Calculated tabulations of H 2 0 line broadening by H 2 0, N 2 , 02, and CO 2 at high temperature Corinne Delaye, Jean-Michel Hartmann, and Jean Taine We present calculations of the temperature dependence of H 2 0 pressure broadening parameters by H 2 0, CO 2 , N 2 , and 02. They were made for Q-lines with a theoretical model which provides a correct treatment of close collisions and has been widely tested. The results should be useful for Raman spectra calculations. A simple law is proposed to deduce halfwidths of P- and R-lines from the Q-line results. The accuracy of this law at high temperature is demonstrated. A simple analytical representation of a constant halfwidth approxima- tion is also given. 1. Introduction Precise knowledge of H 2 0 line broadening by colli- sions is required for synthetic spectra calculations. In the particular case of high temperatures, pressure broadening coefficients have a great influence on the modeling of radiative heat transfer in combustion me- dia; they are also required for temperature measure- ments from infrared and Raman spectra inversion. The room temperature broadening of H 2 0 lines has received considerable experimental and theoretical at- tention (see Ref. 1). Models derived from the ap- proaches of Anderson, 2 Tsao and Curnutte 3 (ATC), and Davies and Oli4,5 (QFT) have been used for sys- tematic predictions of self- and N 2 broadening. 6 - 9 Nevertheless, the available data, which are limited to a few perturbers and restricted temperature ranges, re- main insufficient for most applications. Another model (RB) has been proposed by Robert, Bonamy, and others. 10 ' 1 ' It has been widely and successfully tested on the broadening of C 2 , molecules such as H 2 0 (Refs. 1 and 10-12) and 03 (Ref. 13). Accurate results also obtained for linear molecules have enabled calcu- lation of broadening data bases for CO 2 (Ref. 14) and CO.15 Our model insures 1 ,1 3 a more accurate treat- ment of close collisions than the ATC and QFT ap- The authors are with Laboratoire d'Energetique Moleculaire et Macroscopique-Combustion du CNRS (UPR73) et de 'Ecole Cen- trale des Arts et Manufactures, Grande Voie des Vignes, 92295 Chatenay-Malabry CEDEX, France. Received 7 December 1988. 0003-6935/89/235080-08$02.00/0. ) 1989Optical Society of America. proaches. It has improved theoretical predictions for high rotational quantum number lines, high tempera- tures, and weakly polar perturbers."'121 3 In this paper our model (RB) is used for systematic predictions of the broadening of H 2 0 lines by H 2 0, N 2 , 02, and CO 2 up to 3000 K. The theoretical model is briefly recalled in Sec. II. Due to the great number of H 2 0 lines and computer cost involved by systematic calculations, only the broadening of Q-lines has been calculated; the results are presented in Sec. III. Sim- ple expressions which enable the prediction of the broadening of P- and R-lines are given and tested in Sec. IV. A convenient constant halfwidth approxima- tion for high temperature spectra calculations is also presented. II. Theoretical Model The theoretical model used was initially developed by Robert and Bonamy for linear molecules.' 6 It has enabled the constitution of accurate broadening data bases of CO 2 (Ref. 14) and CO (Ref. 15) for many perturbers and lines and in a wide temperature range. This model was recently adapted to the case of C2v molecules.10"'1It has been tested on the broadening of H 2 0 (Refs. 1 and 10-12) and 03 (Ref. 13) lines for which accurate results are obtained for various per- turbers, lines, and temperatures. Within the RB model, the halfwidth y (HWMH) of the Jir - Jrf line is given by'16 aRB = (N 2 I/2rc) ( 27R[v'(Rv)/i3 2 \JRmin X 1 - exI-S(RivJiiJrf/V 2 J 2 ])dR)V 2 J 2 I (1) where N 2 is the density of perturbers and Rmin is the 5080 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989
Transcript
Calculated tabulations of H2 0 line broadening by H2 0, N2, 02,and CO2 at high temperature
Corinne Delaye, Jean-Michel Hartmann, and Jean Taine
We present calculations of the temperature dependence of H20 pressure broadening parameters by H20, CO2,N2, and 02. They were made for Q-lines with a theoretical model which provides a correct treatment of closecollisions and has been widely tested. The results should be useful for Raman spectra calculations. A simplelaw is proposed to deduce halfwidths of P- and R-lines from the Q-line results. The accuracy of this law athigh temperature is demonstrated. A simple analytical representation of a constant halfwidth approxima-tion is also given.
1. Introduction
Precise knowledge of H2 0 line broadening by colli-sions is required for synthetic spectra calculations. Inthe particular case of high temperatures, pressurebroadening coefficients have a great influence on themodeling of radiative heat transfer in combustion me-dia; they are also required for temperature measure-ments from infrared and Raman spectra inversion.
