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Spectroscopic study of the C(3)1Σ+ ← X1Σ+ and c(2)3Σ+ ← X1Σ+ transitions in KCs molecule
Accepted Manuscript
Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule Jacek Szczepkowski, Anna Grochola, Wlodzimierz Jastrzebski, Pawel Kowalczyk PII: DOI: Reference:
S0022-4073(17)30448-X 10.1016/j.jqsrt.2017.09.013 JQSRT 5840
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
2 June 2017 16 August 2017 12 September 2017
Please cite this article as: Jacek Szczepkowski, Anna Grochola, Wlodzimierz Jastrzebski, Pawel Kowalczyk, Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule, Journal of Quantitative Spectroscopy & Radiative Transfer (2017), doi: 10.1016/j.jqsrt.2017.09.013
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Highlights • The C1 Σ+ ← X 1 Σ+ and c3 Σ+ ← X 1 Σ+ band systems of KCs observed
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by polarisation labelling spectroscopy.
• The pointwise potential energy curve of the C1 Σ+ state constructed from experimental data.
• The pointwise potential energy curve of the c 3 Σ+ state constructed from experimental data.
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• The results compared with the theoretical predictions.
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Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule
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Jacek Szczepkowski∗, Anna Grochola, Wlodzimierz Jastrzebski Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warszawa, Poland
Pawel Kowalczyk
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Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland
Abstract
We present an experimental investigation of the C(3)1 Σ+ and c(2)3 Σ+ states of KCs using two-colour polarisation labelling spectroscopy (PLS) technique. Pointwise potential energy curves are generated for both states using the in-
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verted perturbation approach (IPA) method. The experimental pointwise potentials and molecular constants Te , ωe and Re are compared with theoretical
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calculations.
Keywords: laser spectroscopy, alkali dimers, electronic states, potential energy curves
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PACS: 31.50.Df, 33.20.Kf, 33.20.Vq, 42.62.Fi
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1. Introduction
Molecules with permanent electric dipole moments are a basis for experi-
ments from many different areas of physics. Polar molecules give opportunity
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to control interactions with external fields [1], allow for precision measurements
5
of fundamental constants [2, 3], open the way to exploration of quantum phases in dipolar gases [4, 5] and testing universality in few-body systems [6, 7]. One ∗ Corresponding
author Email address: [email protected] (Jacek Szczepkowski )
Preprint submitted to Journal of Quantitative Spectroscopy and Radiative TransferSeptember 13, 2017
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may also notice a development of so called ultracold chemistry [8]. A particular goal for many groups is to create molecules in their absolute ground state, since it is expected to be the first step in achieving heteronuclear molecular Bose-Einstein condensate. It has been done only for few molecules so far: KRb
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10
[9] and RbCs [10, 11] using Feshbach resonances and stimulated Raman adia-
batic passage (STIRAP) process and for LiCs using photoassociation [12]. KCs
with its permanent electric dipole moment of 1.92 D is the next promising can-
didate. First K-Cs interspecies Feshbach resonances were observed [13] and 15
several optical coherent schemes to create ultracold KCs molecules in their ab-
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solute rovibrational ground state were proposed [14]. Described processes base on spectroscopic studies performed so far.
Spectroscopic data are particularly important for all physicists interested in experiments conducted in an ultracold environment, since they give precise 20
information about energies of rovibrational levels, couplings influencing transition probabilities and possible local perturbations, which may become an ob-
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stacle as well as an opportunity in process of manipulation of molecules. So far several electronic states of KCs molecule were investigated in high precision
25
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spectroscopic experiments: the ground state [15] and a few higher excited states [16, 17, 18, 19]. But particular interest is focused on states dissociating to lower lying asymptotes [20, 21, 22], since most of excitation schemes lead through
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these states [14, 23]. Couplings between these states were investigated and deperturbation procedures were performed [24] to open a possibility of further
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study on cooling and manipulating of KCs molecules. To extend present knowledge of the electronic structure of KCs in the most
30
interesting region we present the first experimental data concerning the c(2)3 Σ+
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state dissociating to the first excited asymptote K(42 S1/2 ) + Cs(62 P3/2 ) and the C(3)1 Σ+ state dissociating to the K(42 P1/2 ) + Cs(62 S1/2 ) asymptote, but interacting with the states lying below. All potential energy curves correlating
35
to both mentioned asymptotes are presented in Fig. 1.
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K +Cs
-1
E[cm ]
3
1
3
2
4p+6s
+
3
16500
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2
4s+6p
15000 1
1
13500
1
3 3
2
+
+
1
2
M
12000
+
10500 3
9000
3
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1
4
5
6
R[Å]
7
8
9
10
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Figure 1: Ab initio potential energy curves of KCs [25] for all states correlating to the K(4s) + Cs(6p) and K(4p) + Cs(6s) asymptotes. Energies are given with respect to the minimum of the electronic ground state. The horizontal lines with atomic products on them indicate
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the energies of the adiabatic dissociation limits.
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2. Experiment Metallic samples of potassium and caesium (5 g each) were used to generate the vapour of KCs molecules in a stainless steel linear heat-pipe oven, heated
40
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to 430 ◦ C. Helium under the pressure of 5 Torr served as a buffer gas to prevent
alkali metal deposition on the heat-pipe windows. The polarisation labelling spectroscopy (PLS) technique was applied to record the KCs excitation spectra.
Details of the method may be found in our previous papers [26, 27]. In general, excitation spectrum was studied using the V-type optical-optical double
resonance method with two independent pump and probe lasers. As a pump
laser served a parametric oscillator/amplifier system (OPO/OPA, Sunlite EX,
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45
Continuum) pumped with the third harmonic of an injection seeded Nd:YAG laser (Powerlite 8000). Either the signal or idler beams of the system were used to scan the region 11800 cm−1 - 14000 cm−1 . The probe beam, of a fixed wavelength controlled with a HighFinesse WS-7 wavemeter, originated from a homemade dye laser operating on Rhodamine 6G and pumped synchronously by the
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50
same Nd:YAG laser. Its wavelength was in resonance with selected transitions in the well known KCs 41 Σ+ ←− X1 Σ+ band system [16]. The rotationally
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resolved C(3)1 Σ+ ←− X1 Σ+ and c(2)3 Σ+ ←− X1 Σ+ excitation spectra were recorded, with the accuracy of observed molecular line positions better than
55
0.1 cm−1 . Calibration of molecular spectra was based on simultaneously gener-
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ated optogalvanic spectra of argon and transmission fringes from a Fabry-Pérot
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interferometer of FSR = 1 cm−1 .
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3. Analysis and results
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In the spectral region examined in the present experiment, we could observe
several characteristic patterns. They consisted either of vibrational progressions of P, Q and R lines (’triplets’) or of P and R lines only (’doublets’). All recorded P, Q, R ’triplets’ were identified as transitions to the B(1)1 Π state, studied
before [20]. The remaining P and R lines from doublets were corresponding to transitions to four different electronic states in KCs. In the lower part of the 5
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65
scanned area traces of the coupled A(2)1 Σ+ and b(1)3 Π states were recognized. In the upper part of the examined region transitions to the c(2)3 Σ+ and C(3)1 Σ+ states overlap with strong lines registered due to a presence of the B(1)1 Π state.
