- Email: [email protected]
Spectroscopic modeling of the CO2+ A2Πu→X2Πg (110-010) Renner-Teller bands in electron beam fluorescence of carbon dioxide
J. Quant.
Spectrosc.
Pergamon PII: !%0224073(97)00130-1
Radial.
Tramfir Vol. 59. No. l/2, pp. 25-31. 1998 Q 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/98 $19.00 + 0.00
SPECTROSCOPIC MODELING OF THE CO: A211,+X21-Ig (110-010) RENNER-TELLER BANDS IN ELECTRON BEAM FLUORESCENCE OF CARBON DIOXIDE DAVID
R. FARLEY
and ROBERT
J. CATTOLICA
Department of Applied Mechanics and Engineering Sciences, University of California at San Diego, 9500 Gilman
Drive.
La Jolla,
(Received
CA 92093-041 I. U.S.A.
May 1997)
Abstract-Improvements have been made to a spectroscopic model of the electron beam fluorescence (EBF) spectra from the COz+ A’JI ,+X%, electronic transition of the (lO&OOO),
(20&100) and (300-200) bands through the inclusion of the (I l&010) Renner-Teller bands. The (010) vibrational state of the CO, molecule is thermally excited at temperatures near 300 K based on an analysis of the vibrational partition function for this molecule. The excitation and emission processes for these Renner-Teller bands were modeled with dipole moments. The inclusion of these Renner-Teller bands into the spectroscopic model produces an improved fit to the measured fluorescence spectrum between 337 nm and 339 nm. This improved model provides for a more accurate measurement of the rotational temperature using the EBF technique. Q 1998 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION To measure rotational temperatures through electron beam fluorescence (EBF) of carbon dioxide a spectroscopic model of the excitation/emission processes has been developed by Farley and Cattolica’ for temperatures up to - 300 K. With this technique, the temperature of the CO, gas can be obtained through varying the rotational temperature of the spectroscopic model until a least-squares fit with the measured spectrum is attained. Therefore, the spectroscopic model must accurately represent measured spectra in order to obtain accurate temperature measurement. The motivation behind this research has been for application to aerobraking or aerocapture in the atmosphere of Mars, where ambient temperatures vary between 150 K and 300 K. However, this type of spectroscopic modeling is useful for other applications involving CO?, and also gives insights into conducting similar analyses for other linear triatomic molecules. 2. BACKGROUND The previous results of Farley and Cattolica’ are shown in Fig. 1. Depicted in Fig. 1 are both the experimentally measured spectrum obtained with the EBF apparatus, and the theoretically modeled spectrum. For the measured spectrum, a 10 keV electron beam (nominal beam diameter of 1.0 mm) was injected into ambient CO, gas at 295 K and maintained at a pressure of 100 PHg. The emitted CO: fluorescence resulting from electron-impact and ionization was recorded using a 0.32 m spectrometer equipped with a 3600 groove/mm grating and a 1024 element image-intensified diode array. The measured spectrum has a corresponding resolution of 0.833 nmjmm. The frequency response of the diode array was normalized through the use of a measured spectrum from a calibrated deuterium source. Five overlapping symmetric-stretch bands are included in the spectroscopic model of Farley and Cattolica.’ These are the (lO&OOO), (200-100) and (300-200) bands from the CO: All-I 3,2”+X2&p electronic transition, and the (100-000) and (200-100) bands from the CO: electronic transition. Since the electron beam energy is much higher than the A*l-I,,+X*II.,, interaction energy of the collision process, Born’s first approximation is assumed and thus the electronic, vibrational and rotational states of the initial and final state wave functions are separable. Dipole moments were used for the excitation and emission rotational transition 25
26
David R. Farley and Robert J. Cattolica
16 Pressure = 100 FHg Temperature = 295 K Remolution = 0.076 nm
337
336
339
338 Wavelength
340
341
(nm)
Fig. 1. Comparison of spectroscopic model’ of the CO: A’II, +X2& (1O&000), (200- 100) and (30&200) bands and a fluorescence emission spectrum taken with a beam energy of 10 keVat 100 pHg and 295 K.
