Reanalysis of the laser-induced (4658 ŏ) B1Πu→X1Σ+g fluorescence of Na2

ÓPTICA PURA Y APLICADA – Vol. 38, núm. 1, - 2005

Reanalysis of the laser-induced (4658

Å)

B

1

Πu→X

1

Σ

+g

fluorescence of

Na2

Reanálisis de la fluorescencia-inducida por láser (4658 Å)

B

1

Π

u

→

X

1

Σ

+g

en Na

2

J.J. Camacho*, A. Pardo, A.M. Polo, E. Martín and J.M.L. Poyato

1. Departamento de Química-Física Aplicada, Facultad de Ciencias, Universidad Autónoma de Madrid. Cantoblanco. 28049-Madrid. Spain.

REFERENCES.

[1] W. Demtröder, M. McClintock and R.N. Zare, J. Chem. Phys., 51, 5495-5508 (1969). [2] W. Demtröder and M. Stock, J. Mol. Spectrosc., 55, 476-486 (1975).

[3] P. Kusch and M.M. Hessel, J. Chem. Phys., 68, 2591-2606 (1978).

[4] K.K. Verma, J.T. Bahns, A.R. Rajaei-Rizi, W.C. Stwalley. and W.T. Zemke, J. Chem. Phys., 78, 3599-3613 (1983).

[5] R.F. Barrow, J. Verges, C. Effantin, K. Hussein and D'Incan, Chem. Phys. Lett.,104, 179-185 (1984). [6] O. Babaky and K. Hussein, Can. J. Phys.,67, 912-919 (1989).

[7] W.T. Zemke and W.C. Stwalley, J. Chem. Phys.,100, 2661-2670 (1994).

[8] K.M. Jones, S. Maleki, S. Bize, P.D. Lett, C.J. Williams, H. Richling, H. Knöckel, E. Tiemman, H. Wang, P.L. Gould and W.C. Stwalley, Phys. Rev. A, 54, 1006-1009 (1996).

[9] J. Keller and J. Weiner, Phys. Rev. A, 29, 2943-2945 (1984). [10] G. Gerber and R. Möller, Phys. Rev. Lett., 55, 814-817 (1985).

[11] H. J. Vedder G.K. Chawla and R. W. Field, Chem. Phys. Lett., 111, 303-308 (1984). [12] G.K. Chawla, H. J. Vedder and R. W. Field, J. Chem. Phys., 86, 3082-3088 (1987).

* _{Corresponding author.Tel.: (34)-914978656; Fax:(34)-914974512; E-mail address: [email protected]}

ABSTRACT:

The fluorescence spectrum of Na2 induced by the 4658 Å line of an argon ion laser has been

reanalyzed. We have observed five fluorescence series for the B1_Π

u→X1Σ+gsystem corresponding

to the excitation transitions: v’=17,J’=38←v”=4,J”=39, v’=25,J’=3←v”=9,J”=3,

v’=28,J’=55←v”=9,J”=55, v’=26,J’=32←v”=9,J”=33 and v’=29,J’=39←v”=10,J”=40. The

two later series are reported for the first time. The radiative transition probabilities for the two new fluorescence series were calculated using hybrid potential energy curves.

Key words: Laser induced fluorescence, transition probabilities, radiative lifetime.

RESUMEN:

Se ha analizado nuevamente el espectro de fluorescencia del Na2 inducida por la línea de

4658 Å de un láser de argon ionizado. Hemos observado cinco series de fluorescencia para el

sistema B1_Π

u→X1Σ+g que corresponden a las transiciones de excitación:

v’=17,J’=38←v”=4,J”=39, v’=25,J’=3←v”=9,J”=3, v’=28,J’=55←v”=9,J”=55,

v’=26,J’=32←v”=9,J”=33 y v’=29,J’=39←v”=10, J”=40. Las últimas dos series se presentan

por primera vez. Utilizando curvas de energía potencial híbridas se calcularon probabilidades de transición radiativa para las dos nuevas series de fluorescencia.

[13] H. Richter. H. J. Vedder and R. W. Field, Chem. Phys., 157, 217-225 (1991).

[14] W.J. Stevens, M.M. Hessel, P.J. Bertoncini and A.C. Wahl, J. Chem. Phys., 66, 1477-1500 (1977). [15] M. M. Hessel, E.W. Smith and R.E. Drullinger, Phys. Rev. Lett., 33, 1251-1254 (1974).

[16] W. Demtröder, W. Stetzenback, M. Stock and J. Witt, J. Mol. Spectrosc., 61, 382-394 (1976).

