Since the electron's electric dipole moment (eEDM) was hypothesized, the prediction and measurement of eEDM have attracted much attention from the scientific community. [1][2][3][4][5] Experimental detection of the eEDM would prove the time-reversal (T) symmetry violation and, hence, may lead to a modification of the Standard Model (SM). PbF, a paramagnetic molecule, has been proposed as an excellent candidate for the detection of eEDM. 6 The eEDM is usually determined from the energy level shift in external fields. Therefore, the energy level structures of PbF, especially its fine and hyperfine structures, are of great interest to molecular spectroscopists and atomic, molecular, and optical (AMO) physicists. The effective spin-rotational Hamiltonian, Hsr, of PbF in an Ω = 1/2 state in the presence of external fields can be written as 7 Hsr = BJ 2
The first two terms of Eq. ( 1) give the J-dependent part of the rotational energy including the Λ-doubling (or Ω-doubling), where B is the rotational constant, J is the total angular momentum excluding the nuclear spin, Δ is the Λ-doubling constant, and S ′ is the effective spin of the electron. 8 The third term describes the hyperfine interaction, where I is the nuclear spin and  is the hyperfine interaction tensor. The fourth term represents the interaction between the eEDM (de) and the effective internal electric field (W d ), where n is the unit vector directed along the internuclear axis from the positively charged atom to the negative one. The fifth term is for the Stark interaction, where D is the permanent dipole moment of
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp the molecule and E is the external electric field. The final term is for the Zeeman interaction, where μ B is the Bohr magneton, B is the external magnetic field, and the tensor Ĝ-which is diagonal in the molecular principal-axis system-determines the strength of the Zeeman interaction. Since this Hamiltonian determines the energy level structure of the PbF molecule in electromagnetic fields, which is of significance for detecting eEDM-sensitive transitions, the determination of molecular parameters in Eq. ( 1) with high precision is essential to improve future eEDM measurements using PbF. Spectroscopic studies on PbF started in the 1930s when Rochester reported the first vibrational spectroscopic study on this molecule. [9][10][11][12][13][14][15][16][17][18][19][20][21][22] Later, Lumley and Barrow analyzed the rotational structure of the 208 PbF and gave the rotational parameters of the v = 0 and 1 levels of each electronic state. 19 The high-resolution spectrum of the X 2 2 Π 3/2 -X 1 2 Π 1/2 transition was studied in 1998. 20 In 2010, fine structure constants of the D, E, and F states of 208 PbF, 207 PbF, and 206 PbF were determined in a mass-resolved resonance-enhanced multiphoton ionization (REMPI) spectroscopic measurement. 21 In 2011, the hyperfine structure of the X 1 state of 208 PbF, 207 PbF, and 206 PbF was determined from pure rotational spectra. 22 However, the high-resolution laser-induced fluorescence (LIF) spectrum of the B 2 Σ + -X 2 Π 1/2 transition and the hyperfine parameters of the B 2 Σ + state have not been reported yet.
In the present work, a supersonic expanded PbF molecular beam was produced, and the spectrum of the (0, 0) band of its B 2 Σ + -X 2 Π 1/2 transition was recorded using the LIF technique. Rotationally resolved transitions of the 208 PbF, 207 PbF, and 206 PbF isotopologues were assigned, and their molecular parameters were determined and compared with previously reported results.
Although the future PbF-based eEDM measurement utilizes its X state, it is critical to investigate its excited electronic states too. First, the energy level structure of the X state is perturbed by low-lying excited electronic states. Although the X state is a relatively isolated state, its molecular constants determined in fitting experimentally obtained spectra, e.g., the Λ-doubling constants, still contain contributions due to interactions with other electronic states, including the B state. Experimentally determined molecular constants can also be used to benchmark relativistic configuration interaction calculations of the low-lying electronic state of PbF. The B 2 Σ + state of PbF is of special interest in that it is the lowest Rydberg state arising from the σ 2 σRπ 4 configuration, where σR is the lowest Rydberg orbital. 23 Therefore, the B state plays an important role in the valence-Rydberg mixing, which shall be considered in both spectroscopic analysis 24 and ab initio calculations. 25 In the present work, the B-state rotational and spin-rotation constants of PbF have been determined with better precision in fitting the high-resolution LIF spectrum, while the hyperfine constants are reported for the first time, adding important, new pieces of information to the study of this promising candidate for the search of eEDM using diatomic radicals.
The PbF molecular beam was produced by the reaction of laser-ablated Pb plasma with SF 6 . The B 2 Σ + -X 2 Π 1/2 transition was measured using the LIF technique with resolved fine and hyperfine structures. A brief description of the experimental setup is given as follows.