probabilities. With this assumption, the intensities of rotational spectral lines of a symmetric-stretch band are given bylm3
I”,~,“,V) Iv Cd0,“‘,u,‘).p(J’).q(U,‘,~,“).S~,~*p/(2J’+ 1) Y,“’
(1)
where I,,,,U,.@‘)is th e intensity of rotational line J’ corresponding to the appropriate R - , Q - , or P - branch within a (u,‘-+u,“) symmetric-stretch vibrational band. No dependence of the intensity distribution on wavenumber, u’, is included in Eq. (1) since the variation is negligible over the transitions that are modeled. Because the symmetric-stretch vibrational bands are composed of rotational transitions, which assume a dipole moment transition, the dipole selection rules AJ = 0, F 1 apply. The standard convention is used where AJ = + 1 corresponds to an R-branch transition, AJ = - 1 for P-branch transitions, and AJ = 0 for Q-branch transitions. q(u,“‘,u,‘) is the excitation Franck-Condon factor for the CO* symmetric-stretch u,“’ state to the excited CO: u,’ state, q(uI’,u,“) is the emission Franck-Condon factor for the (u,‘-+u,“) symmetric-stretch band, and SR*cpare the rotational line strengths, or Honl-London factors, for emission. Published values for the Franck-Condon factors were used.‘s4,5 The forms of rotational line strengths for dipole transitions, SR*Q,p,can be found tabulated in the work of Farley and Cattolica.6 In the symmetric-stretch model it was assumed that only the v;II= 0 ground vibrational state was significantly populated, which is a valid assumption at room temperatures. The excitation rate into excited rotational states p(J), is calculated utilizing a dipole moment in the electron-impact excitation process. The unexcited CO* X’ C,+ molecules are assumed to be in equilibrium at rotational temperature r. The ground state unexcited molecules then follow a Boltzmann distribution n(Ntt,>
_
o(2N”’
+
l)exp
-
N”W; + I)@, I
>
where n is the relative population of molecules of a given ground state rotational quantum number, 0, is the rotational constant of the ground state dimensionalized in terms of temperature (0, = hcB,./k = 0.5614 K), and 0 is a factor to account for nuclear spin statistics which
Spectroscopic modeling of Renner-Teller bands
27
is 1 for even, and 0 for odd. Thus there are no molecules in odd states. Note the common convention that the lowest lying states, those of the unexcited CO,X’ Cl ground state, are denoted with three primes (,“). The excitation rate into an excited rotational state is given by p N n(N”‘).SR.e,P/(2N”’+ 1)
(3)
where SR.c,Pare the electric dipole transition probabilities, or Honl-London factors, for excitation of R, Q, and P branches, respectively. The resulting theoretical spectrum calculated from Eq. (1) was convolved with the line shape of a measured mercury line occurring at 334.1 nm to produce the modeled spectrum of Fig. 1. The measured mercury line had a FWHM linewidth of 0.08 nm. A numerical routine was developed to calculate the convolving of the mercury line transfer function with the idealized line spectrum. Although the symmetric-stretch model fits well with the measured spectrum, there are still several perturbations in the measured spectrum which have not yet been accounted for in the model. Eight of these perturbations are known to be due to the (110-010) and (210-l 10) Renner-Teller bands.’ The four CO: (110-010) Renner-Teller bands originating at 337.25 nm, 337.51 nm, 338.23 nm and 338.64 nm, have been identified and their wavelength locations investigated by Gauyacq et ~1.~The emissions of these four (1 l&010) Renner-Teller bands are summarized below and are shown labeled in Fig. 1. 1. 2. 3. 4.
Cl +Z: emission occurring at 337.25 nm &zg+&zu emission occurring at 337.51 nm &,z, +AJiZUemission occurring at 338.23 nm C, +Z; emission occurring at 338.64 nm
The addition of these bands would improve the spectroscopic model between 337 nm and 339 nm, and therefore result in more accurate rotational temperatures being measured. There are also four (210-l 10) Renner-Teller bands located between 339 nm and 341 nm, but these bands have not yet had their wavelength locations spectroscopically analyzed. If there were accurate line positions for the (210-l 10) bands, however, then a similar analysis as will be done herein for the (110-010) bands could be conducted for the (210-I 10) Renner-Teller bands. Also, between 340 nm and 341 nm are perturbations which are not due to Renner-Teller bands, and which have not been identified,’ thus making it impractical to attempt to add these perturbations to the model. However, if the spectroscopic model can be improved between 337 nm and 339 nm, and the fit between the measured and theoretical spectra is done within this spectral region, then accurate temperatures can be measured without a significant need to improve the model beyond 339 nm. For completeness it would be useful to have all contributions to the spectra included in the spectroscopic model, but towards the goal of accurate temperature measurements such completeness is not required. Therefore, a spectroscopic analysis will be done of the excitation/emission processes for the four (110410) Renner-Teller bands, and these bands will be incorporated into the previous spectroscopic model of Farley and Cattolica.’ 3.
ANALYSIS
To account for the observed intensities of the (110-010) Renner-Teller bands, two approaches are possible. In one approach, it is assumed that the (110) upper states of CO: are excited directly from the (000) vibrational ground state of CO,. This assumes that only the (000) vibrational ground state is significantly populated at ambient temperatures near 300 K, and requires a quadrupole transition moment for the rotational excitation since a (OOO)+(l 10) transition is not allowed with a dipole moment. This is because the excitation of the bending quantum v2 = 1 adds an angular momentum component / = + 1 along the internuclear axis which is coupled to the rotational state of the mo1ecule.‘,6 Therefore, for a transition to occur from the (000) ground vibrational state to the (110) excited state would require a change of AK = 2 in the quantum of angular momentum projected along the internuclear axis, which is forbidden with a dipole moment.6 In the second approach, it is assumed that the (010) vibrational state of the ambient CO, molecules is thermally excited from the (000) state prior to electron-impact excitation. Both of these approaches were
28
David R. Farley and Robert J. Cattolica