[17] J.J. Camacho, J.M.L. Poyato, A.M. Polo and A. Pardo, J. Quant. Spectrosc. Radiat. Tranfer, 56, 353-361 (1996).

[18] J.J. Camacho, J. Santiago, A. Pardo, D. Reyman and J.M.L. Poyato, J. Quant. Spectrosc. Radiat. Tranf., 65, 729-749 (2000).

[19] J.M.L. Poyato J.J. Camacho, A.M. Polo and A. Pardo, Spectrochim. Acta A, 51, 1879-1890 (1995). [20] J.J. Camacho, A. Pardo, A.M. Polo, D. Reyman and J.M.L. Poyato, J. Mol. Spectrosc., 191, 248-257 (1998).

[21] A. Pardo, J. Mol. Spectrosc., 199, 225-229 (2000).

[22] J.J. Camacho, J. Santiago, A. Pardo, D. Reyman and J.M.L. Poyato, Spectrochim. Acta A, 56, 769-781 (2000).

[23] J.J. Camacho, A.M. Polo, A. Pardo and J.M.L. Poyato, J. Quant. Spectrosc. Radiat. Tranfer, 74, 667-681 (2002).

[24] J.M.L. Poyato J.J. Camacho, A.M. Polo and A. Pardo, Spectrochim. Acta A, 52, 409-418 (1996). [25] J.J. Camacho, A. Pardo and I.P. Acín, J. Phys. B: At. Mol. Opt. Phys., 34, 2597-2614 (2001). [26] J.J. Camacho, A. Pardo, and J.M.L. Poyato, Submitted for publication.

[27] M. Lapp and L.P. Harris, J. Quant. Spectry. Radiat. Trans., 6, 169-179 (1966). [28] M. D'Orazio and B. Schrader, J. Raman Spectry., 2, 585-589 (1974).

[29] J. Reader, C.J. Sansonetti and J.M. Bridges, Appl. Opt., 35, 78-83 (1996).

[30] G. Herzberg, Molecular Spectra and Molecular Structure, Van Nostrand, Princeton (1950).

1.- Introduction.

A vast amount of experimental work has been done on the X1Σ+

g and B1Πu states of Na2, some

very precise and elaborate, but its investigation is still far for complete. Through laser-induced fluorescence (LIF) spectroscopy, the Β1Πu →X1Σ+g

band system of sodium dimer has been investigated by Demtröder et al. [1,2], Kusch and Hessel [3], Verma et al. [4], Barrow et al. [5], Babaky and Hussein [6] and Zemke et al. [7]. Both X1_Σ+

g [8] and

Β1_Π

u (see refs. Keller et al. [9], Gerber et al. [10],

Vedder et al. [11], Chawla et al.[12] and Richter et al. [13]) electronic states are known near their dissociation limits. For the Β1_Π

u →X1Σ+g band

system of Na2 there are several theoretical [14] and

experimental studies reporting the variation of the Β1_Π

u →X1Σ+g electronic transition dipole moment

with internuclear distance based on line intensities [15] and from lifetimes measurements [16]. In previous publications, we have reported detailed studies of the fluorescence spectrum of Na2

transition Β1_Π

u →X1Σ+g induced by nine Ar+ laser

lines: 4545.052 [17], 4579.349 [18], 4657.901 [19],

4726.868 [20], 4764.864 [21], 4879.863 [22],

4965.079 [23], 5017.163 [24] and 5145.308[25] Å

(air). An important motivation in these works was the analysis (direct and satellite lines) of LIF spectra with especial emphasis on determination of precise fluorescence line intensities and comparison with calculated accurate radiative transition probabilities

and lifetimes. Another important motivation in this kind of studies was the possibility of new optically pumped laser (OPL) candidates. A detailed study of line intensities for LIF spectra combined with transition probability calculations can be very useful in order to make the proper OPL assignments and to show candidates lines which may lase under certain conditions. Recently we have reported an analysis (107 fluorescence series and rotational relaxation lines for the most intense vibrational bands) of the Β1_Π

u →X1Σ+g band system of sodium dimer

determining molecular constants and potential energy curves up to dissociation limit for both electronic states [26].

