Potential curve of NaK a 3Σ+ state near dissociation limitKiyoshi Ishikawa, Norihito Mukai, and Mari Tanimura Citation: The Journal of Chemical Physics 101, 876 (1994); doi: 10.1063/1.467740 View online: http://dx.doi.org/10.1063/1.467740 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/101/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hyperfine structure of the NaK c 3Σ+ state and the effects of perturbation J. Chem. Phys. 96, 6423 (1992); 10.1063/1.462856 High resolution laser spectroscopy up to the dissociation limit of the NaK B 1Π state, and predissociationnear the dissociation limit J. Chem. Phys. 94, 7713 (1991); 10.1063/1.460157 Bound–free 13Π→13Σ+ emission from the NaK molecule: Determination of the 13Σ+ repulsive wall abovethe dissociation limit J. Chem. Phys. 92, 5801 (1990); 10.1063/1.458400 The a 3Σ+ state of NaK. High resolution spectroscopy using laserinduced fluorescence (LIF) J. Chem. Phys. 71, 1994 (1979); 10.1063/1.438590 A mass spectrometric study of heteronuclear diatomic alkali metal molecules. Dissociation energies andionization potentials of NaLi, KLi, and NaK. J. Chem. Phys. 67, 4603 (1977); 10.1063/1.434621

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Potential curve of NaK a 3I + state near dissociation limit Kiyoshi Ishikawa, Norihito Mukai, and Mari Tanimura Department of Chemistry, Faculty of Science, Kobe University, Nada-ku, Kobe 657, Japan

(Received 22 February 1994; accepted 28 March 1994)

Vibrational levels near the dissociation limit of NaK a 3}; + state are observed with an optical­optical double resonance technique. High resolution spectra are sensitively detected with the combination of perturbation facilitated polarization spectroscopy with frequency modulation spectroscopy. An electronically excited state, the B I II state, is mixed with the c 3}; + state through spin -orbit interaction. The transition from the X I I + state to the a 3I + state through the B I II state is facilitated by the perturbation by the c 3I + state. Hyperfine structures of the a 3I + (v = 3 -16,N=4 - 25) state are resolved with this spectroscopic technique and found to be independent of vibrational and rotational quantum number. The a 3I + state can only be perturbed by the X I I + state through the hyperfine interaction. The vibrational levels (v ~ 16) of the a 3I + state are demonstrably not perturbed by the X I I + state. The potential curve of the a 3I + state is determined by the near-dissociation expansion fitting of molecular constants and the inverse perturbation analysis method. The coefficients C6 , Cs , and C 10 of the potential function are determined to be (12.75±0.15)XI06 cm- I A6, (2.22±0.19)XI0s cm- I AS, and (1.100±0.061)X 10 10 cm- I AIO, respectively. The dissociation energy, De' is obtained to be 207.858 ± 0.019 cm- I , which is 2.8 cm- I above the highest observed vibrational level (v= 16).

I. INTRODUCTION

The lowest triplet ( 3I +) state of diatomic alkali-metal molecules, which dissociates to two ground-state atoms, has been studied by spectroscopic techniques. I

-4 Because the po­

tential is very shallow, the triplet state is not initially popu­lated and has been studied in many works by observing fluo­rescence from electronically excited states. The potential curve of NaK a 3I + (v ~ 11) state was precisely determined by detecting laser induced fluorescence with Fourier­transform spectroscopy. I The hyperfine structure of NaK a 3};+(v=4,5,9,12) state was observed with high resolu­tion spectroscopy,2 and hyperfine constants ANa<a 3I +) and AK(a 3I +) were obtained. The study of the hyperfine inter­action and the potential curve at large internuclear distance contributes to an understanding of chemical reactions of cold or energy selected atoms.

In our previous work a laser diode was used as probe laser.2 This diode laser was very stable in light power but lacked the ability of scanning the frequency over a wide range. Therefore, a limited number of levels were observed and their term values could not be determined. In the present work a single mode Ti:Sapphire laser is used, which has higher power and is easy to tune the frequency. Hyperfine structure can be observed for highly excited vibrational lev­els and the transition frequencies are accurately calibrated by the absorption frequencies of the iodine molecule.

