Observation of Electronic Isotope Shifts in Molecular Bromine

Journal of Molecular Spectroscopy 210, 146–153 (2001) doi:10.1006/jmsp.2001.8437, available online at http://www.idealibrary.com on

Observation of Electronic Isotope Shifts in Molecular Bromine J. L. Booth,∗ I. Ozier,† and F. W. Dalby† ∗ Physics Department, British Columbia Institute of Technology, 3700 Willingdon Avenue, Burnaby, British Columbia V5G 3H2, Canada, and †Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1, Canada Received July 30, 2001; published online October 16, 2001

The B–X electronic system of 79 Br2 , 79,81 Br2 , and 81 Br2 has been investigated with laser-induced fluorescence at a resolution corresponding to 1% of the Doppler width. Hyperfine-free frequency differences between pairs of nearly coincident rovibronic transitions have been measured to an accuracy of ±7 MHz. The data fall into two categories: cases where the two members of the pair arise from different isotopomers, and cases where they arise from the same isotopomer. From the frequency differences, determinations have been made of the electronic isotope shifts arising from terms in the Hamiltonian that lead to the breakdown of the Born–Oppenheimer approximation. In the Born–Oppenheimer approximation, BO for the system B 3 5 1 + the electronic term value TBX 0+u − X 6g is isotopically invariant. However, the measurements show 79,81 79,81 79 79 − T 81 ] = 386(7) MHz. In the next order of 81 that [TBX − TBX ] = 177(10) MHz; [TBX − TBX ] = 209(7) MHz; and [TBX BX β α approximation, [TBX − TBX ] should be proportional to (1/µα − 1/µβ ), where µα is the reduced mass of isotopomer α. The meaβ α − TBX ] are seen to deviate from this dependence by approximately 15%. It is suggested that the deviation sured values of [TBX occurs because the Hamiltonian for the heteronuclear isotopomer includes interactions that are forbidden for the homonuclear C 2001 Elsevier Science species. ° Key Words: electronic isotope shift; bromine; Born–Oppenheimer approximation; multi-isotopomer spectra; isotope dependence.

I. INTRODUCTION

The standard description of the rovibronic energy levels of a diatomic molecule relies on the Born–Oppenheimer approximation to separate the nuclear and electronic degrees of freedom (1). The Born–Oppenheimer approximation predicts simple relationships between the rovibronic constants belonging to different isotopomers. These relationships are extremely useful in the simultaneous fitting of spectroscopic data from different isotopomers. In particular, the number of independent parameters required to describe the data is reduced considerably. However, deviations from this idealized behavior for heavier molecules have been measured as spectroscopic resolution has increased. In order to predict and fit multi-isotopomer data sets correctly, the limitations of the Born–Oppenheimer approximation must be carefully investigated. This problem has been addressed, for example, by Bunker (2–5) and Watson (6, 7). One of the most challenging topics to study is the deviation of the electronic state potential minimum, Te (W ), from the isotopic invariance predicted by the Born–Oppenheimer approximation. In practice, one fits the observed spectra to the difference between the minima of two electronic states, W 0 and W 00 ; namely, TW 0 W 00 = [Te (W 0 ) − Te (W 00 )]. If frequency measurements for several isotopomers are analyzed simultaneously, the small isotopic dependence of TW 0 W 00 is likely to be absorbed by other fitting parameters. To isolate the isotopic variation of 0022-2852/01 $35.00 ° C 2001 Elsevier Science All rights reserved

TW 0 W 00 , one must make high-precision measurements of the frequency separations 1νobs of nearly coincident transitions and compare these to the corresponding Born–Oppenheimer predictions 1νBO . In the current work, high-precision measurements of such frequency separations were made while carrying out a subDoppler molecular beam study of the B–X system of Br2 using laser-induced fluorescence. Bromine occurs naturally in three isotopomers of comparable abundance, 79 Br2 , 79,81 Br2 , and 81 Br2 , and the spectra provided many strong, nearly coincident transitions for study. When frequency separations were examined between two lines arising from a single isotopomer (intra-isotopomer), the discrepancy between the observed and predicted frequency differences was on the order of 5 MHz. These measurements serve to establish the precision and accuracy of the measurement and data reduction procedures. On the other hand, when lines arising from different isotopomers (inter-isotopomer) were examined, discrepancies between the observed frequency separation 1νobs and its calculated counterpart 1νBO greater than 100 MHz were found for each case. The magnitude of the discrepancy depended only on the isotopomers being compared. In addition, the shift observed when the 79 Br2 was compared to 79,81Br2 was approximately 15% less than that observed between 79,81Br2 and 81Br2 . This variation contradicts the reduced mass dependence predicted for the electronic isotope shift by current models used for simultaneous fitting of multi-isotopomer data sets.