The present paper reports our observations of the fluorescence spectrum of Na2 excited by a 4658-Å Ar+ _{laser line. Five fluorescence series in}

the blue-yellow region 4500-5800 Å have been observed and identified. Some series are reported for the first time. Also we report our observations of accurate relative intensities of fluorescence and our intensity calculations for the new reported fluorescence series. For the calculation of wavefunctions and radiative transition probabilities between vibrational-rotational states implied in the transitions we have used precise potential energy curves (basically type-RKR) for both Β1Πu and

X1_Σ+

g electronic states constructed up to dissociation

satisfactory agreement between the experimental measurements and the calculated theoretical values. From the intensity ratio of some satellite lines to parent lines for the most intense fluorescence series, collision-induced rotational transition rates and average cross sections have been obtained for some vibrational bands.

2.- Experimental details.

Sodium metal, which was 99.90 % pure and contains potassium metal as ≈0.05% impurity, was enclosed in a crossed stainless steel heat-pipe oven connected to a vacuum system and with cooled windows in order to prevent condensation of sodium vapor on the cell windows. The relative ratio of molecular concentration of Na2 dimers to atomic

sodium Na vapor concentration is generally very small and increases with temperature (about ≈1% at

300 ºC, ≈2.5 % at 400 ºC, ≈6% at 550 ºC [27]). A

chromel-alumel thermocouple was placed into the central zone of the pipe oven in order to monitor the temperature of fluorescence region. The pressure of the heat-pipe oven was controlled by varying the temperature and the buffer gas pressure. Several experiments using argon as buffer gas and without buffer gas were done. Different spectra were recorded at different temperatures between 300 ºC

and 550 ºC corresponding to a total sodium vapor

pressure in the range 0.01 to 10torr. An argon ion laser (Spectra Physics model 171) was tuned tothe wavelengths of 4657.901 Å in air using an internally reflecting prism as part of the laser cavity.

Fig.1.- A photograph of the system used to produce and to register the LIF spectra.

The fluorescence from the sodium dimer molecules was collected at right angles to the laser propagation direction. A small region of molecular fluorescence was imaged with the help of a lens, onto the entrance slit of different spectrometers. The spectra at moderate resolution were recorded with a

1/8 m spectrometer Oriel (25 µm slit) with an Andor

DU420-OE (open electrode) CCD (charge-coupled

device) camera (1024x256 array with 26 µm2_{) with}

thermoelectric cooling down to –90 ºC. High resolution spectra were recorded on a high light-gathering 3-m modified Huet spectrograph. Both a heavy-flint double prism and a plane reflecting diffraction grating were used as dispersive elements. The reciprocal linear dispersion is approximately constant with wavelength (2.85 Å/mm spectral resolving power about 30000) for the grating, whereas for heavy flint glass it is 3.2 Å/mm at 4000 Å (theoretical maximum resolving power of 70400)

and 11.4Å/mm at 5890 Å (22400).

The intensity response of the detection system was calibrated with a standard (Osram

No.6438, 6.6-A, 200-W) halogen lamp [28] and a

Hg/Ar pencil lamp [29]. The spectral wavelength

calibration of the spectrometers was made using several (Hg/Ar, Neon, Krypton and Xenon) pencil lamps. Our estimate of uncertainty in the experimental intensity measurements is about 20%.

3.- Results and discussion.

Figure 2 shows a LIF spectrum of Na2

(Β1_Π

u →X1Σ+g band system) excited by the

4657.901-Å (air) argon ion laser line. The

densitometric scan corresponds to a exposure time of about five minutes and temperature ∼450 ºC,

recorded from the heavy-flint double prism spectrometer. The prism recording allows to detect very low line intensities and also eliminates overlapping between lines of high diffraction orders when it is used a diffraction grating as dispersive element. Other spectra to high resolution were obtained using a plane reflecting diffraction grating as dispersive element. Five fluorescence series have been observed and identified. Three of these consist of R-P doublets and the rest consist of Q lines. The assignment of the individual lines of such five fluorescence series is indicated plotting a column for each laser induced fluorescence line with a proportional height to the Franck-Condon factor. Also for many fluorescence lines is shown a label indicating the wavenumber and the assignment of the discrete resonance fluorescence line. For example, a label 17901.26-26P(33)40 indicates the fluorescence line: ν(v’=26, J’=32; v”=40,

J”=33)=17901.26 cm-1_{. In fig. 1 one also recognizes}

collision induced satellite lines with ∆J’=±1, ±2…

for the most intense bands of the strongest fluorescence series v’=17, J’=38→v”,J”=39 and