In this paper we present the high resolution spectra of NaK a 3I+(v=3-16,N) state observed by the combina­tion of perturbation facilitated polarization (PFP) spectros­copy with frequency modulation (FM) spectroscopy. An electronically excited state, the B I II state, is mixed with the c 3I + state through spin-orbit interaction. In the whole of this article the mixed level is called B I I II, of which the dominant character is B I II, and the mixed level is called c I 3I +, of which the dominant character is c 3I +. The

transition from the X I};+(V",J") state to the a 3I+(v,N) state through the B I II (v I ,J ') state is facilitated by the per­turbation of the c 3I + (v'" ,N'" ,J"') state. The B I - X transi­tion is due to the electric dipole moment between the pure singlet states, and the B I - a transition is due to the electric dipole moment between the pure triplet states. Because the potential of the c 3I + state is deep and the potential of the a 3I + state is very shallow, Franck-Condon factors (FCF) between the B I I II (v ') and a 3I + (v> 10) states are small. It is needful to detect sensitively the transitions to the a 3I + state near the dissociation limit.

The a 3}; + state can only be perturbed by the X I I +

state through the hyperfine interaction among sodium nuclear spin, potassium nuclear spin, and electron spin whose strength is very weak (perturbation matrix element on the order of 0.01 cm- I ). It is found that the hyperfine structure of the a 3I + state is independent of vibrational and rota­tional quantum numbers. We shall obtain an accurate poten­tial curve for the a 3I + state, which is demonstrated not to be perturbed by the X I I + state, by the following proce­dures. First, the molecular constants, G u and B u , are fitted to observed term values for each vibrational level. Second, G lJ

and B lJ are expanded according to the near-dissociation ex­pansion (NDE) formula.4- 7 Third, a Rydberg-Klein-Rees (RKR) potential is calculated from the NDE formula. Finally, the potential form is directly refined by the method of in­verted perturbation analysis (IPA).s,9 The long-range poten­tial function will be obtained and extrapolated from the IPA potential curve.

II. EXPERIMENT

The experimental setup is shown in Fig. L The main differences from our previous work2 are that the phase of an intense and tunable probe beam is modulated by an electro­optic modulator (EOM) and the axis of a polarization prism

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Ishikawa, Mukai, and Tanimura: Potential curve of NaK 877

BS

j

L --------------1

FIG. I. Experimental setup of PFP FM spectroscopy. The pump and probe beams are collinearly introduced into the heat pipe cell. The probe beam of an one-color polarization spectroscopy counter-propagates against the pump beam. PM is photomultiplier; BS. beam-splitter; EOM, electro-optic modu­lator; DBM, double-balanced mixer; CH, mechanical chopper; P n' polariza­tion prism; PS, phase shifter; SG, signal generator; PC, personal computer; and CF. color filter.

(P 3) is slightly inclined from a right angle with the polariza­tion plane of probe beam. The phase modulation method is called frequency modulation (FM) spectroscopy.1O The com­bination of polarization spectroscopy II with FM spectros­copy is sensitive, background-free, and convenient for ob­serving P or R transitions as described below.

In a polarization spectroscopy experiment, a circularly polarized pump beam and a linearly polarized probe beam are used for P or R transitions. A Lorentzian line shape of transition is obtained by selecting an exactly right angle (8= 1T12) between the polarization plane of the probe beam and the axis of the polarization prism (P 3 ). The signal to noise ratio is best when () takes the value 1T/2 + 80 , where eo is small and depends on experimental conditions. The spectral line shape is dispersive at 1T/2 + 80 , II

In the FM spectroscopy experiment, a heterodyne beat between the carrier and side band frequencies is detected with a double-balanced mixer (DBM). The beat frequency is so high that background noise is reduced efficiently. The spectral line shape is the derivative of the line shape without phase modulation. Because both the carrier and side band irradiate the photodetector through the polarization prism in­clined by 80 , a heterodyne beat can be detected effectively. By the combination of polarization spectroscopy with FM spectroscopy, the shape of a spectral line is like the second derivative of a Lorentzian shape for P or R transitions.