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ELECTRONIC ISOTOPE SHIFTS IN Br2 BO Notice that TBX has no isotopic dependence. In Eq. [4],

II. THEORETICAL BACKGROUND

The rovibronic energy for isotopomer α of a diatomic molecule will be written E α (W, v, J ); here W , v, and J indicate the electronic, vibrational, and rotational quantum numbers, respectively. This energy is generally well represented by the expression (1) E α (W, v, J ) = Teα (W ) + G α (W, v) + F α (W, v, J ), [1] where G α (W, v) and F α (W, v, J ), respectively, are polynomials giving the vibrational and rotational energies specific to electronic state W of the molecule. Within the framework of the Born–Oppenheimer approximation (BOA), G α (W, v) and F α (W, v, J ) are isotopically dependent, while Teα (W ) is isotopically independent. An equivalent description was provided by Dunham (8): E α (W, v, J ) =

X i. j

· ¸ 1 i Yiαj (W ) v + [J (J + 1)] j . 2

[2]

The Dunham coefficients Yiαj (W ) can easily be expressed in terms of molecular constants with more transparent physical α α (W ) = Teα (W ); Y10 (W ) = ωeα (W ); meanings. For example, Y00 α α α Y01 (W ) = Be (W ). Here ωe (W ) is the harmonic vibrational frequency and Bαe (W ) is the equilibrium rotational constant. One advantage of Dunham relationship [2] is that it allows the straightforward prediction of molecular parameters for a second isotopomer, β, based on a knowledge of the Dunham coefficients of isotopomer α, provided, of course, that the BOA is valid. Specifically, if the Dunham coefficients for one isotopomer are known, the coefficients for other isotopomers can be determined as follows: β

Yi j (W ) = Yiαj (W )[µα /µβ ](i+2 j)/2 .

[4]

[5]

The experimentally observed frequency can be written in a similar fashion: α α α = TBX + Hobs (B, X ; v 0 , v 00 ; J 0 , J 00 ). νobs

[6]

α has been expressed in Eq. [6] as the sum For convenience, νobs α from the electronic term values of a constant contribution TBX α and a part Hobs that depends on v 0 , v 00 , J 0 , and J 00 . Several terms in the Hamiltonian that were neglected in the BO approximation can introduce an isotopic dependence in Teα (W ) (2, 6). These lead to a difference between the “true” electronic term value Teα (W ) for electronic state W of isotopomer α, and the corresponding BO prediction TeBO (W ). The matrix elements of the terms diagonal in the electronic state should produce the largest effects. The leading adiabatic contributions arise from the coupling of the electronic orbital anguE and from interactions lar momenta LE with the electron spin S, involving the nuclear or the electronic kinetic energy operators. Each of these effects makes a contribution to the electronic term values that varies as 1/µα . It then follows that in lowest order:

Teα (W ) − TeBO (W ) =

1 0W , µα

[7]

where 0W is a mass-reduced matrix element that does not depend on the quantum numbers v or J . Furthermore,

[3]

In Eq. [3], µα and µβ are the reduced masses of isotopomers α and β, respectively. An extensive study of the B 350+ u and X 16g+ electronic states of 79 Br2 was carried out by Gerstenkorn and Luc (G–L) (9), from which rovibronic constants were derived. Using the 79 Br2 constants and isotopic relations [3], the corresponding constants for 79,81 Br2 and 81 Br2 (10) were evaluated. The BO transition frequencies were then calculated for each of the three cases. The labeling of the electronic states used here is appropriate to the homonuclear molecules, but its use for the heteronuclear isotopomer should not lead to any confusion. For isotopomer α, the BO transition frequencies from the level (X, v 00 , J 00 ) to level (B, v 0 , J 0 ) can be written α BO α = TBX + HBO (B, X ; v 0 , v 00 ; J 0 , J 00 ). νBO

α HBO (B, X ; v 0 , v 00 ; J 0 , J 00 ) ¤ £ = G αBO (B; v 0 ) − G αBO (X ; v 00 ) ¤ £ α α (B; v 0 , J 0 ) − FBO (X ; v 00 , J 00 ) . + FBO

α TBX

−

β TBX

µ =

¶ 1 1 − 0BX , µα µβ

[8]

where 0BX = (0 B − 0 X ) is isotopically invariant. From Eq. [8], it follows that the observed electronic isotope shift between any two isotopomers should be small and the shift between 79 Br2 and 79,81 Br2 should equal that between 79,81 Br2 and 81 Br2 . The models currently in use for the simultaneous fitting of multiisotopomer data sets (11, 12) lead to the same conclusion. Any observed deviations from Eq. [8] can therefore have important consequences. To investigate the small isotopic shift predicted by Eq. [8], α α − νBO ) between it is possible to measure the difference (νobs the observed and predicted frequencies for a single B–X transition in isotopomer α, then make an independent measurement β β of the corresponding difference (νobs − νBO ) for isotopomer β, and finally compare the two. However, a far more sensitive technique is to compare the observed frequency separation

