Analysis and transition probabilities of the B1Πu→X1Σg+ band system of Na2 excited by the 4579 Å Ar+ laser line

Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

Analysis and transition probabilities of the B1% PX1&` band 6 ' system of Na excited by the 4579 As Ar` laser line 2 J.J. Camacho*, J. Santiago, A. Pardo, D. Reyman, J.M.L. Poyato Departamento de Qun& mica-Fn& sica Aplicada, Facultad de Ciencias, Universidad Auto& noma de Madrid. Cantoblanco, 28049-Madrid, Spain

Abstract The #uorescence spectrum of Na induced by the 4579.35 As line of an argon ion laser has been analyzed 2 with special emphasis on determination of accurate relative intensities and on the observation of higher vibrational levels in the B1% excited state close to the dissociation limit. We have observed seven 6 #uorescence series for the B1% PX1&` band system corresponding to the excitation transitions: 6 ' v@"27, J@"31QvA"7, JA"31; v@"28, J@"43QvA"7, JA"43; v@"24, J@"3QvA"6, JA"3; v@ "15, J@"55QvA"0, JA"55; v@"31, J@"42QvA"8, JA"42; v@"29, J@"25QvA"8, JA"24 and v@"23, J@"39QvA"5, JA"38. Some series are reported for the "rst time. Dunham coe$cients and Franck}Condon factors valid for all quasibound vibrational energy levels v@"28}33 of the B1% excited 6 electronic state are also reported. The radiative transition probabilities and radiative lifetimes for the observed #uorescence series were calculated by using hybrid potential energy curves for the B1% and X1&` 6 ' states constructed up to last vibrational levels and a transition dipole moment function. The transition probabilities and lifetimes agree with the observed measurements usually within the experimental uncertainty. A very weak rotational satellite structure with jumps with *J@"$1,$2,2,$20 for some bands of most intense #uorescence series Q(31) at v@"27 was also analyzed. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Transition probabilities; Laser-induced #uorescence; Na

2

1. Introduction Optical spectra of the sodium dimer have been frequently studied theoretically as well as experimentally. Especially since the application of lasers much more accurate spectra have become

* Corresponding author. 0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 4 1 - 7

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available. The laser-induced #uorescence (LIF) spectra of Na for the B1% PX1&` band system 2 6 ' induced by several Ar` laser lines have been analyzed by di!erent authors [1}3]. Using this technique at high resolution, Kusch and Hessel [3] have reported an analysis of this band system determining accurate molecular constants and Franck}Condon factors (FCF) in the range 04vA445 and 04v@429 for both electronic states. A review including more than 300 papers published up to 1982 on Na can be found in Verma et al. [4]. In this review, Verma et al. [4] also 2 reported experimental information of the ground state up to vA"56. Barrow et al. [5] have extended the vibrational information of the X1&` state up to vA"62, at about 5 cm~1 below the ' dissociation limit. Very accurate measurements of the dissociation energy of the ground state have been reported by Zemke and Stwalley [6] and more recently by Jones et al. [7] (DA"6022.029(5) cm~1). Using di!erent techniques such as molecular beam [8}10] or modulated e gain spectroscopy [11,12], the uppermost quasibound vibrational levels v@"27}33 of the B1% 6 state have been studied in order to derive the position and the height of the potential barrier. Di!erent measurements for dipole moment function D(R) for the B!X electronic transition in Na , from line intensities [13], radiative lifetimes [14] and ab initio theoretical calculations [15], 2 have been reported with a good agreement among them. In this paper, we report on our observations of the #uorescence spectrum of Na excited by 2 4579 As Ar` laser line. Fluorescence lines in the spectrum cover the blue}yellow region 17,400}22,000 cm~1. In addition to the #uorescence spectrum, the absorption by atomic sodium, possibly due to the reabsorption of atomic #uorescence Na(32S )PNa(32P ), is also 1@2 1@2,3@2 observed. Seven #uorescence series have been observed and identi"ed, "ve of which consist of Q lines and the rest consist of RP doublets. Four of the observed series were previously reported by DemtroK der and Stock [2] and Kusch and Hessel [3]. Some of these series are excited to high quasibound vibrational levels (v@"27, 28, 29 and 31) in the B1% state and show mostly anti6 Stokes lines on the short-wavelength side of the laser line. The rotationless potential energy curve of the B1% state shows a barrier arising from Na(32S )#Na(32P ) predicted by King and 6 1@2 1@2,3@2 van Vleck [16] and studied both theoretically [17,18] and experimentally [2,3,11,8,12]. A comparative study of the theoretical radiative transition probabilities for the #uorescence series which originate from these high-lying rovibronic quasibound levels with the corresponding experimental intensities will provide a very sensitive test of the accuracy of the Na B1% potential barrier. This 2 6 constituted an important motivation for this research. A new set of Dunham coe$cients for the B1% , valid for v@"0 to 33 and needed for describing 6 the high-lying range of vibrational level v@"29 to 33, are presented. Franck}Condon factors for B1% PX1&` transitions from v@"27 to 33 are also reported. 6 ' Recently, some studies have been made in our laboratory [19}22] on intensities of LIF spectra for Na . The experimental intensities of #uorescence lines in the spectra were corrected by using an 2 halogen lamp for calibration of the spectral response of the entire optical system. Also we checked the densitometry on photoplate in order to assure a linear behavior of opacity versus exposure. This point is particularly important for determining right intensity measurements from photographic plate. We have made an independent theoretical prediction (frequencies as well as intensities and lifetimes) of the entire spectrum of Na excited by 4579 As Ar` laser line. The striking agreement 2 between the experiment and theory has served as a veri"cation of the accuracy of our experimental assignments and the reliability of our theoretical calculations.

