Theory and analysis of sodium dimer Rydberg states observed by all-optical triple ..... 16 P. Labastie, M. C. Bordas, B. Tribollet, and M. Broyer, Phys. Rev. Lett.
JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 14
8 OCTOBER 1999
Theory and analysis of sodium dimer Rydberg states observed by all-optical triple resonance spectroscopy Edward S. Chang,a) Jing Li, Jianming Zhang, and Chin-Chun Tsai Department of Physics, University of Connecticut, Storrs, Connecticut 06269
John Bahns Department of Physics and Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06269
William C. Stwalley Department of Physics, Institute of Materials Science, and Department of Chemistry, University of Connecticut, Storrs, Connecticut 06269
共Received 18 June 1999; accepted 15 July 1999兲 For the nf (l⫽3) series of Na2, quantum defects are calculated from theoretical values of the core quadrupole moment and polarizabilities. They compare favorably with those inferred from our preliminary report of high resolution all-optical triple resonance spectroscopy 关Chem. Phys. Lett. 236, 553 共1995兲兴 and from the full report of such spectra given here. The spectrum is the simplest when the higher intermediate state has J ⬘ ⫽0 which requires a final state J⫽1. We predict the stroboscopic effect should first occur when n⫽52 for the nf series, as we observe, rather than at n⫽69 for the np series. Our data thus confirm that the strongest series is the nf. Hence the ionization potential is not 39 478.75⫾0.04 cm⫺1 as previously reported, but rather 39 478.101⫾0.013 cm⫺1, ⫺1 © 1999 implying a molecular ion dissociation energy of D 00 (Na⫹ 2 )⫽7914.038⫾0.014 cm . American Institute of Physics. 关S0021-9606共99兲01138-1兴
I. INTRODUCTION
we are encouraged to apply LRPT to the nf levels in our all-optical triple resonance 共AOTR兲 data, which should be more valid as l is even higher than for the nd case. As preliminarily reported by Tsai et al.,8 we have previously observed four Na2 Rydberg series by high resolution all-optical triple resonance 共AOTR兲. With the original assignment of the long series as the n p0 series 共where 0 represents the rotational quantum number of the ion core, N⫹兲, the ionization potential was found to be 39 478.75 ⫾0.04 cm⫺1 in agreement with the value found by Bordas et al.7 Using accurate theoretical values of the core parameters to be described in Sec. II, we evaluate the quantum defects of the nf series. They agree rather well with those originally assigned to the np series, suggesting that the np and the nf assignments ought to be interchanged; in that event, the long series would actually be the n f 2 series. Hence the previously observed limit was not T ⬁ but T ⬁ ⫹6B 0 , where the rotational constant of the ion v ⫽0 level, B 0 ⫽0.112 95⫾0.000 13 cm⫺1. 7 The new assignments are based on separate analyses in each of the three regimes of the principal quantum number n: low (n⭐40), intermediate (40⭐n⭐80), and high (n⭓80). For low n, the n f 1 ⌺ ⫹ u and the n f 1 ⌸ u series are identified by matching their quantum defects to those calculated from the LRPT. In the high n regime, there is only one strong series with an approximately constant quantum defect, the value of which agrees with that of the n f 2 series calculated from the above model. Optimizing the conditions for weaker signals in this regime, we have remeasured this series at a much higher gain and with a more careful and reliable I2 calibration spectrum. Finally our spectra show stroboscopic fringes near n⫽52, 65, 75, 82, 86,
Analysis of molecular Rydberg series by multichannel quantum defect theory 共MQDT兲 has been well developed by Jungen and his collaborators.1 In their seminal work on H2, Herzberg and Jungen2 demonstrated how absorption spectra can be fitted with a few quantum defects which explain all the level positions, intensities, and even autoionization widths. Then they suggested that the experimentally determined quantum defects of the np levels could be related to the quadrupole moment of the ionic core, which might be calculated from first principles. Actually, in an earlier work on the 4 f and 5 f complex of NO, Jungen and Miescher3 gave expressions for those levels in terms of the core properties of NO⫹, and used them to find the observed values of the core parameters, specifically the quadrupole moment.4 More recently, these concepts embodied in the long-range potential theory 共LRPT兲 have been developed with everincreasing degrees of sophistication for application to the high-l levels of H2. 5,6 In the simplest form, the LRPT requires only certain core parameters of the molecular ion, the quadrupole moment, and the dipole polarizabilities, which can be obtained from either ab initio calculations or analysis of other Rydberg series. In diatomic sodium, the quantum defects and their derivatives with respect to the internuclear distance R have been extensively measured for the nd series from double resonance experiments by Bordas and collaborators.7 We will show that those values and derivatives are semiquantitatively accounted for in the LRPT. Thus a兲
Present address: Department of Physics and Astronomy, University of Massachusetts, Amherst, MA 01003.