studied. Although the quadrupole modeling produced satisfactory qualitative results, the quantitative magnitudes required for the observed intensities of the Renner-Teller bands were much higher than are typical for quadrupole moments. 9 After further investigation into the vibrational partition function, particularly for the population of the (010) vibrational state of CO?, it was found that an adequate number of CO, molecules are thermally excited into the (010) vibrational state before electron impact and ionization to account for the observed Renner-Teller band intensities. It was therefore decided that the quadrupole excitation of the Renner-Teller bands from the (000) ground state was unlikely. Hence, the (010) state is assumed to be excited thermally, and the excitation mechanism between the CO, (010)-X0:(1 10) excited vibrational states caused by electron-impact ionization can then be modeled with a dipole moment. The vibrational partition function for a linear triatomic molecule is given by
where E,,(u,,u2,uj)= (w,u, + wZu2+ w&rc; o, is the frequency of the symmetric-stretch vibrational mode corresponding to vibrational quantum number u,;02 is the frequency of the bending vibrational mode corresponding to vibrational quantum number u2;w3is the frequency of the anti-symmetric-stretch vibrational mode corresponding to vibrational quantum number 0,; h, c, and k are Planck’s constant, the speed of light, and Boltzmann’s constant, respectively, and r is the vibrational temperature. The energy of the vibrational ground state has been arbitrarily set to (w,/2 + o2 + o$2)hc. Using Eq. (4) for the vibrational partition function, it is then possible to calculate the relative populations of the CO, vibrational states, n(u,,uZ,uJ), as a function of temperature as u2
ntu, 9
9
u3j
=
exp[ - Evtw2~~3Yk~l Qd
(5)
Eqns (4) and (5) were calculated numerically between 100 K and 2000 K, summing from 0 to 10 for all vibrational quantum numbers in Eq. (4). Values for the vibrational frequencies were taken from Herzberg,” and are w, = 1337 cm - ‘, w2 = 667 cm - ‘, and Ok = 2349 cm - ‘. The results of this numerical analysis are shown in Fig. 2, depicting the significantly populated vibrational states between 100 K and 2000 K, which as shown are the (000), (100) (OlO), (020), and (030) states. The (200), (040) and (001) vibrational states had relative populations of the same magnitude or less than the (030) state, but these plots are not depicted in Fig. 2 to avoid cluttering the graph. Note in Fig. 2 that the (100) and (020) vibrational state populations are equal. This is due to the vibrational bending frequency o2 being nearly exactly half that of the symmetric-stretch vibrational frequency 0,. Thus Fermi resonance can occur between the (100) and (020) states, consistent with the observations of Larcher et al” for CO:. The relative population of the (010) vibrational state near 300 K is approximately 4%, which is consistent with the observed relative intensities of the Renner-Teller bands as compared with the symmetric-stretch bands. Therefore, it is assumed that the (010) vibrational state is thermally populated prior to electron impact, and subsequently through electron-impact ionization the molecules can be excited through a dipole moment to CO: (110) vibrational states. The CO: (110-010) Renner-Teller bands can then be modeled using dipole moments for both the excitation and emission processes. The locations of the rotational lines are obtained from the rotational term values for the upper and lower states of the emission. The general equation for a rotational term value is given by F(J) = B;J(J + 1) + D,:J’(J + 1)2 The rotational term value constants B, and D, to be used for bands are those obtained by Gauyacq et al,’ and are given rotational line is then calculated as v. + F(Y) - F(Y), where and J” are the rotational quantum numbers of the upper respectively.
(6)
the CO: (110-010) Renner-Teller in Table 1. The wavenumber of a v. is the origin of the band, and J and lower states of the emission,
29
Spectroscopic modeling of Renner-Teller bands
0.8
0.2 0.1 0.0 0
250
!xO
750
1000
1250
1500
1760
2000
Vibrational Temperature (I() Fig. 2. Relative vibrational populations of CO, vs temperature. Not shown are the (200), (040), and (001) states which had populations of the same magnitude or less than the (030) state.
Analogous to Eq. (I) for the rotational line intensities of symmetric-stretch bands, the line intensities for Renner-Teller bands will also be proportional to the excitation rate, p, and the emission probability based on a dipole moment. However, because the bending component of angular momentum along the internuclear axis (P = i_ 1) is strongly coupled with the electronic angular momentum, the electronic and vibrational contributions of the initial and final state wave functions cannot be separatedP Therefore, Franck-Condon factors, which are overlap integrals of the initial and final state vibrational wave functions, are not appropriate in describing the line intensities of Renner-Teller bands. However, a factor to account for the relative band strengths of the four (110410) Renner-Teller bands is needed. We therefore introduce an overall vibronic scaling parameter C which includes the combined electronic and vibrational transition moments for both the excitation and emission processes. Thus, the intensity of a line of a Renner-Teller band resulting from electron-impact ionization will be given by I(J) * C&I’)~SRQJ/(2J
+ 1)
(7)
where again p(Y) is the rotational excitation rate into upper state J’. The value for C will be determined for each of the (110-010) Renner-Teller bands through a best fit with the measured spectrum. The sum of the four values for C for the four Renner-Teller bands will be normalized to unity.
Table I. Rotational term value constants as taken from Gauyacq ef al.” Transition term value constant Co(cm _‘1 B, ‘(cm - ‘) B, ” (cm - ‘)
D,.’ { x 10’ cm - ‘) a,” ( x 10’ cm - ‘)
*~--z~29,52 1.83 0.3509 0.3847 0.91 0.96
‘A32-‘A<2
?Ai,2-‘As2
‘2 + -T +
29557.20 0.3510 0.3830 0.91 0.96
29,620.Sl 0.3510 0.3830 0.91 0.96
29642.93 0.3509 0.3833 0.91 0.96
David R. Farley and Robert J. Cattolica
30
18 16
T Pre~oure = 100 png Temperature = 295 K Rwolution = 0.076 nm
2
339
338
338
Wavelength
(nm)
Fig. 3. Comparison of the improved spectroscopic model including the (11010) Renner-Teller bands and a fluorescence emission spectrum taken with a beam energy of 10 keV at 100 pHg and 295 K.