The second harmonic output of a Nd:YAG laser (Quantel-Brilliant, 532 nm, 10 Hz repetition rate, pulse energy ∼1 mJ) was focused onto a high-purity Pb rod (Alfa Aesar, 99.999%) that underwent a continuous rotational and translational motion. The Pb rod was polished to remove the oxide coating before it was installed in the source chamber. The gas mixture of SF 6 /Ar (volume ratio, 5:95) passed through a pulsed nozzle valve and expanded into the source chamber to form a free jet expansion. The SF 6 molecule reacted with the ablated Pb plasma to produce the PbF molecule, which was excited from the ground (X 2 Π 1/2 ) state to the B 2 Σ + state by the frequency-doubled output from a tunable continuous-wave (CW) ring dye laser (Sirah, Matisse DS) in the detection chamber. The excitation laser beam was slightly focused to ∼1 mm onto the molecular beam. The fluorescence following the B 2 Σ + -X 2 Π 1/2 excitation of PbF was collected by a lens telescope system onto a photomultiplier tube (PMT, Sofn Instrument Co., 71D101-CR131). The PMT signal was amplified by a preamplifier (Standford Research System, SR240A), integrated using a boxcar system (Standford Research System, SR200 series), and sent to a personal computer (PC). The LIF spectrum was recorded by a home-written LabVIEW program.
The frequency of the CW dye laser was measured by a wavemeter (HighFinesse, WS-U) and calibrated using the Doppler-free saturation absorption spectrum molecular iodine during the spectral scan.
The high-resolution LIF spectrum of the B 2 Σ + -X 2 A portion of the P-branch of the experimental spectrum is shown in Fig. 2, in which transitions assigned to 208 PbF, 207 PbF, and 206 PbF are labeled. The notation used to label the transitions is defined similarly to the one given previously, 26,27
where N = J -S is the total angular momentum excluding both nuclear and electron spins. The first subscript, Fi ′ , with i = 1, 2, corresponds to the upper (J = N + 1/2) and lower (J = N -1/2) spin-rotation components of the B 2 Σ + state, respectively. The second subscript, Fi ′′ , with i = 1, 2, corresponds to the lower (X 1 2 Π 1/2 ) and upper (X 2 2 Π 3/2 ) spin-orbit components of the X 2 Π state, respectively. When Fi ′ = Fi ′′ , only one number is used.
For the ground X 2 Π 1/2 state of 208 PbF and 206 PbF, Hund's case (a) angular momentum coupling scheme is used to describe its energy level structure. 20 The Hamiltonian includes the spin-orbit interaction (with associated molecular constant, Ã) and its centrifugal distortion correction ( ÃD), the rotational Hamiltonian (B), the Λ-doubling (p, q), 26
where J + , J -, S+ and Sare the shift operators and ϕ is the orbital azimuthal angle of the unpaired electron. Centrifugal distortion corrections to the three terms ( ÃD, ÃH, D, q D ) have also been included in the spectral simulation and fitting, although they are not included in Eq. ( 2) for clarity. The readers are referred to Ref. 28 for details. The Λ-doubling parameter q is fixed to zero in the present work due to its negligible effect on the energy level structure. 22 For the B 2 Σ + state of PbF, Hund case (b) is adopted. The Hamiltonian for the B 2 Σ + state of 208 PbF and 206 PbF can be described as
where the first term is the band origin, the second term is the rotation Hamiltonian (the centrifugal distortion correction has also been included in the spectral simulation and fitting), and the final term describes the spin-rotation interaction. There are six transition branches, p P 1 , q P 21 , q Q 1 , r Q 21 , r R 1 , and s R 21 in the (0, 0) band of the B 2 Σ + -X 2 Π 1/2 transition of 208 PbF and 206 PbF.
In the LIF spectrum of 207 PbF, the hyperfine splitting from the nuclear spin of 207 Pb(I = 1/2) is resolved. Hund's case (a βJ ) coupling scheme is used in constructing the effective Hamiltonian of the X 2 Π state of 207 PbF, 26
where a( 207 Pb) and d( 207 Pb) are the nuclear spin-electron orbit interaction constant and the constant for the dipole-dipole coupling between the nuclear spin and the electron spin, respectively. For the B 2 Σ + state of 207 PbF, Hund's case (b βS ) coupling scheme is used, 27
where the Fermi contact constant bF( 207 Pb) and the Frosch and Foley hyperfine constant c( 207 Pb) are adopted for the nuclear spin-electron spin interaction and the nuclear spin-electron spin dipole-dipole interaction, respectively. In this Hund's case (b βS ) coupling scheme, the electron spin angular momentum S(=1/2) is coupled to the nuclear spin of 207 Pb (I = 1/2) to give the total spin angular momentum G, with the approximately good quantum numbers (G) having possible values of 1 and 0. G is coupled with N to give the total angular momentum F. The aforementioned six branches of the 2 Σ + (b)-X 2 Π 1/2 (a) labeling scheme regroup into eight branch features of the 2 Σ + (b βS )-X 2 Π 1/2 (a βJ ) scheme. Therefore, 207 PbF has eight branches, 29 p P 1 , q P 21 , p O 21 , q Q 1 , r Q 21 , r R 1 , s R 21 , and s S 1 in the (0, 0) band of its B 2 Σ + -X 2 Π 1/2 transition (see Fig. 2).