0021-9606/99/111(14)/6247/6/$15.00
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etc., corresponding to the n f 2 series rather than n ⫽69,87,... corresponding to the np0 series. II. THE LONG-RANGE POTENTIAL THEORY „LRPT… IN HUND’S CASE a
The theory for molecular nf (l⫽3) Rydberg levels was first introduced in the context of photoabsorption in NO 共Ref. 3兲 for n⫽4 and 5. The LRPT assumed that the levels were hydrogenic with the addition of ⌬E, representing certain electrostatic long range potential energies evaluated in the molecular frame, E 共 nl⌳ 兲 ⫽T ⬁ ⫺Ry/n 2 ⫺⌬E.
共1兲
In Eq. 共1兲, T ⬁ is the series limit, the Rydberg constant Ry ⫽109 736.009 cm⫺1 for Na2, and n the principal and l the orbital quantum number. In this model, we find that ⌬E ⫽E Q ⫹E S ⫹E T can be evaluated for any arbitrary value of l as follows. The interaction between the electron and Q 共in ea 20 兲 the quadrupole moment of Na ⫹ 2 is given by E Q ⫽eQ 具 l⌳ 兩 P 2 兩 l⌳ 典具 r ⫺3 典 nl 2Q 关 l 共 l⫹1 兲 ⫺3⌳ 2 兴 Ry ⫽ 3 . n l 共 l⫹0.5兲共 l⫹1 兲共 2l⫺1 兲共 2l⫹3 兲
l⌳ ⫽n 3 ⌬E/2Ry, 共2兲
The electric field of the Rydberg electron also induces a dipole moment in the molecular ion, described by the core polarizability. It is customary to separate the core polarizability into a scalar core polarizability ␣ S 共in a 30 兲 and a tensor core polarizability ␣ T 共same units兲, where
␣ S ⫽ 共 ␣ 储 ⫹2 ␣⬜ 兲 /3,
共3兲
␣ T ⫽2 共 ␣ 储 ⫺ ␣⬜ 兲 /3.
Then these core polarizabilities produce second-order energy shifts given by E S ⫽e 2 ␣ S 具 r ⫺4 典 nl /2 ⫽
␣ S 关 3n 2 ⫺l 共 l⫹1 兲兴 Ry , 2n 5 共 l⫺0.5兲 l 共 l⫹0.5兲共 l⫹1 兲共 l⫹1.5兲
共4兲
and E T ⫽e 2 ␣ T 具 l⌳ 兩 P 2 兩 l⌳ 典具 r ⫺4 典 nl
␣ T 关 l 共 l⫹1 兲 ⫺3⌳ 兴 Ry ⫽ 共 2l⫺1 兲共 2l⫹3 兲
Q ⬘ ⫽Q⫹3 ␣ T / 关 4 共 l⫺0.5兲共 l⫹1.5兲兴 .