17400174501750017550176001765017700177501780017850179001795018000 17 99 1.3 1 17 91 8. 15 -28Q(55)4 1 17 848 .23 17 78 1. 64 177 18 .50 1765 8.9 4 17 603 .07 1793 4. 75-2 5Q( 3) 39 1785 1.5 5 17 771 .18 3 17 99 5.2 6-2 6R (3 1)39 17 914 .20 -26 R (3 1)40 17 836 .06 -26 R (3 1)41 17 76 0.9 2-2 6R( 31)4 2 17 982. 03-2 6P(33 )39 17 901 .26 -26 P( 33 )4 0 17 823 .42 -26 P( 33 )4 1 17 74 8.5 9-2 6P(33 )42 18 005 94 29 R( 38) 41 17 93 1. 98 -2 9R (3 8) 42 17 86 1. 18 -2 9R( 38) 43 17 793. 61 -2 9R( 38) 44 17 72 9. 41-29 R (38 )4 5 17 66 8. 66 -2 9R( 38) 46 17 61 1. 49 -29R( 38 )47 17 558 .0 1-29 R( 38) 48 17 990 .7 4-29 P( 40 )4 1 17 91 7. 17 -2 9P (4 0) 42 17 84 6. 77 -2 9P (4 0) 43 17 779. 63 -2 9P (4 0) 44 17 71 5. 87-29 P (40 )4 5 17 65 5. 59 -2 9P (4 0) 46 17 59 8. 91 -29P( 40) 47 17 545 .9 5-29 P( 40 )4 8

Wavenumber / cm-1

18600 18650 18700 18750 18800 18850 18900 18950 19000 19050 19100 19150 19200

191 46. 92 -17R( 37) 21 19024. 96-17 R (37) 22 18904. 91-17R( 37) 23 18786. 80-17 R (37) 24 18670. 69-17R( 37) 25 191 26. 80 -17P (3 9) 21 19005. 04-17 P (39) 22 18885. 19-17P (39) 23 18767. 29-17 P (39) 24 18651. 39-17P (39) 25 19 195 .22 19 08 7.85 18 98 2.71 18 87 9.8 6 18 77 9.3 5 18 68 1.2 5 19 01 6. 62 18 90 6. 74 18 79 9.0 1 18 69 3.47 19 16 8.4 5 19 05 7. 92 18 94 9.5 4 18 84 3. 34 -2 6R (3 1) 30 18 73 9.3 7 18 63 7.7 0-2 6R(3 1)3 2 19 15 2.4 5 19 04 2. 11 18 93 3.9 2 18 82 7. 93 -2 6P (3 3) 30 18 72 4.1 8 18 62 2.7 2-2 6P(33 )32 19 10 8. 17 19 00 2.75 18 89 9.5 8 18 79 8. 73 18 70 0. 26 18 60 4. 22 -29 R( 38 )3 4 19 19 6.7 1 19 08 9. 32 18 98 4.15 18 88 1.2 4 18 78 0. 66 18 68 2. 46

Wavenumber / cm-1

19800 19850 19900 19950 20000 20050 20100 20150 20200 20250 20300 20350 20400

2032 5. 59-17 R( 37 )1 2 20187 .75-17R( 37 )1 3 2005 1. 57 -1 7R( 37) 14 199 17. 08 -17R( 37) 15 ₂₀₃₀ 3. 87-17 P (39) 12 20166 .20-17P( 39) 13 2003 0. 19 -1 7P (3 9) 14 198 95. 87 -17P( 39) 15 20 38 1. 80 20 25 4.4 9 20 12 9. 01 20 00 5. 39 19 88 3. 67 20 35 5. 25 20 22 4. 35 20 09 5. 19 19 96 7. 81 19 84 2. 24 20 38 2. 51 20 25 2.7 3 20 12 4. 73 19 99 8.5 4 19 87 4. 18 20 36 4. 81 20 23 5.1 9 20 10 7. 35 19 98 1.3 1 19 85 7. 11 20 40 4 08 20 27 6.6 7 20 15 1. 07 19 90 5. 46 20 382. 87 20 25 5.6 5 20 13 0. 25 20 00 6.7 0 19 88 5. 03

