- Email: [email protected]
On the 41Σ+ state of the KCs molecule
Journal of Molecular Spectroscopy 276–277 (2012) 19–21
Contents lists available at SciVerse ScienceDirect
Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms
On the 41 Rþ state of the KCs molecule J. Szczepkowski a,⇑, A. Grochola b, W. Jastrzebski a, P. Kowalczyk b a b
Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Institute of Experimental Physics, Department of Physics, University of Warsaw, ul. Hoz_ a 69, 00-681 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2012 In revised form 4 June 2012 Available online 15 June 2012 Keywords: Laser spectroscopy Alkali dimers Electronic states Potential energy curves
a b s t r a c t The polarization labeling spectroscopy technique is used to study high vibrational levels of the 41 Rþ state in KCs molecule up to vibrational level v = 91, located 27 cm1 below the atomic asymptote. Basing on energies of rovibrational levels measured in this work supplemented by the data from earlier experiment by Busevica et al. [J. Chem. Phys. 134 (2011) 104307] and employing the pointwise Inverted Perturbation Approach algorithm, we construct the potential energy curve of the 41 Rþ state in the range 3.61–15.1 Å and extend it to the dissociation limit using a theoretical long-range tail. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Advances in production of ultracold diatomic alkali metal molecules from cold atoms by photoassociation [1] and magnetoassociation (Feshbach resonances) [2] have provided a new challenge to molecular spectroscopy. Spectroscopic measurements are now often oriented to determination of potential energy curves for electronic states under investigation not only around the equilibrium distance, but over the whole range of available energies, with emphasis on the region close to dissociation limit, decisive for cold physics experiments. This concerns the ground and low excited molecular states, directly involved in formation of ultracold molecules, as well as higher excited states which may be instrumental in various schemes of observation and further manipulation of molecules by laser light. The interest is centered on excited electronic states with broad potential wells, such as double minimum and ‘shelf’ states. Such states can be efficiently excited to their bound levels from vibrational levels located near molecular dissociation limits (i.e. levels populated in photoassociation or magnetoassociation) and simultaneously display high probability of transitions to low levels in the ground electronic state, thus enabling formation of stable ultracold molecules or even their transfer to the absolute rovibrational ground state. The 41 Rþ state in KCs is an example of an excited ‘shelf’ state which may be attractive to the cold physics community for several reasons. First, its potential curve is sufficiently broad as a result of crossing of ion-pair and valence diabatic states of 1 Rþ symmetry, which becomes an avoided crossing in adiabatic picture. Since ⇑ Corresponding author. Fax: +48 228430926. E-mail address: [email protected] (J. Szczepkowski). 0022-2852/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jms.2012.06.004
the 41 Rþ state is accessible in transitions from both the ground X1 Rþ state and the B1 P state usually employed for photoassociation, it can serve purposes explained in the previous paragraph. Second, the 41 Rþ state is surprisingly free of local rotational perturbations, as found in former experiments and confirmed in our work. Consequently, it can be described by a single potential energy curve model and positions of rovibrational levels can be retrieved with high accuracy even if not observed before. Finally, the KCs molecule itself is an interesting object to study in cold experiments due to its permanent electric dipole moment, promising possibility of manipulation by external fields. In a recent experiment Busevica et al. [3] have observed the 41 Rþ state at high resolution conditions by recording laser induced fluorescence from this state down to the ground electronic states of both singlet and triplet symmetry with a Fourier transform spectrometer. Although such experiments are primarily suitable for investigation of lower states in fluorescence transitions, collisional energy transfer in the excited state permitted them to observe more than 1600 rovibronic levels in the 41 Rþ state. They were restricted to vibrational quantum numbers v 0 ¼ 2—74, their last level being bound by 226 cm1 and having a classical outer turning point at Rout 10:4 Å. In our present study we have chosen an alternative way to characterize the 41 Rþ state, namely observation of its excitation spectrum from the ground state X1 Rþ by a polarization labeling spectroscopy technique. The measurements, oriented to supplement the existing data with energies of levels closer to the atomic asymptote, provided about 570 new rovibrational term values and reached the level v 0 ¼ 91, located only 27 cm1 below the asymptote and with Rout 12:6 Å. In the following sections we present details of the measurements and construction of the potential energy curve of the 41 Rþ state which