The room temperature broadening of H20 lines hasreceived considerable experimental and theoretical at-tention (see Ref. 1). Models derived from the ap-proaches of Anderson,2 Tsao and Curnutte3 (ATC),and Davies and Oli4,5 (QFT) have been used for sys-tematic predictions of self- and N2 broadening.6 -9Nevertheless, the available data, which are limited to afew perturbers and restricted temperature ranges, re-main insufficient for most applications. Anothermodel (RB) has been proposed by Robert, Bonamy,and others.10'1' It has been widely and successfullytested on the broadening of C2, molecules such as H20(Refs. 1 and 10-12) and 03 (Ref. 13). Accurate resultsalso obtained for linear molecules have enabled calcu-lation of broadening data bases for CO2 (Ref. 14) andCO.15 Our model insures 1,13 a more accurate treat-ment of close collisions than the ATC and QFT ap-
The authors are with Laboratoire d'Energetique Moleculaire etMacroscopique-Combustion du CNRS (UPR73) et de 'Ecole Cen-trale des Arts et Manufactures, Grande Voie des Vignes, 92295Chatenay-Malabry CEDEX, France.
Received 7 December 1988.0003-6935/89/235080-08$02.00/0.
) 1989 Optical Society of America.
proaches. It has improved theoretical predictions forhigh rotational quantum number lines, high tempera-tures, and weakly polar perturbers."'1213
In this paper our model (RB) is used for systematicpredictions of the broadening of H20 lines by H20, N2,02, and CO2 up to 3000 K. The theoretical model isbriefly recalled in Sec. II. Due to the great number ofH20 lines and computer cost involved by systematiccalculations, only the broadening of Q-lines has beencalculated; the results are presented in Sec. III. Sim-ple expressions which enable the prediction of thebroadening of P- and R-lines are given and tested inSec. IV. A convenient constant halfwidth approxima-tion for high temperature spectra calculations is alsopresented.
II. Theoretical Model
The theoretical model used was initially developedby Robert and Bonamy for linear molecules.'6 It hasenabled the constitution of accurate broadening databases of CO2 (Ref. 14) and CO (Ref. 15) for manyperturbers and lines and in a wide temperature range.This model was recently adapted to the case of C2vmolecules.10"'1 It has been tested on the broadening ofH20 (Refs. 1 and 10-12) and 03 (Ref. 13) lines forwhich accurate results are obtained for various per-turbers, lines, and temperatures.
Within the RB model, the halfwidth y (HWMH) ofthe Jir - Jrf line is given by'16
aRB = (N2 I/2rc) ( 27R[v'(Rv)/i32\JRmin
X 1 - exI-S(RivJiiJrf/V 2 J2 ])dR)V 2 J2 I (1)
where N2 is the density of perturbers and Rmin is the
distance of closest approach1 6 ; v' and v are the equiva-lent straight trajectory velocity'6 and average initialrelative velocity, respectively. The S-function re-sults' from the broadening contribution of the aniso-tropic potential; (... ) V2 J2 is an average on the rovibra-tional level v2J2 of the perturber. The mainimprovements of this model, when compared to theATC and QFT approaches, consist of a correct model-ing of close collisions. This is achieved' 0 by account-ing for contributions of a short range potential to boththe anisotropic interaction (in addition to the electro-static part of S) and classical trajectory calculation(introduction of the v' velocity).
The data required include molecular and spectro-scopic data for the active molecule and perturber aswell as parameters of the interaction. Those used hereare given in Refs. 1 and 17 (except for the H20-N 2
Lennard-Jones parameters of the isotropic potentialwhich are e = 96.6 K and af = 3.28 A). The spectroscop-ic data and transition probabilities have been calculat-ed by Flaud and Camy-Peyret.18
111. Broadening of O-Lines
We have calculated the pressure broadening coeffi-cients (HWMH) of H20 Q-lines by H20, N2, 02, andCO2 in the 300-2000-K temperature range. Equation(1) was used and the average on the Boltzmann veloci-ty distribution was not performed (only the averagevelocity was considered). Calculations have beenmade for lines involving the ground vibrational state inboth the upper and lower states of the transition. Letus note that such lines are fictive and do not exist;nevertheless, as shown later, the calculated data can beused to deduce broadening of other lines. Only thelines for which the rotational energy of the level islower than 2900 cm-' have been calculated. As shownin Ref. 19, the vibrational dependence of linewidths issmall and the present results can be used for anyvibrational band. The room temperature halfwidths-y(300 K) and temperature exponents N, such as,
y(7) = y(300 K) [300/TIN, (2)
are given in Table I. They have been deduced fromcalculations for temperatures of 300, 750, and 1825 K.Tests have shown that Eq. (2) is accurate within 10% atthe worst (for high rotational quantum number lines).Detailed analysis of the broadening mechanism wasmade previously.1'2 Resonance overtaking and kinet-ic effects are responsible for the great variations of Nand y(300 K) with respect to the perturber and linequantum numbers.1' 2 They explain' the increase ofsome linewidths with temperature. Figure 1 demon-strates that line- and perturber-dependent data are tobe used.