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1νobs (α, 1; β, 2) of nearly coincident rovibronic transitions, one in isotopomer α and the other in isotopomer β, £ α 0 00 0 00 ¤ β (v1 , v1 ; J1 , J1 ) − νobs (v20 , v200 ; J20 , J200 ) , 1νobs (α, 1; β, 2) = νobs [9] to the corresponding predicted BO frequency separation, £ α 0 00 0 00 ¤ β (v1 , v1 ; J1 , J1 ) − νBO (v20 , v200 ; J20 , J200 ) . 1νBO (α, 1; β, 2) = νBO

independent of the transitions (1) and (2) being compared. However, d(α, 1; α, 2) might also reveal a vibrational and/or rotational state dependence which can be attributed to extrapolation errors in the rovibronic fitting parameters for the different isotopomers, or to further deviations of the rovibronic constants from the BO relationship [3]. In any case, a systematic study of the intra-isotopomer second differences d(α, 1; α, 2) can play a crucial role in setting the confidence level for measurements of the inter-isotopomer second differences d(α, 1; β, 2).

[10] The difference between these two frequency separations will be referred to as the second difference: α α (α, 1; β, 2) − 1νBO (α, 1; β, 2). [11] d(α, 1; β, 2) = 1νobs

On the one hand, if α 6= β, d(α, 1; β, 2) provides a direct measure of any electronic isotope shift present, £ α β ¤ − TBX + ε, d(α, 1; β, 2) = TBX

[12]

where ε is the contribution to d from all sources except electronic isotope shift. This contribution ε can be estimated as p (ε M )2 + (εBO )2 . The uncertainty ε M arises in the determination of 1νobs from errors in the measurements and in the data reduction procedures; the value of ε M is discussed in Section IV. The uncertainty εBO arises in the determination of 1νBO primarily from the BO rovibronic constants of G–L (9, 10). These constants were reported as capable of reproducing the absolute frequency of any transition to within ±48 MHz. It will be assumed that this error is uniformly distributed over the electronic, vibrational, and rotational energies. Then from Eq. [1], it follows that εBO (in MHz) can be estimated as (

β

BO − T BO G α (B, X ; v10 , v100 ) − G BO (B, X ; v20 , v200 ) TBX BX + BO νBX νBX ) β α FBO (B, X ; v10 , v100 ; J10 , J100 ) − FBO (B, X ; v20 , v200 ; J20 , J200 ) , + νBX

εBO ≈ 48

[13]

where νBX is the approximate frequency of the transition under study; for the B–X transitions observed, νBX can be taken as 17 000 cm−1 . Since the numerators (in cm−1 ) are much smaller than 17 000, the value of εBO will be much less than 48 MHz. The first term in εBO obviously vanishes, but it is included here BO does not dependent on α or β. to emphasize the fact that TBX On the other hand, if α = β, the second difference d(α, 1; α, 2) given in Eq. [12] is independent of the electronic term value contribution. Nonetheless, such intra-isotopomer measurements can be very useful, in particular, to estimate the value of ε. If εBO vanishes, then d(α, 1; α, 2) should be the same

III. EXPERIMENTAL METHODS

The B–X system of 79 Br2 , 79,81 Br2 , and 81 Br2 was studied at high resolution using laser-induced fluorescence. The fluorescing molecules were in a molecular beam produced by injecting a 20% mixture (by pressure) of bromine into argon through a 30-µm-diameter nozzle at a backing pressure of 300 Torr. The beam was collimated using a 1-mm-diameter skimmer located 2 cm from the nozzle and an adjustable stainless steel slit placed 10 cm further downstream. The molecules were excited with the output beam of a Coherent CR699-21 tunable ring dye laser operating between 560 and 600 nm with Rhodamine 6G dye. The dye laser was powered by a Coherent Innova 420 argon ion laser. The dye laser radiation had a spectral width of approximately 2 MHz. The fluorescence was spatially filtered and sent through a redpass filter (Corning CS2-62) to remove any scattered laser light. The detector was a photomultiplier (RCA model 3034A) held at −30◦ C to reduce thermal noise. For phase-sensitive detection, the molecular beam was modulated at a frequency of 200 Hz with a mechanical chopper. The signal from the photomultiplier was sent to a lock-in amplifier and the output recorded in a computer file. The observed full width at half maximum for isolated spectral lines was approximately 8 MHz. A reduction by a factor of 100 in the Doppler width was achieved by having the laser and molecular beams perpendicular to one another. The data were calibrated using a system designed and built by Adam et al. (13) In a single scan over a range of 30,000 MHz, the experimental uncertainties were ±25 MHz and ±1 MHz for absolute and relative measurements, respectively. A full description of the experimental equipment and procedures is given elsewhere (14). IV. HYPERFINE-FREE FREQUENCY DIFFERENCES