共8兲
From Eq. 共7兲, we find the quantum defect is given by
l⌳ ⫽
3 ␣ S ⫹ 关 l 共 l⫹1 兲 ⫺3⌳ 2 兴 Q ⬘ . l 共 l⫹0.5兲共 l⫹1 兲共 2l⫺1 兲共 2l⫹3 兲
共9兲
Equation 共9兲 shows that the quantum defects decrease rapidly as l increases, so for high l Rydberg series, channel interactions are always weak since their quantum defects are all near zero. For the present values of core parameters and l values of interest, the second term is typically one order-ofmagnitude smaller than the first. Therefore, Q ⬘ is quite insensitive to the value of ␣ T adopted. Specifically for the sodium dimer, we find for nd levels, 共10兲
and for nf levels
3⌳ ⫽0.154⫹0.038共 4⫺⌳ 2 兲 .
2
共5兲
Of course when l⫽3, our results coincide with those from Jungen and Lefebvre-Brion.4 For the Na⫹ 2 ion, values for Q, ␣ 储 , and ␣⬜ have been calculated9 for internuclear distance R ranging from 4 to 10a 0 , in a model potential approach. At the equilibrium distance R e ⫽6.8a 0 , the value for Q is 21.95 ea 20 . Actually Q can also be inferred from the expectation values of certain one-electron operators evaluated by Cerjan et al.,10 where one nucleus is taken to be the origin. It is found that Q⫽ZR 2 ⫺ 共 R/2兲 2 ⫺ 具 3z 2 ⫺r 2 典 ,
共7兲
which becomes independent of n for nⰇl. In this limit, it is convenient to combine the two anisotropic interactions represented by Eq. 共2兲 and Eq. 共5兲 into one identical to Eq. 共2兲 except that Q is replaced by the effective quadrupole moment Q ⬘ given by
2⌳ ⫽0.927⫹0.250共 2⫺⌳ 2 兲 ,
2
关 3n ⫺l 共 l⫹1 兲兴 ⫻ 5 . 2n 共 l⫺0.5兲 l 共 l⫹0.5兲共 l⫹1 兲共 l⫹1.5兲
where Z⫽11 is the nuclear charge and the bracketed quantity has been tabulated by Cerjan et al. up to R⫽20a 0 . Thus Q can be calculated for R e by a linear interpolation of the values given at R⫽6 and 7. However, the value given for 具 3z 2 ⫺r 2 典 differs considerably from the one inferred from 具 z 2 典 and 具 r 2 典 , tabulated separately. Taking the latter as correct, we find Q⫽21.7ea 20 , in remarkable agreement with Alikacem and Aubert-Frecon.9 For the core polarizabilities, the Alikacem and AubertFrecon values are ␣ S ⫽97.3 and ␣ T ⫽30.0a 30 at R e . Using a pseudopotential local spin-density method, Moullet et al.11 found ␣ S ⫽89 and ␣ T ⫽32a 30 . They have also calculated the scalar polarizabilities for the neutral sodium monomer, dimer, and trimer, which are typically 10% lower than the corresponding experimental values. Since they conclude that their values tend to be too low, we adopt the larger value from Alikacem and Aubert-Frecon. With these values for the molecular ion, we can readily calculate the long range energy shift ⌬E. However, for our Rydberg levels where n⭓10, it is more convenient to use the quantum defect
共6兲
共11兲
Equation 共11兲 shows that the difference between the n f ⌺ u and the n f ⌸ u quantum defects in Na2 is only 0.038; hence perturbation between these two channels can be expected to be very weak. III. ANALYSIS OF THE AOTR DATA A. Low n levels in Hund’s case a
We have previously presented a preliminary report8 of our observation of four main series, tentatively identified as the n p ⌺, n p ⌸, n f ⌺, and n f ⌸ in the AOTR data for the sodium dimer. Of particular interest are the data sets which
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J. Chem. Phys., Vol. 111, No. 14, 8 October 1999
FIG. 1. Quantum defects of the strongest lines in the AOTR spectra at low n 共⭐40兲, organized into four series. Theoretical values of the n f ⌳ series are shown as horizontal lines: ⌺, ⌸, ⌬, ⌽.