The spectrum of the (0, 0) band of the B 2 Σ + -X 2 Π 1/2 system was simulated using the PGOPHER program. 29 Molecular constants of both the X 2 Π 1/2 ground state and the B 2 Σ + excited state of the three isotopologues of PbF determined in fitting the spectrum are listed in Tables I and II in comparison with previous values, wherever available. 19,20,22,[30][31][32][33] Observed and calculated transition frequencies and residuals of the frequency fit are given in the supplementary material. The standard deviations of the fits for the 208 0.0011 cm -1 , respectively, which are significantly smaller than the FWHM of the LIF spectrum. A global fit involving the field-free emission spectrum of the X 2 2 Π 3/2 -X 1 2 Π 1/2 transition in the mid-infrared region, 20 the pure rotational microwave spectrum of the X 1 2 Π 1/2 state, 22 and the LIF spectrum of the B 2 Σ + -X 2 Π 1/2 recorded in this work has been carried out. Transition frequencies from different spectra were weighted according to their reported accuracies. Molecular constants of the ground (X 2 Π 1/2 ) and the excited (B 2 Σ + ) electronic states determined in the global fit are given in Table I. The spin-orbit constant ( Ã) and its centrifugal distortion correction parameter ( ÃD) of the X a The relativistic coupled-clusters method combined with the generalized relativistic effective core potential approach and nonvariational one-center restoration technique. b The ab initio relativistic correlation method. c The self-consistent field method. d The ab initio effective core potential calculation method.
Based on our experimental results, the energy level diagrams of B 2 Σ + and X 2 Π 1/2 states of PbF isotopologues can be derived. The energy level associated with the LIF transitions assigned in Fig. 2 are illustrated in Fig. 3 (for 208 PbF and 206 PbF) and Fig. 4 (for 207 PbF). The energy level diagrams for the X 2 Π 1/2 state of the 208 PbF and 207 PbF show that these two isotopologues have small Λ-doubling splittings. In the eEDM measurement, a sufficient external field is required to mix the ground state Λ-doubling levels. Therefore, the nearly degenerated lowest-lying Λ-doubling levels of the 208 PbF and 207 PbF could have advantages for the PbF-based eEDM measurement.
The hyperfine interaction tensor  [see Eq. ( 1)] is diagonal in the molecular frame and can be parameterized as A and A ∥ by combining the Frosch and Foley hyperfine constants. 6 For the X 2 Π 1/2 state of 207 PbF, A ( 207 Pb) = -d and A ∥ ( 207 Pb) = 2abc. 22 We derived A ( 207 Pb) = -0.2423(7) cm -1 from our experimental results (see Table I). The Frosch and Foley hyperfine constants, b and c, in the X 2 Π 1/2 state are negligible and, hence, were set to zero in our fits. As a result, A ∥ ( 207 Pb) is given as 0.3380(12) cm -1 . The A ( 207 Pb) and A ∥ ( 207 Pb) of the X 2 Π 1/2 state of 207 PbF are listed and compared with previously determined values in Table II. Our A and A ∥ values are in good agreement with the experimental results derived for the X 2 Π 1/2 state in Ref. 22. The discrepancy in the sign of A in Ref. 20 is caused by different sign conventions in the expression of the energy levels. 22 The theoretically predicted values are slightly larger than ours and previously reported experimental ones. Moreover, we have derived the values of the A ( 207 Pb) = 0.1584 (17) and A ∥ ( 207 Pb) = 0.1644(21) cm -1 for the B 2 Σ + state of 207 PbF from our experimental results (see Table II). Currently, only McRaven et al. 17 and Baklanov et al. 31 have reported the experimental and theoretical results of the A ( 207 Pb) and A ∥ ( 207 Pb) of the A 2 Σ + state. There are no reported values of the A ( 207 Pb) and A ∥ ( 207 Pb) of the B 2 Σ + state in the literature.
In the fourth term of Eq. ( 1), (W d de)S ′ ⋅ n, the nonzero value of de can be measured experimentally. However, the effective internal electric field (W d ) can only be predicted by high-level electronic
In summary, we recorded the high-resolution LIF spectrum of the (0, 0) band of the B 2 Σ + -X 2 Π 1/2 transition of PbF under jetcooled conditions and studied the fine and hyperfine interactions in its B 2 Σ + and X 2 Π 1/2 states. The spectral lines of 208 PbF, 207 PbF, and 206 PbF were assigned and analyzed to derive the fine and hyperfine molecular parameters of the v = 0 levels of the B 2 Σ + and X 2 Π 1/2 states of all three isotopologues. The hyperfine parameters, A and A ∥ , of the X 2 Π 1/2 and B 2 Σ + states of 207 PbF were given and discussed. The Fermi contact interaction constant (b F ) of the B 2 Σ + state of 207 PbF was determined for the first time in the present work. These parameters are essential in determining energy shifts in eEDM measurements and can be used to test theoretical methods in predicting the internal electric field of PbF.
See the supplementary material for the rotational assignment of experimentally observed spectral lines in the (0, 0) band of the B 2 Σ + -X 2 Π 1/2 transition of 208 PbF, 207 PbF, and 206 PbF (in units of cm -1 ).
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