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731

The radiative transition probabilities for the most intense #uorescence series were calculated by using hybrid Rydberg}Klein}Rees (RKR)}long range potential energy curves for the B1% and 6 X1&` states previously reported [22] and ab initio transition dipole moment function [15]. ' Radiative lifetimes of the rovibronic levels of the upper state pumped by the laser line have also been calculated. Measured relative #uorescence intensities corrected for the spectral response of spectrograph and plate agree well with calculated radiative transition probabilities.

2. Experimental details A schematic diagram of the experimental setup is shown in Fig. 1. An argon ion laser (Spectra Physics Model 170) was tuned to 4579 As (Ar`(4p2S P4s2P )) using an internally re#ecting 1@2 1@2 prism as part of the laser cavity. The maximum output power is about 0.7 W at this wavelength. The sodium vapor was produced in a stainless-steel heat pipe oven (&60 cm of total length and ¹"450$103C) with cooled glass windows in order to prevent condensation of sodium vapor on the cell windows. The vapor pressure of atomic sodium at this temperature is about 1.2 Torr. No bu!er gases were used in our experiments in order to avoid rotational satellite lines induced by collision around the most intense #uorescence series. A small region (a few millimeters in length) of molecular #uorescence was imaged, with the help of a lens, onto the entrance slit of the spectrometer. A chromel}alumel thermocouple was placed in the observation region (central zone of the heat pipe oven) in order to monitor the temperature of the #uorescence region. The molecular #uorescence was observed in a way opposite to that way of the laser beam. The observation of the #uorescence along the line of the laser beam yields a much higher intensity of #uorescence radiation on the slit of the spectrograph than does observation transverse to the laser beam. This way of observation parallel to the laser beam may lead to Doppler shifts on the #uorescence lines. The Doppler width of sodium lines at 4503C is about 4 GHz (&0.13 cm~1) and for the laser line is about 5 GHz (&0.17 cm~1). The detection system consisted of a high light-gathering 3 m modi"ed

Fig. 1. Experimental setup.

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Huet spectrograph. Both a heavy #int double prism (3.2 As /mm at 400 nm and resolving power of 70,400 and 11.4 As /mm at 589 nm, 22,400) and a plane re#ecting di!raction grating (2.85 As /mm and resolving power about 30,000) were used as the dispersive elements. Spectral lines of Ar` (the laser has a multiline spectrum [23]), neon and xenon from spectroscopic pen lamps were used as spectral standards for the calibration of the #uorescence line positions. The line positions were measured on photoplates recorded using the grating as the dispersive element by means a comparator (uncertainty about $0.05 As ). In contrast, intensity measurements were obtained from the prism spectrometer. The prism recording allows one to detect very low line intensities. A program for least-squares "tting to a third-order polynomial was used to interpolate the #uorescence line positions from the calibration standards. The relative intensity distribution of the resonance #uorescence spectrum was determined from the microdensitometry of the photographic plate. These measurements were corrected for the spectral response of the spectrograph and the plate. In order to make this calibration we used an Osram No. 6438; 6.6 A; 200 W halogen lamp. The relative spectral irradiance of this lamp has been calibrated by D'orazio and Schader [24] in the range 400}700 nm within $5%. The spectra were taken at di!erent exposure times in the range from 30 s to 2 h. The microdensitometry was made by area integration in a rectangular region for which the exposure time was 30 s, in order to make sure that the work area remains within the linear part of the emulsion characteristic curve (graphical representation of the photographic densities against the logarithm of exposure). We estimate an uncertainty in the intensity measurement of about 10}20% for most of the lines. Only for some lines was the uncertainty beyond of this interval.

3. Results and discussion Fig. 2 shows the assignment of Na (B!X) #uorescence spectrum (blue}yellow region 2 17,400}22,000 cm~1) obtained by excitation with the 4579 As argon ion laser line. The densitometric scan corresponds to the shorter exposure time of 30 s recorded from the prism spectrometer. In addition to the #uorescence spectrum, the absorption by atomic sodium, possibly due to the reabsorption of atomic #uorescence Na (32S )PNa (32P ) is also observed (see Fig. 2). 1@2 1@2,3@2 The Na atomic #uorescence can be produced by photodissociation via absorption of the laser line to high-lying vibrational levels above the dissociation limit of the B1% state, and also by the 6 collision processes NaH#NaPNa #NaH via dimer-atom exchange or by normal inelastic 2 2 collisions. As mentioned earlier, seven #uorescence series have been observed and identi"ed, "ve of which consist of Q lines and the rest consist of RP doublets. Table 1 summarizes the excitation transitions that originate from these series and the range of observed vA values. A further con"rmation of the correct assignment is provided below by the good agreement between the measured and calculated radiative transition probabilities. The assignments were done by matching the measured wavenumbers of the spectral lines with the wavenumbers calculated from the accurate molecular constants from Kusch and Hessel (analysis III in Table VII) [3] for both the X and the B electronic states. Four of the observed series were previously reported by DemtroK der and Stock [2] and Kusch and Hessel [3]. Some of these series are excited to high quasibound vibrational levels (v@"27, 28, 29 and 31) in the B1% state and show mostly Stokes lines on the short-wavenumber side of the laser 6 line. The molecular constants from Kusch and Hessel [3] cover the vibrational region 04vA445

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Fig. 2. Spectrum and assignment of #uorescence series of Na induced by the 4579 As line of the argon ion laser. 2

Table 1 Some parameters of the laser-induced #uorescence transitions for Na due to selective excitation by means of the 2 4579.35 As (air) (l "21831.05 cm~1 (vacuum) and a power output of about 0.5 W) line of an argon ion laser -!4%3 Excitation v@, J@QvA, JA

Intensity!

vA .*/

vA .!9

N A A /N v , J 0,0 (¹"4503C)

B #0%&&. (]107 s/g)

q (ns) v{, J{

q (ns)" %91.