originate from the 3 1 ⌺ ⫹ g ( ⬘ ,J ⬘ ) state, with ⬘ ⫽0 and J ⬘ ⫽0, 1, 2, and 3. In the J ⬘ ⫽0 set, the long series previously identified as the np series has been extrapolated to the series limit, yielding T ⬁ ⫽39 558.09⫾0.04 cm⫺1. In Fig. 1, we show the measured quantum defects of these four series, in the low n regime, plotted against n from 11 to 40. From theory given by Eq. 共11兲, we obtain the values 0.307, 0.268, 0.154, and ⫺0.036 for ⌳⫽0, 1, 2, and 3, respectively. Except for the first case where ⌳⫽0, the calculated values, shown as horizontal lines, run through the midst of the experimental data, resembling a least-square fit. These theoretical results strongly suggest that the four series should be labeled n f ⌺, n f ⌸, n f ⌬, and n f ⌽, rather than n p ⌺, np ⌸, n f ⌺, and n f ⌸ as in Ref. 8. Actually there is a good reason why the theoretical value for the n f ⌺ series may not fit the data. As noted by Jungen3 for his case of NO, the n f ⌺ quantum defect was not accurately predicted by the long-range potential theory because its wavefunction has to be orthogonalized to those of certain electrons in the core. The same argument applies to the present case of Na2, and can be strengthened with the following elaboration. In the ‘‘United Atom’’ designation, the electronic configuration of 2 2 2 2 4 2 Na⫹ 2 is (1s g ) (2p u ) (2s g ) (3 p u ) (2 p u ) (3d g ) 4 2 (3d g ) (4 f u ) (3s g ). Thus our Rydberg n f u orbital has to be orthogonalized to the core 4 f u orbital, mixing their wavefunctions and energies. Note that this mixing of Rydberg and core orbitals does not occur for the other three values of ⌳, although one can argue that there is weak mixing in the case of the n f u orbitals with the core 2p u , which itself contains a small admixture of 4 f u . So the agreement between theory and experiment identifies the n f ⌸ series and even the n f ⌺ series, which is further verified in the next two sections. Unfortunately, the apparent identification of the next two series proves to be erroneous. Since the intermediate state has J ⬘ ⫽0, the Rydberg states must all have J⫽1. Hence it is impossible to observe the ⌬ and the ⌽ series in J ⬘ ⫽0 spectra. Thus the appearance of agreement with those two series with quantum defects near 0.15 and ⫺0.04 is coincidental and they must be assigned instead to the np ⌸ and the n p ⌺ series. It is apparent that our situation is different from that of Bordas et al.7 who have observed excitation of the nd ⌬ g
Sodium dimer Rydberg states
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states from the A 1 ⌺ ⫹ u state in a high J ⬘ state in their double resonance experiment. From a MQDT analysis of their double resonance experiment on the sodium dimer, Bordas et al.7 have determined the nd quantum defects to be 0.215, ⫺0.035, and 0.42 for ⌺ g , ⌸ g , ⌬ g , respectively, all with a standard deviation of 0.01. Since the LRPT has been successfully employed in the analysis of the 4d complex of H2 共Ref. 5兲, we feel justified in exploring how the LRPT applies to the larger sodium molecule. The corresponding values calculated from Eq. 