Pump and probe beams were collinearly introduced into

the heatpipe cell. A PFP FM spectrum was observed by fix­ing the pump frequency and scanning the probe frequency. According to Eq. (4) in Ref. 2 which accounts for Doppler shifts, the resonance frequency of the probe beam is repre­sented as follows:

(1)

where 0B' and oa are, respectively, small splittings of the B' I IT and a 3k + states due to the hyperfine interaction, and higher orders of o's are neglected. Ex, EB" and Ea are term values of the X I k +, B' I IT, and a 3k + states, respectively. Because the coefficient of the third term, (Ea-Ex)/(EB,-Ex), is smaller than 1, the splitting of an excited rotational level, 0B' , is observed to be small in spec­tra. The pump laser (around 17386 or 17450 cm- I ) was a single mode dye laser (Coherent 699-29), and the probe laser (12 200 - 12 350 cm - I) was a single mode Ti:Sapphire la­ser (Coherent 899-29). The amplitude of the pump beam was modulated by a mechanical chopper (CH) at about 3.0 kHz. The phase of the probe beam was modulated by an EOM, and modulation index was about 0.5 at a modulation fre­quency of 12.9 MHz.

The frequency of the pump beam was tuned to the tran­sition between the B' I IT and X I k + states. For the purpose of accurate determinations of transition frequencies, the pump frequency had to be fixed at center of the Doppler line. The signal intensity of an one-color polarization spectros­copy was monitored by using the pump and probe beams which came from the same dye laser and counter-propagated each other in the heat pipe cell. Light frequency was cali­brated by markers from a 150 MHz free spectral range etalon (Burleigh eFT-SaO). The temperature of the etalon is con­trolled and the etalon cavity is sealed to insure against fre­quency shift. There are hot bands of the iodine molecule in the infrared region. The absorption spectrum of the iodine molecule was used as a frequency standard for the Ti:Sap­phire laser. The iodine cell was heated to 600 K and its optical path length was 8 m.

III. EXPERIMENTAL RESULTS

An example of a PFP FM spectrum is shown in Fig. 2. The pump frequency was resonant with the transition between the B,IIT(v'=7,j'=23e) and X 1!+(v"=O,J"=22) levels. Because the B I IT ( v ' = 7 ,J' = 23 e) level mixes with the c 3k +(v lll =21,NIII =23F2) level through spin-orbit interaction, 12 the tranSitIOns between the B' 'IT(v'=7,J'=23e) and a 3k +(v,N=22,24) levels are allowed. The probe frequency was scanned around the transition frequency between the B' 'IT(v'=7,j'=23e) and a 3k + (v = 4,N = 24) levels. The shape of the PFP FM spectrum is similar to the second derivative of a Lorentzian as described above. It was found that the hyperfine splitting of the F 2(NIII = Jill) levels of the c' 3! + state was smaller than that of the F,(NIII=jlll-l) and F3(NIII = jill + 1) levels of the c' 3k + stateY·'4 In addition, the splitting of the ex­cited state was compressed in the observed spectrum, as

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878 Ishikawa, Mukai, and Tanimura: Potential curve of NaK

BfiII(v' = 7, JI = 23 e) - a3I;+(v = 4, N = 24)

12319.8 12319.9 em-I

FIG. 2. Example of PFP FM spectrum. The pump frequency is resonant with a transltJon between the X Il +(v"=0,J"=22) and B' lII(v'=7,J'=23e) levels. The B lII(v'=7,J'=23e) level is mixed with c 'l+(v"'=2I,N"'=23F2 ) level through spin-orbit interaction. The abscissa is the probe wave number, which is scanned around the transition between the B' 'II(v'=7,J'=23e) and a 3l+(v=4,N=24) levels. Hy­perfine structure is observed to be identical with that of other levels of the a 3l + state.

shown in Eq. (1). The structure of spectral line is due to the hyperfine interaction in the a 3I + state.2 The hyperfine structure of other levels of the a 3I + state was observed to be identical with that shown in Fig. 2.