Similar to Eq. (3), the excitation rate of rotational given by
transitions, assuming a dipole moment, is
p N n(J”‘).SR,Q,P/(2J’”+ 1)
(8)
where n is the population distribution of CO, ‘TI, molecules in the (010) vibrational state and in rotational state J. Rotational quantum number J is used for the molecule before electron-impact ionization. The population distribution of the (010) vibrational state is given by n(P)
N
exp( - o,hc/kT) QLih
xr(2J”’ + 1)exp
- J”‘(Y” + l)O, T
1
(9)
The rotational and vibrational modes are assumed to be in thermal equilibrium at a common temperature T. The thermal excitation of the (010) vibrational state is accounted for through the exp( - ozhc/kT)/Q,, term in Eq. (9). The results of Eqns (7) - (9) will thus give the intensities of the rovibronic lines of the CO: (110-010) Renner-Teller bands. 4. RESULTS
Using Eqns (6) - (9) the four (1 l&010) Renner-Teller bands have been added to the model of the CO: A2111,+X2TI, symmetric-stretch model of Farley and Cattolica.’ The results of this improved model are shown in Fig. 3. Through comparison of Figs. 1 and 3, the improvement in the model of the fluorescence is apparent. The present model produces a much better fit with the experimentally measured EBF spectrum between 337 nm and 339 nm. The vibronic scaling factors C of Eq. (7) were adjusted to obtain the best fit of the model with the measured spectrum. These values are summarized in Table 2.
Table 2. Vibronic scaling parameters for (11@-010) Renner-Teller Renner-Teller Band Vibronic scaling parameter
bands.
z: -+z:
AJ!u+Auzu
A~zg+Axzu
0.14
0.20
0.20
z; -ix; 0.46
Spectroscopic modeling of Renner-Teller bands
31
The addition of the vibrational partition function to Eq. (9) should account for the intensity of the (110-010) Renner-Teller bands at varying temperatures. Thus, as temperatures are lowered below 300 K, the relative intensities of these bands will decrease. Similarly, at temperatures larger than 300 K, the relative intensities of these bands will increase and thus become more prominent at these elevated temperatures. The model should still be valid at elevated temperatures, but the vibronic scaling parameters of Table 2 may need to be modified at temperatures significantly above 300 K ( N 700 K). This would be due to other contributions to the CO: (110) vibrational state from vibrational levels other than the CO2 (010) vibrational state, such as the (020) or (100) states. The relative populations of these other vibrational states can be seen in Fig. 2, where these states become significantly populated near 700 K. It should also be noted that the use of Eq. (9) depends on the assumption of thermal equilibrium between the vibrational and rotational modes of the CO> molecules. This may not be necessarily true in fluid applications where strong gradients in the flow exist. These assumptions used in the present spectroscopic model must be considered for specific applications of the model. 5. CONCLUSIONS
The four CO: (110-010) Renner-Teller bands (Cl +&+, As,2g-+As,2U, A3,2g-tAJ,zU, and C, +Z; transitions) have been added to the previous model of Farley and Cattolica,’ which included the (lO&OOO), (200-100) and (300-200) symmetric-stretch vibrational bands. This model assumes that the (010) vibrational state of CO, is thermally excited, as calculated through the vibrational partition function. The excitation and emission processes were then modeled using a dipole moment approximation. The vibronic scaling parameters used to adjust the Renner-Teller bands to fit the spectrum have been obtained. This improved model produces a better fit to the measured spectrum, which will allow more accurate temperature measurements to be made. Further improvement of the model could be obtained through including the four (210-l 10) Renner-Teller bands which are overlapping the (200-100) and (30%200) symmetric-stretch bands in the wavelength region from 339-341 nm. The intensity analysis for these bands would follow the same procedure as outlined in the present analysis, however accurate rotational term value constants for these bands have not been published. However, inclusion of the (210-l 10) Renner-Teller bands are not necessary to obtain accurate rotational temperature measurements since it has been shown that the improved spectroscopic model including the (110-010) Renner-Teller bands accurately fits measured spectra between 337 nm to 339 nm. Therefore, if just the spectral region covered by the (lO&OOO)symmetric-stretch and (110-010) Renner-Teller bands (337-339 nm) are used in a least-squares fit with the measured spectra, sufficient accuracy in rotational temperature measurements can be attained. REFERENCES 1. Farley, D. R. and Cattolica, R. J., JQSRT, 1996, 56, 83. 2. Muntz, E. P., Phys. Fluids, 1962, 5, 80. 3. Robben, F. and Talbot, L., Phys. Fluids, 1966, 9, 644. 4. Tokue, I., Shimada, H., Masuda, A. and Ito, Y., J. Chem. Phys., 1990, 93, 4812. 5. Poulizac, M. C. and Dufay, M., Astrophys. Lett., 1967, 1, 17. 6. Farley, D. R. and Cattolica, R. J., JQSRT, 1996, 56, 753. 7. J. Rostas, private communication (1994). 8. Gauyacq, D., Larcher, C. and Rostas, J., Can. J. Phys., 1979, 57, 1634. 9. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 1, Krieger, Malabar, FL (1950). 10. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2, Krieger, Malabar, FL (1950). Il. Larcher, C., Gauyacq, D. and Rostas, J., J. Chem. Phys., 1980, 77, 655.