共10兲 are 1.426, 1.177, and 0.427. It is seen that for the last case of the ⌬ series, the agreement is amazingly good. Further, Bordas et al. have determined experimental first derivatives of the nd quantum defects from their excited vibrational levels. These can also be calculated since we have the R dependence of ␣ S , ␣ T , and Q. For the derivative of 2 ␦ , the calculated value of 0.03 Å⫺1 agrees with and is probably superior to the experimental value of 0.00 ⫾0.03 Å ⫺1 . Theoretical argument based on the relative interaction strengths is consistent with results from the LRPT in predicting that the quantum defects should decrease monotonically as ⌳ increases for the same l. Therefore the above result suggests that the absolute quantum defects for the other two values of ⌳, measured only to modulo one, requires the addition of unity, i.e., 1.215 for ⌺ g and 0.965 for ⌸ g . Now the discrepancies between the calculated and the experimental values are only 20%, not unlike our case of the n f ⌺ u series discussed above. Similarly, the nd g orbital has to be orthogonal to the core 3d g orbital and the nd g orbital to the core 3d g orbital, so their quantum defects may depart from the predictions of the LRPT. Calculated values for the derivatives are 0.24 and 0.18 Å⫺1, which compare reasonably well with the experimental values of 0.15 ⫾0.2 and 0.21⫾.01 Å ⫺1 for the nd ⌺ g and the ⌸ g series, respectively. Thus we see that the LRPT is appropriate even for the relatively low l⫽2 and for a relatively large molecule like Na2, as evidenced by the data for the nd ⌬ g series. However, for series whose wavefunctions need to be orthogonalized to core orbitals such as the nd ⌺ g and ⌸ g series, values from the LRPT are not expected to be very accurate, but nevertheless should serve as useful guides for assignment by comparison to the measured values. B. High n levels in Hund’s case d and the ionization potential
In our previous preliminary report,8 the signals for the high n levels were very weak. Thus it was desirable to remeasure the AOTR spectrum for higher laser frequency 共⭓13 780 cm⫺1, corresponding to n⭓61兲 to more precisely determine the term limit and ionization potential. The gas pressure and laser intensities were optimized for the weaker lines expected in this high n range. The experimental setup was the same as in Ref. 10. Three coherent 899-29 single mode ring lasers 共R-6G, middle wavelength Ti:sapphire, short wavelength Ti:sapphire corresponding to lasers L1, L2, and L3 respectively兲 were used as shown in Fig. 2. An ultrasensitive space–charge limited diode ion detector was used to detect the ions. Residual beams were sent into an I2 cell for frequency calibration. When the probe laser was scanned, the system accumulated the ion signals and I2 spec-
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TABLE I. Measured n f 2 levels (61⭐n⭐115) and their quantum defects. ‘‘Present’’ refers to measurements reported here, which are compared to the slightly recalibrated results of Ref. 8. T e 共cm⫺1兲
FIG. 2. Relevant potential curves of the AOTR experiment.