20
J. Szczepkowski et al. / Journal of Molecular Spectroscopy 276–277 (2012) 19–21
reproduces term energies cumulated from both experiments to an accuracy of 0.02 cm1. 2. Apparatus KCs molecules were generated in a stainless steel linear heat pipe oven 46 cm long and 3 cm in outer diameter. Approximately equal amounts of potassium and cesium (about 5 g each) were inserted to the center of the heat pipe and heated to 400 °C. Argon at pressure of a few millibar was used as a buffer gas to prevent interaction of the hot alkali metal vapor with the quartz windows. The temperature gradient along the heat pipe resulted in partial separation of both metals, due to their different melting temperatures. However, potassium and cesium are miscible with each other and visual inspection of the inside of the oven revealed that close to the borders of the heated zone liquid metal droplets were formed, apparently consisting of eutectic alloy of K and Cs. We believe that these droplets were the main source of KCs vapor, resulting in stable operation of the oven and strong molecular spectra. To study the 41 Rþ X1 Rþ excitation spectrum of KCs under rotational resolution, we employed the V-type optical–optical double resonance polarization labeling spectroscopy method with two independent pump and probe lasers [4]. Both of them were dye lasers pumped synchronously by an excimer laser (LightMachinery IPEX-848). The weaker, home built probe laser (operated on Rhodamine 6G in ethanol, linewidth less than 0.5 cm1) was set at fixed frequencies resonant with transitions from the low vibrational levels in the X1 Rþ state to low levels in the 41 Rþ state (i.e. the transitions known from previous measurements and listed in Ref. [3]), thus labeling selected rovibrational levels in the ground electronic state. To control the probe laser frequency during the measurements we used a HighFinesse WS-7 wavemeter. The strong pump laser (Lumonics HD500 with Coumarine 540 in methanol as laser dye) was scanned in the spectral range 17550–18300 cm1, across the part of the 41 Rþ X1 Rþ band system corresponding to transitions to high vibrational levels in the 41 Rþ state. A fraction of the pump laser beam was directed to the same wavemeter. The absolute frequency calibration during the scan was achieved by periodic reading of its measurements, while linearity of the scan was controlled by recording transmission fringes of a Fabry–Pérot etalon 0.5 cm long. The estimated uncertainty of the laser wave numbers determined this way was below 0.05 cm1. Crossed polarizers were placed at both sides of the heat pipe oven in the path of the probe beam. At the frequencies, at which transitions induced by the pump beam shared the same lower level with any of the probe transitions, the probe light passed through the analyzer. It was subsequently detected by a photomultiplier placed behind a 0.3 m monochromator centered at the probe laser wavelength. As the pump laser was scanned, the photoelectric current representing the polarization spectrum of KCs, as well as calibration signals, were averaged with boxcar integrators, recorded in a digital form with a personal computer and stored for further analysis. The same computer was used simultaneously to trigger the excimer laser and to control the tuning of the pump dye laser. 3. Results and analysis As the polarization labeling technique allows to observe molecular spectra against dark background, it is well suited for detection of weak absorption lines. Therefore we were able to record transitions to vibrational levels in the 41 Rþ state up to v0 = 91. The highest levels have been reached from the v0 = 4 level in the ground X1 Rþ state, which ensures simultaneously a decent thermal population at the oven temperature and sufficient Franck–Condon factors for transitions terminating that high (Fig. 1). Attempts to
Fig. 1. A portion of the polarization spectrum of KCs. The assigned vibrational progression corresponds to excitation of subsequent levels of the 41 Rþ state from the ground X1 Rþ state level v 00 ¼ 4; J00 ¼ 70 labeled by the probe laser set at the wave number 16864.1 cm1.