IV.1. Intra-isotopomer Frequency Differences One of the motivations in measuring frequency separations between nearly coincident lines in the same isotopomer was to determine the uncertainty ε to be assigned to the interisotopomer measurements; see Eq. [12]. The first set of intraisotopomer measurements is summarized in Table 1. These nearcoincidences appeared for each vibrational band investigated because a bandhead systematically occurred between transitions

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TABLE 1 Frequency Differencesa for Intra-isotopomer Close Coincidences near the R-Branch Head Isotopomer

Vibrational Band

Rotational Transitions

79 Br

130 − 000 170 − 200 170 − 200 110 − 000 120 − 000 120 − 000 130 − 000 130 − 000 140 − 100 140 − 100 150 − 100 150 − 100 160 − 100 160 − 100 160 − 100 170 − 200 130 − 000 130 − 000 170 − 200

R(1) − R(2) R(2) − R(0) R(1) − R(2) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(4) − P(1) R(1) − R(2) R(2) − R(0) R(1) − R(2) R(2) − R(0)

2

79,81 Br

81 Br

2

2

Observed Frequency Difference 1νobs b 538.0(4.0) 46.6(1.4) 1054.2(1.2) 276.2(0.8) 947.3(2.3) 396.2(1.2) 756.7(5.1) 516.3(0.8) 593.8(4.6) 614.7(1.3) 377.0(1.4) 740.4(1.4) 150.8(3.0) 873.0(2.2) 456.2(1.4) 21.0(0.4) 754.1(1.0) 514.5(2.1) 6.2(1.6)

Calculated Frequency Difference 1ν B O c 538.2 48.2 1052.1 278.2 954.6 397.8 749.2 521.0 594.4 610.1 376.3 741.0 151.5 876.0 452.4 22.8 757.1 504.2 2.0

Second Difference d(α, 1; β, 2)d −0.2 −1.6 1.9 −2.0 −7.3 −1.6 7.5 −4.7 −0.6 4.6 0.7 −0.6 −0.7 −3.0 3.8 −1.8 −3.0 10.3 4.2

a

All frequencies are in MHz. For each row, the error in 1νobs listed below reflects only the scatter in the individual measurements averaged to form the entry in that row. A value of ±5.0 MHz is a conservative estimate of the error for each entry in this column. c The error in 1ν BO is estimated to be 2 kHz; see Section IV.1. d The error in d = [1ν obs − 1νBO ] is estimated to be ±5.0 MHz; see Section IV.1. b

R(1) and R(2). For each isotopomer, spectra could be recorded within a single laser scan either for R(1) and R(2), or for R(0) and R(2). For the 79,81 Br2 (160 –100 ) band, an additional nearcoincidence between transitions R(4) and P(1) was observed. To illustrate the reduction of the data, consider the first line of Table 1. This involves the B–X (130 –000 ) R(1) and R(2) rovibronic transitions in 79 Br2 . Each of the two rovbronic transitions is split into many components by hyperfine effects. In each case, the frequency separations between these components were recorded. Using a standard model for the hyperfine interactions (15, 16), the measured components were converted to the hyperfine parameters for the molecule and the hyperfine-free frequency separation was determined. The R(1) to R(2) separation 1νobs (79, R(1); 79, R(2)) was found to be 538.0 MHz. Next, the G–L rovibronic molecular parameters (9, 10) were used to determine the corresponding difference in the BO prediction; it was found that 1νBO (79, R(1); 79, R(2)) = 538.2 MHz. The measured second difference d(79, R(1); 79, R(2)) = −0.2 MHz; see Eq. [11]. The same procedure was used to determine the other entries in Table 1. This first data set compared intra-isotopomer frequencies belonging to the same vibrational bands for low rotational states. Consequently, the second differences d(α, 1; α, 2) are independent of the G α (W, v) and are insensitive to the F α (W, v, J ) involved. This makes these d(α, 1; α, 2) ideally suited for esti-