27,31Q7,31# 28,43Q7,43# 24,3Q6,3 15,55Q0,55 31,42Q8,42 29,25Q8,24# 23,39Q5,38#

vs s m m m w w

6 6 5 0 7 8 5

44 45 40 29 52 46 39

6.9 7.5 1.3 46.8 5.7 4.6 12.7

3.65 2.86 3.86 0.161 * 4.52 2.41

7.30 7.35 7.20 7.12 * 7.34 7.22

7.42 7.40 * * * 7.41 7.20

!vs, very strong; s, strong; m, medium; w, weak. "Reported by DemtroK der et al. [14]. #Reported by DemtroK der and Stock [2].

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for the X1&` and 04v@429 for the B1% . In order to determine frequencies for transitions with ' 6 v@'29 (for example, the red end of the #uorescence spectrum shows Q(42), v@"31 series starting at vA"7 and continuing all the way up to vA"52) an extended set of Dunham coe$cients for the B1% , valid for v@"0 to 33 has been determined. The data "elds of vibrational and rotational 6 values used in the calculation are from the results obtained in this work, as well as the results obtained in other experiments carried out in our laboratory [19}22]; the term values ¹@(v@, J@) were derived starting from the Dunham coe$cients (analysis III in Table VII) in the ranges of v@ and J@ reported by Kusch and Hessel [3] in their Table V; the term values from the experimental information for all quasibound vibrational levels v@"27 to 33 (44J@417) were reported by Vedder et al. [11] and the term values for the vibrational levels v@"30 to 33 (14J@447) by Richter et al. [10]. The term values of the B1% state are expressed by means of a Dunham-type 6 expansion: ¹@(v@, J@)"+ >@ (v@#1/2)i[J@(J@#1)!"2]k (1) ik i,k " being equal to 1 for the excited electronic state. The molecular constants and their standard errors are given in Table II. The number of digits reported does not imply precision to that degree. The precision is indicated by the quoted standard error. However, enough digits have been reported to insure that there is no loss in the replication of data due to roundo!. No energy terms were eliminated from the input data. The most striking deviations correspond to the energy terms of the higher rotational levels. The root mean square (RMS) error in the least-squares "t is 0.088 cm~1, which is somewhat large. Obviously, Dunham coe$cients presented by Kusch and Hessel [3] are more accurate for vibrational transitions with v@(29 of the B1% state than those 6 presented in this work, which are valid particularly for transitions with v@529. The fundamental goal to obtain these molecular constants was the possibility of building the B potential energy curve close to the dissociation region as well as to determine Franck}Condon factors for transitions with v@529. Before proceeding to the calculation of radiative transition probabilities, it is necessary to select adequate potential energy functions for both the B1% and the X1&` electronic states and the 6 ' B!X dipole strength function. The RKR potential energy curves for both B1% and X1&` 6 ' electronic states used in the present calculation were taken from Ref. [22] although we have modi"ed slightly the RKR region for the B1% state for v@'29, calculated starting from the 6 spectroscopic constants presented in Table 2. The RKR X1&` potential was constructed from ' the vibrational G(vA) and rotational B A energies reported by Kusch and Hessel (analysis III in v Table VIII of Ref. [3]), valid up to vA"45, and combined with the molecular constants reported by Barrow et al. [5], which extended the vibrational information up to vA"62 at about 5 cm~1 below the dissociation limit at DA"6022.029(5) cm~1 [7]. The RKR potential for the B1% state e 6 was determined from the spectroscopic constants reported by Kusch and Hessel [3] for the B1% 6 state of Na valid up to v@"29 and combined with the experimental information reported in 2 Table 2, valid especially for the uppermost quasibound vibrational levels v@"29}33. A long-range potential with a barrier of 370.7 cm~1 with respect to the dissociation energy to Na(32P )# 3@2 Na(32S ) [10] was used for the B1% state. In addition, at small internuclear distances 1@2 6 (r(r ) an analytic extrapolation to a ; (r)"A exp(!Br)#C function is -!45v*//%3vRKRv563/*/'v10*/5 *//%3 used to extend the curves. The RKR points were interpolated by a spline "t to obtain a smooth

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735

Table 2 Dunham coe$cients > for the B1% state of Na which accurately represent the high vibrational levels v@"29}33. All i,k 6 2 values are in cm~1 i, k

>@ (B1% ) i,k 6

Standard error

i, k

>@ (B1% ) i,k 6

Standard error

1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 0,1 1,1 2,1 3,1

124.45696231(#03) !0.7400536906(#00) !1.6171155460(!02) 3.8760779952(!03) !5.1526569920(!04) 3.5041636811(!05) !1.3088905325(!06) 2.5387259317(!08) !2.00706126062(!10) 0.12581086514(#00) !9.09585828167(!04) 6.06477425073(!06) !5.56242357782(!06)

1.82(!02) 1.26(!02) 3.53(!03) 5.13(!04) 4.31(!05) 2.16(!06) 6.39(!08) 1.03(!09) 6.90(!12) 6.22(!06) 5.42(!06) 1.83(!06) 2.95(!07)

4,1 5,1 6,1 7,1 0,2 1,2 2,2 3,2 4,2 5,2 0,3 1,3 2,3

6.82973173148(!07) !4.15051432339(!08) 1.20445184865(!09) !1.36481744833(!11) !4.98470500494(!07) !1.47595626673(!08) 1.91260046094(!09) !1.69652870871(!10) 7.61588329072(!12) !1.59796171221(!13) 6.97193532589(!13) 2.69187686622(!13) !3.3858736301(!14)

