Vol. 6, No. 12/December 1989/J. Opt. Soc. Am. B 2309

Experimental observation of the (3) 12+ state of Na2 bydeperturbation of the Cll-X'2; system

G.-Y. Yan and A. L. Schawlow

Department of Physics, Stanford University, Stanford, California 94305

Received June 19, 1989; accepted August 24, 1989

Doppler-free UV-excitation spectra, obtained by modulated population spectroscopy, reveal perturbations of a fewhundred e levels of the C 'II,, state in Na2. We ascribe the perturbations to the (3)12' state. After deperturbation,information about 15 vibrational levels of the (3)12+ state was extracted. This information was used to deduce a setof Dunham coefficients and the Rydberg-Klein-Rees potential for the state. The experimental results show thatthe potential of the (3)12:' state is one with a double minimum or an inflection. In addition, it has a flattenedbottom with Te = 31 247.82 cm-1 . As a consequence, the vibrational spacing at first increases and then decreases asa function of v. Comparison of the experimental results and theoretical calculations yielded fairly good agreement.

INTRODUCTION

As for H2, some Rydberg states of Na2 exhibit interestingforms of potential, such as a double minimum or an inflec-tion, in accordance with predictions by Jeung' as well asthose by Valance and Tuan. 2 Only two of these, the (2)174+and (5)1k states, have been observed in experiments re-cently.3 -5 The (3)1;'4 state is another such state. No vi-bronic level of this state has been observed previously. In astudy of Doppler-free UV-excitation spectra of the C 'lH-X 12g system, we found that a few hundred levels of e parityin the C 'II state, located near 31 400-32 500 cm-', are per-turbed by one or more 'Z+ states.6 In this paper we presentan indirect observation of the (3)1;'4 state through depertur-bation of the UV-excitation spectra of the C 'llu-X lZsystem, after ascribing the perturbations to the (3)'4,+ state.This state is found to have an inflection potential curve withan unusual flattened bottom in the inner section.

EXPERIMENT AND ANALYSIS

The experiment has been described in detail in a previouspaper.6 A cross oven containing Na was heated to 400-450°C with Ar buffer gas at a pressure of 1 Torr. A modifiedCoherent 699-29 dye laser with intracavity frequency dou-bling, operating in AutoScan mode, provided cw UV radia-tion with an output power of a few milliwatts. A strongvisible beam from a Coherent 599-21 dye laser, counterprop-agating with respect to the UV beam, was directed throughthe oven. The visible beam was tuned in resonance with atransition of the A 12+-X 1t system and chopped. Thisprocedure modulated the population on the lower level X1;(v", J") of the transition. UV-excitation spectra of C'll,(v', J')-X 9(v", J"), originating on the lower level, wererecorded when the 699-21 laser scanned. These spectrawere then simplified by modulated population spectroscopy.Transitions into those dramatically perturbed levels areweakened, and deviations of transition frequencies fromtheir expected values are sometimes more than 1 cm-'. Be-cause of the modulation, identification of the transitions ofthe simplified spectra was still unambiguous. A measure-

ment accuracy of the energies of the observed levels wasestimated conservatively to be 0.03 cm-'.

Figure 1 shows deviations of experimental energies fromcalculated energies versus J'(J' + 1) - 1 for the vibroniclevels with v' = 15 and v' = 16 of the C 'lHu state; thecalculations use our new constants.6 Only levels with eparity are perturbed. For 14 < v' < 26, 18 crossing pointswere found. For each of the crossing points, maximumshifts of perturbed levels ranged from 0.1 to 1.7 cm-'. Fig-ure 2 shows energies of vibronic levels for the C 'lHu state,located near 31 300-32 600 cm-', versus J'(J' + 1) - 1. Thecircles in the figure represent the measured crossing pointsbetween levels of the C flu state and perturbing levels. Wealso observed extra lines for some points with higher v'values. For those points differences between rotationalconstants for perturbing levels and their corresponding per-turbed levels were deduced directly and were found to beconsistently -0.02 cm-'.