Wavenumber / cm-1

18000 18050 18100 18150 18200 18250 18300 18350 18400 18450 18500 18550 18600

18 556. 60 -17R (37) 26 18444. 58-17 R( 37) 27 183 34. 66 -17R (37) 2 1822 6. 91 -17R (37) 29 18121. 37-17R (3 7) 30 18 537. 52 -17P (3 9) 26 18425. 72-17 P (39) 27 183 16. 04 -17P (3 9) 28 1820 8. 52 -17P (3 9) 29 18103. 22-17P (39) 30 18585.61 18492.50 18402.00 18229.10 18146.88 18067.58 18590.18 18489.1 8-25Q (3 )33 18294.30-25Q (3) 35 18020.69-25Q (3)3 18538.38-26R (31)33 18441.46-26 R (31)34 18347.02-26R (31)35 18255.10-26R (31)36 18165.79-26R (31)37 995 6 6 (3 )39 18523.62-26P (3 3)33 18426.94-26 P( 33)34 18332.73-26P (33) 35 18241.07-26P( 33)36 18152.01-26P( 33)37 18604 22 29R (38) 34 18510.6 8 18419.7 1 18331.37 18245.74 18162.91 1 8082.94-29R (3 8)40 18005 .94 -29R (38)41 18586 .71-29P (40)34 18493.4 6 18402.8 0 18314.77 18229.48 18146.98-29P( 40)39 1 8067.37-29P (40) 40

Wavenumber / cm-1

192001925019300 19350 19400 19450 19500 19550 1960019650197001975019800

1978 4. 3-17R( 37) 16 19 653. 25-17R( 37) 17 19523 .96-17R( 37) 18 19396 .45-17 R (37) 19 192 70. 76 -1 7R (3 7) 20 1976 3. 26 -1 7P (3 9) 16 19 632. 38-17P (39) 17 19503 .27-17P( 39 )1 8 19375 .95-17 P (39 )1 9 192 50. 45 -1 7P (39) 20 19763.88 19646.05 19530.23 19416.46 19304.77 99 5 19718.51 19596.64-25Q (3 )23 19476.68 19358.66-25Q (3 )25 19242.62 19751.69 19631.10 19512.44 1939 5.75 19281.08 19734.79 19614.36 19495.88 1937 9.37 19264.88 19785.51 19667.51 19551.50 19 325.60

19215.81 19647.50 19765.29

19531.71

19417.95

19

306.27

19196.71

Wavenumber / cm-1

20400 20450 20500 20550 20600 20650 20700 20750 20800 20850 20900 20950 21000

20893. 15-17 R (37) 8 2074 8. 87 -17R (3 7) 9 20606. 16-17R (37) 10 20465. 07-17R (37) 11 20870. 81-17 P (39) 8 2072 6. 68 -17P (39) 9 20584. 13-17P (3 9) 10 20443. 19-17P (3 9) 11 20908.77 20774.42 20641.79 20510.91 20895.91 20758 .24 20622.23 20487.89 20918.85 20782.22 20647.28 20514.03 20900.58 20764.10 20629.29 20496.19 20931.35 20796.95 20664.25 20533.29 20404 .08 20909.42 20775.19 20642.67 20511.89

21000 21050 21100 21150 21200 21250 21300 21350 21400

21

335

.3

5-17

R(

37

)5

211

86.

41

-1

7R

(3

7)

6

210

39.

00

-1

7R

(3

7)

7

21

312

.5

6-17

P

(39)

5

211

63.

77

-1

7P

(3

9)

6

210

16.

51

-1

7P

(3

9)

7

21321.

89

2118

2.53

21

04

4.

81

2131

8.65

2117

6.14

210

35.22

21

338.55

2

1197.0

3

210

57.12-26

R

(31)

12

21

319.88

2

1178.4

9

210

38.72

213

44.60

21205

.20

210

67.45

213

22.17

21182

.93

210

45.35

Wavenumber / cm-1

21400 21450 21500 21550 21600 21650 21700 21750 21800 21850 21900

21

79

1.

25

-1

7R

(3

7)

2

21

63

7.

78

-1

7R

(3

7)

3

21

48

5.

81

-17R

(3

7)

4

21

76

8.

03

-1

7P

(3

9)

2

21

61

4.

71

-1

7P

(3

9)

3

21

46

2.

88

-1

7P

(3

9)

4

2

189

5.48

-2

8Q(

55

)6

217

49.7

0-28Q

(5

5)

7

21

605.

50

2

1462

.89

217

55.6

2

216

08.

41

21

462

.75

2

177

2.62

21

626

.37

21

481.

67

2

175

3.56

21

607

.43

21

462.