2.54(!08) 1.21(!09) 2.99(!11) 2.93(!13) 6.47(!10) 3.85(!10) 7.99(!11) 8.94(!12) 4.43(!13) 7.81(!15) 1.75(!14) 9.13(!15) 5.57(!16)

potential curvature. Thus, the resulting hybrid potentials are then de"ned at all internuclear distances although the energy levels depend basically on the RKR zone and only slightly on the exterior regions of extrapolation. "(:=( (r)( A A (r)dr)2 for B1% PX1&` The Franck}Condon factors (FCF) q 0 v{, J{ v ,J 6 ' v{, J{?vA, JA transitions for J"0 (i.e. J@"JA"0) within a range from v@"27 to 33 are also computed. Although the lowest rotational level for the B1% excited electronic state is J@"1 the di!erences 6 with the FCF calculated from rotationless wavefunctions are negligible. The calculated Franck}Condon factors are shown in Tables 3 and 4 and can been used to determine the intensities of #uorescence series in an approximate way. For determining accurate intensities of spectral lines it is necessary to calculate transition probabilities (Eq. (7)) starting from the wavefunctions of the implied states as well as the B!X electronic transition dipole moment function D(r). Only when D(r) does not change signi"cantly with internuclear distance r in the regions of appreciable integrand, the intensity is proportional to the Franck}Condon factor. To provide a qualitative idea of the B1% PX1&` band intensity distribution for higher vibrational levels of the B1% elec6 ' 6 tronic state, Fig. 3 shows a three-dimensional plot of the FCFs. Obviously, from this graph one sees that higher vibrational X1&`-state levels cannot be directly populated by #uorescence from ' levels of the B1% excited electronic state because of vanishing small Franck}Condon factors. In 6 the same way, high vibrational B1% -state levels can be directly excited from thermally populated 6 levels of the ground electronic state. Given the full hybrid potential without rotation ; (R) for the B1% and X1&` electronic states 0 6 ' of Na , we solve the one-dimensional radial SchroK dinger equation (energy and potential in cm~1) 2

C

D

h d2( h[J(J#1)!"2] v, J # ; (r)# ! ( (r)"E ( (r), 0 v, J v, J v, J 8p2ck dr2 8p2ckr2

(2)

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J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749 Table 3 Franck}Condon factors (]10~3) for B1% !X1&` transitions of Na for J"0 from v@"27 to 33 6 ' 2 vA/v@ 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

27

28

29

30

31

32

33

1 3 9 22 32 26 5 3 19 13 * 15 12 1 18 6 7 18 * 20 4 15 12 9 20 5

2 6 16 28 28 11 * 13 18 2 7 17 1 10 14 * 17 5 8 16 1 21 1 21 5

1 4 12 23 27 16 1 6 18 7 1 15 7 2 16 3 8 14 * 17 3 12 12 5 20

1 3 8 18 25 19 4 2 14 12 * 10 12 * 11 9 1 15 3 8 12 1 17 1 17

2 6 13 21 20 7 * 10 13 2 4 13 2 5 12 * 10 8 1 14 1 11 8 5

1 4 10 17 18 9 * 6 13 4 1 10 5 1 11 3 4 11 * 10 6 3 12 *

1 3 7 13 16 10 1 3 10 6 * 7 7 * 8 5 1 9 2 5 8 * 11 2

where the term h[J(J#1)!"2]/8p2ckr2 is referred to as the centrifugal potential and E is the v, J eigenvalue of the rotation}vibration eigenfunction ( (r). Eq. (2) was solved by a "nite-di!erence v, J method using a computer program developed in our laboratory [25]. Derivatives of the unknown eigenfunction in the di!erential equation are replaced by di!erences of the function at some chosen set of grid points. For linear homogeneous boundary value problems the "nite di!erence method is very easy to use because it leads to a matrix eigenvalue problem. The potential energy curves for both electronic states were checked by solving numerically the radial SchroK dinger equation. The RMS di!erences between the experimental vibrational energies and the corresponding quantum mechanical energy eigenvalues calculated are 0.4 and 0.3 cm~1 for the B and X states, respectively. The interpretation of diatomic spectral line intensities has been well developed and thoroughly discussed in the literature [26]. The theoretical intensity of a spectral #uorescence emission line, de"ned as the energy emitted per second, is given by I5)3: A A "hclN A , v{, J{?v , J v{, J{ v{, J{?vA, JA

(3)

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Table 4 Franck}Condon factors (]10~3) for B1% !X1&` transitions of Na for J"0 from v@"27 to 33 6 ' 2 (continued) vA/v@ 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

27

28

29

30

31

32

33

27 4 34 6 40 19 34 71 1 137 204 102 23 3

20 11 20 15 25 16 43 7 84 7 84 213 141 40 6

1 25 * 30 1 38 * 53 1 70 39 31 198 183 67 12 1

6 12 14 9 21 7 29 9 36 23 32 76 1 145 218 110 26 3

16 * 20 * 22 3 24 7 31 8 49 2 86 16 66 222 170 58 11 1

14 3 11 9 7 17 3 24 2 33 3 44 13 46 67 4 158 225 120 33 5

7 8 2 14 * 17 1 18 5 20 8 28 9 48 2 85 23 47 209 197 86 22 5 3 1

where l is the #uorescence wavenumber (in cm~1) for the corresponding transition v@, J@PvA, JA, N is the population density in the excited level and A is the radiative transition v{, J{ v{, J{?vA, JA probability or Einstein A coe$cient of spontaneous emission. The relative intensity of the di!erent #uorescence series observed is proportional to the population density of the upper level v@, J@ populated by the laser. This population of the upper level is proportional to N JN A A B f (*l), v{, J{ v , J v{, J{0vA, JA

(4)

where N A A is the population density of the lower vibrational}rotational level, B is the v ,J v{, J{0vA, JA absorption Einstein coe$cient and f (*l) is the relative overlap of the absorption line centered l with the Ar` laser line. At thermal equilibrium, the population of the lower rota0 tional}vibrational levels follows a Boltzmann distribution: N A A J(2JA#1)e~*GA(vA)`FvA (JA)+hc@kT. v ,J

(5)

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Fig. 3. Three-dimensional plot of Franck}Condon factors (FCF) for B1% PX1&` vibrational transitions for J"0 6 ' from v@"27 to 33 and vA"0 to 62.