extend the range of observed levels by starting from v0 = 5 have failed, providing unacceptable signal to noise ratio. All spectral lines assigned in the present experiment correspond to the most abundant isotopologue, 39K133Cs. Assignment of quantum numbers to the observed 41 Rþ state levels was straightforward since the recorded spectra partially overlapped the energy range studied in the previous experiment. From the measured wave numbers m of the observed lines, the term values T0 of levels in the 41 Rþ state could be calculated as
T 0 ðv 0 ; J 0 Þ ¼ T 00 ðv 00 ; J 00 Þ þ m:
ð1Þ 00
The ground state term values T were calculated by numerically solving the radial Schrödinger equation with experimental potential of the X1 Rþ state given by Ferber et al. [5], resulting from high resolution Fourier transform spectroscopy measurements. As a reference zero of energy for the term values we have chosen the minimum of the ground state potential well, determined in Ref. [5] with an uncertainty of 0.04 cm1. As it is about 10 times lower than our experimental uncertainty, we assume that no additional errors were introduced into our analysis of the 41 Rþ state levels. The experimental term values of rovibrational levels resulting from the present measurements and those by Busevica et al. [3] are listed in Supplementary material. The potential energy curve of the 41 Rþ state provided by Busevica et al. [3] served us as a starting point in construction of the extended potential. We have chosen the pointwise representation of the molecular potential, in which we define it as a set of points (Ri, U(Ri)) interpolated using a natural spline algorithm [6]. At large internuclear distances the potential has been extended analytically using the theoretical dispersion coefficients, as described below. We find such representation more direct and easier to handle numerically than expansion of the U(R) dependence in a series of Chebyshev polynomials proposed in Ref. [3]. The potential curve was optimized by a fitting routine based on the Inverted Perturbation Approach (IPA) method [7] to represent the full set of 2196 rovibrational energies (2 6 v 0 6 91; 1 6 J 0 6 188) resulting from the present measurements as well as measurements of Busevica et al. [3], to an accuracy of 0.02 cm1 (see Fig. 2 for the data field used for fitting). Note that the 41 Rþ state is apparently free of local rotational perturbations and all the observed levels were included in the final fit. The pointwise potential energy curve was extended to the atomic asymptote (placed at U(1) = 18568.466 cm1, as calculated from the ground state dissociation energy of KCs [5] and
21
J. Szczepkowski et al. / Journal of Molecular Spectroscopy 276–277 (2012) 19–21 Table 1 Parameters defining the IPA potential energy curve of the 41 Rþ state of KCs.
Fig. 2. The range of levels in the 41 Rþ state of KCs used in the fit of the potential energy curve. Solid circles represent the measurements of Busevica et al. [3], open circles (red online) – the present work, crosses (blue online) – levels observed in both experiments. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the well known distance between 62 S1=2 and 52 D3=2 levels in cesium atom [8]) by the analytical potential of the form
UðRÞ ¼ Uð1Þ C 6 =R6 C 8 =R8 C 10 =R10
for R > Rmatch :
ð2Þ
The C6 and C8 coefficients were fixed at values resulting from the ab initio calculations by Bussery et al. [9], the only ones taking into account the fine structure splitting of the 52 D state in cesium. On the other hand, C10 and Rmatch were treated as free parameters used to match smoothly the analytical long-range curve with the numerical IPA potential, with Rmatch limited to values close to the modified LeRoy radius [10], for the 41 Rþ state equal to 15.1 Å. As a result the potential curve is smooth together with its first derivative over the entire presented range, R P 3:61 Å. The final potential curve is defined in the region 3.61 Å 6 R 6 15.10636 Å by 61 parameters U(Ri ) and for R > 15.10636 Å by four long range parameters of Eq. (2), Cn and U(1), all of them listed in Table 