Since only e levels are perturbed, any perturbing statemust be a 'Z+ state. According to the differences of rota-tional constants, tentative lines for rotational energies of theperturbing levels versus J(J + 1), which passed through thecrossing points, were plotted. Being vibrational spacings,distances between consecutive lines were approximately 50-60 cm-l and decreased monotonically as energies of thecrossing points increased, except for a few lines with lowerenergies. These results indicated that all the perturbationscould be caused by one 1';+ state. Moreover, we noticed thatthe (2)12;+ state has a dissociation limit of 31 767 cm-' andthat its vibrational spacings of levels beneath this limit aremuch less than 50 cm-. 3 Other 1+ states, except the(3)14+ state, are located above the energy regime that weobserved. Thus we concluded that all the perturbations arecaused by crossing between levels of the C 'llu state and the(3)124+ state.

For some vibrational levels, there is more than one cross-ing point, as shown in Fig. 2. Spacing AvP between twovibrational levels of the perturbing state, which cross overone vibrational level of the C 'lHu state, can be expressedapproximately as

0740-3224/89/122309-04$02.00 © 1989 Optical Society of America

G.-Y. Yan and A. L. Schawlow

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2310 J. Opt. Soc. Am. B/Vol. 6, No. 12/December 1989

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A = AB[J2(J2 + 1) - J1(J, + 1),

where AB = B(C lII) - B[(3)1'+] and J, and J2 are fraction-al J values of the consecutive crossing points in one of thevibrational levels. If we assume that AB does not changemuch for the adjacent vibrational levels, the value of theexpression in the brackets in the equation above is a measureof vibrational spacing for the perturbing state to a first-orderapproximation. The vibrational spacing between perturb-ing levels, which cross with the level v' = 16, is greater thanthat between levels passing through the level v' = 15, al-though the former have higher energies than the latter. It isanomalous that vibrational spacing increases with v. Inaddition the vibrational spacings between perturbing levels,crossing with the level v' = 24, decreased rapidly, and pertur-bations of the corresponding crossing points were weaker.The maximum shifts of levels around the highest crossingpoints in Fig. 2 were only a few tenths of a wavenumber.These perturbing levels may be close to an inflection ofpotential for the perturbing state.

All the perturbations are heterogenous. The matrix ele-ment of perturbation, namely, the perturbation integralW12, is proportional to [J(J + 1)]1/2 and related to the over-lap integral of vibrational wave function for the perturbedlevel and the perturbing level.7 If all perturbed levels be-long to one electronic state and all perturbing levels belongto another electronic state, and also the electronic parts inW12 vary slowly with the nuclear separation, the matrixelements will be proportional to the overlap integrals. Aleast-squares fitting with four parameters, ATe, ABv, ADD,and the coefficient of [J(J + 1)]1/2 in the perturbation matrixelement, was used to fit the shifts of levels of the C 'lH, statefor each of the crossing points. The fitting within the ex-perimental uncertainty was not unique for a certain point.Sometimes the best fitting even presented parameters with-out any significance. However, the estimates of the rota-tional constants for perturbing levels were useful for gettingmore satisfactory fittings. Fittings for a certain pointshowed almost the same energies and fractional J values forthe crossing point, although parameters for the fittings weredifferent. In other words, the positions and the fractional Jvalues for the crossing points were more reliable and wereused in the following to determine constants for the state.

For v' = 14, rotational levels with 12 S J' < 56 weremeasured. No noticeable shifts of those levels with e paritywere observed. We then tentatively numbered the perturb-ing levels such that the perturbing level of the (3)12;+ statewith v = 0 crosses over the level of the C llIu state with v' =15 at the fractional J of 39.55. The highest v value assignedwas 16. The other three points, which have v values greaterthan 16 in this assignment, correspond to minor perturba-tions and are close to the inflection. As shown above, thevibrational spacing versus v for the perturbing state hasanomalous behavior. We did not attempt to fit all the 18points observed. The 15 crossing points with v < 16 wereused to deduce the Dunham coefficients and the Rydberg-Klein-Rees (RKR) potential for the (3)1+ state. To pro-duce a good reproduction, 12 parameters were needed in aleast-squares fitting for these points. We were not surprisedin a sense; we expected the vibrational spacing versus v to bepeculiar for the (3)1'+ state in that the vibrational spacing

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G.-Y. Yan and A. L. Schawlow

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Vol. 6, No. 12/December 1989/J. Opt. Soc. Am. B 2311

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increases at the beginning and then decreases as v increasesfrom v = 0. The reproduction of energies of the crossingpoints was within a few tenths of a wavenumber. Overlapintegrals between wave functions of (3)12k+ and C 'I, stateswere calculated and compared with the experimentally de-rived coefficients of perturbation matrix elements, as shownin Fig. 3(a). The comparison was unsatisfactory. We thenrenumbered vibrational quantum numbers by an incrementof one and lowered the tentative potential. The overlapintegrals of wave functions between the C II, state and the(3)'Z+ state with the lowered potential were recalculated.The comparison is agreeable and is shown in Fig. 3(b). Wedid not lower the potential further, because no perturbationoccurred in the rotational e levels with J' 56 for thevibrational level v' = 14.