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Observation of five electronic states in the same spectral region resulted in complexity of the spectra. Moreover the probe laser was typically labelling more 70
than one rovibrational level in the ground state, what additionally increased the number of the recorded spectral lines. In such situation, despite the expected
simplification of experimental spectra due to the polarisation labelling method, they remained highly congested. As a solution to this problem we recorded two
75
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or more spectra containing progressions originating from the same rovibrational
level in the ground state, but labelled with different wavelengths of the probe laser. An example of two spectra originating from the same rovibrational level in the ground state is presented in Fig. 2a and 2b. Then we looked for common features in the spectra, taking numerical product of two or three of them. Such procedure allowed for unambiguous assignment of spectral lines, as shown in Fig. 2c.
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Transitions to the C(3)1 Σ+ state were relatively easy to identify, since ob-
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servation of several rovibrational levels of this state has been already reported [20]. The remaining spectral lines, much weaker, could correspond to transitions to either upper part of the b(1)3 Π and A(2)1 Σ+ states or to the c(2)3 Σ+ state. The first two states were described with high precision before [21] and elimina-
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85
tion of the corresponding spectral lines [24, 28] allowed for conclusion that we
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recorded mostly transitions to the latter, i.e. the c(2)3 Σ+ state. In the case of KCs molecule direct optical excitation from the singlet ground state to the excited triplet states is possible due to strong spin-orbit coupling and mixing of singlet and triplet rovibrational levels [15, 21]. Such spin-forbidden transitions
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90
have been observed before in KCs [29], as well as in heavier (RbCs [30]) but also lighter (LiCs [31] and NaCs [32]) alkali metal diatomics containing caesium atoms.
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a)
c) 3
27
v'=26
13480
13500
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13460
8
7
6
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b)
13520
13540
v'=9
13560
13580
1
13600
-1
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[cm ]
Figure 2: Exemplary spectrum of KCs after taking numerical product of two experimental polarisation spectra (c). The assigned P,R progressions correspond to transitions to consecu-
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tive vibrational levels in the c(2)3 Σ+ and C(3)1 Σ+ states when the (v 00 = 3, J 00 = 87) ground state level is labelled by the probe laser. In the two upper panels spectra before multiplication procedure are presented, recorded with the circularly polarised probe laser set at wavelengths
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corresponding nominally to transitions 41 Σ+ (v 0 = 16, J 0 = 86) ←− X1 Σ+ (v 0 = 3, J 0 = 87) (a) and 41 Σ+ (v 0 = 16, J 0 = 88) ←− X1 Σ+ (v 0 = 3, J 0 = 87) (b). The unassigned, randomly distributed lines in (c) are artefacts resulting from accidental overlap of different transitions
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in (a) and (b).
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18
16
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14
v'
12
10
8
4
2 20
40
60
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6
80
100
120
J'
Figure 3: The range of rovibrational levels of the C(3)1 Σ+ state covered in the present experiment (dots, black in colour). Triangles (red in colour) correspond to the data from the numbering.
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3.1. C(3)1 Σ+ state
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experiment performed in the Riga group [20], included in the fit with corrected vibrational
Altogether 441 spectral lines were recorded corresponding to transitions to
95
rovibrational levels of the C state in the range v 0 = 3 − 18, J 0 = 38 − 127.
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Distribution of the experimental data is shown in Fig.3. Knowing energies of the ground state levels [15], we converted measured transition energies into
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390 C(3)1 Σ+ state term values T (v 0 ,J 0 ). A potential energy curve for the 100
C(3)1 Σ+ state was built using the pointwise Inverted Perturbation Approach (IPA) method [33]. As a starting potential the theoretical curve from [25] was
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used, but shifted in the energy scale to correlate to the asymptote K(42 P1/2 ) + Cs(62 S1/2 ) at 16964.398 cm−1 . To obtain this value, the potassium excitation
energy 42 P1/2 ←− 42 S1/2 , equal to 12895.19 cm−1 [34] was added to the De
105
value of the ground X1 Σ+ state, 4069.208 cm−1 [15].
To establish a proper vibrational numbering of energy levels, experimental data obtained by the Riga group [20] were used. In their paper a progression of 8
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lines corresponding to transitions from a single vibrational level of the C(3)1 Σ+ state, assumed by the authors to be v 0 = 20, to consecutive levels of the ground 110
state was presented, including their relative intensities. As the same level could
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be included to our experimental database, we tried several vibrational numberings for the observed levels, constructed a different potential curve of the C
state for each numbering and calculated the corresponding Franck-Condon fac-
tors (FCFs) for transitions to the ground state levels. Then the FCF values were 115
compared with relative intensities of lines from the Riga experiment. Results presented in Fig.4 indicate a unique vibrational numbering of the C(3)1 Σ+ state
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levels, differing from the one assumed in Ref. [20] by three vibrational quanta. In addition the Franck-Condon minima calculated with the chosen vibrational numbering coincide also with the intensity minima obtained in our experiment. All term values found in the present analysis combined with a few term
120
values measured in the Riga experiment [20] were used in our fit. A full list of the experimental C state level energies can be found in the supplementary
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materials [35]. It turned out that a direct observation of the v 0 = 0, 1, 2 vibrational levels was impossible in our experiment due to unfavourable values of Franck-Condon factors. According to theoretical predictions excitation from
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125
the ground state vibrational levels with v 00 values higher than 10 were needed to detect transitions to v 0 = 0 in the C(3)1 Σ+ state. Population of these levels at
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temperatures used in our experiment was too low, while increasing temperature caused instability in the KCs heat-pipe operation. Absence of the lowest levels v 0 = 0 − 2 in the recorded spectra influences
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130
the accuracy of determination of the Te value, i.e. the minimum of the C state potential energy curve. To improve this accuracy, we followed the recent sug-
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gestion presented in ref. [36]. First, we represented the observed level energies of the C state with the conventional Dunham expansion:
T (v, J) = Te +
X
Ymn (v + 0.5)m [J(J + 1)]n ,
(1)
mn
135
with Te and Ymn symbols having their usual meaning. Description of the C state
9
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35
40
45
50
55
a) 1.0
FCF v'=18 Experiment R lines
Int [a.u]
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Experiment P lines
0.8
0.6
0.4
0.2
0.0
FCF v'=17 Experiment R lines Experiment P lines
Int [a.u]
0.8
0.6
0.4
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0.2
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b) 1.0
0.0
c) 1.0
FCF v'=16
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Experiment R lines Experiment P lines
Int [a.u]
0.8
0.6
PT
0.4
0.2
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0.0
40
45
50
55
v"
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35
Figure 4: Comparison between line intensities in the progression starting from a single vibrational level (v 0 = ?, J 0 = 94) in the C state to consecutive levels in the ground state
(v 00 , J 00 = 93, 95) recorded in the Riga group experiment [20] (dashed bars corresponding to pairs of P and R lines) and Franck-Condon factors (FCF) calculated assuming three different vibrational numberings for v 0 (solid bars). Panels a), b), c) correspond to v 0 = 18, 17 and 16, respectively, pointing out that for v 0 = 17 the best agreement is obtained.
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Table 1: The experimentally determined Dunham coefficients for the 31 Σ+ state in
39 KCs
with corresponding fitting errors. σrms is the root mean square deviation of the fit to 390
value [cm−1 ]
Te
13535.89
Y10
0.042
39.365
Y20 × 10
3
Y01 × 10
2
Y30 × 104 Y11 × 10
5
Y02 × 108 σrms
error [cm−1 ]
0.013
2.0
1.3
−7.81
0.41
1.9864
0.00027
−2.80 −1.82
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Yij
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term values.