The Einstein transition probability of absorption can be obtained as 1 2J@#1 B A A A" A A. v{, J{0v , J 8phcl3 2JA#1 v{, J{?v , J

(6)

The values of relative population densities for the initial rovibronic level vA, JA which originates the #uorescence series with respect to the population density of lower level vA"0, JA"0 at one working temperature ¹K4503C and the Einstein B absorption coe$cients v{, J{0vA, JA for the observed excitation transitions are shown in Table 1. The di!erence l !l varies -!4%3 0 for di!erent excitation transitions but due to the large Doppler width (&0.3 cm~1), this factor can be considered as a constant and the relative intensity of the #uorescence series should be proportional to the product N A A .B . As seen from data of Table 1, there is v , J v{, J{0vA, JA satisfactory agreement between theoretical and experimental relative intensities for all #uorescence series.

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The Einstein A coe$cients of spontaneous emission for each v@, J@PvA, JA transition v{, J{?vA, JA were calculated as 64p4l3 A S A A A" v{, J{?v , J J{, J 3h

KP

=

0

K

2 ( (r)D(r)( A A (r) dr , v{, J{ v ,J

(7)

where D(r) is the dipole strength function (transition moment) for the B!X band system, S A is J{, J the line strength HoK nl}London factor [27] and ( (r) and ( A A (r) are the radial wavefunctions v{, J{ v ,J for the upper B1% and lower X1&` states, respectively. For the B!X electronic transition dipole 6 ' moment function, we used the MCSCF calculations reported by Stevens et al. [15]. These values of electronic transition moment agree with the experimental results reported by Hessel [13] and DemtroK der [14] for our work range. Tensioned spline "ts were used to make the electronic transition dipole moment function continuous. By using these equations, we have obtained Einstein A coe$cients for the #uorescence series observed in our spectrum. The measured frequencies and calculated Einstein A coe$cients of spontaneous emission for the "ve Q series: v@"27!Q(31), v@"28!Q(43), v@"24!Q(3) and v@"15!Q(55) and the two RP #uorescence series: v@"29!R(24)P(26) and v@"23!R(38)P(40) are shown in Tables 5, 6, 7 and 8, respectively. The intensity of a emission line is proportional to the Einstein A coe$cient of spontaneous emission from the upper level to the lower level (Eq. (7)). Thus, in order to compare measured and theoretical relative intensities, we have considered the Einstein A coe$cient as the calculated intensity. The relative areas of the #uorescence 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 the values of intensity of the laser transition of the R!P #uorescence series we considered the corresponding value summing over the doublet. For the values of intensity of the laser transition of Q #uorescence series, an estimation was made taking into account the corresponding theoretical values. These assumptions were necessary because seven emission peaks, at least, coincide in laser wavelength, and the relative intensities of the #uorescence lines were normalized to the sum of all observed lines. In Figs. 4 and 5 we present a comparison of the experimental (hollow bars) and calculated (solid bars) values for some intense #uorescence series. As seen from these "gures, the agreement between experiment and theory is good, usually within the experimental uncertainty. In the spectrum shown in Fig. 2, from which the experimental intensities have been determined, the two or three more intense lines of the most intense series v@"27, J@"31PvA, JA"31 and v@"28, J@"43PvA, JA"43 show a smaller intensity than one would expect theoretically due to saturation phenomenon. The over-exposed lines are indicated in Fig. 4. All lines with an appreciable intensity have been detected because the sum of Franck}Condon factors is close to 1 for all #uorescence series. For the remaining Q series v@"31!Q(42), it was not possible to obtain the transition probabilities but FCF because the vibrational}rotational level v@"31, J@"42, which has an energy of ¹@(31,42)"3074.51 cm~1(taking as zero of energy the minimum of the B electronic state potential curve), is only 29.78 cm~1 below the e!ective maximum of the potential barrier that is at ; (r@ )#; (r@ )"3046.19 cm~1#58.10 cm~1 being r@ "6.750 As . 0 .!9 42 .!9 .!9 The rotationless B1% excited electronic state shows a maximum at r@ "6.8382 As (10). The 6 .!9 e!ective J@"42 B1% potential barrier cuts o! the line representing the highest vibrational level 6 populated by the 4579 As laser line, v@"31, J@"42, at the internuclear distances r "6.2632 As and a

740

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Table 5 Frequency and Einstein A coe$cients of spontaneous emission for the Q #uorescence series: v@"27, J@"31PvA, JA"31; v@"28, J@43PvA, JA"43; v@"24, J@"3PvA, JA"3 and v@"15, J@"55PvA, JA"55

vA

Q(31) l (cm~1)

Q(31) A #0%&&. (]107 s~1)

Q(43) l (cm~1)

Q(43) A #0%&&. (]107 s~1)

Q(3) l (cm~1)

Q(3) A #0%&&. (]107 s~1)

Q(55) l (cm~1)

Q(55) A #0%&&. (]107 s~1)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

* * 22585.2 22431.4 22279.0 22128.2 21978.9 21831.1 21684.8 21540.1 21397.0 21255.5 21115.5 20977.3 20840.7 20705.7 20572.5 20440.9 20311.2 20183.2 20057.0 19932.6 19810.1 19689.5 19570.9 19454.2

* * 0.00002 0.00026 0.00224 0.01390 0.06122 0.1897 0.4005 0.5318 0.3579 0.04228 0.08316 0.3198 0.1492 0.01730 0.2646 0.1252 0.04210 0.2745 0.03595 0.1575 0.1987 0.02066 0.2839 0.01160

* * 22580.9 22427.9 22276.4 22126.4 21977.9 21831.0 21685.6 21541.8 21399.6 21259.0 21120.0 20982.7 20847.0 20713.1 20580.9 20450.4 20321.6 20194.7 20069.6 19946.4 19825.1 19705.7 19588.2 19472.8