tra simultaneously. The absolute frequencies of the Ti:sapphire lasers were calibrated with Ne atomic transitions using a hollow cathode tube 共Jarrell Ash, Type 45439, Cathode Na–K, Gas Ne兲. To minimize the Stark effect and field ionization, platinum metal, which has a large work function, was used as a cylindrical shield for the field-free region. The lasers were all operated at relatively low powers 共typically 100 mW兲, and the gas pressure was 0.1 torr. In Table I, we show the average measured term values of the long series 共three independent measurements兲 along with the preliminary values measured by Tsai et al.8 The slight recalibration of the lines used in Ref. 8 is within the uncertainty reported therein.8 After applying the calibration corrections from the I 2 cell, they generally agree to within 0.01 cm⫺1 as would be expected since each of the three frequencies associated with a term value is accurate to ⬃0.005 cm⫺1. Weak lines with questionable identifications are labeled with an asterisk 共*兲 and are not used in the fits below. Beyond 13 800 cm⫺1, corresponding to n⭓104, the signals become quite weak. For our three independent spectra for this region, we found that the peaks are clearly reproducible up to 13 801.8 cm⫺1 (n⫽115). In Fig. 3, we show one of our three spectra for the higher n levels originating from the 3 1⌺ ⫹ g (0,0) intermediate state. The lines marked with a solid circle are found to yield a near constant quantum defect, around 0.26, when we take the series limit to be near our preliminary value8 of 39 558.09⫾0.04 cm⫺1. In the high n regime, it is appropriate to transform the foregoing theoretical results in Sec. II to Hund’s coupling case d via MQDT or alternatively to reformulate the theory starting in case d as was done by Eyler and Pipkin.5 The
a
a
n
Present
Recalibrated
Present
Recalibrateda
61 62 63 64 65 66 67 68* 69* 70* 71* 72 73 74 75 76* 77 78 79 80 81 82 83 84 85 86 87* 88 89 90 91 92 93 94 95* 96* 97* 98 99 100* 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115
39 528.4423 39 529.3047 39 530.2600 39 531.1447 39 531.9778 39 532.7634 39 533.4571 39 534.1891 39 534.9160 39 535.6022 39 536.1923 39 536.7559 39 537.3516 39 537.9238 39 538.4690 39 539.0307 39 539.4952 39 539.9712 39 540.4103 39 540.8629 39 541.3099 39 541.6610 39 542.0680 39 542.4535 39 542.8326 39 543.1601 39 543.5381 39 543.8635 39 544.1825 39 544.4883 39 544.7980 39 545.1014 39 545.3629 39 545.6494 39 545.9244 39 546.1657 39 546.4051 39 546.6424 39 546.8821 39 547.1610 39 547.2984 39 547.5108 39 547.7341 39 547.9340 39 548.1026 39 548.2992 39 548.4922 39 548.6912 39 548.8241 39 549.0085 39 549.1816 39 549.3366 39 549.5006 39 549.6550 39 549.8099
39 528.4555 39 529.3165 39 530.2780 39 531.1455 39 531.9848 39 532.7718 39 533.4724 39 534.2031
0.22 0.31 0.27 0.25 0.24 0.25 0.33 0.32 0.27 0.23 0.30 0.37 0.35 0.33 0.32 0.40 0.29 0.29 0.34 0.31 0.26 0.41 0.38 0.37 0.34 0.42 0.32 0.34 0.34 0.36 0.32 0.27 0.34 0.28 0.23 0.285 0.31 0.32 0.29 0.04 0.41 0.41 0.33 0.33 0.46 0.42 0.37 0.26 0.49 0.40 0.35 0.38 0.32 0.30 0.25
0.20 0.30 0.25 0.25 0.23 0.23 0.31 0.30
39 536.1215 39 536.7676 39 537.3557 39 537.9285 39 538.4763 39 538.9222 39 539.4968 39 539.9773 39 540.4173 39 540.8714 39 541.3110 39 541.6697 39 542.0787 39 542.4661 39 542.8393 39 543.1726 39 543.8734 39 544.1869 39 544.4932 39 544.7957 39 545.0874 39 545.3675 39 545.6088 39 545.8944
39 546.6485 39 546.8951 39 547.2869 39 547.5137 39 547.6358 39 547.9293 39 548.0961 39 548.2896 39 548.4896 39 548.6832 39 548.8301 39 549.0101 39 549.1836 39 549.2970 39 549.4972 39 549.6506 39 549.8106
0.41 0.35 0.35 0.32 0.30 0.44 0.29 0.28 0.32 0.29 0.26 0.38 0.35 0.34 0.32 0.39 0.31 0.33 0.34 0.33 0.32 0.32 0.44 0.35
0.30 0.23 0.47 0.40 0.26 0.35 0.50 0.47 0.39 0.30 0.46 0.39 0.34 0.63 0.35 0.33 0.25
Reference 8.