1. To calculate the potential energy for an arbitrary internuclear distance R 6 15.10636 Å, a natural cubic spline interpolation through all 61 points of the short range potential should be applied. Actually the points U(R) given in Table 1 extend beyond the matching point Rmatch . Due to properties of cubic spline interpolation all of them must be used to calculate the potential energy curve but it must be stressed that this curve is valid only for internuclear distances below Rmatch. For R > 15.10636 Å only the analytical formula (2) should be employed for calculation of the molecular potential. The numerical potential and the term values included in the fitting procedure are listed in the Supplementary material accompanying this paper. To reproduce energies of rovibrational levels a numerical mesh of at least 10000 points between 3.61 and 19.6 Å and a natural spline algorithm should be used. The results presented in this work supplement the measurements of Busevica et al. [3]. The error analysis specific to the pointwise IPA method [11] shows that the energies of all the observed levels determine reliably the potential of the 41 Rþ state constructed here between 3.9 and 13.5 Å, i.e. up to about 20 cm1 from the dissociation limit. Although the modified LeRoy radius is still not reached, the region particularly interested in cold physics experiments is approached. Once again we recall the lack of rotational perturbations in the 41 Rþ state within the accuracy of our measurements. This should allow to use the constructed potential energy curve to determine positions of rovibrational levels
R (Å)
U (cm1)
R (Å)
U (cm1)
3.61 3.71 3.81 3.91 4.06 4.17 4.28 4.39 4.50 4.61 4.72 4.84 4.95 5.06 5.17 5.28 5.39 5.50 5.61 5.72 5.84 5.95 6.06 6.28 6.50 6.62 6.73 6.84 6.95 7.06 7.28
20217.1409 19637.2976 19120.4441 18658.0234 18165.1085 17890.9898 17667.1412 17483.6859 17334.1704 17212.4852 17114.4300 17030.4836 16972.3051 16930.0256 16901.6281 16885.4368 16880.6047 16886.4523 16901.2526 16922.5210 16952.8204 16985.4012 17021.3295 17098.6064 17177.7579 17219.9553 17257.4050 17293.4645 17327.9570 17360.8725 17422.0432
7.50 7.73 7.95 8.17 8.39 8.62 8.84 9.06 9.28 9.51 9.73 9.95 10.20 10.51 10.65 10.80 11.00 11.20 11.43 11.70 12.00 12.30 12.60 13.55 15.10 15.35 16.60 17.60 18.60 19.60
17478.1398 17533.1420 17584.5263 17636.8365 17692.1881 17755.1617 17821.3721 17893.5043 17970.3876 18053.3252 18131.8881 18206.4522 18282.8795 18361.7655 18391.2270 18418.5805 18448.9568 18472.8643 18494.0847 18511.9546 18525.8451 18535.4703 18542.1708 18554.9247 18563.5743 18564.1603 18566.0812 18566.9119 18567.4187 18567.7396
Rmatch=15.10636 Å U(1)=18568.466 cm1 C 6 ¼ 3:045851 107 cm1 Å6
C 8 ¼ 9:772757 108 cm1Å8
C 10 ¼ 1:206881 1012 cm1 Å10
with an accuracy of 0.02 cm1 not only in the 39K133Cs molecule observed here, but, after appropriate mass scaling, also in other isotopologues, 40K133Cs and 41K133Cs . Acknowledgment This work was partially supported by the Polish Ministry of Science and Higher Education (Grant No. N202 203938). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jms.2012.06.004. References [1] K.M. Jones, E. Tiesinga, P.D. Lett, P.S. Julienne, Rev. Mod. Phys. 78 (2006) 483–535. [2] C. Chin, R. Grimm, P. Julienne, E. Tiesinga, Rev. Mod. Phys. 82 (2010) 1225–1286. [3] L. Busevica, I. Klincare, O. Nikolayeva, M. Tamanis, R. Ferber, V.V. Meshkov, E.A. Pazyuk, A.V. Stolyarov, J. Chem. Phys. 134 (2011). art. no. 104307. [4] A. Pashov, W. Jastrze ß bski, P. Kowalczyk, Chem. Phys. Lett. 292 (1998) 615–620. [5] R. Ferber, I. Klincare, O. Nikolayeva, M. Tamanis, H. Knöckel, E. Tiemann, A. Pashov, Phys. Rev. A80 (2009). art. no. 062501. [6] C. De Boor, A Practical Guide to Splines, Springer, Berlin, 1978. [7] A. Pashov, W. Jastrzebski, P. Kowalczyk, Comput. Phys. Commun. 128 (2000) 622–634. [8] K.B.S. Eriksson, I. Wenåker, Phys. Scr. 1 (1970) 21–24. [9] B. Bussery, Y. Achkar, M. Aubert-Frécon, Chem. Phys. 116 (1987) 319–338. [10] B. Ji, C.-C. Tsai, W.C. Stwalley, Chem. Phys. Lett. 236 (1995) 242–246. [11] A. Pashov, W. Jastrze ß bski, P. Kowalczyk, J. Chem. Phys. 113 (2000) 6624–6628.