* * 0.00002 0.00020 0.00167 0.01044 0.04676 0.1488 0.3267 0.4614 0.3495 0.06666 0.04069 0.2616 0.1784 0.00066 0.1966 0.1667 0.00496 0.2217 0.09271 0.06221 0.2291 0.00242 0.2121 0.0883

* * 22441.0 22286.2 22133.0 21981.2 21831.0 21682.2 21535.0 21389.4 21245.3 21102.7 20961.8 20822.5 20684.8 20548.8 20414.5 20281.9 20151.0 20021.8 19894.4 19768.8 19645. 19523.3 19403.3 19285.3

* * 0.00013 0.00149 0.01117 0.05725 0.2006 0.4669 0.6690 0.4762 0.05902 0.1109 0.4097 0.1523 0.05241 0.3593 0.08076 0.1440 0.3097 0.00063 0.3332 0.07139 0.2228 0.2069 0.1188 0.3191

21831.0 21676.1 21522.7 21370.7 21220.3 21071.4 20924.1 20778.3 20634.1 20491.5 20350.5 20211.1 20073.4 19937.4 19803.0 19670.4 19539.5 19410.4 19283.1 19157.6 19034.0 18912.3 18792.5 18674.7 18558.8 18445.1

0.00084 0.01667 0.1317 0.5128 0.9839 0.7303 0.02851 0.3904 0.5376 0.00157 0.5513 0.1225 0.3532 0.2866 0.2398 0.3615 0.2443 0.3382 0.3891 0.1863 0.7064 0.00058 0.8802 0.5450 0.1108 1.597

r "7.4283 As . Within the JWKB approximation, the half-width ! (in cm~1) is given by b

C

D

~1 1 1 q (+J@)e4pJ2kc@h : rrba JUJ{ (r)~T{(v{, J{) $r !" pc 2 7*"

(8)

where q (J@)"[L¹@(v, J@)/cLv] is the vibrational collisional rate with the barrier, ; (r) is the 7*" v/v{ J{ e!ective potential, ¹(v@, J@) is the term value of the quasibound rovibrational level (v@, J@) (both energies are in cm~1) and r and r are the classical turning points inside and outside the barrier, a b where the e!ective potential and the term values are equal, i.e. ; (r)"¹@(v@, J@). The reciprocal of J{ the exponential factor in Eq. (8) is the transmission coe$cient which indicates the probability of passage through the potential barrier. The term between brackets to the power !1 in Eq. (8) is the mean life q of the v@, J@ state and the exponent is the JWKB phase integral / (integral over the .v{, J{

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

741

Table 6 Frequency and Einstein A coe$cients of spontaneous emission for the Q #uorescence series: v@"27, J@"31PvA, JA"31; v@"28, J@43PvA, JA"43; v@"24, J@"3PvA, JA"3 and v@"15, J@"55PvA, JA"55 (continued)

vA

Q(31) l (cm~1)

Q(31) A #0%&&. (]107 s~1)

Q(43) l (cm~1)

Q(43) A #0%&&. (]107 s~1)

Q(3) l (cm~1)

Q(3) A #0%&&. (]107 s~1)

Q(55) l (cm~1)

Q(55) A #0%&&. (]107 s~1)

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

19339.5 19226.9 19116.4 19008.0 18901.8 18797.8 18696.1 18596.8 18499.9 18405.4 18313.5 18224.2 18137.6 18053.7 17972.6 17894.5 17819.3 17747.3 17678.4 *

0.2570 0.09412 0.1853 0.1979 0.1289 0.2904 0.1120 0.3699 0.1576 0.4124 0.3693 0.2939 1.026 0.00229 1.529 2.755 1.517 0.3747 0.04679 *

19359.4 19248.1 19138.9 19031.9 18927.1 18824.6 18724.5 18626.7 18531.4 18438.6 18348.5 18260.9 18176.2 18094.2 18015.2 17939.2 17866.2 17796.5 17730.0 17667.0

0.1098 0.2092 0.02674 0.2910 0.00008 0.3361 0.01175 0.3830 0.03034 0.4799 0.02368 0.6896 0.00247 0.9658 0.4258 0.4831 2.621 2.339 0.8452 0.1530

19169.2 19055.2 18943.2 18833.4 18725.6 18620.1 18516.8 18415.8 18317.1 18220.9 18127.1 18035.9 17947.3 17861.4 17778.2 * * * * *

0.06356 0.3977 0.05460 0.4582 0.1059 0.4754 0.3286 0.3029 0.9803 0.00808 1.296 2.558 1.541 0.4175 0.05596 * * * * *

18333.4 18223.8 18116.5 18011.3 17908.5 17808.0 * * * * * * * * * * * * * *

2.153 1.209 0.3606 0.06096 0.00584 0.00030 * * * * * * * * * * * * * *

area of the barrier which the molecule has to tunnel). By Eq. (8), for the v@"31, J@"42 quasibound level, we have determined a half-width of 101 MHz, q (J@"42)"9.30]10~13 s, a transmission 7*" coe$cient of 0.030%, a mean life of 1.57]10~9 s and a phase integral / "8.13. The 31,42 calculated half-width is similar to the value reported by Gerber and MoK ller [9] and it is approximately two times the value reported by Richter et al. [10]. Although for such rovibrational level there is a certain probability that the system penetrates into the barrier, classically forbidden region, this probability exponentially decreases going to zero at r&6.9 As . Therefore, the probability of passage through the barrier is negligible and no broadening of the lines of the #uorescence series starting from this level occurs at least because of this tunneling. In Fig. 6 we present a comparison of the experimental and calculated (Franck}Condon factors) intensities for the #uorescence series v@"31, J@"42PvA, JA"42. As is apparent in Fig. 6, there is only a qualitative agreement between the measured line intensities and calculated Franck}Condon factors indicating that the variation of the B!X transition dipole moment with internuclear distance is signi"cant. The biggest discrepancies are found for the higher vibrational levels of the ground state (vA"46!52).