0.022 0.014
0.041 cm
−1
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by Dunham coefficients is well justified by a Morse-like shape of its potential. Then we supplemented our database, used in the IPA fit of the potential curve,
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with the energies of several rovibrational levels corresponding to v 0 = 0 − 2,
calculated from the Dunham coefficients. According to discussion in [36], by 140
this method the uncertainty in position of the minimum of the IPA potential
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was reduced. The Dunham coefficients mentioned above are given in Table 1, with their values rounded by the procedure suggested by Le Roy [37]. The final potential energy curve of the C(3)1 Σ+ state, shown in Fig.5 and
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Table 2, is defined by 36 non-equidistant grid points. The natural cubic spline
145
procedure should be used for interpolation purposes. Beyond the range cov-
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ered by the experimental data (4.42 Å – 6.48 Å) the curve is extended with the theoretical potential [25] to ensure proper boundary conditions for solving the radial Schrödinger equation. When the Schrödinger equation with the IPA potential is solved in a mesh of 10000 points between 3.21 Å and 8.37 Å, the ex-
150
perimental energy levels are reproduced with a standard deviation of 0.04 cm−1 . Only for a few levels their experimental positions differ from the predicted ones 11
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-1
E[cm
]
14600
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14400
v'=18
14200
14000
13800
v'=3
4
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13600
5
6
7
R[Å]
Figure 5: The IPA potential curve of the C(3)1 Σ+ state in KCs molecule (circles) compared with the calculations (triangles) [25]. The theoretical curve was shifted in the energy scale to correlate to the proper atomic asymptote. The horizontal lines mark the borders of the region
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covered by the experimental data. The part of the potential below v 0 = 3 is extrapolated using Dunham coefficients in the way described in the text, above v 0 = 18 the potential is
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extended by matching it smoothly with the properly shifted theoretical potential.
by more than 0.2 cm−1 . We assume that the observed deviations are due to
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perturbations caused by interaction with the neighbouring states. Comparison of the experimental curve with the theoretical result [25] shows 155
generally good agreement. The experimental and all available theoretical values
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of Te , ωe and Re for the C(3)1 Σ+ state are presented in Table 3. As the recommended experimental value of Te we give here the minimum of the IPA
potential curve in respect to the bottom of the ground state potential. We
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estimated the uncertainty of the Te to about 0.2 cm−1 , five times larger than
160
the error of this parameter obtained in Dunham coefficients fit [36].
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Table 2: The rotationless IPA potential energy curve for the C(3)1 Σ+ state in KCs; σrms indicates root mean square deviation of the fit to all observed levels.
V [cm−1 ]
R [Å]
V [cm−1 ]
3.21
20664.0999
5.67
13620.5025
3.38
18927.0604
5.79
13680.0095
3.55
17605.5353
5.91
13753.5786
3.72
16564.9855
6.03
13840.2220
3.89
15764.4816
6.15
13938.4331
4.06
15155.7356
6.27
14046.6375
4.23
14674.4462
6.39
14163.6573
4.35
14398.5341
6.51
14288.3672
4.47
14167.4465
6.63
14420.1156
4.59
13980.3846
6.93
14767.9659
4.71
13831.3831
7.11
14978.0578
4.83
13716.9772
7.29
15185.3250
4.95
13632.6236
7.47
15386.0479
5.07
13576.1102
7.65
15578.6355
5.19
13545.1240
7.83
15761.5114
5.31
13535.7225
8.01
15933.4216
5.43
13546.6006
8.19
16092.3027
5.55
13575.0582
8.37
16238.2845
σrms
0.04 cm
−1
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R [Å]
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Table 3: Comparison between the experimental and theoretical values of the salient molecular
Te [cm−1 ]
ωe [cm−1 ]
Re [Å]
Ref.
13535.7
39.365
5.307
exp.
13561
38.99
5.261
[38]
13617
39.16
5.231
[39]
13469
40.84
5.219
[25]
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3.2. c(2)3 Σ+ state
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constants for the C(3)1 Σ+ state in KCs.
The elimination of spectral lines corresponding to transitions to A(2)1 Σ+ , b(1)3 Π and C(3)1 Σ+ states simplified the spectra and allowed to analyse the c(2)3 Σ+ state. Altogether 766 spectral lines corresponding to transitions to the 165
c(2)3 Σ+ state were recorded. The observed spectra revealed irregular spacings between consecutive vibrational levels of the c state, presented in Fig.6c, what
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signalizes irregular shape of the c(2)3 Σ+ state potential energy curve. On the other hand, theoretical calculations [25] (see Fig.1) predict a regular, Morse-
170
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like shape for this curve. The only way to reconcile theoretical predictions with our experimental observations is to assume that the c(2)3 Σ+ state in KCs approaches Hund’s case (c) of angular momenta coupling and should be rather
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considered as a pair of (3)Ω = 1 and (2)Ω = 0− states (see Fig.6a and Fig.6b). The Ω = 0− component cannot be excited directly from the ground electronic
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state and will be left out from further discussion. The potential curve of the Ω = 175
1 component can be deformed by the anticrossing with the Ω = 1 component
of the b(1)3 Π state (Fig.1) resulting in irregular spacings between consecutive
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vibrational levels. The fact that in this spectral region we did not record any Q lines in the (3)Ω = 1 ←− X1 Σ+ transition suggests that the observation of
the (3)Ω = 1 state is possible due to the mixing of wavefunctions only for its e
180
parity levels with these of the A(2)1 Σ+ ∼ b(1)3 Π and C(3)1 Σ+ ∼ b(1)3 Π states. This concept is supported at least partially by large values of spin-orbit coupling
matrix elements between the b(1)3 Π and c(2)3 Σ+ states, predicted theoretically 14
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[38]. The dominant character of the f parity levels remains triplet and they are not accessible from the ground state. Apparently the potential curve of the 185
B(1)1 Π state is above the studied energy region and thus the interaction with
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the c(2)3 Σ+ state is weak. For consistency with terminology in this paper we decided to continue referring to the investigated state as c(2)3 Σ+ in the future
analysis, but whether this is more appropriate than the (3)Ω = 1 notation should be clarified in future theoretical studies.
Since the potential energy curve of the investigated state substantially dif-
190
fers from a Morse-like potential, a power series representation of level energies
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was impossible. We used solely the model-free pointwise IPA method, which is able to represent potential energy functions even of an exotic shape. The known energies of the X state levels [15] were used to convert wavenumbers into term 195
values T (v 0 ,J 0 ) of rovibrational levels in the c(2)3 Σ+ state, referred to the bottom of the ground state potential well. As a starting potential curve we used the theoretical (3)Ω = 1 potential from Ref. [40]. Also in this case we shifted the
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theoretical curve to ensure the proper asymptotic behaviour. Since this state correlates to the K(42 S1/2 ) + Cs(62 P3/2 ) asymptote we added excitation energy of the 62 P3/2 state in the caesium atom, 11732.31 cm−1 [41], to the depth of the
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200
KCs ground state potential to obtain the final value of the asymptotic energy, 15801.518 cm−1 .