Table I shows the map of the observed a 3I + (v ,N) lev­els. Vibrational levels (v = 0 - 2) cannot be detected because of small FCF between B lin (v I = 7,8) and a 3I + (v = 0 - 2) levels. Pump transitions between the B I I n and X I I + states are in the (8-0) band for low-J levels (J~ 11) and in the (7-0) band for high-J levels (J = 22 - 25). Term values of observed levels are determined by using the molecular constants for the X 1 I + state in Ref. 1. The accuracy of the observed term values depends on the

TABLE 1. Map of the observed a 3l +(v,J) levels. All of the observed 71 levels are shown in this map. Vibrational levels (v =0- 2) are not observed because of small FCF between the vibrational levels of the a 3l + and B' 'II states.

v Observed rotational level (1)

3 22 23 24 25 4 5 7 9 11 22 23 24 25 5 4 5 6 7 8 9 10 11 22 23 24 25 6 5 9 22 24 7 4 5 6 7 9 11 22 23 24 25 8 22 24 9 4 6 7 9 11

10 22 23 24 25 11 22 23 24 25 12 4 5 6 7 22 24 13 4 5 6 7 14 4 5 6 7 15 5 7 16 5 7

molecular constants of the X 1 I + (v /I = 0) state and the cali­bration method for the pump and probe frequencies in this experiment.

IV. DISCUSSION

Hyperfine structure was observed for 71 spectra, which sample 14 vibrational quantum numbers and 12 rotational quantum numbers (see Table I). Hyperfine structure is due to the spin coupling among the sodium nuclear spin INa' the potassium nuclear spin IK , and the electron spin S. Because the a 3I + state is composed of the 3s orbital of the sodium atom and the 4s orbital of the potassium atom, the spin-spin and spin-rotation interactions are very small.2,lS These mechanisms are the origins of N-dependent spin splitting for a 3I + [case(b)] state. The hyperfine structure was actually found to be independent of rotational angular momentum N.

However, the hyperfine structure is expected to change as v increases because the a 3I + state mixes with the X 1 I + state through hyperfine interaction at large internu­clear distance. The mixing arising from the hyperfine inter­action was observed in the case of the CS2 X II; state. IS

But the hyperfine structure with the NaK a 3I + state was observed to be invariant for vibrational levels (v = 3 - 16). Hyperfine constants, ANaCa 3I +) and AK(a 3I +), were de­termined with the accuracy of ± 1 % and the same as ones reported previously.2

It is concluded from the above discussion that the a 3I + state is not affected by the X 1 I + state and the rep­resentation as a pure 3I + state is good for the observed levels (v~ 16). Molecular constants, G v and B v , are deter­mined for each vibrational level by fitting to the formula

(2)

D v is fixed to zero in the first iteration. The Dunham expan­sion of G v and B v is not adequate in this case which includes near dissociation levels. We adopt the NDE fitting method.4

-7 The number of fit parameters used is 4 for G v

and 3 for Bv , the same as in Ref. 4:

G(v)=XO(VD-V) 3+Ko(VD-V) 4+De , (3)

(4)

where v D, the vibrational quantum number at the dissocia­tion limit, is the parameter which couples Eqs. (3) and (4). Term values of unobserved levels (v = 0 - 2 ,N = 0) are cal­culated from the molecular constants in Ref. 1, and are used for NDE fitting. If these constraints are not used, the poten­tial form around the minimum deviates from the true one.

There remains systematic errors larger than the estimated errors in observation (~ 0.005 cm - 1) in this simple fit pro­cedure, and the standard deviation is about 0.22 cm - I. If more fit parameters are used, the fitting error will be smaller.6

,7 However, we choose to obtain directly the poten­tial curve, not molecular constants. The RKR potential is approximately obtained by initially using the six parameters of the above NDE fit. The eigenvalues G v and vibrational wave function if!v for the nonrotating molecule can be calcu­lated from the RKR potential by the Cooley method. 16 Mo­lecular constants Bv and Dv are calculated from G v and

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Ishikawa, Mukai, and Tanirnura: Potential curve of NaK 879

TABLE II. IPA potential of NaK a 3~ + state. The region R < 4.25 A is extrapolated by fitting the C 121 R 12

fonn to the IPA potential at R =4.25 - 5.0 A. The region R> 14.0 A is extrapolated by fitting Eq. (5) to the IPA potential at R = 7.6 - 14.0 A. The energy at the potential minimum is 5065.8579 em - 1 above the potential minimum of the X 1 ~ + state.