Contents lists available at SciVerse ScienceDirect
Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms
On the 41 Rþ state of the KCs molecule J. Szczepkowski a,⇑, A. Grochola b, W. Jastrzebski a, P. Kowalczyk b a b
Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Institute of Experimental Physics, Department of Physics, University of Warsaw, ul. Hoz_ a 69, 00-681 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2012 In revised form 4 June 2012 Available online 15 June 2012 Keywords: Laser spectroscopy Alkali dimers Electronic states Potential energy curves
a b s t r a c t The polarization labeling spectroscopy technique is used to study high vibrational levels of the 41 Rþ state in KCs molecule up to vibrational level v = 91, located 27 cm1 below the atomic asymptote. Basing on energies of rovibrational levels measured in this work supplemented by the data from earlier experiment by Busevica et al. [J. Chem. Phys. 134 (2011) 104307] and employing the pointwise Inverted Perturbation Approach algorithm, we construct the potential energy curve of the 41 Rþ state in the range 3.61–15.1 Å and extend it to the dissociation limit using a theoretical long-range tail. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Advances in production of ultracold diatomic alkali metal molecules from cold atoms by photoassociation [1] and magnetoassociation (Feshbach resonances) [2] have provided a new challenge to molecular spectroscopy. Spectroscopic measurements are now often oriented to determination of potential energy curves for electronic states under investigation not only around the equilibrium distance, but over the whole range of available energies, with emphasis on the region close to dissociation limit, decisive for cold physics experiments. This concerns the ground and low excited molecular states, directly involved in formation of ultracold molecules, as well as higher excited states which may be instrumental in various schemes of observation and further manipulation of molecules by laser light. The interest is centered on excited electronic states with broad potential wells, such as double minimum and ‘shelf’ states. Such states can be efficiently excited to their bound levels from vibrational levels located near molecular dissociation limits (i.e. levels populated in photoassociation or magnetoassociation) and simultaneously display high probability of transitions to low levels in the ground electronic state, thus enabling formation of stable ultracold molecules or even their transfer to the absolute rovibrational ground state. The 41 Rþ state in KCs is an example of an excited ‘shelf’ state which may be attractive to the cold physics community for several reasons. First, its potential curve is sufficiently broad as a result of crossing of ion-pair and valence diabatic states of 1 Rþ symmetry, which becomes an avoided crossing in adiabatic picture. Since ⇑ Corresponding author. Fax: +48 228430926. E-mail address: [email protected] (J. Szczepkowski). 0022-2852/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jms.2012.06.004
the 41 Rþ state is accessible in transitions from both the ground X1 Rþ state and the B1 P state usually employed for photoassociation, it can serve purposes explained in the previous paragraph. Second, the 41 Rþ state is surprisingly free of local rotational perturbations, as found in former experiments and confirmed in our work. Consequently, it can be described by a single potential energy curve model and positions of rovibrational levels can be retrieved with high accuracy even if not observed before. Finally, the KCs molecule itself is an interesting object to study in cold experiments due to its permanent electric dipole moment, promising possibility of manipulation by external fields. In a recent experiment Busevica et al. [3] have observed the 41 Rþ state at high resolution conditions by recording laser induced fluorescence from this state down to the ground electronic states of both singlet and triplet symmetry with a Fourier transform spectrometer. Although such experiments are primarily suitable for investigation of lower states in fluorescence transitions, collisional energy transfer in the excited state permitted them to observe more than 1600 rovibronic levels in the 41 Rþ state. They were restricted to vibrational quantum numbers v 0 ¼ 2—74, their last level being bound by 226 cm1 and having a classical outer turning point at Rout 10:4 Å. In our present study we have chosen an alternative way to characterize the 41 Rþ state, namely observation of its excitation spectrum from the ground state X1 Rþ by a polarization labeling spectroscopy technique. The measurements, oriented to supplement the existing data with energies of levels closer to the atomic asymptote, provided about 570 new rovibrational term values and reached the level v 0 ¼ 91, located only 27 cm1 below the asymptote and with Rout 12:6 Å. In the following sections we present details of the measurements and construction of the potential energy curve of the 41 Rþ state which