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Also in case of the c(2)3 Σ+ state establishing of the absolute vibrational numbering was a problem, because we have not observed transitions from the labelled ground state levels to the lowest vibrational levels of the c state due
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205
to low values of the transition probability and we have not observed transitions other than in major isotopologue (39 KCs). The standard procedure in such case,
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based on comparison of the experimentally observed line intensities with those predicted from the calculation, in this particular case cannot give us absolute
210
vibrational numbering because the observed line intensity patterns result mainly from mixing of the c3 Σ+ state wavefunctions with those of the neighbouring states and a degree of mixing cannot be estimated with the present data. Since we were not able to determine the numbering unambiguously, we decided for an 15
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a)
44
42
40
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36
34
v
-1
G [cm ]
38
32
30
28
26
24
0
10
20
30
40
50
V'
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b)
60
M
v
-1
G [cm ]
40
39
15
20
25
30
35
V'
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c)
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v
-1
G [cm ]
41
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40
39
15
20
25
30
35
V'
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10
Figure 6: (a) Theoretical v’-dependence of vibrational spacings ∆Gv for the c(2)3 Σ+ state (red line with triangles) [25] and (3)Ω = 1 state (black dotted line) [40], enlarged in the region covered by the experimental data (b), compared with ∆Gv values obtained in the present experiment (blue squares) (c).
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30
v'
20
40
60
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10
80
100
J'
Figure 7: Range of rovibrational levels of the c(2)3 Σ+ state covered in the present experiment.
215
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arbitrary choice of numbering which provides the Te value for the c state close to the mean value of Te obtained from the four available theoretical calculations [25, 38, 39, 40]. Certainly this vibrational numbering should be treated as
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provisional with an error amounting even to a few vibrational quanta. The final potential bases on 646 term values, which span the range v 0 =
220
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5 − 29, J 0 = 38 − 106, what is presented in Fig.7. A full list of the experimental c(2)3 Σ+ state level energies can be found in the supplementary materials
[35]. The highest observed vibrational level v 0 = 30 was not included in the
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IPA procedure because of a strong local perturbation. The beginning of the perturbed area may be noticed in Fig.6c, where the last point in the graph is shifted from the expected position. The final potential curve is given in Table 4 as set of 40 non-equidistant grid points. Fig.8 shows the comparison between
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experimentally determined potential and the theoretical curves. In regions outside the experimental range (marked with the horizontal lines in Fig.8), i.e. for internuclear distance values R < 4.26 Å and R > 6.83 Å, the IPA curve was extrapolated with the theoretical one [40] to ensure proper boundary conditions 17
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-1
E[cm
]
14400
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(4)
14000
v'=29
13600
3
+
2
13200
v'=5
(2)
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12800
(3)
4
5
6
7
R[Å]
Figure 8: The pointwise IPA potential energy curve of the c(2)3 Σ+ state in KCs (solid line, red in colour) compared with theoretical potentials: calculated without spin-orbit coupling [25] (dash-dot line, green in colour) and with spin-orbit coupling included [40] (dashed lines, indicated with horizontal lines.
for solving the radial Schrödinger equation. The shape of the curve obtained
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230
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black in colour). The limits of vibrational levels observed in our experiment (v’=5 to 29) are
in the IPA procedure explains irregular vibrational spacings observed in the
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spectra. Experimental values of the main equilibrium constants Te , ωe and Re are presented in Table 5 together with theoretical values calculated either for
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c(2)3 Σ+ or (3)Ω = 1 state. 4. Conclusions
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235
The recorded spectra allowed for the first experimental determination of
potential energy curves for two electronic states of KCs, C(3)1 Σ+ and c(2)3 Σ+ . The C(3)1 Σ+ state with a regular shape of the potential can be described also by a set of molecular constants. Unfortunately observation of bottoms of both
240
states was impossible because of unfavourable Franck-Condon factors. This problem can be overcome in future by using different experimental techniques, 18
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indicates root mean square deviation of the fit.
V [cm−1 ]
R [Å]
V [cm−1 ]
3.67
16050.3628
5.47
12780.9191
3.78
15527.6294
5.58
12830.1175
3.88
15139.3164
5.68
12880.2897
3.95
14908.6456
5.79
12946.5020
3.99
14796.5568
5.90
13021.3724
4.05
14646.8207
6.00
13094.6344
4.10
14500.9346
6.11
13180.6432
4.14
14355.4116
6.21
13263.0075
4.20
14122.8082
6.32
13358.3711
4.31
13766.3678
6.42
13448.7350
4.41
13510.0578
6.53
13558.1939
4.52
13266.2316
6.64
13679.8800
4.63
13075.0320
6.74
13808.7289
4.73
12937.4106
6.85
13947.3577
4.84
12825.4089
6.95
14075.8971
4.94
12765.1975
7.06
14212.4364
5.05
12730.6381
7.17
14339.1427
5.15
12720.8110
7.27
14444.5979
5.26
12727.4657
7.38
14550.5086
5.37
12749.9269
7.48
14638.4891
σrms
0.051 cm−1
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R [Å]
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Table 4: The rotationless IPA potential energy curve for the c(2)3 Σ+ state in KCs. σrms
19
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Table 5: Comparison between the experimental and theoretical values of the salient molecular constants of the c(2)3 Σ+ state in KCs. Theoretical values for both c(2)3 Σ+ and (3)Ω = 1
ωe [cm−1 ]
Re [Å]
Ref.
12720
41.02
5.15
exp.
12845
42.69
5.237
[38] (c(2)3 Σ+ )
12570
46.75
5.109
[39] (c(2)3 Σ+ )
12762
44.52
5.106
[25] (c(2)3 Σ+ )
12643
45.4
5.10
[40] ((3)Ω = 1)
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Te [cm−1 ]
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states are cited.
in particular by observation of laser induced fluorescence from suitably chosen vibrational levels of higher excited electronic states, possibly states of triplet symmetry excited via perturbation facilitated optical-optical double resonance. Determination of higher parts of both potentials also turned out to be impossible
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245
due to a spectral overlap by strong transitions to the B(1)1 Π state, unavoidable in the corresponding spectral region. A transition dipole moment for the B ←−
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X transition is typically five times larger than for the C ←− X transition in the
range of internuclear distances covered by this experiment [39] and a probability for nominally forbidden singlet-triplet c ←− X transition is even lower.
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250
As it has been predicted during preliminary analysis, the experimental po-
tential curve of the c(2)3 Σ+ state follows the theoretical potential calculated
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with spin-orbit coupling taken into account. However, as discussed above, we cannot assume a pure Hund’s case (c) coupling here, since in the spectra related to the c state we could observe only P and R branches, while for transitions to
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255
the Ω = 1 state also Q branch should be visible. Observation of only e parity levels may be explained by indirect mixing of the c(2)3 Σ+ state wavefunction
with that of both states of 1 Σ+ symmetry: A(2)1 Σ+ and C(3)1 Σ+ through the b(1)3 Π state, while the coupling to the B(1)1 Π state is apparently low in this
260
region. But a full coupled channels deperturbation procedure for the whole set 20
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of the involved electronic states is required to clear up this problem. Acknowledgements
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This work was partially supported by the National Science Centre (Grant
no. 2016/21/B/ST2/02190). The authors thank Dr Asen Pashov for his help 265
in preparation of the revised version of the paper. References
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Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule Jacek Szczepkowski, Anna Grochola, Wlodzimierz Jastrzebski, Pawel Kowalczyk PII: DOI: Reference:
S0022-4073(17)30448-X 10.1016/j.jqsrt.2017.09.013 JQSRT 5840
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
2 June 2017 16 August 2017 12 September 2017
Please cite this article as: Jacek Szczepkowski, Anna Grochola, Wlodzimierz Jastrzebski, Pawel Kowalczyk, Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule, Journal of Quantitative Spectroscopy & Radiative Transfer (2017), doi: 10.1016/j.jqsrt.2017.09.013
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Highlights • The C1 Σ+ ← X 1 Σ+ and c3 Σ+ ← X 1 Σ+ band systems of KCs observed
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by polarisation labelling spectroscopy.