mating the uncertainty ε M associated with the experimental technique and the data reduction. Each of the observed frequency differences 1νobs listed in Table 1 was determined by averaging from 3 to 5 independent determinations. The statistical error listed varies from 0.4 to 5.1 MHz. The value of ε M will be conservatively taken to be ±5 MHz. In order to evaluate the overall uncertainty ε, the value of the uncertainty εBO in the determination of the calculated 1νBO is also required. Since intra-isotopomer transitions arising from the same vibrational band are being considered, the uncertainties introduced by the published BO parameters will be determined by the differences in rotational state energy polynomial i.e., by the last term in Eq. [13]. The resulting estimate for εBO is 2 kHz. Clearly, the expected uncertainty ε in d(α, 1; α, 2) for this data set is dominated by ε M . This agrees with the results listed in Table 1. The difference between the observed and calculated frequency separations is less than 5 MHz in magnitude for 16 out of the 19 entries. A second motivation for the current measurements of frequency separations between nearly co-incident lines in the same isotopomer was to provide constraints which will lead to improvement in the molecular parameters of molecular bromine. With an experimental error of only ±5 MHz, the addition of the measurements 1νobs (α, 1; α, 2) given in Table 1 to a global fit of the bromine data should aid in this purpose.

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TABLE 2 Frequency Differencesa for Intra-isotopomer Close Coincidences between P(J00 ) and R(J00 + 3) in the (170 –200 ) Vibrational Band 79 Br 2 1νobs 144.2(0.6) 192.6(1.2) 243.0(0.7) 294.3(1.3) 346.0(1.6) 451.6(0.9)

Transition P(1) − R(4) P(2) − R(5) P(3) − R(6) P(4) − R(7) P(5) − R(8) P(7) − R(10) a

79,81 Br

1ν B O 145.2 194.4 244.4 295.3 347.3 455.3

1νobs 68.1(0.6) 92.1(0.4) 115.4(0.8) 140.8(0.6) 170.2(0.9) 223.6(0.3)

1ν B O 69.1 92.9 117.5 142.9 169.4 226.3

81 Br 2 1νobs 1ν B O −6.2(1.6) −5.4 −6.6(0.6) −6.1 −8.7(0.2) −6.8 −7.2(1.0) −6.2 −7.5(0.6) −4.7 −1.8(0.6) +2.2

See the Footnotes to Table 1.

The second set of intra-isotopomer measurements summarized in Table 2 should provide additional useful constraints. Here, the near coincidences are between the P(J 00 ) and R(J 00 + 3) transitions of the (170 –200 ) band for each isotopomer. As was the case for Table 1, the same vibrational bands are involved for both members of the nearly co-incident pair. The observed separations of the P(J 00 ) and R(J 00 + 3) lines can be analyzed as 1νobs (α, P(J 00 ); α, R(J 00 + 3)) = c0 + c2 (J 00 + 2)2 , [14] (J 00 + 2) where ¡ ¢ ¡ ¢ c0 = −10 Bα170 + 6 Bα200 + 120 Dα170 − 24 Dα200 ; ¡ ¢ c2 = 20 Dα170 − 12 Dα200 .

[15] [16]

Here Bαv and Dαv are, respectively, the rotational constant and the quartic distortion constant for isotopomer α corresponding to vibrational state v. The sextic and higher-order distortion constants can be neglected for the range of J -values of interest here. For each isotopomer, values for c0 and c2 were obtained by fitting 1νobs to Eq. [14]. The results are listed in Table 3 along TABLE 3 Parametersa Defining the J00 -Dependence of να [P(J00 )] − να [R(J00 + 3)] in the (170 –200 ) Vibrational Band 79 Br

2

79,81 Br

81 Br

2

2

c0 (c0 )G L c c0 − (c0 )G L

47.87(13) 48.12 −0.25(13)

22.45(4) 22.7 −0.31(4)

−2.38(6) −2.06 −0.32(6)

c2 b (c2 )G L c c2 − (c2 )G L

0.0288(23) 0.0304 −0.0016(23)

0.0298(6) 0.0294 0.0004(6)

0.0268(13) 0.0284 −0.0016(13)