20
J. Szczepkowski et al. / Journal of Molecular Spectroscopy 276–277 (2012) 19–21
reproduces term energies cumulated from both experiments to an accuracy of 0.02 cm1. 2. Apparatus KCs molecules were generated in a stainless steel linear heat pipe oven 46 cm long and 3 cm in outer diameter. Approximately equal amounts of potassium and cesium (about 5 g each) were inserted to the center of the heat pipe and heated to 400 °C. Argon at pressure of a few millibar was used as a buffer gas to prevent interaction of the hot alkali metal vapor with the quartz windows. The temperature gradient along the heat pipe resulted in partial separation of both metals, due to their different melting temperatures. However, potassium and cesium are miscible with each other and visual inspection of the inside of the oven revealed that close to the borders of the heated zone liquid metal droplets were formed, apparently consisting of eutectic alloy of K and Cs. We believe that these droplets were the main source of KCs vapor, resulting in stable operation of the oven and strong molecular spectra. To study the 41 Rþ X1 Rþ excitation spectrum of KCs under rotational resolution, we employed the V-type optical–optical double resonance polarization labeling spectroscopy method with two independent pump and probe lasers [4]. Both of them were dye lasers pumped synchronously by an excimer laser (LightMachinery IPEX-848). The weaker, home built probe laser (operated on Rhodamine 6G in ethanol, linewidth less than 0.5 cm1) was set at fixed frequencies resonant with transitions from the low vibrational levels in the X1 Rþ state to low levels in the 41 Rþ state (i.e. the transitions known from previous measurements and listed in Ref. [3]), thus labeling selected rovibrational levels in the ground electronic state. To control the probe laser frequency during the measurements we used a HighFinesse WS-7 wavemeter. The strong pump laser (Lumonics HD500 with Coumarine 540 in methanol as laser dye) was scanned in the spectral range 17550–18300 cm1, across the part of the 41 Rþ X1 Rþ band system corresponding to transitions to high vibrational levels in the 41 Rþ state. A fraction of the pump laser beam was directed to the same wavemeter. The absolute frequency calibration during the scan was achieved by periodic reading of its measurements, while linearity of the scan was controlled by recording transmission fringes of a Fabry–Pérot etalon 0.5 cm long. The estimated uncertainty of the laser wave numbers determined this way was below 0.05 cm1. Crossed polarizers were placed at both sides of the heat pipe oven in the path of the probe beam. At the frequencies, at which transitions induced by the pump beam shared the same lower level with any of the probe transitions, the probe light passed through the analyzer. It was subsequently detected by a photomultiplier placed behind a 0.3 m monochromator centered at the probe laser wavelength. As the pump laser was scanned, the photoelectric current representing the polarization spectrum of KCs, as well as calibration signals, were averaged with boxcar integrators, recorded in a digital form with a personal computer and stored for further analysis. The same computer was used simultaneously to trigger the excimer laser and to control the tuning of the pump dye laser. 3. Results and analysis As the polarization labeling technique allows to observe molecular spectra against dark background, it is well suited for detection of weak absorption lines. Therefore we were able to record transitions to vibrational levels in the 41 Rþ state up to v0 = 91. The highest levels have been reached from the v0 = 4 level in the ground X1 Rþ state, which ensures simultaneously a decent thermal population at the oven temperature and sufficient Franck–Condon factors for transitions terminating that high (Fig. 1). Attempts to
Fig. 1. A portion of the polarization spectrum of KCs. The assigned vibrational progression corresponds to excitation of subsequent levels of the 41 Rþ state from the ground X1 Rþ state level v 00 ¼ 4; J00 ¼ 70 labeled by the probe laser set at the wave number 16864.1 cm1.