• The pointwise potential energy curve of the C1 Σ+ state constructed from experimental data.
• The pointwise potential energy curve of the c 3 Σ+ state constructed from experimental data.
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• The results compared with the theoretical predictions.
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Spectroscopic study of the C(3)1 Σ+ ← X1 Σ+ and c(2)3 Σ+ ← X1 Σ+ transitions in KCs molecule
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Jacek Szczepkowski∗, Anna Grochola, Wlodzimierz Jastrzebski Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warszawa, Poland
Pawel Kowalczyk
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Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland
Abstract
We present an experimental investigation of the C(3)1 Σ+ and c(2)3 Σ+ states of KCs using two-colour polarisation labelling spectroscopy (PLS) technique. Pointwise potential energy curves are generated for both states using the in-
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verted perturbation approach (IPA) method. The experimental pointwise potentials and molecular constants Te , ωe and Re are compared with theoretical
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calculations.
Keywords: laser spectroscopy, alkali dimers, electronic states, potential energy curves
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PACS: 31.50.Df, 33.20.Kf, 33.20.Vq, 42.62.Fi
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1. Introduction
Molecules with permanent electric dipole moments are a basis for experi-
ments from many different areas of physics. Polar molecules give opportunity
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to control interactions with external fields [1], allow for precision measurements
5
of fundamental constants [2, 3], open the way to exploration of quantum phases in dipolar gases [4, 5] and testing universality in few-body systems [6, 7]. One ∗ Corresponding
author Email address: [email protected] (Jacek Szczepkowski )
Preprint submitted to Journal of Quantitative Spectroscopy and Radiative TransferSeptember 13, 2017
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may also notice a development of so called ultracold chemistry [8]. A particular goal for many groups is to create molecules in their absolute ground state, since it is expected to be the first step in achieving heteronuclear molecular Bose-Einstein condensate. It has been done only for few molecules so far: KRb
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10
[9] and RbCs [10, 11] using Feshbach resonances and stimulated Raman adia-
batic passage (STIRAP) process and for LiCs using photoassociation [12]. KCs
with its permanent electric dipole moment of 1.92 D is the next promising can-
didate. First K-Cs interspecies Feshbach resonances were observed [13] and 15
several optical coherent schemes to create ultracold KCs molecules in their ab-
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solute rovibrational ground state were proposed [14]. Described processes base on spectroscopic studies performed so far.
Spectroscopic data are particularly important for all physicists interested in experiments conducted in an ultracold environment, since they give precise 20
information about energies of rovibrational levels, couplings influencing transition probabilities and possible local perturbations, which may become an ob-
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stacle as well as an opportunity in process of manipulation of molecules. So far several electronic states of KCs molecule were investigated in high precision
25
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spectroscopic experiments: the ground state [15] and a few higher excited states [16, 17, 18, 19]. But particular interest is focused on states dissociating to lower lying asymptotes [20, 21, 22], since most of excitation schemes lead through
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these states [14, 23]. Couplings between these states were investigated and deperturbation procedures were performed [24] to open a possibility of further
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study on cooling and manipulating of KCs molecules. To extend present knowledge of the electronic structure of KCs in the most
30
interesting region we present the first experimental data concerning the c(2)3 Σ+
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state dissociating to the first excited asymptote K(42 S1/2 ) + Cs(62 P3/2 ) and the C(3)1 Σ+ state dissociating to the K(42 P1/2 ) + Cs(62 S1/2 ) asymptote, but interacting with the states lying below. All potential energy curves correlating
35
to both mentioned asymptotes are presented in Fig. 1.
3
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K +Cs
-1
E[cm ]
3
1
3
2
4p+6s
+
3
16500
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2
4s+6p
15000 1
1
13500
1
3 3
2
+
+
1
2
M
12000
+
10500 3
9000
3
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1
4
5
6
R[Å]
7
8
9
10
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Figure 1: Ab initio potential energy curves of KCs [25] for all states correlating to the K(4s) + Cs(6p) and K(4p) + Cs(6s) asymptotes. Energies are given with respect to the minimum of the electronic ground state. The horizontal lines with atomic products on them indicate
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the energies of the adiabatic dissociation limits.
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2. Experiment Metallic samples of potassium and caesium (5 g each) were used to generate the vapour of KCs molecules in a stainless steel linear heat-pipe oven, heated
40
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to 430 ◦ C. Helium under the pressure of 5 Torr served as a buffer gas to prevent
alkali metal deposition on the heat-pipe windows. The polarisation labelling spectroscopy (PLS) technique was applied to record the KCs excitation spectra.
Details of the method may be found in our previous papers [26, 27]. In general, excitation spectrum was studied using the V-type optical-optical double
resonance method with two independent pump and probe lasers. As a pump
laser served a parametric oscillator/amplifier system (OPO/OPA, Sunlite EX,
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45
Continuum) pumped with the third harmonic of an injection seeded Nd:YAG laser (Powerlite 8000). Either the signal or idler beams of the system were used to scan the region 11800 cm−1 - 14000 cm−1 . The probe beam, of a fixed wavelength controlled with a HighFinesse WS-7 wavemeter, originated from a homemade dye laser operating on Rhodamine 6G and pumped synchronously by the
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same Nd:YAG laser. Its wavelength was in resonance with selected transitions in the well known KCs 41 Σ+ ←− X1 Σ+ band system [16]. The rotationally
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resolved C(3)1 Σ+ ←− X1 Σ+ and c(2)3 Σ+ ←− X1 Σ+ excitation spectra were recorded, with the accuracy of observed molecular line positions better than
55
0.1 cm−1 . Calibration of molecular spectra was based on simultaneously gener-
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ated optogalvanic spectra of argon and transmission fringes from a Fabry-Pérot
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interferometer of FSR = 1 cm−1 .
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3. Analysis and results
60
In the spectral region examined in the present experiment, we could observe
several characteristic patterns. They consisted either of vibrational progressions of P, Q and R lines (’triplets’) or of P and R lines only (’doublets’). All recorded P, Q, R ’triplets’ were identified as transitions to the B(1)1 Π state, studied
before [20]. The remaining P and R lines from doublets were corresponding to transitions to four different electronic states in KCs. In the lower part of the 5
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65
scanned area traces of the coupled A(2)1 Σ+ and b(1)3 Π states were recognized. In the upper part of the examined region transitions to the c(2)3 Σ+ and C(3)1 Σ+ states overlap with strong lines registered due to a presence of the B(1)1 Π state.
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Observation of five electronic states in the same spectral region resulted in complexity of the spectra. Moreover the probe laser was typically labelling more 70
than one rovibrational level in the ground state, what additionally increased the number of the recorded spectral lines. In such situation, despite the expected
simplification of experimental spectra due to the polarisation labelling method, they remained highly congested. As a solution to this problem we recorded two
75
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or more spectra containing progressions originating from the same rovibrational
level in the ground state, but labelled with different wavelengths of the probe laser. An example of two spectra originating from the same rovibrational level in the ground state is presented in Fig. 2a and 2b. Then we looked for common features in the spectra, taking numerical product of two or three of them. Such procedure allowed for unambiguous assignment of spectral lines, as shown in Fig. 2c.