JOSRT W-2
B
Spectrosc.
Pergamon PII: !%0224073(97)00130-1
Radial.
Tramfir Vol. 59. No. l/2, pp. 25-31. 1998 Q 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/98 $19.00 + 0.00
SPECTROSCOPIC MODELING OF THE CO: A211,+X21-Ig (110-010) RENNER-TELLER BANDS IN ELECTRON BEAM FLUORESCENCE OF CARBON DIOXIDE DAVID
R. FARLEY
and ROBERT
J. CATTOLICA
Department of Applied Mechanics and Engineering Sciences, University of California at San Diego, 9500 Gilman
Drive.
La Jolla,
(Received
CA 92093-041 I. U.S.A.
May 1997)
Abstract-Improvements have been made to a spectroscopic model of the electron beam fluorescence (EBF) spectra from the COz+ A’JI ,+X%, electronic transition of the (lO&OOO),
(20&100) and (300-200) bands through the inclusion of the (I l&010) Renner-Teller bands. The (010) vibrational state of the CO, molecule is thermally excited at temperatures near 300 K based on an analysis of the vibrational partition function for this molecule. The excitation and emission processes for these Renner-Teller bands were modeled with dipole moments. The inclusion of these Renner-Teller bands into the spectroscopic model produces an improved fit to the measured fluorescence spectrum between 337 nm and 339 nm. This improved model provides for a more accurate measurement of the rotational temperature using the EBF technique. Q 1998 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION To measure rotational temperatures through electron beam fluorescence (EBF) of carbon dioxide a spectroscopic model of the excitation/emission processes has been developed by Farley and Cattolica’ for temperatures up to - 300 K. With this technique, the temperature of the CO, gas can be obtained through varying the rotational temperature of the spectroscopic model until a least-squares fit with the measured spectrum is attained. Therefore, the spectroscopic model must accurately represent measured spectra in order to obtain accurate temperature measurement. The motivation behind this research has been for application to aerobraking or aerocapture in the atmosphere of Mars, where ambient temperatures vary between 150 K and 300 K. However, this type of spectroscopic modeling is useful for other applications involving CO?, and also gives insights into conducting similar analyses for other linear triatomic molecules. 2. BACKGROUND The previous results of Farley and Cattolica’ are shown in Fig. 1. Depicted in Fig. 1 are both the experimentally measured spectrum obtained with the EBF apparatus, and the theoretically modeled spectrum. For the measured spectrum, a 10 keV electron beam (nominal beam diameter of 1.0 mm) was injected into ambient CO, gas at 295 K and maintained at a pressure of 100 PHg. The emitted CO: fluorescence resulting from electron-impact and ionization was recorded using a 0.32 m spectrometer equipped with a 3600 groove/mm grating and a 1024 element image-intensified diode array. The measured spectrum has a corresponding resolution of 0.833 nmjmm. The frequency response of the diode array was normalized through the use of a measured spectrum from a calibrated deuterium source. Five overlapping symmetric-stretch bands are included in the spectroscopic model of Farley and Cattolica.’ These are the (lO&OOO), (200-100) and (300-200) bands from the CO: All-I 3,2”+X2&p electronic transition, and the (100-000) and (200-100) bands from the CO: electronic transition. Since the electron beam energy is much higher than the A*l-I,,+X*II.,, interaction energy of the collision process, Born’s first approximation is assumed and thus the electronic, vibrational and rotational states of the initial and final state wave functions are separable. Dipole moments were used for the excitation and emission rotational transition 25
26
David R. Farley and Robert J. Cattolica
16 Pressure = 100 FHg Temperature = 295 K Remolution = 0.076 nm
337
336
339
338 Wavelength
340
341
(nm)
Fig. 1. Comparison of spectroscopic model’ of the CO: A’II, +X2& (1O&000), (200- 100) and (30&200) bands and a fluorescence emission spectrum taken with a beam energy of 10 keVat 100 pHg and 295 K.