extend the range of observed levels by starting from v0 = 5 have failed, providing unacceptable signal to noise ratio. All spectral lines assigned in the present experiment correspond to the most abundant isotopologue, 39K133Cs. Assignment of quantum numbers to the observed 41 Rþ state levels was straightforward since the recorded spectra partially overlapped the energy range studied in the previous experiment. From the measured wave numbers m of the observed lines, the term values T0 of levels in the 41 Rþ state could be calculated as
T 0 ðv 0 ; J 0 Þ ¼ T 00 ðv 00 ; J 00 Þ þ m:
ð1Þ 00
The ground state term values T were calculated by numerically solving the radial Schrödinger equation with experimental potential of the X1 Rþ state given by Ferber et al. [5], resulting from high resolution Fourier transform spectroscopy measurements. As a reference zero of energy for the term values we have chosen the minimum of the ground state potential well, determined in Ref. [5] with an uncertainty of 0.04 cm1. As it is about 10 times lower than our experimental uncertainty, we assume that no additional errors were introduced into our analysis of the 41 Rþ state levels. The experimental term values of rovibrational levels resulting from the present measurements and those by Busevica et al. [3] are listed in Supplementary material. The potential energy curve of the 41 Rþ state provided by Busevica et al. [3] served us as a starting point in construction of the extended potential. We have chosen the pointwise representation of the molecular potential, in which we define it as a set of points (Ri, U(Ri)) interpolated using a natural spline algorithm [6]. At large internuclear distances the potential has been extended analytically using the theoretical dispersion coefficients, as described below. We find such representation more direct and easier to handle numerically than expansion of the U(R) dependence in a series of Chebyshev polynomials proposed in Ref. [3]. The potential curve was optimized by a fitting routine based on the Inverted Perturbation Approach (IPA) method [7] to represent the full set of 2196 rovibrational energies (2 6 v 0 6 91; 1 6 J 0 6 188) resulting from the present measurements as well as measurements of Busevica et al. [3], to an accuracy of 0.02 cm1 (see Fig. 2 for the data field used for fitting). Note that the 41 Rþ state is apparently free of local rotational perturbations and all the observed levels were included in the final fit. The pointwise potential energy curve was extended to the atomic asymptote (placed at U(1) = 18568.466 cm1, as calculated from the ground state dissociation energy of KCs [5] and
21
J. Szczepkowski et al. / Journal of Molecular Spectroscopy 276–277 (2012) 19–21 Table 1 Parameters defining the IPA potential energy curve of the 41 Rþ state of KCs.
Fig. 2. The range of levels in the 41 Rþ state of KCs used in the fit of the potential energy curve. Solid circles represent the measurements of Busevica et al. [3], open circles (red online) – the present work, crosses (blue online) – levels observed in both experiments. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the well known distance between 62 S1=2 and 52 D3=2 levels in cesium atom [8]) by the analytical potential of the form
UðRÞ ¼ Uð1Þ C 6 =R6 C 8 =R8 C 10 =R10
for R > Rmatch :
ð2Þ
The C6 and C8 coefficients were fixed at values resulting from the ab initio calculations by Bussery et al. [9], the only ones taking into account the fine structure splitting of the 52 D state in cesium. On the other hand, C10 and Rmatch were treated as free parameters used to match smoothly the analytical long-range curve with the numerical IPA potential, with Rmatch limited to values close to the modified LeRoy radius [10], for the 41 Rþ state equal to 15.1 Å. As a result the potential curve is smooth together with its first derivative over the entire presented range, R P 3:61 Å. The final potential curve is defined in the region 3.61 Å 6 R 6 15.10636 Å by 61 parameters U(Ri ) and for R > 15.10636 Å by four long range parameters of Eq. (2), Cn and U(1), all of them listed in Table 