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Transitions to the C(3)1 Σ+ state were relatively easy to identify, since ob-
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servation of several rovibrational levels of this state has been already reported [20]. The remaining spectral lines, much weaker, could correspond to transitions to either upper part of the b(1)3 Π and A(2)1 Σ+ states or to the c(2)3 Σ+ state. The first two states were described with high precision before [21] and elimina-
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85
tion of the corresponding spectral lines [24, 28] allowed for conclusion that we
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recorded mostly transitions to the latter, i.e. the c(2)3 Σ+ state. In the case of KCs molecule direct optical excitation from the singlet ground state to the excited triplet states is possible due to strong spin-orbit coupling and mixing of singlet and triplet rovibrational levels [15, 21]. Such spin-forbidden transitions
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have been observed before in KCs [29], as well as in heavier (RbCs [30]) but also lighter (LiCs [31] and NaCs [32]) alkali metal diatomics containing caesium atoms.
6
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a)
c) 3
27
v'=26
13480
13500
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13460
8
7
6
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b)
13520
13540
v'=9
13560
13580
1
13600
-1
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[cm ]
Figure 2: Exemplary spectrum of KCs after taking numerical product of two experimental polarisation spectra (c). The assigned P,R progressions correspond to transitions to consecu-
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tive vibrational levels in the c(2)3 Σ+ and C(3)1 Σ+ states when the (v 00 = 3, J 00 = 87) ground state level is labelled by the probe laser. In the two upper panels spectra before multiplication procedure are presented, recorded with the circularly polarised probe laser set at wavelengths
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corresponding nominally to transitions 41 Σ+ (v 0 = 16, J 0 = 86) ←− X1 Σ+ (v 0 = 3, J 0 = 87) (a) and 41 Σ+ (v 0 = 16, J 0 = 88) ←− X1 Σ+ (v 0 = 3, J 0 = 87) (b). The unassigned, randomly distributed lines in (c) are artefacts resulting from accidental overlap of different transitions
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in (a) and (b).
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18
16
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14
v'
12
10
8
4
2 20
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80
100
120
J'
Figure 3: The range of rovibrational levels of the C(3)1 Σ+ state covered in the present experiment (dots, black in colour). Triangles (red in colour) correspond to the data from the numbering.
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3.1. C(3)1 Σ+ state
M
experiment performed in the Riga group [20], included in the fit with corrected vibrational
Altogether 441 spectral lines were recorded corresponding to transitions to
95
rovibrational levels of the C state in the range v 0 = 3 − 18, J 0 = 38 − 127.
PT
Distribution of the experimental data is shown in Fig.3. Knowing energies of the ground state levels [15], we converted measured transition energies into
CE
390 C(3)1 Σ+ state term values T (v 0 ,J 0 ). A potential energy curve for the 100
C(3)1 Σ+ state was built using the pointwise Inverted Perturbation Approach (IPA) method [33]. As a starting potential the theoretical curve from [25] was
AC
used, but shifted in the energy scale to correlate to the asymptote K(42 P1/2 ) + Cs(62 S1/2 ) at 16964.398 cm−1 . To obtain this value, the potassium excitation
energy 42 P1/2 ←− 42 S1/2 , equal to 12895.19 cm−1 [34] was added to the De
105
value of the ground X1 Σ+ state, 4069.208 cm−1 [15].
To establish a proper vibrational numbering of energy levels, experimental data obtained by the Riga group [20] were used. In their paper a progression of 8
ACCEPTED MANUSCRIPT
lines corresponding to transitions from a single vibrational level of the C(3)1 Σ+ state, assumed by the authors to be v 0 = 20, to consecutive levels of the ground 110
state was presented, including their relative intensities. As the same level could
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be included to our experimental database, we tried several vibrational numberings for the observed levels, constructed a different potential curve of the C
state for each numbering and calculated the corresponding Franck-Condon fac-
tors (FCFs) for transitions to the ground state levels. Then the FCF values were 115
compared with relative intensities of lines from the Riga experiment. Results presented in Fig.4 indicate a unique vibrational numbering of the C(3)1 Σ+ state
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levels, differing from the one assumed in Ref. [20] by three vibrational quanta. In addition the Franck-Condon minima calculated with the chosen vibrational numbering coincide also with the intensity minima obtained in our experiment. All term values found in the present analysis combined with a few term
120
values measured in the Riga experiment [20] were used in our fit. A full list of the experimental C state level energies can be found in the supplementary
M
materials [35]. It turned out that a direct observation of the v 0 = 0, 1, 2 vibrational levels was impossible in our experiment due to unfavourable values of Franck-Condon factors. According to theoretical predictions excitation from
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125
the ground state vibrational levels with v 00 values higher than 10 were needed to detect transitions to v 0 = 0 in the C(3)1 Σ+ state. Population of these levels at
PT
temperatures used in our experiment was too low, while increasing temperature caused instability in the KCs heat-pipe operation. Absence of the lowest levels v 0 = 0 − 2 in the recorded spectra influences
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130
the accuracy of determination of the Te value, i.e. the minimum of the C state potential energy curve. To improve this accuracy, we followed the recent sug-
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gestion presented in ref. [36]. First, we represented the observed level energies of the C state with the conventional Dunham expansion:
T (v, J) = Te +
X
Ymn (v + 0.5)m [J(J + 1)]n ,
(1)
mn
135
with Te and Ymn symbols having their usual meaning. Description of the C state
9
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35
40
45
50
55
a) 1.0
FCF v'=18 Experiment R lines
Int [a.u]
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Experiment P lines
0.8
0.6
0.4
0.2
0.0
FCF v'=17 Experiment R lines Experiment P lines
Int [a.u]
0.8
0.6
0.4
M
0.2
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b) 1.0
0.0
c) 1.0
FCF v'=16
ED
Experiment R lines Experiment P lines
Int [a.u]
0.8
0.6
PT
0.4
0.2
CE
0.0
40
45
50
55
v"
AC
35
Figure 4: Comparison between line intensities in the progression starting from a single vibrational level (v 0 = ?, J 0 = 94) in the C state to consecutive levels in the ground state
(v 00 , J 00 = 93, 95) recorded in the Riga group experiment [20] (dashed bars corresponding to pairs of P and R lines) and Franck-Condon factors (FCF) calculated assuming three different vibrational numberings for v 0 (solid bars). Panels a), b), c) correspond to v 0 = 18, 17 and 16, respectively, pointing out that for v 0 = 17 the best agreement is obtained.
10
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Table 1: The experimentally determined Dunham coefficients for the 31 Σ+ state in
39 KCs
with corresponding fitting errors. σrms is the root mean square deviation of the fit to 390
value [cm−1 ]
Te
13535.89
Y10
0.042
39.365
Y20 × 10
3
Y01 × 10
2
Y30 × 104 Y11 × 10
5
Y02 × 108 σrms
error [cm−1 ]
0.013
2.0
1.3
−7.81
0.41
1.9864
0.00027
−2.80 −1.82
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Yij
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term values.
