Investigation on the fine structure of the B 1∏ − c 3∑+ complex in KRb

The fine structure of the transitions in the (X, υ″ = 0 → (B, c), υ′ = 2) band system in K85Rb is studied by using selective Doppler-free spectroscopy techniques. Energy shifts of the perturbed levels due to spin-spin, spin-rotation and spin-orbit interactions were analyzed and modeled within the effective Hamiltonian approach. Molecular and coupling constants were determined, which successfully model the experimental observations.


Introduction
The KRb molecule has been subject of many different investigations, both theoretical and experimental.Following the first experimental characterization of the ground state X 1 Σ + at high resolution in [1] and [2], by using optical-optical double resonance polarization spectroscopy Okada et al. [3] and Kasahara et al. [4] have studied the excited B(1) 1 Π and 2 1 Π electronic states.The B 1 Π state was studied also in References [5,6].These reports have shown occurrence of many perturbation, however they were not thoroughly studied afterwards.Such a multitude of perturbations is observed due to: (i) the close proximity of the lowest excited 2 P atomic levels in Rb and K, leading to electronic states with potential curves close to each other (ii) the spin-orbit coupling between the triplet and singlet states correlating to the first excited asymptote (n 2 S+m 2 P) and (iii) because of the avoided crossing between the two Π states.In a previous study [7] the (B 1 Π, c 3 Σ + ) complex was used to observe Laser-Induced-Fluorescence (LIF) to the triplet a 3 Σ + state, following excitation from the singlet X 1 Σ + state.In this study it was reported that the appearance of the hyperfine structure (HFS) of the (B, c) → a 3 Σ + transitions changes by tuning the laser across the X 1 Σ + → (B, c) Doppler profile.The only plausible explanation was the HFS of the (B, c) levels.The still missing understanding of the coupling between the B and the c states and the HFS of theirs levels was the main motivation to undertake this initial study of the fine structure.
Based on the previous research [4] and on the diode laser source available in our laboratory (∼ 15100 cm −1 ), we have chosen to analyze the perturbation between B 1 Π and c 3 Σ + in the (0, 2) band occurring around J ′ = 46.In Section 2 details on the experiments performed are given and in Section 3 the model which was used to explain the observations is presented.

Experiment
The KRb molecules were produced in a stainless steel heat pipe in approximately 1:1 mass ratio of the ingredients in natural isotopic composition and Ar as a buffer gas saturation spectroscopy was employed.The beam (10 mW) of a diode laser (HL6544FM) with extended cavity (FWHM about 5 MHz) was split in two parts with an intensity ratio of 1:9.The intensity of the stronger beam was modulated by mechanical chopper.The beams entered the HP in opposite directions and were overlapped in its center.The absorption of the weaker beam was registered through a lock-in amplifier.After optimization of experimental conditions it was possible to reduce the spectral resolution down to about 50 MHz at T = 280 • C and about 0.5 Torr initial pressure of Ar.Part of the spectrum is shown in figure 1 (upper trace).The saturated absorption spectra were very complex due to the presence of strong K 2 and KRb B-X bands in multiple isotopologues.There were more than fifty lines on average per 1 cm −1 .Nevertheless, it was possible to assign most of the strong and nonoverlapping transitions, for much of them have already been observed in [4,10].However, in the region of perturbation identification was difficult, so it was decided to employ filtered laser excitation (FLE) spectroscopy technique.The experimental setup is shown in figure 2. The diode laser was scanned in the range of the KRb B(v ′ = 2) -X(v ′′ = 0) band, 15105 − 15125 cm −1 .The whole range was covered by overlapping ≈ 0.8 cm −1 scans.For calibration of the spectrum, Dopplerlimited absorption of iodine was monitored simultaneously, along with the fringes from a confocal interferometer with a free spectral range of 748 ± 1 MHz.To overcome the Doppler broadening of the FLE lines, the laser beam was split by a 50/50 beam splitter.The two counterpropagating beams were modulated separately with a dual blade chopper (MC2F5360) at 6 kHz and 5.3 kHz and were overlapped in the center of the heat pipe.The emitted fluorescence was collected by a pierced mirror in front of the heat pipe and then focused on an avalanche photodiode (APD440A).In front of the detector an interference band pass filter was mounted with 900 nm central wavelength and 10 nm FWHM.With a lock-in amplifier the sum frequency signal was extracted.The filter transmits only the KRb (B, c) -a 3 Σ + fluorescence, thus not only the Doppler broadening was overcome but also all transitions in K 2 and transitions in KRb, whose upper levels are of predominantly singlet character and their associated fluorescence decays mainly to the ground X state, were eliminated.Part of the spectrum is shown in figure 1   This type of spectroscopy proved useful to isolate the P, Q and R transitions to the B 1 Π state in and around the perturbation region.Lines, for which the assignment of quantum numbers was still not certain, were separately examined by LIF spectroscopy.The laser frequency was tuned to these lines and the following (B, c) -a fluorescence was analyzed with a Brucker V-80 spectrometer.As already mentioned X 1 Σ + and a 3 Σ + states for KRb were studied [7], so from the fluorescence to the singlet and to the triplet state it was possible to unambiguously assign quantum numbers of the lines in question.
However, transitions to levels which have mainly c 3 Σ + state character were much more difficult to register and only few of them were identified at this stage.In order to find even more transitions to the c 3 Σ + state, optical-optical double resonance saturation spectroscopy in V-type configuration with two diode lasers was used.The frequency of the first laser was fixed at selected Doppler free X -(B, c) transition (through saturation spectroscopy) thus labeling the corresponding ground state level.The second laser (Toptica, LD-0655-0050-1) was scanned in region where transitions to the mixed upper states sharing the labeled lower state are expected to be found, e.g. for J ′′ = 46 -(15110 − 15118 cm −1 ).These spectra were useful in identifying a few more excited energy terms of the triplet state.