R(A) E(em- 1) R(A)

4.10000 785.143 5.058306 4.13845 696.281 5.097287 4.16804 634.986 5.137561 4.19885 576.801 5.179195 4.23095 521.831 5.222257 4.26443 469.659 5.266822 4.29938 420.854 5.312971 4.33589 375.119 5.360790 4.37407 332.376 5.410370 4.41405 292.574 5.461812 4.45594 255.624 5.515221 4.49990 221.422 5.570714 4.522702 205.3197 5.628413 4.546078 189.8658 5.688455 4.570050 175.0515 5.750983 4.594643 160.8594 5.816156 4.619881 147.2890 5.884146 4.645789 134.3482 5.955138 4.672395 121.7884 6.029337 4.699727 109.9523 6.106965 4.727815 99.0185 6.188264 4.756692 88.7822 6.273502 4.786390 79.2431 6.362972 4.816945 70.3419 6.456998 4.848395 62.0491 6.555935 4.880781 54.3278 6.660180 4.914143 47.1467 6.770172 4.948527 40.4828 6.886398 4.983982 34.3143 7.009405 5.020556 28.6251 7.139804

!/IV .17 We refit Gu , Bu, and Du of Eq. (2) for vibrational levels (v =4,5,7), and refit Gu and Bu values with the calcu­lated Du value for the vibrational levels (v =3,6,8-16) where few rotational levels are observed (see Table I). NDE fitting is done for G v and B u again, and an improved RKR potential is obtained for NDE-fitted G(v) and B(v) values. The ei­genvalues of rotational-vibrational levels calculated from the RKR potential systematically deviate from observed term values, because the number of parameters are too few to fit adequately in the NDE method.

The form of the RKR potential is changed as errors are minimized by the IPA method.8

•9 Small changes of the po­

tential are expanded as Legendre polynomials P m(x) (m = 0 - 16) weighted with exp( - ~n) (n = 2 - 4), where x is the R -dependent variable defined in Ref. 8 and R is the in­ternuclear distance. The number of Legendre polynomials is changed every few iterations in order to minimize systematic error. After many iterations we get the potential curve which is listed in Table II and shown in Fig. 3. The standard devia­tion is 0.022 cm- I for 74 data including three levels (v = 0 - 2, N = 0), and is much smaller than that calculated from the RKR potential. There are many points used in the IPA potential curve in addition to the classical turning points of each vibrational level. We can obtain the form of the long range potential of the a 3! + state, which is represented as follows: 1.18

E(em- 1) R(A) E(em- 1)

23.4174 7.278281 115.3682 18.7003 7.425612 123.6540 14.4831 7.582674 131.7908 10.7700 7.750463 139.7167 7.5694 7.930117 147.3683 4.8998 8.122941 154.6824 2.7861 8.330439 161.5981 1.2528 8.554351 168.0592 0.3206 8.796706 174.0173 0.0000 9.059879 179.4366 0.2963 9.346675 184.2949 1.2202 9.660422 188.5891 2.7829 10.005107 192.3302 4.9849 10.385544 195.5491 7.8173 10.807600 198.2794

11.2663 11.278498 200.5536 15.3150 11.807238 202.4086 19.9451 12.405176 203.8835 25.1343 13.086845 205.0049 30.8509 13.871159 205.7972 37.0515 14.78319 206.517 43.6935 15.85689 206.989 50.7376 17.13945 207.320 58.1343 18.69835 207.542 65.8328 20.63369 207.685 73.7836 23.10057 207.771 81.9297 26.35280 207.819 90.2178 30.83643 207.843 98.5916 37.41415 207.853

106.9943 42.00736 207.855

v=16

,­, ' . , '

}1~~~~:1r~ f\! \ .... -----------------------------------,~/"III, II I ,

• ~ • V I t

" cm-1

200

100

o~~--L-__ ~ __ _L __ _J ____ L_ __ ~ __ ~

10 20 30 A 40

FIG. 3. Potential curve and wave functions of NaK a 3~ + state. Wave functions for the v = 1 9 and v = 16 levels are represented by solid and dashed lines. respectively. The vibrational wave function (v = 19) is ob­tained from the extrapolated potential curve, which is calculated from Eq. (5), and spreads over a wide range of R. On the other hand, the vibrational wave function (v= 16) is localized at R < 15.0 A.

