HYPERFINE INTERACTION IN DIATOMICS AS A TOOL FOR VERIFICATION OF THEORETICAL VALUES FOR THE EFFECTIVE ELECTRIC FIELD ON ELECTRON A.N.Petrov PNPI QChem Group: B.P. Konstantinov PNPI, St.-Petersburg State University, St.-Petersburg, RUSSIA L.V. Skripnikov, N.S. Mosyagin, and A.V.Titov

Effective electric field (E eff ) on the electron is one of the most important parameters for EDM search E eff can not be obtained in an experiment  Challenging molecular calculation is required What should be calculated ?

Hyperfine structure constant(s), A ‖ (A ┴ ) can be obtained both in an experiment and in calculation Similarly to E eff hyperfine structure constant(s) is(are) determined by wave function near heavy nuclei the Ω=1/2 (YbF, PbF …)molecules have an advantage here since their hyperfine structure is determined by two constants, A ‖ and A ┴, whereas hyperfine structure for Ω=1 (HfF +,WC,…) molecules is mainly determined by only one constant, A ‖ How to check accuracy of the E eff ?

YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5

YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ]

YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] YbF: 2 ∑ 1/2 ; config.: […]  

YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] YbF: 2 ∑ 1/2 ; config.: […]   = A6s + B6p z 

YbF molecule studies for eEDM experiment

M.G.Kozlov, JPB 30, L607 (1997):

YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, (2007):

YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, (2007):

YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, (2007):

HfF + studies for eEDM experiment

3 Δ 1 state, Ω=1

HfF + studies for eEDM experiment

For Ω=1 molecules A ┴ ≡ 0

HfF + studies for eEDM experiment

3 Δ 2 state of HfF +, WC … is close to 3 Δ 1 In this work we take into account interaction between 3 Δ 1 and 3 Δ 2

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 [outer core] [ valence ]

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 [outer core] [ valence ] Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ]

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2 

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z 

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z  ↓ = C5d +2 ↓ + D6p +1 ↑ 

HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z  ↓ = C5d +2 ↓ + D6p +1 ↑  ↑ = E5d +2 ↑ 

HFS (MHz) HF Atomic matrix elements

HFS (MHz) HF EDM(GV/cm) Atomic matrix elements

HF Molecular matrix elements A ‖ = (-2298A B C D 2 )MHz A ┴ /√2 = (2298A B EC)MHz E eff = ( -125AB + 12CD)GV/cm

HF Molecular matrix elements A ‖ = (-2298A B C D 2 )MHz A ┴ /√2 = (2298A B EC)MHz Using of two constants A ‖ and A ┴ is important for the accuracy check of E eff

Off-diagonal matrix elements

HFS splitting (A ┴ =0)

Hyperfine energy splitting between F=J+1/2 and F=J-1/2 levels of 3 Δ 1 WC (MHz), theoretical data [J. LEE et al. PRA 87, (2013)] JA ┴ =0A ┴ ≠0Δ −

Observed rotational transition frequencies γ obs (MHz) [R. J. Mawhorter et al. PRA 84, (2011) ] of 208 Pb 19 F. The values in parentheses give the 1σ experimental error of the last digit of precision(100 Hz) levelsγ obs (20) (7) (8) (10) (20) (5) (5) (5)

Hamiltonian of PbF( 2 Π) molecule First term describes the rotational motion Second term describes the hyperne structure Therd term gives small corrections to the hyperne structure

The subscripts 1 and 2 refer to nuclear spin of the fluorine and lead respectively

First line – nuclear spins – rotational interaction Second line - nuclear magnetic dipole-dipole interaction Third line - takes into account interactions with other electronic states

Observed rotational transition frequencies γ obs (MHz) [R. J. Mawhorter et al. PRA 84, (2011) ] of 208 Pb 19 F. The deviation of fit is given by Δ = γ fit - γ obs in units of the last digit of precision (100 Hz) levelsγ obs Δ1Δ (20) (7) (8) (10) (20) (5) (5) (5)8

Observed rotational transition frequencies γ obs (MHz) [ PRA 84, (2011) ] of 208 Pb 19 F. The deviation of fit is given by Δ = γ fit - γ obs in units of the last digit of precision (100 Hz) levelsγ obs Δ1Δ1 Δ2Δ (20) (7) (8) (10) (20) (5) (5) (5)8

From experimental data  A D ┴ (F) = 0.53 kHz A D ┴ (Pb) = -6.4 kHz Ab initio calculation  A D ┴ (F) = 0.71 kHz A D ┴ (Pb) = -5.9 kHz Centrifugal correction to hyperfine constant in 208,207 PbF [ 2 Π 1/2 ] [A.N. Petrov, L.V. Skripnikov, A.V. Titov, and R. J. Mawhorter, to be published]

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