probabilities. With this assumption, the intensities of rotational spectral lines of a symmetric-stretch band are given bylm3
I”,~,“,V) Iv Cd0,“‘,u,‘).p(J’).q(U,‘,~,“).S~,~*p/(2J’+ 1) Y,“’
(1)
where I,,,,U,.@‘)is th e intensity of rotational line J’ corresponding to the appropriate R - , Q - , or P - branch within a (u,‘-+u,“) symmetric-stretch vibrational band. No dependence of the intensity distribution on wavenumber, u’, is included in Eq. (1) since the variation is negligible over the transitions that are modeled. Because the symmetric-stretch vibrational bands are composed of rotational transitions, which assume a dipole moment transition, the dipole selection rules AJ = 0, F 1 apply. The standard convention is used where AJ = + 1 corresponds to an R-branch transition, AJ = - 1 for P-branch transitions, and AJ = 0 for Q-branch transitions. q(u,“‘,u,‘) is the excitation Franck-Condon factor for the CO* symmetric-stretch u,“’ state to the excited CO: u,’ state, q(uI’,u,“) is the emission Franck-Condon factor for the (u,‘-+u,“) symmetric-stretch band, and SR*cpare the rotational line strengths, or Honl-London factors, for emission. Published values for the Franck-Condon factors were used.‘s4,5 The forms of rotational line strengths for dipole transitions, SR*Q,p,can be found tabulated in the work of Farley and Cattolica.6 In the symmetric-stretch model it was assumed that only the v;II= 0 ground vibrational state was significantly populated, which is a valid assumption at room temperatures. The excitation rate into excited rotational states p(J), is calculated utilizing a dipole moment in the electron-impact excitation process. The unexcited CO* X’ C,+ molecules are assumed to be in equilibrium at rotational temperature r. The ground state unexcited molecules then follow a Boltzmann distribution n(Ntt,>
_
o(2N”’
+
l)exp
-
N”W; + I)@, I
>
where n is the relative population of molecules of a given ground state rotational quantum number, 0, is the rotational constant of the ground state dimensionalized in terms of temperature (0, = hcB,./k = 0.5614 K), and 0 is a factor to account for nuclear spin statistics which
Spectroscopic modeling of Renner-Teller bands
27
is 1 for even, and 0 for odd. Thus there are no molecules in odd states. Note the common convention that the lowest lying states, those of the unexcited CO,X’ Cl ground state, are denoted with three primes (,“). The excitation rate into an excited rotational state is given by p N n(N”‘).SR.e,P/(2N”’+ 1)
(3)
where SR.c,Pare the electric dipole transition probabilities, or Honl-London factors, for excitation of R, Q, and P branches, respectively. The resulting theoretical spectrum calculated from Eq. (1) was convolved with the line shape of a measured mercury line occurring at 334.1 nm to produce the modeled spectrum of Fig. 1. The measured mercury line had a FWHM linewidth of 0.08 nm. A numerical routine was developed to calculate the convolving of the mercury line transfer function with the idealized line spectrum. Although the symmetric-stretch model fits well with the measured spectrum, there are still several perturbations in the measured spectrum which have not yet been accounted for in the model. Eight of these perturbations are known to be due to the (110-010) and (210-l 10) Renner-Teller bands.’ The four CO: (110-010) Renner-Teller bands originating at 337.25 nm, 337.51 nm, 338.23 nm and 338.64 nm, have been identified and their wavelength locations investigated by Gauyacq et ~1.~The emissions of these four (1 l&010) Renner-Teller bands are summarized below and are shown labeled in Fig. 1. 1. 2. 3. 4.
Cl +Z: emission occurring at 337.25 nm &zg+&zu emission occurring at 337.51 nm &,z, +AJiZUemission occurring at 338.23 nm C, +Z; emission occurring at 338.64 nm
The addition of these bands would improve the spectroscopic model between 337 nm and 339 nm, and therefore result in more accurate rotational temperatures being measured. There are also four (210-l 10) Renner-Teller bands located between 339 nm and 341 nm, but these bands have not yet had their wavelength locations spectroscopically analyzed. If there were accurate line positions for the (210-l 10) bands, however, then a similar analysis as will be done herein for the (110-010) bands could be conducted for the (210-I 10) Renner-Teller bands. Also, between 340 nm and 341 nm are perturbations which are not due to Renner-Teller bands, and which have not been identified,’ thus making it impractical to attempt to add these perturbations to the model. However, if the spectroscopic model can be improved between 337 nm and 339 nm, and the fit between the measured and theoretical spectra is done within this spectral region, then accurate temperatures can be measured without a significant need to improve the model beyond 339 nm. For completeness it would be useful to have all contributions to the spectra included in the spectroscopic model, but towards the goal of accurate temperature measurements such completeness is not required. Therefore, a spectroscopic analysis will be done of the excitation/emission processes for the four (110410) Renner-Teller bands, and these bands will be incorporated into the previous spectroscopic model of Farley and Cattolica.’ 3.
ANALYSIS
To account for the observed intensities of the (110-010) Renner-Teller bands, two approaches are possible. In one approach, it is assumed that the (110) upper states of CO: are excited directly from the (000) vibrational ground state of CO,. This assumes that only the (000) vibrational ground state is significantly populated at ambient temperatures near 300 K, and requires a quadrupole transition moment for the rotational excitation since a (OOO)+(l 10) transition is not allowed with a dipole moment. This is because the excitation of the bending quantum v2 = 1 adds an angular momentum component / = + 1 along the internuclear axis which is coupled to the rotational state of the mo1ecule.‘,6 Therefore, for a transition to occur from the (000) ground vibrational state to the (110) excited state would require a change of AK = 2 in the quantum of angular momentum projected along the internuclear axis, which is forbidden with a dipole moment.6 In the second approach, it is assumed that the (010) vibrational state of the ambient CO, molecules is thermally excited from the (000) state prior to electron-impact excitation. Both of these approaches were
28
David R. Farley and Robert J. Cattolica