b

2

with the corresponding values (c0 )GL and (c2 )GL calculated from Eqs. [15] and [16] using the G–L molecular parameters (9, 10). For all three isotopomers, the observed and calculated values of c2 are in good agreement. However, for each isotopomer, the calculated value (c0 )G L is larger than the observed value by about 0.3 MHz. This suggests that some revision of the molecular rotational parameters is required for this vibrational state. Overall, the agreement between the observed and predicted values in Tables 1, 2, and 3 is excellent. This provides a stringent test of the measurements themselves, of the hyperfine analysis, and of the calculation of the frequency differences 1νBO (α, 1; α, 2) from the G–L molecular parameters. In addition, the internal consistency of the procedure used by Gerstenkorn and Luc to generate the molecular parameters for 79,81 Br2 and 81 Br2 using the BO isotopic relations has been tested and the calculation of the isotopic dependence of F α (v, J ) has been verified for the (170 –200 ) vibrational band. The intraisotopomer data indicate that the measurement and analysis uncertainties are of the order of ε M = ±5 MHz. Any larger discrepancies discovered during the analysis of the inter-isotopomer frequency separations must be attributed to some factor outside of measurement and data reduction errors. IV.2. Inter-isotopomer Frequency Differences The pairs of near-coincidences for different isotopomers were measured and analyzed following the procedures outlined in Section IV.1. The results are summarized in Table 4. Each pair of near-coincidences in Table 4 has been assigned a row index k for reference purposes. These have been divided into three categories according to the isotopomers involved: 79 Br2 with 79,81 Br2 ; 79,81 Br2 with 81 Br2 ; and 79 Br2 with 81 Br2 . In sharp contrast to the intra-isotopomer case, the inter-isotopomer second difference d(α; β) between the observed frequency difference 1νobs and its calculated counterpart 1νBO is the order of 1νBO itself; see Eq. [11]. For each isotopomer pair, the second differences obtained have been averaged:

a

See Eqs. [15] and [16]. All the parameters are in MHz. This is the experimental value. c This is the value calculated from the molecular parameters given in Refs. (9) and (10). b

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d(79; 79, 81) = 177(10) MHz;

[17]

d(79, 81; 81) = 209(7) MHz;

[18]

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ELECTRONIC ISOTOPE SHIFTS IN Br2

TABLE 4 Frequency Differencesa for Inter-isotopomer Close Coincidences Row Index k

Nearly Coincident Transitions

1 2 3 4 5 6 7 8 9 10 11 12

− 79,81 Br2 0 00 (11 -0 )R(18) − (110 -000 )R( 0) (110 -000 )R(18) − (110 -000 )R( 2) (130 -000 )P(15) − (130 -000 )R( 3) (130 -000 )P(16) − (130 -000 )P( 4) (130 -000 )P(17) − (130 -000 )R(10) (170 -200 )R(15) − (170 -200 )R( 4) (170 -200 )P(12) − (170 -200 )R( 4) (140 -100 )P( 7) − (170 -200 )R( 3) (140 -100 )R(11) − (170 -200 )R( 5) (140 -100 )R(12) − (170 -200 )R( 7) (140 -100 )R(14) − (170 -200 )R(10) (190 -200 )P( 6) − (130 -000 )P( 2) 79 Br

Observed Frequency Difference 1νobs b

Calculated Frequency Difference 1ν B O c

427.9(2.8) −720.1(4.4) 1038.6(5.2) 511.0(1.4) 1219.5(1.6) 299.6(0.6) 1057.6(2.3) 1071.1(1.9) 663.1(1.6) 971.9(4.0) 829.8(2.2) 459.4(2.2)

251.8 −902.1 854.7 330.8 1031.6 117.8 875.0 897.8 525.4 807.7 662.7 283.9

Second Difference d (α, 1; β, 2) d

2

176.1 182.0 183.9 180.2 187.9 181.8 182.8 173.3 167.7 164.2 167.1 175.5

i h = 177(10) d(79; 79, 81) = TB79X − TB79,81 X 2 − (140 -100 )R( 79,81 Br

13 14 15 16

Subgroup Average (MHz)

k = 1 to 2 179(7) k = 3 to 5 184(7) k = 6 to 7 182(7) k = 8 to 11 168(7)

k = 12 176(7)

81 Br

2

8) − (170 -200 )R( 1) (140 -100 )P( 7) − (170 -200 )P( 4) (140 -100 )P( 5) − (170 -200 )R( 3) (140 -100 )R(11) − (170 -200 )R( 8)

−106.7(3.5) 337.8(0.5) −458.9(3.0) 1086.1(2.2)

−317.4 134.4 −669.0 874.4

210.7 203.4 210.1 211.7

i h d(79, 81; 81) = TB79,81 − TB81X = 209(7) X 17

79 Br − 81 Br 2 2 (140 -100 )P(17) − (170 -200 )R(10)

126.1(2.1)

−260.0

386.1

¤ £ d(79; 81) = TB79X − TB81X = 386(7) a

The hyperfine contribution has been removed. All frequencies are in MHz. For each row, the error in 1νobs listed below reflects only the scatter in the individual measurements averaged to form the entry in that row. A value of ±5.0 MHz is a conservative estimate of the error for each entry in this column. c The error in 1ν BO is estimated to be ±5.0 MHz; see Section IV.2. d The error in d (α, 1; β, 2) for each individual row is estimated to be ±7.0 MHz; see Section IV.2. b

d(79; 81) = 386(7) MHz. To within the uncertainties specified here, £ α β ¤ d(α; β) = TBX − TBX .