87

2

1772

.46

216

28.2

5

214

85.6

2

2

1749

.55

216

05.4

925

214

63.0

3

Wavenumber / cm-1 Laser: 4657.901 (air) 21462.89 cm-1

(vacuum)

Fig. 2.- The Na2 Β1Πu →X1Σ+g fluorescence spectrum

excited by the Ar+ _{laser line at 4657.901 Å.}

Table I lists the five fluorescence series from Β1_Π

u

levels (v’=17, J’=38), (28, 55), (25, 3), (26, 32) and (29, 39) originating from (v”=4, J’=39), (9, 55), (9,

3), (9, 33) and (10, 40), respectively.

TABLE I.- Na2 (Β1Πu →X1Σ+g) transitions excited at

4657.901 Å-air (21462.89 cm-1 _{in vacuum). The transition}

frequencies (in wavenumber in vacuum) are calculated from the Dunham coefficients.

ν (cm-1) v' , J’ v" , J”

21462.88 17 , 38 4 , 39

21462.89 28 , 55 9 , 55

21462.75 25 , 3 9 , 3

21462.87 26 , 32 9 , 33

21463.03 29 , 39 10 , 40

These series are excited to high vibrational levels

(v’=17, 25, 26, 28 and 29) in the Β1Πu state and

show mostly Stokes lines on the short-wavenumber side on the laser line. Taking into account that

G’(v’=10)<νlaser + G"(v"=0) -Te < G’(v’=11), the

laser line could presumably excite vibrational levels

of the excited electronic state with v'≥11. Besides, in this case, the excitation transitions will take place from high v" values of the X1_Σ+

g state and thus the

fluorescence spectrum will present a great number of anti-Stokes lines. The assignments were done by matching the measured wavenumbers of the fluorescence spectral lines with the wavenumbers calculated from the Dunham coefficients reported in Refs. [3,26]. A further confirmation of the correct assignment of the fluorescence series is also provided by the good agreement between the measured and calculated radiative transition probabilities.

In order to calculate the intensities in the Β1_Π

u →X1Σ+g fluorescence transitions, hybrid

potentials for both electronic states were constructed (see Refs. 20, 25). As example, figure 3 shows the potential energy curves obtained from the rotationless molecule U”0(R) and U’0(R) and for the

rotating molecule U”39(R) with J”=39 of the X1Σ+g

state and U’38(R) with J’=38 of the Β1Πu state. The

effective potential energy curves (in cm-1_{) for the}

rotating molecule were obtained from

[

2

]

2

0 ( 1)

4 ) ( )

( = + J J+ −Λ

R c R U R

UJ _πh_µ (1)

where Λ is the component of the electronic orbital angular momentum along the internuclear axis, being Λ=1 for the B state and Λ=0 for the X state. Plotted in addition to the potential energy curves are the probability density distributions involved, in the laser transition v’=17, J’=38←v”=4, J”=39 which produces the most intense fluorescence series and, in the most intense P fluorescence lines v’=17,

J’=38→v”=27, J”=39, v’=17, J’=38→v”=28,

J”=39, v’=17, J’=38→v”=29, J”=39 and v’=17,

J’=38→v”=30,J”=39.

2.5 3.0 3.5 4.0 4.5 5.0

0 2500 5000 7500 10000 12500 15000 17500 20000 22500

U'₀(R)

U'₃₈(R)

ν17, 38

-->

27

, 39

ν17, 38

-->

30

, 39

U''₃₉(R)

Ψ2 v''=30,J''=39 Ψ2

v''=29,J''=39 Ψ2

v''=28,J''=39

U''₀(R)

Ψ2 v'=17,J'=38

Ψ2 v''=27,J''=39 ν17, 3

8 <

4, 39

Na

₂

B1Π_u

X1Σ+_g

Internuclear distance (Å)

Ψ2

v''=4,J''=39(R)

E

ner

gy

(

cm

-1)

Fig. 3.- Potential energy curves and probability distribution functions implied in the laser transition v’=17,

J’=38←v”=4, J”=39 of the most intense fluorescence

series (in the spectrum excited by the Ar+ _{laser line at}

It is obvious that the intensities of the transitions

v’=17,J’=38→v”=27-30,J”=39 will be maximum

because the right terminal maximum of the upper probability density distribution

Ψ

2v'=17,J'=38

(

R

)

lies vertically above the right terminal maximum of the lower probability density distributions

)

(

39 " , 30 27 "

2

_R

J v= − =

Ψ

.