742

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

Table 7 Frequency and Einstein A coe$cients of spontaneous emission for the RP #uorescence series: v@"29, J@"25PvA, JA"24 and 26; v@"23, J@"39PvA, JA"38 and 40

vA

R(24) l (cm~1)

R(24) A #0%&&. (]107 s~1)

P(26) l (cm~1)

P(26) A #0%&&. (]107 s~1)

R(38) l (cm~1)

R(38) A #0%&&. (]107 s~1)

P(40) l (cm~1)

P(40) A #0%&&. (]107 s~1)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

* * * 22579.4 22426.7 22275.5 22125.8 21977.6 21831 21685.9 21542.4 21400.5 21260.2 21121.5 20984.4 20849.1 20715.4 20583.4 20453.2 20324.7 20198 20073.2 19950.2 19829.1 19709.9 19592.7

* * * 0.00006 0.00060 0.00441 0.02293 0.08557 0.2257 0.4033 0.4402 0.2157 0.00311 0.1366 0.2763 0.07028 0.04861 0.2386 0.06372 0.06824 0.2277 0.01363 0.1563 0.1529 0.02305 0.2392

* * * 22564.1 22411.4 22260.3 22110.7 21962.6 21816.1 21671.1 21527.7 21385.9 21245.7 21107.1 20970.2 20834.9 20701.3 20569.5 20439.4 20311 20184.5 20059.7 19936.9 19815.9 19696.9 19579.8

* * * 0.00005 0.00055 0.00406 0.02140 0.08101 0.2171 0.3958 0.4443 0.2303 0.00623 0.1239 0.2782 0.08242 0.03861 0.2364 0.07627 0.05585 0.2312 0.02148 0.1417 0.1677 0.01427 0.2394

* 22441.8 22286.9 22133.5 21981.6 21831.2 21682.4 21535 21389.3 21245 21102.4 20961.4 20822 20684.2 20548.1 20413.7 20281 20150.1 20020.9 19893.5 19767.9 19644.1 19522.3 19402.3 19284.3 19168.3

* 0.00003 0.00044 0.00436 0.02827 0.1221 0.3478 0.6223 0.6055 0.1890 0.02127 0.3589 0.2867 0.00125 0.3163 0.1997 0.04590 0.3591 0.02661 0.2639 0.1756 0.1200 0.3118 0.03623 0.3953 0.00815

* 22417.8 22263.1 22109.9 21958.1 21807.9 21659.1 21512 21366.3 21222.3 21079.8 20939 20799.7 20662.1 20526.2 20392 20259.5 20128.7 19999.7 19872.4 19747 19623.5 19501.8 19382.1 19264.3 19148.5

* 0.00002 0.00038 0.00388 0.02563 0.1130 0.3298 0.6085 0.6205 0.2177 0.01131 0.3381 0.3123 0.00011 0.2941 0.2293 0.02759 0.3584 0.04600 0.2336 0.2103 0.08781 0.3391 0.01664 0.4094 0.00048

The total emission Einstein coe$cient A from a given vibrational}rotational level is v{, J{ connected to the radiative lifetime of this level by

N

q "1 v{, J{

+ A "1/A . v{, J{?vA, JA v{, J{ vA, JA

The calculated radiative lifetime q arising from all bound}bound transitions in the B!X band v{, J{ system of Na , excited by 4579 As Ar` laser line, are listed in Table 1. As experimental lifetimes for 2 di!erent vibrational (04v@429) and rotational level (124J@4152) in the B state, reported in Ref. [14] using a modi"ed delayed coincidence single-photon counting technique, range from 7.1 to 7.5 ns the agreement with calculated lifetimes (Table 1) is good, usually within the experimental uncertainty.

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743

Table 8 Frequency and Einstein A coe$cients of spontaneous emission for the RP #uorescence series: v@"29, J@"25PvA, JA"24 and 26; v@"23, J@"39PvA, JA"38 and 40 (continued)

vA

R(24) l (cm~1)

R(24) A #0%&&. (]107 s~1)

P(26) l (cm~1)

P(26) A #0%&&. (]107 s~1)

R(38) l (cm~1)

R(38) A #0%&&. (]107 s~1)

P(40) l (cm~1)

P(40) A #0%&&. (]107 s~1)

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

19477.5 19364.3 19253.2 19144.2 19037.4 18932.8 18830.5 18730.5 18632.8 18537.7 18445 18354.9 18267.5 18182.8 18100.9 18021.8 17945.7 17872.7 17802.8 17736.2 17613.1

0.01211 0.2040 0.1034 0.1154 0.2158 0.04422 0.3077 0.01074 0.3902 0.00209 0.4956 0.00602 0.6510 0.08341 0.7447 0.6956 0.1910 2.339 2.585 1.072 0.527

19464.7 19351.7 19240.7 19131.9 19025.2 18920.8 18818.6 18718.8 18621.4 18526.4 18433.9 18344 18256.8 18172.3 18090.6 18011.8 17936 17863.2 17793.6 17727.2 17664.2

0.02100 0.1884 0.1234 0.09502 0.2353 0.02861 0.3203 0.00308 0.3962 0.00008 0.5009 0.00022 0.6735 0.04828 0.8227 0.6089 0.2789 2.461 2.506 0.9818 0.489

19054.3 18942.4 18832.6 18725 18619.5 18516.4 18415.5 18317.1 18221 18127.5 18036.5 17948.2 17862.5 17779.7 17699.7 17622.7 * * * * *

0.4556 0.00557 0.5280 0.03062 0.5910 0.2027 0.4611 0.8902 0.00427 1.533 2.528 1.393 0.3544 0.04523 0.00292 0.00010 * * * * *

19034.7 18923.1 18813.5 18706.1 18600.9 18498 18397.5 18299.3 18203.5 18110.3 18019.6 17931.6 17846.3 17763.8 17684.2 17607.5 * * * * *

