / / . . . − / − / . . − . / − / . . − . / / . . . .
902 1/2 3/2 − / − / − . − . . − / − / − . a − . − . / − / − . a − . − . − / / . . . / / . . . / / . . − . .
749 1/2 1/2 1 / − / − . − . . − / / . . . a Typographic error in [20], corrected here -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4Frequency - 22541.911 / MHz00.020.040.060.080.10.120.14 S i gna l / m V (r m s ) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4Frequency - 22541.911 / MHz00.020.040.060.080.10.120.140.160.18 S i gna l / m V (r m s ) M F = 1B =0.465G M F =0B =0.458G FIG. 1. (Color online) Rotational Zeeman spectra for the
PbF J L = 3 / → J U = 5 / M F = ± = 0.465G) and (right) four ∆ M F = 0 (B = 0.458 G) splittings,respectively. Note the doubled transitions due to the COBRADoppler effect. magnetic field calibration becomes the primary factordetermining the uncertainty of the molecule-fixed g − factor, G ⊥ . The currents in the 3 pairs of Helmholtz coils surrounding the chamber were independentlyvaried to null out the magnetic field. This was doneby adjusting the Helmholtz coil currents for all threepairs until all Zeeman splittings are minimized. Havingfull 3-axis control enabled the application of magneticfields either perpendicular along two different axes orparallel to the radiation polarization. Having individualaxis control, we can verify that the magnetic field inthe sample region was indeed determined from thechange in current in each coil. This was done bymaking independent experimental determinations of G ⊥ , in both parallel and perpendicular configurations.They agreed to within about 2.5%, indicating that theuncertainty in our magnetic field calibration is approx-imately equal to the statistical error of our measurement.Note that the initial experimental level assignments inRef. [20] have been reversed, resulting in the completelyconsistent set of experimental and theoretical G − factorspresented here. These are a factor of 1.45 smaller thanused in Refs. [7, 8], and result in the prediction for theavoided level crossing discussed in Ref. [8] to occur at amagnetic field of approximately 1190 ±
80 G.
METHODS
We represent the Hamiltonian for , PbF as H = H mol + H B + H E . (1) H mol includes rotational energy, hyperfine interactionbetween electrons and nuclei, nuclear spin – rotationalinteraction, and nonadiabatic and hyperfine interactionsbetween Π / and other electronic states. Parametersfor H mol , were very carefully determined in Ref. [10]. H B describes the interaction of the molecule with anexternal magnetic field B including the small nuclearZeeman effect. It is determined by the body-fixed g − factors G (cid:107) = 0 . G ⊥ = − .
27 obtained inRef. [11] and nuclear g − factors g F = 5 . µ N , g P b = 1 . µ N . Interaction with an externalelectric field, H E , is determined by the body-fixedmolecular dipole moment D = 1 .
38 a.u. [7]. Furtherdetails are provided in Refs. [7, 21].Following Refs. [10, 22] eigenvalues and eigenfunc-tions of the lead monofluoride molecule were obtained bynumerical diagonalization of the Hamiltonian ( ˆH ) overthe basis set of the electronic-rotational and nuclear spinwavefunctions Ψ Ω θ JM, Ω ( α, β ) U Pb I M U F I M . (2)Here Ψ Ω = Π / , Π / is the electronic wavefunc-tion, θ JM, Ω ( α, β ) = (cid:112) (2 J + 1) / πD JM, Ω ( α, β, γ = 0) isthe rotational wavefunction, α, β, γ are Euler angles, U Pb I M ( ≡ PbF) and U F I M are the Pb and Fnuclear spin wavefunctions, M (Ω) are the projectionsof the sum of the projections of the molecule’s electronicspin and orbital angular momenta on the lab ˆ z (internu-clear ζ ) axis and M , are the projections of the nuclearspin angular momenta on the laboratory ˆ z − axis.We define the g − factors such that the Zeeman shift isequal to E Zeeman = gµ B BM F , (3)where M F is projection of the total angular momentum F (including nuclear spin) on the direction of B and theelectric field, E . RESULTS AND DISCUSSION
The energy levels of interest for potential eEDMexperiments on
PbF are the F p = 1 − and F p = 1 + Lab Electric field (kV/cm) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91 ∆ E / ( E e ff d e ) (a) Lab Electric field (kV/cm) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91 ∆ E / ( E e ff d e ) (b) Lab Electric field (kV/cm) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91 ∆ E / ( E e ff d e ) (c) FIG. 2. (Color online) The eEDM induced Stark splitting(∆ E ) between ± M F pairs of hyperfine states. (a) PbF.The solid (red) line corresponds to the | M F | =1 lower lyingΩ-doublet states whereas the dashed (green) line correspondsto the higher lying | M F | =1 states. (b) PbF. The solid(red) line corresponds to the | M F | =3/2 lower lying Ω-doubletstates whereas