Tables II and III give the relative theoretical intensities of the two new fluorescence series reported in this paper. The method of computing these intensities has been described before (see for example Ref. [25]). Essentially, given the full hybrid potential energy curves for both X1_Σ+

g and Β1Πu



electronic states of Na2 (Refs. 20, 25 and 26), we

solve numerically the radial Schrödinger equation to obtain the energy eigenvalues Ev,J and the

rovibrational eigenfunction

v

,

J

≡

Ψ

_v,J. The

calculated rovibrational eigenfunctions

v

,'

J

'

for the upper Β1_Π

u state and

v

"

,

J

"

for the lower

X1_Σ+

g state are further combined with the B-X

transition dipole moment function D(R) (Refs. 14-16) to calculate the Einstein

A

_{v,'J'→v",J"} coefficient of spontaneous emission (in s-1_{) for each fluorescence}

transition. The Einstein A coefficient can be written as

" , " ) ( ' ,' 3

64

" ,' 3 4 " , " '

,' S v J D R v J

h

Av J v J J J

ν π

=

→ (2)

where

S

_J,'J" is the line strength or Hönl-London factor. By using equation (2), we have obtained Einstein A coefficient for the two new assigned fluorescence series observed in our spectrum. Tables II and III give the frequencies and Einstein A

coefficient of spontaneous emission for the fluorescence series: v’=26, J’=32 → v”, J”=31;33

and v’=29, J’=39 → v”, J”=38;40.

TABLE II.- Frequency and Einstein A coefficients of spontaneous emission for the fluorescence series v’=26,

J’=32 → v”, J”=31 and 33 of Na2 (Β1Πu →X1Σ+g)

excited at 4657.901 Å. *Laser line.

R(31) R(31) P(33) P(33)

v" ν

cm-1 ₁₀A 7 _s-1 _cmν-1 ₁₀A 7 _s-1

5 22069.8 0.0225 22050.5 0.0206

6 21920.4 0.0935 21901.2 0.0873

7 21772.6 0.2656 21753.6 0.2530

8 21626.4 0.4930 21607.4 0.4816

9 21481.7 0.5341 21462.9* 0.5410

10 21338.6 0.2357 21319.9 0.2576

11 - 0.0007 - 0.0010

12 21057.1 0.2166 21038.7 0.1975

13 20918.8 0.3207 20900.6 0.3301

14 20782.2 0.0325 20764.1 0.0458

15 20647.3 0.1368 20629.3 0.1169

16 20514.0 0.2838 20496.2 0.2928

17 20382.5 0.0106 20364.8 0.0199

18 20252.7 0.2016 20235.2 0.1810

19 20124.7 0.1943 20107.3 0.2150

20 19998.5 0.0225 19981.3 0.0118

21 19874.2 0.2958 19857.1 0.2931

22 19751.7 0.0256 19734.8 0.0413

23 19631.1 0.2241 19614.4 0.2003

24 19512.4 0.1511 19495.9 0.1787

25 19395.8 0.1067 19379.4 0.0808

26 19281.1 0.2716 19264.9 0.2938

27 19168.5 0.0316 19152.4 0.0158

28 19057.9 0.3513 19042.1 0.3623

29 - 0.0048 - 0.0012

30 18843.3 0.4178 18827.9 0.4205

31 - 0.0006 - 0.0002

32 18637.7 0.5072 18622.7 0.5107

33 - 0.0006 - 0.0000

34 18441.5 0.6486 18426.9 0.6759

35 18347.0 0.1043 18332.7 0.0596

36 18255.1 0.6656 18241.1 0.7569

37 18165.8 0.7515 18152.0 0.6573

38 18079.1 0.0996 18065.6 0.1763

39 17995.3 2.0263 17982.0 2.1745

40 17914.2 2.5338 17901.3 2.4661

41 17836.1 1.1694 17823.4 1.0619

42 17760.9 0.2560 17748.6 0.2181

43 17688.9 0.0273 17676.9 0.0216

TABLE III.- Frequency and Einstein A coefficients of spontaneous emission for the fluorescence series v’=29,

J’=39 → v”, J”=38 and 40 of Na2 (Β1Πu →X1Σ+g)

excited at 4657.901 Å. *Laser line.