Based on the renumbering, a new set of the Dunhamconstants and the RKR potential for the (3)1';+ state werededuced. The constants for vibrational levels with v 17are listed in Table 1. Energies of vibronic levels for the(3)'Z+ state versus J(J + 1) are shown in Fig. 2. Figure 4shows the experimental potentials for (3)12'+ and C IIustates. The circles in Fig. 4 represent Jeung's theoretical

calculations for the inner well of the (3)1'+ state.' Takinginto account the fact that the theoretical results are system-atically high,15 the experimental potential was in closeagreement with the theoretical one. In particular, both po-tentials have flattened bottoms. Vibrational spacing versusv is shown in Fig. 5. The figure also depicts vibrationalspacings calculated by using Jeung's potential for the state.Both figures consistently show anomalous behavior of thevibrational spacings. The local maximum for the potentialcurve, obtained with the constants in Table 1, was near32 385 cm-' at Rout = 6.6 A. Like the (5)1'g state, exploredin our previous work,5 this maximum energy corresponds to

Table 1. Dunham Coefficients for the (3)12+ State

Te = 31 247.82 (5.5 E-1)Y10 = 3.5703269 E+1 Y01 = 8.4503921 E-2 (4.0 E-4)

(4.5 E-1)Y20 = 3.5719380 (1.3 E-1) Yi = -4.8709994 E-5 (7.4 E-5)Y30 = -2.577052 E-1 Y21 = -3.275882 E-6 (9.5 E-6)

(1.7 E-2)Y40 = 9.7711572 E-3 Y31 = -2.636101 E-6 (3.5 E-7)

(9.5 E-4)Y50 = -1.828006 E-4

(2.0 E-5)Y02 = -1.052010 E-6 (1.4 E-7)Y03 = 1.018682 E-10 (2.0 E-11)

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G.-Y. Yan and A. L. Schawlow

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2312 J. Opt. Soc. Am. B/Vol. 6, No. 12/December 1989

a barrier or a hump of potential rather than a dissociationlimit. The exact location of the inflection point should bederived from a detailed study of nearby levels on both sidesof the inflection. Beyond the inflection point, Rout of thepotential for the (3)1',+ state extends suddenly to R > 10 A.Franck-Condon factors between levels of the C 'lII and(3)12+ states above the inflection were calculated by usingour experimental potential for the C '1IU state and the theo-retical potential for the (3)1';+ state. The calculationshowed that perturbations between the two states beyondthe inflection become weaker. Improvement of measure-ment accuracy is needed if one is to observe those minorperturbations on levels beyond the inflection.

We intended to observe perturbations on vibronic levelswith J values near 66 in the vibrational level v' = 14 of the C'll, state, predicted by the experimental potentials for boththe C 'lIIu state and the (3)1';+ state. Unfortunately theFranck-Condon factor for the C 'IIkU-X 12T (14-0) band isless than that for the (14-1) band by almost 1 order ofmagnitude. Signals of transitions with J' > 56 in the (14-0)band could not be detected. An attempted observation ofthe (14-1) band with the higher J' values was unsuccessfulbecause wavelengths of the transitions fall outside the tun-ing range of our system.

An absorption of the (3)12+-X 12T system was observedto be very weak compared with that of the C 'llu-X tsystem.8 In addition, UV fluorescence was detected in ourexperimental scheme, and a sodium molecule in the (3)1;+Ustate may decay to low-lying (3)12gT and (2)1 I1g states withgreater probabilities than decay to the ground state, becausetheir potentials have almost the same equilibrium nuclearseparations as that for the (3)1';+ state. 9 UV signals result-ing from the (3)12+ state were weakened. The weakeningcould explain why direct transitions between the (3)'Z+state and the ground state were not observed in our experi-ment, although the transitions are not forbidden by selec-tion rules.