1. To calculate the potential energy for an arbitrary internuclear distance R 6 15.10636 Å, a natural cubic spline interpolation through all 61 points of the short range potential should be applied. Actually the points U(R) given in Table 1 extend beyond the matching point Rmatch . Due to properties of cubic spline interpolation all of them must be used to calculate the potential energy curve but it must be stressed that this curve is valid only for internuclear distances below Rmatch. For R > 15.10636 Å only the analytical formula (2) should be employed for calculation of the molecular potential. The numerical potential and the term values included in the fitting procedure are listed in the Supplementary material accompanying this paper. To reproduce energies of rovibrational levels a numerical mesh of at least 10000 points between 3.61 and 19.6 Å and a natural spline algorithm should be used. The results presented in this work supplement the measurements of Busevica et al. [3]. The error analysis specific to the pointwise IPA method [11] shows that the energies of all the observed levels determine reliably the potential of the 41 Rþ state constructed here between 3.9 and 13.5 Å, i.e. up to about 20 cm1 from the dissociation limit. Although the modified LeRoy radius is still not reached, the region particularly interested in cold physics experiments is approached. Once again we recall the lack of rotational perturbations in the 41 Rþ state within the accuracy of our measurements. This should allow to use the constructed potential energy curve to determine positions of rovibrational levels
R (Å)
U (cm1)
R (Å)
U (cm1)
3.61 3.71 3.81 3.91 4.06 4.17 4.28 4.39 4.50 4.61 4.72 4.84 4.95 5.06 5.17 5.28 5.39 5.50 5.61 5.72 5.84 5.95 6.06 6.28 6.50 6.62 6.73 6.84 6.95 7.06 7.28
20217.1409 19637.2976 19120.4441 18658.0234 18165.1085 17890.9898 17667.1412 17483.6859 17334.1704 17212.4852 17114.4300 17030.4836 16972.3051 16930.0256 16901.6281 16885.4368 16880.6047 16886.4523 16901.2526 16922.5210 16952.8204 16985.4012 17021.3295 17098.6064 17177.7579 17219.9553 17257.4050 17293.4645 17327.9570 17360.8725 17422.0432
7.50 7.73 7.95 8.17 8.39 8.62 8.84 9.06 9.28 9.51 9.73 9.95 10.20 10.51 10.65 10.80 11.00 11.20 11.43 11.70 12.00 12.30 12.60 13.55 15.10 15.35 16.60 17.60 18.60 19.60
17478.1398 17533.1420 17584.5263 17636.8365 17692.1881 17755.1617 17821.3721 17893.5043 17970.3876 18053.3252 18131.8881 18206.4522 18282.8795 18361.7655 18391.2270 18418.5805 18448.9568 18472.8643 18494.0847 18511.9546 18525.8451 18535.4703 18542.1708 18554.9247 18563.5743 18564.1603 18566.0812 18566.9119 18567.4187 18567.7396
Rmatch=15.10636 Å U(1)=18568.466 cm1 C 6 ¼ 3:045851 107 cm1 Å6
C 8 ¼ 9:772757 108 cm1Å8
C 10 ¼ 1:206881 1012 cm1 Å10
with an accuracy of 0.02 cm1 not only in the 39K133Cs molecule observed here, but, after appropriate mass scaling, also in other isotopologues, 40K133Cs and 41K133Cs . Acknowledgment This work was partially supported by the Polish Ministry of Science and Higher Education (Grant No. N202 203938). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jms.2012.06.004. References [1] K.M. Jones, E. Tiesinga, P.D. Lett, P.S. Julienne, Rev. Mod. Phys. 78 (2006) 483–535. [2] C. Chin, R. Grimm, P. Julienne, E. Tiesinga, Rev. Mod. Phys. 82 (2010) 1225–1286. [3] L. Busevica, I. Klincare, O. Nikolayeva, M. Tamanis, R. Ferber, V.V. Meshkov, E.A. Pazyuk, A.V. Stolyarov, J. Chem. Phys. 134 (2011). art. no. 104307. [4] A. Pashov, W. Jastrze ß bski, P. Kowalczyk, Chem. Phys. Lett. 292 (1998) 615–620. [5] R. Ferber, I. Klincare, O. Nikolayeva, M. Tamanis, H. Knöckel, E. Tiemann, A. Pashov, Phys. Rev. A80 (2009). art. no. 062501. [6] C. De Boor, A Practical Guide to Splines, Springer, Berlin, 1978. [7] A. Pashov, W. Jastrzebski, P. Kowalczyk, Comput. Phys. Commun. 128 (2000) 622–634. [8] K.B.S. Eriksson, I. Wenåker, Phys. Scr. 1 (1970) 21–24. [9] B. Bussery, Y. Achkar, M. Aubert-Frécon, Chem. Phys. 116 (1987) 319–338. [10] B. Ji, C.-C. Tsai, W.C. Stwalley, Chem. Phys. Lett. 236 (1995) 242–246. [11] A. Pashov, W. Jastrze ß bski, P. Kowalczyk, J. Chem. Phys. 113 (2000) 6624–6628.