[19]

[20]

In order to determine the uncertainties listed in Eqs. [17]–[19], ε as defined in connection with Eq. [12] was evaluated using the procedures discussed in Section IV.1. The measurement/data reduction uncertainty ε M is conservatively taken to be ±5 MHz, based on the reproducibility of the 1νobs quoted in Table 4. The uncertainty εBO based on the accuracy of the rovibronic constants can be estimated from Eq. [13]. Because vibrational contributions now enter and are larger than their rotational counterparts, εBO is larger than in the intra-isotopomer case. Here εBO ≤ 5 MHz. By combining ε M and εBO in quadrature, one obtains ε ≤ 7 MHz. The uncertainties listed in Eqs. [17]–[19]

were obtained by combining the scatter in the data points being averaged with this estimate of 7 MHz for ε. While the main objective of the analysis is to determine the electronic isotope shift, the data can also be inspected for evidence of vibrational contributions to d(α; β). Rows 1 through 12 in Table 4 report second differences involving 79 Br2 and 79,81 Br2 . Averages of d(79; 79, 81) have been calculated for subgroups of rows which involve the same vibrational quantum numbers (e.g., Rows 3 through 5). These subgroup averages are listed in Table 4. The errors listed are determined in the same manner as were the errors given in Eqs. [17]–[19]. To within these errors, it appears that all the subgroup averages are equal, with one possible exception. For the subgroup involving Rows 8 through 11, the average d[79, (140 –100 ); 79, 81, (170 –200 )] seems to be lower than the rest, but the effect is marginal. It is clear that any vibrational contribution to the second differences d(α; β) is less than 10 MHz in magnitude.

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BOOTH, OZIER, AND DALBY

The conclusion that d(α; β) equals the electronic isotope shift β α –TBX ] can be subjected to a stringent internal consistency [TBX check. It is clear that £

£ 79,81 ¤ £ 79 ¤ 79,81 ¤ 79 81 81 − TBX − TBX + TBX − TBX − TBX = 0. [21] TBX

If Eq. [20] applies, then d(α; β) must satisfy the same condition. This loop test can be applied using the average values d(α; β) given in Eqs. [17]–[19]. The result is that d(79 : 79, 81) + d(79, 81 : 81) − d(79 : 81) = [0 ± 14] MHz. [22] The agreement is excellent. A very similar loop test can be made using only the data involving the (170 –200 ) to (140 –100 ) transitions. From Table 4, d(79 (140 , 100 ); 79, 81(170 , 200 )) + d(79, 81(140 , 100 ); 81(170 , 200 )) − d(79(140 , 100 ); 81(170 , 200 )) = −[9 ± 10] MHz.

[23]

The agreement in this case is good. The loop test [23] differs from the loop test [22] only in that the subgroup average for rows 8 through 11 was used in the former where d(79 : 79, 81) is used in the latter. Nevertheless, loop test [23] is useful because most of the vibrational contributions cancel. The only vibrational contribution to the loop value of εBO in this case is an intra-isotopomer 79,81 0 00 0 00 term due to [G 79,81 BO (B, X ; 14 , 1 )–G BO (B, X ; 17 , 2 )]; see Eq. [13]. V. DISCUSSION

The inter-isotopomer data presented here provide a direct measurement of the electronic isotope shift of the B 350+ u – X 16g+ electronic state separation of molecular bromine. The secβ α ond difference d(α, 1; β, 2) has been shown to equal [TBX − TBX ] β α to within 10 MHz for the cases studied. Values of [TBX − TBX ] have been obtained for each of the three inter-isotopomer comparisons possible. See Table 4 and Eqs. [17]–[20]. This indicates that the electronic term value in the Gerstenkorn-Luc form has a significant isotopic dependence the order of 100 MHz. The reduced mass dependence of the isotopic shift observed here is somewhat unusual. Eqs. [8] and [20] lead to the prediction that · d(α; β) =

¸ 1 1 − 0BX . µα µβ

[24]