0.022 0.014
0.041 cm
−1
M
by Dunham coefficients is well justified by a Morse-like shape of its potential. Then we supplemented our database, used in the IPA fit of the potential curve,
ED
with the energies of several rovibrational levels corresponding to v 0 = 0 − 2,
calculated from the Dunham coefficients. According to discussion in [36], by 140
this method the uncertainty in position of the minimum of the IPA potential
PT
was reduced. The Dunham coefficients mentioned above are given in Table 1, with their values rounded by the procedure suggested by Le Roy [37]. The final potential energy curve of the C(3)1 Σ+ state, shown in Fig.5 and
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Table 2, is defined by 36 non-equidistant grid points. The natural cubic spline
145
procedure should be used for interpolation purposes. Beyond the range cov-
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ered by the experimental data (4.42 Å – 6.48 Å) the curve is extended with the theoretical potential [25] to ensure proper boundary conditions for solving the radial Schrödinger equation. When the Schrödinger equation with the IPA potential is solved in a mesh of 10000 points between 3.21 Å and 8.37 Å, the ex-
150
perimental energy levels are reproduced with a standard deviation of 0.04 cm−1 . Only for a few levels their experimental positions differ from the predicted ones 11
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-1
E[cm
]
14600
CR IP T
14400
v'=18
14200
14000
13800
v'=3
4
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13600
5
6
7
R[Å]
Figure 5: The IPA potential curve of the C(3)1 Σ+ state in KCs molecule (circles) compared with the calculations (triangles) [25]. The theoretical curve was shifted in the energy scale to correlate to the proper atomic asymptote. The horizontal lines mark the borders of the region
M
covered by the experimental data. The part of the potential below v 0 = 3 is extrapolated using Dunham coefficients in the way described in the text, above v 0 = 18 the potential is
ED
extended by matching it smoothly with the properly shifted theoretical potential.
by more than 0.2 cm−1 . We assume that the observed deviations are due to
PT
perturbations caused by interaction with the neighbouring states. Comparison of the experimental curve with the theoretical result [25] shows 155
generally good agreement. The experimental and all available theoretical values
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of Te , ωe and Re for the C(3)1 Σ+ state are presented in Table 3. As the recommended experimental value of Te we give here the minimum of the IPA
potential curve in respect to the bottom of the ground state potential. We
AC
estimated the uncertainty of the Te to about 0.2 cm−1 , five times larger than
160
the error of this parameter obtained in Dunham coefficients fit [36].
12
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Table 2: The rotationless IPA potential energy curve for the C(3)1 Σ+ state in KCs; σrms indicates root mean square deviation of the fit to all observed levels.
V [cm−1 ]
R [Å]
V [cm−1 ]
3.21
20664.0999
5.67
13620.5025
3.38
18927.0604
5.79
13680.0095
3.55
17605.5353
5.91
13753.5786
3.72
16564.9855
6.03
13840.2220
3.89
15764.4816
6.15
13938.4331
4.06
15155.7356
6.27
14046.6375
4.23
14674.4462
6.39
14163.6573
4.35
14398.5341
6.51
14288.3672
4.47
14167.4465
6.63
14420.1156
4.59
13980.3846
6.93
14767.9659
4.71
13831.3831
7.11
14978.0578
4.83
13716.9772
7.29
15185.3250
4.95
13632.6236
7.47
15386.0479
5.07
13576.1102
7.65
15578.6355
5.19
13545.1240
7.83
15761.5114
5.31
13535.7225
8.01
15933.4216
5.43
13546.6006
8.19
16092.3027
5.55
13575.0582
8.37
16238.2845
σrms
0.04 cm
−1
AC
CE
PT
ED
M
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R [Å]
13
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Table 3: Comparison between the experimental and theoretical values of the salient molecular
Te [cm−1 ]
ωe [cm−1 ]
Re [Å]
Ref.
13535.7
39.365
5.307
exp.
13561
38.99
5.261
[38]
13617
39.16
5.231
[39]
13469
40.84
5.219
[25]
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3.2. c(2)3 Σ+ state
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constants for the C(3)1 Σ+ state in KCs.
The elimination of spectral lines corresponding to transitions to A(2)1 Σ+ , b(1)3 Π and C(3)1 Σ+ states simplified the spectra and allowed to analyse the c(2)3 Σ+ state. Altogether 766 spectral lines corresponding to transitions to the 165
c(2)3 Σ+ state were recorded. The observed spectra revealed irregular spacings between consecutive vibrational levels of the c state, presented in Fig.6c, what
M
signalizes irregular shape of the c(2)3 Σ+ state potential energy curve. On the other hand, theoretical calculations [25] (see Fig.1) predict a regular, Morse-
170
ED
like shape for this curve. The only way to reconcile theoretical predictions with our experimental observations is to assume that the c(2)3 Σ+ state in KCs approaches Hund’s case (c) of angular momenta coupling and should be rather
PT
considered as a pair of (3)Ω = 1 and (2)Ω = 0− states (see Fig.6a and Fig.6b). The Ω = 0− component cannot be excited directly from the ground electronic
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state and will be left out from further discussion. The potential curve of the Ω = 175
1 component can be deformed by the anticrossing with the Ω = 1 component
of the b(1)3 Π state (Fig.1) resulting in irregular spacings between consecutive
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vibrational levels. The fact that in this spectral region we did not record any Q lines in the (3)Ω = 1 ←− X1 Σ+ transition suggests that the observation of
the (3)Ω = 1 state is possible due to the mixing of wavefunctions only for its e
180
parity levels with these of the A(2)1 Σ+ ∼ b(1)3 Π and C(3)1 Σ+ ∼ b(1)3 Π states. This concept is supported at least partially by large values of spin-orbit coupling
matrix elements between the b(1)3 Π and c(2)3 Σ+ states, predicted theoretically 14
ACCEPTED MANUSCRIPT
[38]. The dominant character of the f parity levels remains triplet and they are not accessible from the ground state. Apparently the potential curve of the 185
B(1)1 Π state is above the studied energy region and thus the interaction with
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the c(2)3 Σ+ state is weak. For consistency with terminology in this paper we decided to continue referring to the investigated state as c(2)3 Σ+ in the future
analysis, but whether this is more appropriate than the (3)Ω = 1 notation should be clarified in future theoretical studies.
Since the potential energy curve of the investigated state substantially dif-
190
fers from a Morse-like potential, a power series representation of level energies
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was impossible. We used solely the model-free pointwise IPA method, which is able to represent potential energy functions even of an exotic shape. The known energies of the X state levels [15] were used to convert wavenumbers into term 195
values T (v 0 ,J 0 ) of rovibrational levels in the c(2)3 Σ+ state, referred to the bottom of the ground state potential well. As a starting potential curve we used the theoretical (3)Ω = 1 potential from Ref. [40]. Also in this case we shifted the
M
theoretical curve to ensure the proper asymptotic behaviour. Since this state correlates to the K(42 S1/2 ) + Cs(62 P3/2 ) asymptote we added excitation energy of the 62 P3/2 state in the caesium atom, 11732.31 cm−1 [41], to the depth of the
ED
200
KCs ground state potential to obtain the final value of the asymptotic energy, 15801.518 cm−1 .