Results and discussion
The energy structure of the (B, c) levels results from an interplay of several interactions.In the zero order Born-Oppenheimer approximation, for each J ′ the B state levels are twofold degenerated -the so called e and f symmetry components.The c state levels are usually considered in Hund's coupling case (b).Each rotational quantum number N ′ is three fold degenerated, the so called F 1 , F 2 and F 3 components, which correspond to J ′ = N ′ − 1, N ′ and N ′ + 1 respectively.F 1 and F 3 components are of f symmetry, while F 2 -of e.The degeneracy is lifted by the spin-rotation and spin-spin interactions.The B and c levels are then coupled due to the spin-orbit interaction, which acts only between the levels of the same symmetry (e-e, f -f ).These three interactions are mainly responsible for the fine structure of the (B, c) levels.
In this section we will relate the experimental observations to the theoretical model in order to determine the constants for the fine structure of the (X, v ′′ = 0; (B, c), v ′ = 2) band of K 85 Rb.This will be done through the use of the effective Hamiltonian.Of the complete Hilbert space of eigenstates for the total Hamiltonian of a diatomic molecule, the effective Hamiltonian acts only on a subspace.For the current analysis this subspace consists of two sets of rotational terms.One is for the B 1 Π electronic state with vibrational quantum number v ′ = 2 and the other for the c 3 Σ + electronic state with v 0 (≈ 35, estimated from the theoretical potentials [5,6,8], although the exact value is not required for the model).This subspace is even further restricted for rotational values J ′ ∈ [22, 61] such that the region of perturbation is included and isolated from other perturbation centers.The effective Hamiltonian connects only states with equal J ′ values, and of same symmetry.An explicit J ′ dependent form of the matrix is presented in figure 3. Details on the Matrix elements can be found in [9] where analysis of a similar problem in the NaK molecule is presented.Figure 4 displays the difference between the observed and the unperturbed term energies of the singlet state, calculated by ignoring the interaction with the c 3 Σ + state.One can see that the single state model is unable to explain the observed deviations which exceed the experimental uncertainty about 100 times.For the correct description, the matrix from figure 3 is diagonalized sequentially for every J ′ and the nine model parameters are fitted.From experimental measurements there are in total 89 terms energies serving as an input data.Thereafter a non-linear Levenberg-Marquardt least square fit is performed using the scipy buildin routine (scipy.optimize.leastsquares).The final fitted parameters are listed in table 1.They reproduce the experimental line positions with a rms of 0.0038 cm −1 .The uncertainties in the fitted parameters are calculated based on the 0.004 cm −1 estimation of the line frequency uncertainty (coming from the calibration of the spectra by the Doppler-limited I 2 lines) .
As a future step we plan to extend the present model by including data for the K 87 Rb isotopologue and possibly to study other vibrational bands of the (B, c) complex.More data on several bands may allow to treat the problem via a global model (e.g.[11]), going beyond the effective Hamiltonian approach.

Figure 1 .
Figure 1.Part of the spectrum from saturation spectroscopy (upper trace) and filtered laser excitation spectroscopy (lower trace).Assignments of some transitions are shown above the lines. Photodiode

Figure 2 .
Figure 2. Schematic diagram of the experimental setup.

Figure 3 .
Figure 3. Effective Hamiltonian matrix for the subspace of interacting B 1 Π and c 3 Σ + states.

Figure 4 .
Figure 4. Difference between observed and unperturbed term energies (∆E) of the B 1 Π(v = 2) levels for the e− (a) and the f −symmetry (b).Unperturbed term energies are calculated from the molecular constants listed in table 1 by setting γ, λ and η to zero.

Table 1 .
Molecular and coupling constants for the B 1 Π(v = 2) and c 3 Σ + (v 0 ) states.All values are in units of cm −1 .