resultant nf levels are labeled by N⫹, the rotational quantum number of the molecular ion, and their quantum defects in case d are almost identical: f 2 ⫽0.284 and f 4 ⫽0.291. We therefore surmise that this long series which has started out
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J. Chem. Phys., Vol. 111, No. 14, 8 October 1999
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as the n f ⌺ series in the low n regime has transformed into the n f 2 series in the high n limit. Using the theoretical value for f 2 , we perform a least-square fit to the Rydberg formula for the levels in Table I 共fit B1兲 and found the series limit T to be 39 558.128⫾0.004 cm⫺1 共See Table II兲. If we regard the quantum defect as an additional fitting parameter, we find that f 2 ⫽0.255⫾0.017, which appears to be consistent with the theoretical value of 0.284. The limit remains essentially the same for this fit 共B2兲 at 39 558.116 ⫾0.008 cm⫺1, consistent with 39 558.09⫾0.04 cm⫺1, the preliminary value we previously recommended.8 Both values are given in Table II. Our experiments are carried out in what is nominally an electric field-free region of our space–charge limited ionization detector.12 From our observation of the line width of the n⫽113 lines in our spectra 共Fig. 3兲, we calculate from quasistatic theory13 a maximum electric field of 0.06 V/cm 关attributing somewhat 共unreasonably兲 all line-width to Stark broadening兴. If we fix the quantum defect at its theoretical value and fit our term values, we find a very small maximum electric field of 0.001 V/cm also in Table II. Thus we believe our term value must be between the lowest value permitted by fit B2 and the highest value permitted by fit B1, i.e., 39 558.120⫾0.012 cm⫺1 as recommended in C of Table II. It should be noted that the term limit T of the n f 2 series is not T ⬁ but T ⬁ ⫹6B 0 . Hence given 6B 0 ⫽0.6770 ⫾0.000 78 cm⫺1, we infer that T ⬁ ⫽39 557.442 ⫾0.013 cm⫺1. With the ground state Na2 zero-point energy of 79.3406⫾0.0004 cm⫺1, 14 this yields an ionization potential of 39 478.101⫾0.013 cm⫺1. Our current value is at variance with the most recent experimental value of 39 478.7 ⫾0.1 cm⫺1 of Bordas et al.7 Our measured ionization potential, with the value D 00 (Na 2 )⫽5942.6880⫾0.0049 cm⫺1 共Ref. 14兲 and the atomic ionization potential of 41 449.451 ⫾0.002 cm⫺1 共Ref. 15兲, yields D 00 (Na ⫹ 2 )⫽7914.038 ⫾0.014 cm⫺1. FIG. 3. Spectrum of the high n regime (61⭐n⭐115) near the ionization limit. The frequency 共cm⫺1兲 of the third laser 共L3 in Fig. 2兲 is shown. The term values reported in Table I can be obtained by adding the energy of the ⫺1 3 1⌺ ⫹ g (0,0) level 共25 747.9475 cm 兲 to the L3 frequency.
C. Intermediate n levels and the stroboscopic effect
The stroboscopic effect in the sodium dimer has been explained by classical mechanics for the high J nd Rydberg levels16 and for the np Rydberg levels through the interme8 diate resonance 3 1 ⌺ ⫹ g (0,0). At high J, this effect occurs when the electronic angular frequency matches that of molecular rotation as given by
TABLE II. Results of successive fits for the term value of the ionization limit, using the term value in Table I, the quantum defect f 2 , and the residual electric field 共V/cm兲. Bracketed values are assumed values in the corresponding fit. Parameters ⫺1
Description
T(cm ) a
A. Preliminary report : B. Present work 1. Fixed ,⫽0 2. Fit , ⫽0 3. Fixed , fit C. Recommended
共V/cm兲
39 558.09⫾0.04
关0兴
关0兴
39 558.120⫾0.012 39 558.116⫾0.008 39 558.128⫾0.004 39 558.120⫾0.012
关0.284兴 0.255⫾0.017 关0.284兴 0.284
关0兴 关0兴 0.0095⫾0.0012 0
a
Reference 8.
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n 3 ⬇2kB 共 J⫹1/2兲 .