IV. Broadening of P- and R-Lines
Following Refs. 12 and 20, the halfwidth of the i fline can be approximated by
yRB(q) - (N2j/2c)Roisf)2[v,(Rov)1v]2, (3)
where uM(Rv) is the velocity at distance R (Ref. 16);R,(i,f) is defined by
expJ-S[R0(i),v,ifv 2 = 0, J2 = a, (4)
where a is a constant adjustable parameter and J2 isthe averaged rotational level.1220 When assuming apredominant anisotropic potential in R-n and reso-nant collisions (i.e., neglecting all collision-inducedrotational energy transfers), Eq. (4) leads to12
S(R,vifv2 = ,J2) n(a) [R(if)/R](2n2)
(5)
By using the relation
S(R,vifv 2 ,J2) [S(R,viiv 2 = 0,J2 )
+ S(R,5,ffv 2 = 0,J2)1/2,
Eq. (4), and neglecting kinetic effects [v'(R,v) = vu(R,v)= u1, one gets
Rj~if) (2n-2) [Rjii)(2n-2) + R (f &2n-2) /2 (7)
This result relates the halfwidth of any OJ-r -> 'r'line to those of the corresponding OJr - OJr and OJ'i-'
OJ'r' Q-lines, i.e.,
,Y9J-,9'JSr , TYOi-OJ'r' t (nOR -1)
+ 'YOJg-OJtt(n-1/211/(n-l) (8)
The values of n to be used are given in Table II. Let usnote that Eq. (8) results from two main approxima-tions which assume that nonresonance and kinetic ef-fects1 2 are negligible. This is accurate at high tem-perature but fails at room temperature for highrotational quantum number lines.
Table III gives the relative differences between thebroadenings of P-lines calculated from Eq. (1) andthose predicted by Eq. (8) and Tables I and II. It is atest of the accuracy of both the exponential tempera-ture dependence in Eq. (2) and the approximation inEq. (8). It shows that these approximations give accu-rate results at elevated temperature but fail at roomtemperature when high rotational quantum numberlines are considered. This is to be expected sincekinetic effects, which are responsible for the systemat-ic underestimation of linewidths, are all the more im-portant when the initial velocity is small and broaden-ing occurs at short distances. Let us note that thelargest discrepancies will slightly affect spectra calcu-lations; indeed, they occur for lines of high initial levelenergy which are related to little-populated levels andcontribute little to spectra.
The results of our calculation [through Eq. (1)] arecompared with previous ones in Table IV. The overallagreement on y(3 00 K) is quite good; on the otherhand, the ATC calculation for H20-N 2 (Ref. 6) leads tomuch smaller values ofNfor the narrowest lines. Thisis due to the inaccuracy of the ATC approach for shortdistance collisions and the fact that N has been de-duced6 from low temperature results.
At high temperature, all the lines have similarbroadening values as shown in Fig. 2. We have calcu-lated the averaged halfwidth yav(T) from Table I and
ya(7T) = J-J, (9)JT
where pj, is the relative population of level Jr. Theresults have been approximated by the law
,av(T) = yaV(300 K)[3001TNv(T;
Nav(7) = Nav(300 K)[300/71. (10)
The parameters yav(3 0 0 K), Nav(300 K), and fi deducedfrom Eqs. (9) and (10) are given in Table V. Thevalues of Nav(300 K) are generally larger than those ofindividual lines due to the population term pjT(T) in Eq.(9).
At high temperature (above 1500 K) the data inTable I may be insufficient for spectra calculationssince lines of high initial level energy (>2900 cm-') aremissing; nevertheless, the constant halfwidth approxi-mation is then accurate and can be used to deduce theirbroadening parameters.
V. Conclusion
We have calculated the broadening coefficients ofH2 0 Q-lines by H2 0, N2, 02, and CO2 in the 300-2000-K range with a model which provides a correct treat-ment of close collisions. The data should be useful forRaman spectra calculations. A simple analytic lawhas been proposed to predict the broadening of P- andR-lines which is accurate at elevated temperature. Asimple representation of the constant halfwidth ap-proximation has also been given.
References
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