the dashed (green) line corresponds to thehigher lying | M F | =3/2 states. (c) PbF. The solid (red) linecorresponds to the lower lying F =3/2, | M F | =1/2 Ω-doubletstates, the dashed (green) line corresponds to the higher ly-ing F =3/2, | M F | =1/2 states, the dotted (blue) line corre-sponds to the lower lying F =1/2, | M F | =1/2 states, whereasthe dashed-dotted (violet) line corresponds to the higher lying F =1/2, | M F | =1/2 states states which are the first and fourth energy levels inzero field. p = ± PbF, the levels of interest are the closely spacedΩ-doublet states F p = 3 / − , F p = 1 / − , F p = 1 / + and F p = 3 / + , which are the second, third, fourth andfifth energy levels. The relevant energy levels can beseen in Fig. 1 of Ref. [7]. In an eEDM search experimentopposite parity levels are mixed in an electric field topolarize the molecule. As the molecule becomes fullypolarized the splitting ∆ E between ± M F levels due toeEDM related Stark shift reaches the maximum value2 d e E eff , where E eff = 40 GV / cm [23] is the effectiveinternal electric field. For any real electric field thesplitting is less than 2 d e E eff by an absolute value. InFig. 2 the calculated eEDM induced Stark splittings for , PbF are presented.The calculated and observed[20] Zeeman shifts of the J = 1 / → J = 3 / PbF are givenin Table I and graphically in Fig. 3. The deviationsbetween calculated and observed Zeeman shifts areconsistent with the estimated experimental accuracy. InFig. 4, the calculated electric field dependent g − factoris presented. From Fig. (4) (a) one can see that takinginto account the mixing of different rotational levels bythe electric field is important for accurate evaluation ofthe g − factors.As can be seen in Fig. (2) the advantage of the PbFmolecule is that it is polarized at a lower electric field andhas smaller absolute g − factors than does PbF. Thisis important for the eEDM experiment as larger fieldsand g − factors lead to greater systematic uncertaintiesin experimental measurements. For E = 5 kV/cm theeEDM Stark shift reaches 80% of the maximum valuefor PbF, whereas for
PbF, | M F | =3/2 the same effi-ciency is achieved at E = 1 kV/cm, and for E = 2 kV/cmit is 90%. The values for the g − factors vary from 0 .
04 to0 .
01. As a comparative example, the YbF molecule with g = 2 the efficiency is only about 55% for E = 10 kV/cm.The complex hyperfine structure of PbF makes thedependence on electric field for both the eEDM Starkshift and g − factor not regular. There are several fieldsfor which the g − factors take zero or near-zero values.However they are strongly correlated with zero valuesfor the eEDM Stark shift. Therefore, we conclude the | M F | = 1 / | M F | = 3 / CONCLUSIONS
Experimental data and theoretical calculations for g − factors of Pb F for the electric field-free caseare reported and found to be in a very good agree- -1/2 1/2 3/2-3/2 5/2-1/2 1/2-5/2 3/2-3/2 -1/2 1/2 D E , M H z MF L (F L =3/2) -3/2 FIG. 3. Calculated (circles) and experimental (horizontalbands, bandwidths corresponding to two standard deviationuncertainty) shifts for the J L = 3 / → J U = 5 / ν = 22541 .
912 MHz). MF L values are on the x-axis, and MF U values are marked in the figure. Note the excellentagreement and the eight/four-fold natures of the ∆ M F = ± M F = 0 transitions clearly apparent in Fig. 1. ment with each other and with the body-fixed g − factors G (cid:107) = 0 . G ⊥ = − .
27 obtained in Ref. [11]. Thecalculated sensitivity to electron electric dipole momentshows that an electric field of 1 − g − factors provide the information needed to control sys-tematic effects related to stray magnetic fields in futureexperiments such as those capitalizing on the coincidentalnear-degeneracy of levels of opposite parity in PbF.
ACKNOWLEDGEMENTS
The authors would like to thank A.L. Baum for his ini-tial preparation of the experimental data and acknowl-edge Neil Shafer-Ray as a source of inspiration for thisseries of studies of PbF, dedicating this work in his mem-ory. Molecular calculations were supported by the Rus-sian Science Foundation grant No. 18-12-00227. Workby T. J. Sears was supported by the U.S. Department ofEnergy, Office of Science, Division of Chemical Sciences,Geosciences and Biosciences within the Office of BasicEnergy Sciences, under Award Number DE-SC0018950.R.J. Mawhorter is grateful for research support providedby a Pomona College Sontag Fellowship and Hirsch Re-search Initiation Grant. P. M. Rupasinghe is grateful forresearch support provided by SUNY-Oswego Office of Re-search and Sponsored Programs (ORSP). J.-U. Grabowacknowledges support from the Deutsche Forschungsge-meinschaft grant No. GR 1344 and the Land Niedersach-sen.