R(38) R(38) P(40) P(40)

v" ν

cm-1 ₁₀A 7 _s-1 _cmν-1 ₁₀A 7 _s-1

7 21918.2 0.0946 21895.2 0.0918

8 21772.5 0.2399 21749.5 0.2346

9 21628.2 0.4037 21605.5 0.3986

10 21485.6 0.3991 21463.0* 0.3995

11 21344.6 0.1563 21322.2 0.1615

12 - 0.0008 - 0.0003

13 21067.4 0.1748 21045.3 0.1684

14 20931.4 0.2482 20909.4 0.2496

15 20796.9 0.0305 20775.2 0.0342

16 20664.2 0.0897 20642.7 0.0835

17 20533.3 0.2298 20511.9 0.2301

18 20404.1 0.0260 20382.9 0.0299

19 20276.7 0.1122 20255.6 0.1053

20 20151.1 0.1958 20130.2 0.1989

21 - 0.0008 - 0.0005

22 19905.5 0.1962 19885.0 0.1902

23 19785.5 0.0969 19765.3 0.1038

24 19667.5 0.0646 19647.5 0.0576

25 19551.5 0.2208 19531.7 0.2236

26 - 0.0001 - 0.0002

27 19325.6 0.2368 19306.3 0.2313

28 19215.8 0.0492 19196.7 0.0568

30 19002.7 0.1509 18984.1 0.1614

31 18899.6 0.0969 18881.2 0.0855

32 18798.7 0.2543 18780.7 0.2643

33 18700.3 0.0460 18682.5 0.0363

34 18604.2 0.3582 18586.7 0.3671

35 18510.7 0.0281 18493.5 0.0193

36 18419.7 0.4721 18402.8 0.4831

37 18331.4 0.0466 18314.8 0.0333

38 18245.7 0.5943 18229.5 0.6174

39 18162.9 0.1917 18147.0 0.1624

40 18082.9 0.5733 18067.4 0.6212

41 18005.9 0.8832 17990.7 0.8419

42 17932.0 0.0532 17917.2 0.0794

43 17861.2 2.0259 17846.8 2.1090

44 17793.6 2.7311 17779.6 2.7063

45 17729.4 1.2911 17715.9 1.2395

46 17668.7 0.7452 17655.6 0.4521

The intensity of an emission line is proportional to the Einstein A coefficient of spontaneous transition from the upper level to the lower level. Thus, in order to compare measured and theoretical relative intensities, we have considered the Einstein A

coefficient as the calculated intensity. The relative areas of the fluorescence lines were used for their measured intensities. The experimental and theoretical intensities are normalized to the sum of the intensities of all the observed lines. For laser transition intensities of the R-P fluorescence series we considered the corresponding doublet value.

The total emission Einstein coefficient Av',J'

from a given vibrational-rotational level is connected to the radiative lifetime of this level by τv',J'=1/Σv",J"Av',J'→v",J" =1/Av',J'. Therefore the

radiative lifetimes τv',J' of the upper levels Β1Πu

(v’=26, J’=32) and (v’=29, J’=39) can be readily

calculated by summing the tabulated Av',J'→v",J”

coefficients (tables II and III) to 7.28 and 7.27 ns, respectively. As experimental lifetimes for different vibrational (0≤v'≤ 29) and rotational (12 ≤ J' ≤ 152)

levels in the B1_Π

u state, reported in Ref. [16] using a

modified delayed coincidence single-photon counting technique, range from 7.1 to 7.5 ns, the agreement with calculated lifetime is reasonably good.

On the other hand, in fig. 1 one recognizes collision-induced rotational satellite lines with jumps

∆J' = ±1, ±2, ... ±20 for some v" bands of the most

intense fluorescence series v’=17, J’=38→v”,

J”=37 and 39. The prohibition of intercombination

between symmetric and antisymmetric states holds not only for radiative transitions but also for collision-induced rotational transitions [30]. Thus, the collision-induced rotational transitions with ∆J’

odd produce Q branches if the excitation is by an R

or P line or RP branches if the excitation is by a Q

line. In contrast, jumps with ∆J’ even produce Q

branches if the excitation is by a Q line or RP

branches if the excitation is by an R or P line. Unlike collision-induced vibrational transitions for collision-induced rotational transitions, energy restriction plays no role, since the energy transferred is always much smaller than kT. It is therefore a surprise to find that often the probabilities of upward jumps ∆J'=+1 and downward jumps ∆J'=-1 differ strongly. For the v'=17, J'=38→ v"=(8, 9, 11, 12, 13, 14, 16, 18, 20, 22, 23, 25, 27, 28, 29, 30), J"=37

and 39 RP fluorescence doublets (see figure 1) the collision induced rotational transitions with ∆J'=-1

(Q-1 line) are more favored than ∆J'=+1 (Q+1 line).

Additionally, there is an asymmetry in the intensities since the intensity of the line ∆J'=-1 (Q-1 line) is

bigger than the ∆J'=+1 (Q+1 line) In other analyzed

fluorescence spectra on Na2, we observe the same behaviour.

Acknowledgment