studied. Although the quadrupole modeling produced satisfactory qualitative results, the quantitative magnitudes required for the observed intensities of the Renner-Teller bands were much higher than are typical for quadrupole moments. 9 After further investigation into the vibrational partition function, particularly for the population of the (010) vibrational state of CO?, it was found that an adequate number of CO, molecules are thermally excited into the (010) vibrational state before electron impact and ionization to account for the observed Renner-Teller band intensities. It was therefore decided that the quadrupole excitation of the Renner-Teller bands from the (000) ground state was unlikely. Hence, the (010) state is assumed to be excited thermally, and the excitation mechanism between the CO, (010)-X0:(1 10) excited vibrational states caused by electron-impact ionization can then be modeled with a dipole moment. The vibrational partition function for a linear triatomic molecule is given by
where E,,(u,,u2,uj)= (w,u, + wZu2+ w&rc; o, is the frequency of the symmetric-stretch vibrational mode corresponding to vibrational quantum number u,;02 is the frequency of the bending vibrational mode corresponding to vibrational quantum number u2;w3is the frequency of the anti-symmetric-stretch vibrational mode corresponding to vibrational quantum number 0,; h, c, and k are Planck’s constant, the speed of light, and Boltzmann’s constant, respectively, and r is the vibrational temperature. The energy of the vibrational ground state has been arbitrarily set to (w,/2 + o2 + o$2)hc. Using Eq. (4) for the vibrational partition function, it is then possible to calculate the relative populations of the CO, vibrational states, n(u,,uZ,uJ), as a function of temperature as u2
ntu, 9
9
u3j
=
exp[ - Evtw2~~3Yk~l Qd
(5)
Eqns (4) and (5) were calculated numerically between 100 K and 2000 K, summing from 0 to 10 for all vibrational quantum numbers in Eq. (4). Values for the vibrational frequencies were taken from Herzberg,” and are w, = 1337 cm - ‘, w2 = 667 cm - ‘, and Ok = 2349 cm - ‘. The results of this numerical analysis are shown in Fig. 2, depicting the significantly populated vibrational states between 100 K and 2000 K, which as shown are the (000), (100) (OlO), (020), and (030) states. The (200), (040) and (001) vibrational states had relative populations of the same magnitude or less than the (030) state, but these plots are not depicted in Fig. 2 to avoid cluttering the graph. Note in Fig. 2 that the (100) and (020) vibrational state populations are equal. This is due to the vibrational bending frequency o2 being nearly exactly half that of the symmetric-stretch vibrational frequency 0,. Thus Fermi resonance can occur between the (100) and (020) states, consistent with the observations of Larcher et al” for CO:. The relative population of the (010) vibrational state near 300 K is approximately 4%, which is consistent with the observed relative intensities of the Renner-Teller bands as compared with the symmetric-stretch bands. Therefore, it is assumed that the (010) vibrational state is thermally populated prior to electron impact, and subsequently through electron-impact ionization the molecules can be excited through a dipole moment to CO: (110) vibrational states. The CO: (110-010) Renner-Teller bands can then be modeled using dipole moments for both the excitation and emission processes. The locations of the rotational lines are obtained from the rotational term values for the upper and lower states of the emission. The general equation for a rotational term value is given by F(J) = B;J(J + 1) + D,:J’(J + 1)2 The rotational term value constants B, and D, to be used for bands are those obtained by Gauyacq et al,’ and are given rotational line is then calculated as v. + F(Y) - F(Y), where and J” are the rotational quantum numbers of the upper respectively.
(6)
the CO: (110-010) Renner-Teller in Table 1. The wavenumber of a v. is the origin of the band, and J and lower states of the emission,
29
Spectroscopic modeling of Renner-Teller bands
0.8
0.2 0.1 0.0 0
250
!xO
750
1000
1250
1500
1760
2000
Vibrational Temperature (I() Fig. 2. Relative vibrational populations of CO, vs temperature. Not shown are the (200), (040), and (001) states which had populations of the same magnitude or less than the (030) state.
Analogous to Eq. (I) for the rotational line intensities of symmetric-stretch bands, the line intensities for Renner-Teller bands will also be proportional to the excitation rate, p, and the emission probability based on a dipole moment. However, because the bending component of angular momentum along the internuclear axis (P = i_ 1) is strongly coupled with the electronic angular momentum, the electronic and vibrational contributions of the initial and final state wave functions cannot be separatedP Therefore, Franck-Condon factors, which are overlap integrals of the initial and final state vibrational wave functions, are not appropriate in describing the line intensities of Renner-Teller bands. However, a factor to account for the relative band strengths of the four (110410) Renner-Teller bands is needed. We therefore introduce an overall vibronic scaling parameter C which includes the combined electronic and vibrational transition moments for both the excitation and emission processes. Thus, the intensity of a line of a Renner-Teller band resulting from electron-impact ionization will be given by I(J) * C&I’)~SRQJ/(2J
+ 1)
(7)
where again p(Y) is the rotational excitation rate into upper state J’. The value for C will be determined for each of the (110-010) Renner-Teller bands through a best fit with the measured spectrum. The sum of the four values for C for the four Renner-Teller bands will be normalized to unity.
Table I. Rotational term value constants as taken from Gauyacq ef al.” Transition term value constant Co(cm _‘1 B, ‘(cm - ‘) B, ” (cm - ‘)
D,.’ { x 10’ cm - ‘) a,” ( x 10’ cm - ‘)
*~--z~29,52 1.83 0.3509 0.3847 0.91 0.96
‘A32-‘A<2
?Ai,2-‘As2
‘2 + -T +
29557.20 0.3510 0.3830 0.91 0.96
29,620.Sl 0.3510 0.3830 0.91 0.96
29642.93 0.3509 0.3833 0.91 0.96
David R. Farley and Robert J. Cattolica
30
18 16
T Pre~oure = 100 png Temperature = 295 K Rwolution = 0.076 nm
2
339
338
338
Wavelength
(nm)
Fig. 3. Comparison of the improved spectroscopic model including the (11010) Renner-Teller bands and a fluorescence emission spectrum taken with a beam energy of 10 keV at 100 pHg and 295 K.