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880 Ishikawa, Mukai, and Tanimura: Potential curve of NaK

TABLE m. Molecular constants and classical turning points of NaK a 3~ + state. These values are calculated from the IPA potential. The values for the levels (v = 17 -19) are calculated from the extrapolated potential. Internuclear distances are in A, and energies in crn - I .

v Gv Bv110- 2

0.0 3.88013 0 11.2879 3.85801

33.0279 3.75227 2 53.5248 3.63444 3 72.7820 3.50581 4 90.7844 3.36896 5 107.5141 3.22547 6 122.9539 3.07673 7 137.0910 2.92272 8 149.9158 2.76276 9 161.4167 2.59568 10 171.5808 2.41977 11 180.3965 2.23312 12 187.8613 2.03422 13 193.9941 1.82275 14 198.8474 1.59957 15 202.5040 1.36271 16 205.0529 1.10470 17 206.623 0.8449 18 207.480 0.5711 19 207.805 0.2993

(5)

Equation (5) is fitted to the IPA potential over the range of R = 7.6 - 14.0 A.. The exchange interaction, represented as exponential term, is small (.;;; O. 14 cm - I) in the region longer than the LeRoy-Bernstein distance,18 RLB , which is 10.8 A. for NaK a 3I + state. We obtain De=207.858±0.019 cm- I

, C6 =(l2.75±0.15)X10 6

cm- I A.6, C8=(2.22±0.19)XI08 cm- I A. 8, and C\O=(1.100±0.06l)X10 10 cm- I N°, where Ce and C r are fixed at values from Ref. 1. De is determined more accurately than one in Ref. 1. But the standard deviations of C 6, C 8' and C \0 are larger than the ones of Ref. 1. This is because a small but oscillatory error remains in the resultant IPA potential. Molecular constants and classical turning points are calculated from the extrapolated IPA potential and listed in Table III. Molecular constants are smoothly extrapo­lated above the highest observed vibrational level. The dif­ference of these values from the molecular constants G v and Bv shown in Eq. (2) is found to be within standard deviation.

Finally the potential curve and the vibrational wave functions are shown in Fig. 3. The vibrational wave function (v = 19) calculated from the extrapolated potential spreads over a wide range of R. G v =19- GV =18 is shown in Table ill to be comparable to the magnitude of the hyperfine splitting of Fig. 2. Between the a 3I + (v = 19) level and the nearest vibrational level of the X I I + state, the energy difference is guessed to be small and the FCF is expected to be large. The hyperfine structure of the a 3I + (v = 19) level must be dif­ferent from the observed ones because of the mixing with the X 1 I + state. It is, however, difficult to observe higher vibra­tionallevels (v ~ 17) of the a 3I + state because NaK mol-

Turning point

5.461812 5.172940 5.816538 4.991891 6.135772 4.884344 6.398339 4.808264 6.646899 4.750813 6.894531 4.705789 7.148133 4.669876 7.412712 4.640087 7.693036 4.614878 7.994637 4.593643 8.324616 4.575903 8.692607 4.561215 9.112081 4.549228 9.602099 4.539678 10.189535 4.532303 10.912398 4.526847 11.839870 4.523093 13.124959 4.52080 14.98735 4.51956 18.15251 4.51908 25.08573

ecule is excited to the B' I II or c' 3I + state from the X II+(v"~O) state which is localized around the equilib­rium internuclear distance, and because the FCF is very small between a 3I + (v ~ 17) and the vibrational levels of the B' I II or c' 3I + state.

Recently, photoassociation spectroscopy of molecular excited states has been demonstrated in an optical trap.19,20 During the collision, cold atoms can absorb a photon to pro­duce a bound excited molecule. The transition is of bound­free type but spectra are observed with high resolution be­cause the initial nonbonded state of the cold atoms is selected in energy. Very long range potentials of diatomic alkali-metal molecules will be obtained by photoassociation spectroscopy where the initial state is at large internuclear separation. PFP FM spectroscopy is sensitive enough to de­tect the highly excited vibrational levels where molecular spectroscopy connects to photoassociation spectroscopy.

ACKNOWLEDGMENTS

This work is supported by a Grant-in-Aid for Specially Promoted Research from the Ministry of Education, Science, and Culture of Japan.

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Ishikawa, Mukai, and Tanimura: Potential curve of NaK 881

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