Ionic states cross with a series of Rydberg states in Na2.Avoided crossings between a Rydberg state and ionic statesor valence states produce an interesting form of potential forthe Rydberg state. The (3)1'+ state dissociates to Na(3s) +Na(3d). The dissociation limit is below that of the groundionic state of 43 068 cm-'. Avoiding a crossing with thisionic state causes the inflection in the potential of the (3)1;+Ustate. The peculiar inner well is formed by a strong interac-tion between the (3)1';+ state and other valence states.Thus vibrational and rotational constants at equilibrium forthe state are completely different from those for ionic states.The atomic nature of the (3)'Z+ state varies as a function ofthe internuclear separation. It is necessary to study thestate in detail to determine the interactions.

Infrared Fourier-transform spectroscopy following two-photon excitations has been used by Cooper et al.

3 to explorelevels in the outer well and beyond the barrier of the (2)1;+Ustate. Three vibronic levels in the (5)12g state were excitedby the two-photon transitions in their experiment. Thehighest one of those levels, which was identified as (5)12g(35, 63) in our previous paper, 5 is situated at 35 015.3 cm-'.

Wave numbers of transitions from those levels, terminatingon levels beyond the inflection of the (3)'Z+ state, are lessthan 2800 cm-'. These wave numbers were just at the edgeof the observable regime in Cooper's experiment. Transi-tions between the excited levels of the (5)1'g state with thehigher vibrational quantum numbers and levels inside theinner well of the (3)3Z+ state may have unfavorable Franck-Condon factors, much like transitions between (5)12' and(2)1';+ states. Infrared Fourier-transform spectroscopycovering a wider-wavelength regime and UV-excitationspectroscopy, followed by visible or infrared detection, willbe helpful for observing the (3)1';+ state directly.

CONCLUSIONS

The observations provided information about 15 vibrationallevels in the (3)1;+ state with 1 S v S 17. These data wereused to deduce a set of Dunham coefficients and the RKRpotential for the state. The assignment of vibrational quan-tum numbers of the state was confirmed by a comparison ofobserved matrix elements of perturbation between C IIUand (3)1+ states with calculations of the overlap integrals ofwave functions based on the RKR potentials for the twostates. The observation showed that the (3)1'+ state has adouble minimum or an inflection. The internuclear separa-tion of the inner well at equilibrium is -4.2 A. The bottomof the inner well is located at 31 247.82 cm'1. The bottom isflattened owing to avoided crossings of two diabatic states.This flattening leads to the vibrational spacing's first in-creasing and then decreasing as the vibrational quantumnumber increases. Agreement between the experimentalobservation and the theoretical calculation of the potentialfor (3)12+ state is fairly good.

ACKNOWLEDGMENTS

This research was supported in part by the National ScienceFoundation under grant NSF PHY-86-04441 and in part bythe U.S. Office of Naval Research under contract ONRN00014-87-K-0265.

REFERENCES

1. G. H. Jeung, Phys. Rev. A 35, 26-35 (1987).2. A. Valance and Q. N. Tuan, J. Phys. B 15, 17-33 (1982).3. D. L. Cooper, R. F. Barrow, J. Verges, C. Effantin, and J. d'Incan,

Can. J. Phys. 62,1543-1562 (1984).4. G. Delacretaz and L. Woste, Chem. Phys. Lett. 120, 342-348

(1985).5. G.-Y. Yan, B. W. Sterling, and A. L. Schawlow, J. Opt. Soc. Am. B

5, 2305-2310 (1988).6. G.-Y. Yan, B. W. Sterling, T. Kalka, and A. L. Schawlow, J. Opt.

Soc. Am. B 6, 1975-1978 (1989).7. G. Herzberg, Molecular Spectra and Molecular Structure I.

Spectra of Diatomic Molecules (Van Nostrand Reinhold, NewYork, 1950), p. 288.

8. J. Schlejen, C. J. Jalink, J. Korving, J. P. Woerdman, and W.Miller, J. Phys. B 20,2691-2711 (1987).

9. C. Effantin, J. d'Incant, A. J. Ross, R. F. Barrow, and J. Verges, J.Phys. B 17, 1515-1523 (1984).

G.-Y. Yan and A. L. Schawlow