0.4625 * 0.5421 0.00883 0.6379 0.1372 0.5630 0.8099 0.03186 1.715 2.492 1.268 0.3001 0.03554 0.00212 0.00006 * * * * *

Because of their simple molecular structure, stability, ease of production and high absorption and emission cross sections, the sodium dimers are of particular interest for laser systems in the visible and near-infrared regions. Pulsed laser oscillation in Na has been observed by Man-Pichot 2 and Brillet [28], Wellegehausen [29] and Bahns et al. [30]. The "ve known optically pumped laser (OPL) lines for Na from the 4579 As Ar` wavelength pump laser can be readily assigned to the two 2 most intense #uorescence series Q(31): v@"27PvA"38, 40 and 41 and Q(43): v@"28PvA"42 and 43. Signi"cant progress has been made in the study of state-speci"c collision-induced rotational and vibrational energy transfer in diatomic molecules. Some examples of works on rotational}vibrational energy transfer can be found in Refs. [31}41]. For some bands, specially for the most intense #uorescence series v@"27, J@"31PvA, JA"31, a satellite structure around #uorescence parent lines was observed. As examples, in Figs. 7 and 8 are shown two portions of the Na (B!X) 2 #uorescence spectrum, obtained by excitation with the 4579 As argon ion laser line, and the line positions of the individual satellite lines of the vA"8 and vA"22 bands of the Q(31): v@"27 #uorescence series. This satellite structure is due to the #uorescence from adjacent rotational levels that have been populated by collision-induced rotationless transitions. In our spectrum, processes

744

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

Fig. 4. Comparison of average relative intensities (theoretical}solid bars and experimental}hollow bars) for the Q #uorescence series: v@"27, J@"31PvA"6!45, JA"31, v@"28, J@"43PvA"6!45, JA"45, v@"24, J@"3PvA"5!40, JA"3 and v@"15, J@"55PvA"0!29, JA"55.

of vibrational energy transfers were not detected because the pressure of sodium vapor inside the heat pipe oven is low and no gas was used as bu!er. The prohibition of intercombination between symmetric and antisymmetric states holds not only for radiative transitions but also for collisioninduced rotational transitions [26]. Thus, due to the selection rules of a homonuclear diatomic with nonzero nuclear spin, if the excitation is by a Q line, the collision-induced rotational transitions with *J"$1,$3,2 (odd) produce R!P branches and transitions with *J"$2,$4,2 (even) produce Q branches. On the contrary, if the excitation is by an R!P line, the collision-induced rotational transitions with *J odd produce Q branches and transitions with *J even produce R!P branches [31]. From the intensity ratio of satellite lines to parent lines it is possible to obtain the collision-induced rotational transitions rate, provided one knows the spontaneous lifetime of the level populated by collision-induced transition and the ratio of spontaneous transition probabilities. Also, from the collision-induced transition rate, provided the pressure and temperature are known, it is possible to approximate the average cross sections for collision-induced rotational transitions.

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

745

Fig. 5. Comparison of average relative intensities (theoretical}solid bars and experimental}hollow bars) for the R and P #uorescence series: v@"29, J@"25PvA"8!46, JA"24 and 26 and v@"23, J@"39PvA"5!39, JA"38 and 40.

Such measurements have been reported by us in other #uorescence experiments [20,22]. However for the collision-induced satellite lines observed in this laser-induced #uorescence spectrum (see Figs. 7 and 8), for instance, it is not possible to measure the experimental intensity of some rotational satellites (for jumps *J@"$1 and some other) owing to the overlapping with lines of di!erent #uorescence series. Nevertheless, by considering for the overlapped lines an experimental intensity near the intensity of nonoverlapped lines with the same *J@, we have estimated a mean total average cross section of 550$60 As 2 for the most intense #uorescence series v@"27, J@"31PvA, JA"31. This value is similar to some values reported by Bergmann et al. [33] for Na #Na collisions from lifetime measurements in a Stern}Vollmer plot (for example, some 2 measurements of average cross section for rotational energy transfer transitions are p6 "517 As 2 or p6 "539 As 2). For this calculation we assumed that the pressure v{/9, J{/56 v{/6, J{/27

746

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

Fig. 6. Comparison of measured intensities (hollow bars) and calculated Franck}Condon factors (solid bars) for the Q #uorescence v@"31, J@"42PvA"7!52, JA"42.

Fig. 7. Part of the observed #uorescence spectrum and line positions of the collision-induced rotational satellite lines in the vibrational band vA"8 for the #uorescence series originating from the level v@"27, J@"31.

J.J. Camacho et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 729}749

747

Fig. 8. Part of the observed #uorescence spectrum and line positions of the collision-induced rotational satellite lines in the vibrational band vA"22 for the #uorescence series originating from the level v@"27, J@"31.

equalled the value of the vapor pressure of atomic sodium at the working temperature (1.2 torr at 4503C).

4. Conclusion In summary, we have analyzed seven #uorescence series of Na excited by the 4579 As line of an 2 Ar` ion laser. Dunham coe$cients and Franck}Condon factors valid for the quasibound vibrational levels v@"29 to 33 of the B1% excited electronic state are also reported. The calculated 6 radiative transition probabilities and radiative lifetimes are in good agreement with the corresponding experimental values. These calculations include the rotational dependence and the variation of the B!X electronic transition moment with internuclear distance (no Franck}Condon or r-centroid approximation). The good agreement between the measurements and the calculations for the #uorescence series from the quasibound v@"27, J@"31; v@"28, J@"43; v@"29, J@"25 and v@"31, J@"42 rovibrational levels, very near to the dissociation limit, shows that the potential barrier estimated by Chawla et al. [12] and Richter et al. [10], the last included in our calculations, is very accurate. Finally, from a very weak rotational

748

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satellite, corresponding to the most intense #uorescence series, average cross sections of some vA bands are reported.

Acknowledgement We gratefully acknowledge the support received from the DGICYT (Spain) (Project number PB96-0046) for this research.

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