Similar to Eq. (3), the excitation rate of rotational given by
transitions, assuming a dipole moment, is
p N n(J”‘).SR,Q,P/(2J’”+ 1)
(8)
where n is the population distribution of CO, ‘TI, molecules in the (010) vibrational state and in rotational state J. Rotational quantum number J is used for the molecule before electron-impact ionization. The population distribution of the (010) vibrational state is given by n(P)
N
exp( - o,hc/kT) QLih
xr(2J”’ + 1)exp
- J”‘(Y” + l)O, T
1
(9)
The rotational and vibrational modes are assumed to be in thermal equilibrium at a common temperature T. The thermal excitation of the (010) vibrational state is accounted for through the exp( - ozhc/kT)/Q,, term in Eq. (9). The results of Eqns (7) - (9) will thus give the intensities of the rovibronic lines of the CO: (110-010) Renner-Teller bands. 4. RESULTS
Using Eqns (6) - (9) the four (1 l&010) Renner-Teller bands have been added to the model of the CO: A2111,+X2TI, symmetric-stretch model of Farley and Cattolica.’ The results of this improved model are shown in Fig. 3. Through comparison of Figs. 1 and 3, the improvement in the model of the fluorescence is apparent. The present model produces a much better fit with the experimentally measured EBF spectrum between 337 nm and 339 nm. The vibronic scaling factors C of Eq. (7) were adjusted to obtain the best fit of the model with the measured spectrum. These values are summarized in Table 2.
Table 2. Vibronic scaling parameters for (11@-010) Renner-Teller Renner-Teller Band Vibronic scaling parameter
bands.
z: -+z:
AJ!u+Auzu
A~zg+Axzu
0.14
0.20
0.20
z; -ix; 0.46
Spectroscopic modeling of Renner-Teller bands
31
The addition of the vibrational partition function to Eq. (9) should account for the intensity of the (110-010) Renner-Teller bands at varying temperatures. Thus, as temperatures are lowered below 300 K, the relative intensities of these bands will decrease. Similarly, at temperatures larger than 300 K, the relative intensities of these bands will increase and thus become more prominent at these elevated temperatures. The model should still be valid at elevated temperatures, but the vibronic scaling parameters of Table 2 may need to be modified at temperatures significantly above 300 K ( N 700 K). This would be due to other contributions to the CO: (110) vibrational state from vibrational levels other than the CO2 (010) vibrational state, such as the (020) or (100) states. The relative populations of these other vibrational states can be seen in Fig. 2, where these states become significantly populated near 700 K. It should also be noted that the use of Eq. (9) depends on the assumption of thermal equilibrium between the vibrational and rotational modes of the CO> molecules. This may not be necessarily true in fluid applications where strong gradients in the flow exist. These assumptions used in the present spectroscopic model must be considered for specific applications of the model. 5. CONCLUSIONS
The four CO: (110-010) Renner-Teller bands (Cl +&+, As,2g-+As,2U, A3,2g-tAJ,zU, and C, +Z; transitions) have been added to the previous model of Farley and Cattolica,’ which included the (lO&OOO), (200-100) and (300-200) symmetric-stretch vibrational bands. This model assumes that the (010) vibrational state of CO, is thermally excited, as calculated through the vibrational partition function. The excitation and emission processes were then modeled using a dipole moment approximation. The vibronic scaling parameters used to adjust the Renner-Teller bands to fit the spectrum have been obtained. This improved model produces a better fit to the measured spectrum, which will allow more accurate temperature measurements to be made. Further improvement of the model could be obtained through including the four (210-l 10) Renner-Teller bands which are overlapping the (200-100) and (30%200) symmetric-stretch bands in the wavelength region from 339-341 nm. The intensity analysis for these bands would follow the same procedure as outlined in the present analysis, however accurate rotational term value constants for these bands have not been published. However, inclusion of the (210-l 10) Renner-Teller bands are not necessary to obtain accurate rotational temperature measurements since it has been shown that the improved spectroscopic model including the (110-010) Renner-Teller bands accurately fits measured spectra between 337 nm to 339 nm. Therefore, if just the spectral region covered by the (lO&OOO)symmetric-stretch and (110-010) Renner-Teller bands (337-339 nm) are used in a least-squares fit with the measured spectra, sufficient accuracy in rotational temperature measurements can be attained. REFERENCES 1. Farley, D. R. and Cattolica, R. J., JQSRT, 1996, 56, 83. 2. Muntz, E. P., Phys. Fluids, 1962, 5, 80. 3. Robben, F. and Talbot, L., Phys. Fluids, 1966, 9, 644. 4. Tokue, I., Shimada, H., Masuda, A. and Ito, Y., J. Chem. Phys., 1990, 93, 4812. 5. Poulizac, M. C. and Dufay, M., Astrophys. Lett., 1967, 1, 17. 6. Farley, D. R. and Cattolica, R. J., JQSRT, 1996, 56, 753. 7. J. Rostas, private communication (1994). 8. Gauyacq, D., Larcher, C. and Rostas, J., Can. J. Phys., 1979, 57, 1634. 9. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 1, Krieger, Malabar, FL (1950). 10. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2, Krieger, Malabar, FL (1950). Il. Larcher, C., Gauyacq, D. and Rostas, J., J. Chem. Phys., 1980, 77, 655.
JOSRT W-2
B