It is easily shown that (1/µ79,81 − 1/µ81 ) = (1/µ79 − 1/µ79,81 ). Since 0BX is an isotopically independent parameter (2), it follows from [24] that d(79, 81 : 81) should equal d(79 : 79, 81). The results obtained here contradict this. As can be seen from Table 4,

d(79, 81 : 81) − d(79 : 79, 81) ¤ £ 79 £ 79,81 79,81 ¤ 81 − TBX − TBX − TBX = 32(12) MHz. [25] = TBX The error limit quoted is an estimate of the maximum error possible, rather than an estimate of the standard deviation. This result is significantly different from zero. Similar results are obtained from Table 4 if other combinations of the second differences are used (rather than the averages). The cause of the anomaly in the isotopic dependence of the electronic terms values may well lie in the fact that a heteronuclear isotopomer, 79,81 Br2 , is being compared with two homonuclear isotopomers, 79 Br2 and 81 Br2 . It is assumed in Eq. [8] that the same mass-reduced matrix element 0BX applies to each isotopomer. However, there are interactions which are allowed in the heteronuclear case that are forbidden in the homonuclear case. In order to allow for such effects, 0BX will be taken as the mass-reduced matrix element for 79 Br2 and 81 Br2 , and [0BX + 1BX ] will be taken as the mass-reduced matrix element for 79,81 Br2 . Here (1/µ79,81 )1BX is the matrix element of these extra heteronuclear Hamiltonian terms. It follows then that 79,81 81 − TBX = (1/µ79,81 − 1/µ81 )0BX + (1/µ79,81 )1BX ; TBX 79,81 79 − TBX = (1/µ79 − 1/µ79,81 )0BX − (1/µ79,81 )1BX . TBX

[26]

From Eqs. [25] and [26], it is easily shown that the extra matrix element [1BX /µ79,81 ] = [16 ± 6] MHz and that 1BX = [1.06 ± 0.40] × 10−24 MHz-kg. Using the value for d(79; 81) given in Table 4, one obtains 0BX = [1.024 ± 0.019] × 10−21 MHz-kg. Note that 1BX is three orders of magnitude smaller than 0BX . This anomalous isotope dependence in the electronic term values has an important consequence for the models currently used to fit several isotopomers simultaneously (5, 7, 11). The expression used to fit the Dunham coefficients of several isotopomers is usually written as · ¸ me B me A j)/2 U 1 + 1 1 + , Yiαj = µ−(i+2 ij α MA i j MB i j

[27]

where M A and M B are the masses of the two atoms in isotopomer α, and me is the mass of the electron. In Eq. [27], Ui j , 1iAj , and 1iBj are isotopically independent parameters. Equation [27] does not make any distinction between heteronuclear and homonuclear isotopomers, and leads to the same reduced mass dependence in the electronic isotope shifts as that given in Eq. [8]. Recently, LeRoy (12) has cast Eq. [27] into a more convenient format for fitting experimental data. It appears to lead again to the same reduced mass dependence as that given in Eq. [8]. These currently accepted models should be modified reflect the anomaly reported here in the isotopic dependence of the electronic terms values for the B–X electronic system of diatomic bromine.

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ELECTRONIC ISOTOPE SHIFTS IN Br2

ACKNOWLEDGMENTS The authors thank Dr. A. J. Merer and Dr. A. G. Adam for the use of valuable equipment and for many helpful suggestions. I. O. and F. W. D. express their gratitude for financial support to the Natural Sciences and Engineering Research Council of Canada.

6. 7. 8. 9. 10. 11.

REFERENCES

12. 13.

1. G. Herzberg, “Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules,” 2nd ed. Van Nostrand, New York, 1950. 2. P. R. Bunker, J. Mol. Spectrosc. 28, 422–443 (1968). 3. P. R. Bunker, J. Mol. Spectrosc. 35, 306–313 (1970). 4. P. R. Bunker, J. Mol. Spectrosc. 42, 478–494 (1972). 5. P. R. Bunker, J. Mol. Spectrosc. 68, 367–371 (1977).

14. 15. 16.

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J. K. G. Watson, J. Mol. Spectrosc. 45, 99–113 (1973). J. K. G. Watson, J. Mol. Spectrosc. 80, 411–421 (1980). J. L. Dunham, Phys. Rev. 41, 721–731 (1932). S. Gerstenkorn and P. Luc, J. Phys. (Paris) 48, 1685–1696 (1987). S. Gerstenkorn and P. Luc, J. Phys. (Paris) 50, 1417–1432 (1989). A. H. Ross, R. S. Eng, and H. Kildal, Opt. Commun. 12, 433–438 (1974). R. J. LeRoy, J. Mol. Spectrosc. 194, 189–196 (1999). A. G. Adam, A. J. Merer, D. M. Steunenberg, M. C. L. Gerry, and I. Ozier, Rev. Sci. Instrum. 60, 1003–1007 (1989). J. L. Booth, Ph.D. thesis, The University of British Columbia, Vancouver, Canada, 1994. J. Vigu´e, Ph.D. thesis, L’Ecole Normale Sup´erieure, Universit´e de Paris, Paris, France, 1978. M. Broyer, J. Vigu´e, and J. C. Lehmann, J. Phys. (Paris) 39, 591–609 (1978).

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