PT
Also in case of the c(2)3 Σ+ state establishing of the absolute vibrational numbering was a problem, because we have not observed transitions from the labelled ground state levels to the lowest vibrational levels of the c state due
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205
to low values of the transition probability and we have not observed transitions other than in major isotopologue (39 KCs). The standard procedure in such case,
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based on comparison of the experimentally observed line intensities with those predicted from the calculation, in this particular case cannot give us absolute
210
vibrational numbering because the observed line intensity patterns result mainly from mixing of the c3 Σ+ state wavefunctions with those of the neighbouring states and a degree of mixing cannot be estimated with the present data. Since we were not able to determine the numbering unambiguously, we decided for an 15
ACCEPTED MANUSCRIPT
a)
44
42
40
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36
34
v
-1
G [cm ]
38
32
30
28
26
24
0
10
20
30
40
50
V'
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b)
60
M
v
-1
G [cm ]
40
39
15
20
25
30
35
V'
ED
c)
PT
v
-1
G [cm ]
41
CE
40
39
15
20
25
30
35
V'
AC
10
Figure 6: (a) Theoretical v’-dependence of vibrational spacings ∆Gv for the c(2)3 Σ+ state (red line with triangles) [25] and (3)Ω = 1 state (black dotted line) [40], enlarged in the region covered by the experimental data (b), compared with ∆Gv values obtained in the present experiment (blue squares) (c).
16
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30
v'
20
40
60
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10
80
100
J'
Figure 7: Range of rovibrational levels of the c(2)3 Σ+ state covered in the present experiment.
215
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arbitrary choice of numbering which provides the Te value for the c state close to the mean value of Te obtained from the four available theoretical calculations [25, 38, 39, 40]. Certainly this vibrational numbering should be treated as
ED
provisional with an error amounting even to a few vibrational quanta. The final potential bases on 646 term values, which span the range v 0 =
220
PT
5 − 29, J 0 = 38 − 106, what is presented in Fig.7. A full list of the experimental c(2)3 Σ+ state level energies can be found in the supplementary materials
[35]. The highest observed vibrational level v 0 = 30 was not included in the
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IPA procedure because of a strong local perturbation. The beginning of the perturbed area may be noticed in Fig.6c, where the last point in the graph is shifted from the expected position. The final potential curve is given in Table 4 as set of 40 non-equidistant grid points. Fig.8 shows the comparison between
AC 225
experimentally determined potential and the theoretical curves. In regions outside the experimental range (marked with the horizontal lines in Fig.8), i.e. for internuclear distance values R < 4.26 Å and R > 6.83 Å, the IPA curve was extrapolated with the theoretical one [40] to ensure proper boundary conditions 17
ACCEPTED MANUSCRIPT
-1
E[cm
]
14400
CR IP T
(4)
14000
v'=29
13600
3
+
2
13200
v'=5
(2)
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12800
(3)
4
5
6
7
R[Å]
Figure 8: The pointwise IPA potential energy curve of the c(2)3 Σ+ state in KCs (solid line, red in colour) compared with theoretical potentials: calculated without spin-orbit coupling [25] (dash-dot line, green in colour) and with spin-orbit coupling included [40] (dashed lines, indicated with horizontal lines.
for solving the radial Schrödinger equation. The shape of the curve obtained
ED
230
M
black in colour). The limits of vibrational levels observed in our experiment (v’=5 to 29) are
in the IPA procedure explains irregular vibrational spacings observed in the
PT
spectra. Experimental values of the main equilibrium constants Te , ωe and Re are presented in Table 5 together with theoretical values calculated either for
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c(2)3 Σ+ or (3)Ω = 1 state. 4. Conclusions
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235
The recorded spectra allowed for the first experimental determination of
potential energy curves for two electronic states of KCs, C(3)1 Σ+ and c(2)3 Σ+ . The C(3)1 Σ+ state with a regular shape of the potential can be described also by a set of molecular constants. Unfortunately observation of bottoms of both
240
states was impossible because of unfavourable Franck-Condon factors. This problem can be overcome in future by using different experimental techniques, 18
ACCEPTED MANUSCRIPT
indicates root mean square deviation of the fit.
V [cm−1 ]
R [Å]
V [cm−1 ]
3.67
16050.3628
5.47
12780.9191
3.78
15527.6294
5.58
12830.1175
3.88
15139.3164
5.68
12880.2897
3.95
14908.6456
5.79
12946.5020
3.99
14796.5568
5.90
13021.3724
4.05
14646.8207
6.00
13094.6344
4.10
14500.9346
6.11
13180.6432
4.14
14355.4116
6.21
13263.0075
4.20
14122.8082
6.32
13358.3711
4.31
13766.3678
6.42
13448.7350
4.41
13510.0578
6.53
13558.1939
4.52
13266.2316
6.64
13679.8800
4.63
13075.0320
6.74
13808.7289
4.73
12937.4106
6.85
13947.3577
4.84
12825.4089
6.95
14075.8971
4.94
12765.1975
7.06
14212.4364
5.05
12730.6381
7.17
14339.1427
5.15
12720.8110
7.27
14444.5979
5.26
12727.4657
7.38
14550.5086
5.37
12749.9269
7.48
14638.4891
σrms
0.051 cm−1
M
ED
PT
CE AC
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R [Å]
CR IP T
Table 4: The rotationless IPA potential energy curve for the c(2)3 Σ+ state in KCs. σrms
19
ACCEPTED MANUSCRIPT
Table 5: Comparison between the experimental and theoretical values of the salient molecular constants of the c(2)3 Σ+ state in KCs. Theoretical values for both c(2)3 Σ+ and (3)Ω = 1
ωe [cm−1 ]
Re [Å]
Ref.
12720
41.02
5.15
exp.
12845
42.69
5.237
[38] (c(2)3 Σ+ )
12570
46.75
5.109
[39] (c(2)3 Σ+ )
12762
44.52
5.106
[25] (c(2)3 Σ+ )
12643
45.4
5.10
[40] ((3)Ω = 1)
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Te [cm−1 ]
CR IP T
states are cited.
in particular by observation of laser induced fluorescence from suitably chosen vibrational levels of higher excited electronic states, possibly states of triplet symmetry excited via perturbation facilitated optical-optical double resonance. Determination of higher parts of both potentials also turned out to be impossible
M
245
due to a spectral overlap by strong transitions to the B(1)1 Π state, unavoidable in the corresponding spectral region. A transition dipole moment for the B ←−
ED
X transition is typically five times larger than for the C ←− X transition in the
range of internuclear distances covered by this experiment [39] and a probability for nominally forbidden singlet-triplet c ←− X transition is even lower.
PT
250
As it has been predicted during preliminary analysis, the experimental po-
tential curve of the c(2)3 Σ+ state follows the theoretical potential calculated
CE
with spin-orbit coupling taken into account. However, as discussed above, we cannot assume a pure Hund’s case (c) coupling here, since in the spectra related to the c state we could observe only P and R branches, while for transitions to
AC
255
the Ω = 1 state also Q branch should be visible. Observation of only e parity levels may be explained by indirect mixing of the c(2)3 Σ+ state wavefunction
with that of both states of 1 Σ+ symmetry: A(2)1 Σ+ and C(3)1 Σ+ through the b(1)3 Π state, while the coupling to the B(1)1 Π state is apparently low in this
260
region. But a full coupled channels deperturbation procedure for the whole set 20
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of the involved electronic states is required to clear up this problem. Acknowledgements
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This work was partially supported by the National Science Centre (Grant
no. 2016/21/B/ST2/02190). The authors thank Dr Asen Pashov for his help 265
in preparation of the revised version of the paper. References
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PT
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