共12兲
For the J ⬘ ⫽0 set of data, the only possible value for J is 1, and therefore Eq. 共12兲 is inappropriate. Instead, in quantum mechanics the stroboscopic effect occurs when the rotational level spacing equals k times the electronic level spacing, leading to the condition n 3 ⬇2kB 共 2l⫹1 兲 .
共13兲
For the np series Eq. 共13兲 predicts this effect when n ⬇69,87,100,..., but for the nf series, n⬇52,65,75,82,86,... . Our spectra 共shown in Fig. 2 of Tsai et al.兲8 show the periodic return to the evenly spaced strong line pattern at the n values predicted for the nf series, further confirming our reassignments here. IV. CONCLUSIONS
We have developed long range potential theory 共LRPT兲 for the case of the Rydberg states of Na2. We have carefully examined our AOTR spectra, especially those originating from the 3 1 ⌺ ⫹ g (0,0) intermediate state, in light of the results of the LRPT. For the low n regime, we have calculated theoretical quantum defects for the n f ⌺ and the n f ⌸ series in Hund’s case a, and used them to reassign the various Rydberg series. In the high n regime, we have made improved measurements of these two series which merge into one strong series, where the quantum defect remains in good agreement with the theoretical n f 2 value in Hund’s case d. In the intermediate n regime, our data exhibit the stroboscopic effect consistent with the assignment to the nf but not to the np series. However the new identification leads to an
ionization potential which is 6B 0 lower than the value of Bordas et al.7 We are currently studying the higher J spectra where the n f ⌬ u and the n f ⌽ u series may appear. ACKNOWLEDGMENTS
Partial support of this work by the National Science Foundation is gratefully acknowledged. We thank the New England Land-Grant Universities Faculty Exchange Program for enabling this collaboration. Ch. Jungen and S. C. Ross, Phys. Rev. A 55, R2503 共1997兲. G. Herzberg and Ch. Jungen, J. Mol. Spectrosc. 41, 425 共1972兲. 3 Ch. Jungen and E. Miescher, Can. J. Phys. 47, 1769 共1969兲. 4 Ch. Jungen and H. Lefebvre-Brion, Mol. Phys. 33, 520 共1970兲. 5 E. Eyler and F. Pipkin, Phys. Rev. A 27, 2462 共1983兲. 6 P. W. Arcuni, E. A. Hessels, and S. R. Lundeen, Phys. Rev. A 41, 3648 共1990兲. 7 M. C. Bordas, M. Broyer, J. Chevaleyre, P. Labastie, and S. Martin, Chem. Phys. 129, 21 共1989兲. 8 C. C. Tsai, J. T. Bahns, and W. C. Stwalley, Chem. Phys. Lett. 236, 553 共1995兲. 9 A. Alikacem and M. Aubert-Frecon, Indian J. Phys. 63B, 301 共1989兲. 10 C. J. Cerjan, K. K. Docken, and A. Dalgarno, Chem. Phys. Lett. 38, 401 共1976兲. 11 I. Moullet, J. L. Martins, F. Reuse, and J. Buttet, Z. Phys. D 12, 353 共1989兲. 12 C. C. Tsai, J. T. Bahns, and W. C. Stwalley, Rev. Sci. Instrum. 83, 5576 共1992兲. 13 E. S. Chang and W. G. Schoenfeld, Astrophys. J. 383, 450 共1991兲. 14 K. M. Jones, S. Maleki, S. Bize, P. D. Lett, C. J. Williams, H. Richling, H. Knockel, E. Tiemann, H. Wang, P. L. Gould, and W. C. Stwalley, Phys. Rev. A 54, R1006 共1996兲. 15 M. Ciocca, C. E. Burkhardt, J. J. Leventhal, and T. Bergeman, Phys. Rev. A 45, 4720 共1992兲. 16 P. Labastie, M. C. Bordas, B. Tribollet, and M. Broyer, Phys. Rev. Lett. 52, 1681 共1984兲. 1 2
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