Lab Electric field (kV/cm) -0.100-0.0500.0000.0500.100 g f ac t o r (a) Lab Electric field (kV/cm) -0.0500.0000.050 g f ac t o r (b) Lab Electric field (kV/cm) -0.100-0.0500.0000.0500.100 g f ac t o r (c) FIG. 4. (Color online) Calculated g -factors. (a) PbF. Thesolid (red) and dotted (blue) lines correspond to the | M F | =1lower lying Ω-doublet states whereas the dashed (green) andthe dashed-dotted (violet) lines correspond to the higher ly-ing | M F | =1 states. The dotted (blue) and the dashed-dotted(violet) lines were calculated without interaction with otherrotational levels taken into account. (b) PbF. The solid(red) line corresponds to the | M F | =3/2 lower lying Ω-doubletstates whereas the dashed (green) line corresponds to thehigher lying | M F | =3/2 states. (c) PbF. The solid (red) linecorresponds to the lower lying F =3/2, | M F | =1/2 Ω-doubletstates, the dashed (green) line corresponds to the higher ly-ing F =3/2, | M F | =1/2 states, the dotted (blue) line corre-sponds to the lower lying F =1/2, | M F | =1/2 states, whereasthe dashed-dotted (violet) line corresponds to the higher lying F =1/2, | M F | =1/2 states. ∗ , 4939 (1987).[2] K. Ziebarth, K. D. Setzer, O. Shestakov, and E. H. Fink,J. Mol. Spectrosc. , 108 (1998), ISSN 0022-2852.[3] Y. Y. Dmitriev, Y. G. Khait, M. G. Kozlov, L. N. Lab-zovsky, A. O. Mitrushenkov, A. V. Shtoff, and A. V.Titov, Phys. Lett. A , 280 (1992).[4] N. E. Shafer-Ray, Phys. Rev. A , 034102 (2006).[5] C. P. McRaven, P. Sivakumar, and N. E. Shafer-Ray,Phys. Rev. A , 054502(R) (2008), erratum: Phys. Rev.A , 029902(E) (2009).[6] K. I. Baklanov, A. N. Petrov, A. V. Titov, and M. G.Kozlov, Phys. Rev. A , 060501(R) (2010).[7] R. J. Mawhorter, B. S. Murphy, A. L. Baum, T. J. S. T.Yang, P. M. Rupasinghe, C. P. McRaven, N. E. Shafer-Ray, L. D. Alphei, and J.-U. Grabow, Phys. Rev. A ,022508 (2011).[8] L. D. Alphei, J.-U. Grabow, A. N. Petrov, R. Mawhorter,B. Murphy, A. Baum, T. J. Sears, T. Z. Yang, P. M.Rupasinghe, C. P. McRaven, et al., Phys. Rev. A ,040501(R) (2011).[9] T. Yang, J. Coker, J. E. Furneaux, and N. E. Shafer-Ray,Phys. Rev. A , 014101 (2013). URL http://link.aps.org/doi/10.1103/PhysRevA.87.014101 .[10] A. N. Petrov, L. V. Skripnikov, A. V. Titov, and R. J.Mawhorter, Phys. Rev. A , 010501(R) (2013).[11] L. V. Skripnikov, A. N. Petrov, A. V. Titov, R. J.Mawhorter, A. L. Baum, T. J. Sears, and J.-U. Grabow,Phys. Rev. A , 032508 (2015).[12] C. P. McRaven, P. Sivakumar, and N. E. Shafer-Ray,Phys. Rev. A , 054502(R) (2008).[13] A. Borschevsky, M. Iliaˇs, V. A. Dzuba, V. V. Flambaum,and P. Schwerdtfeger, Phys. Rev. A , 022125 (2013).[14] V. V. Flambaum, Y. V. Stadnik, M. G. Kozlov, and A. N.Petrov, Phys. Rev. A , 052124 (2013).[15] A. N. Petrov, L. V. Skripnikov, A. V. Titov, N. R. Hut-zler, P. W. Hess, B. R. O’Leary, B. Spaun, D. DeMille,G. Gabrielse, and J. M. Doyle, Phys. Rev. A , 062505(2014).[16] A. N. Petrov, L. V. Skripnikov, and A. V. Titov, Phys.Rev. A , 022508 (2017).[17] V. Andreev, D. Ang, D. DeMille, J. Doyle, G. Gabrielse,J. Haefner, N. Hutzler, Z. Lasner, C. Meisenhelder,B. O’Leary, et al., Nature , 355 (2018).[18] J. U. Grabow, W. Stahl, and H. Dreizler, Rev. Sci. Instr. , 4072 (1996).[19] B. M. Giuliano, L. Bizzocchi, and J. U. Grabow, J. Mol.Spectrosc. , 261 (2008).[20] A. L. Baum, B.A. thesis, Pomona College, (2010).[21] P. M. Rupasinghe, Ph.D. thesis, University of Oklahoma(2011).[22] A. N. Petrov, Phys. Rev. A , 024502 (2011).[23] L. V. Skripnikov, A. D. Kudashov, A. N. Petrov, andA. V. Titov, Phys. Rev. A90