Theoretical line widths in N2O-N2O and N2O-air collisions

J. Quonr.

Spec~msc.Rodiar.

Transfer. Vol.

THEORETICAL

I I, pp.1659-1664.

Pergamon Press 1971. Printed in Great Britain

LINE WIDTHS IN N,O-N,O N,O-AIR COLLISIONS*

AND

G. D. T. TEJWANIand P. VARANASI Department of Mechanics, State University of New York, Stony Brook, N.Y. 11790, U.S.A. (Received 1 February 1971)

Abstract-Rotational half-widths have been computed at 220”K, 250”K, 280”K, and 300°K for air-broadened absorption lines of NrO using the Anderson-TsaoCurnutte theory. The new results are compared with previously available estimates at 300°K. Self-broadened line widths have also been computed at 300°K; good agreement has been obtained with Goody’s data. INTRODUCTION IN A RECENT paper on the abundance of N,O in the atmosphere, GOLDMAN et cd.“’ have pointed out the wide discrepancies in the published estimates for the half-widths of N,O lines. A value of y” = 0.157 f 0.0075 cm- ‘atm- 1 was derived by GOODY and WORMELL@) as a single estimate for all of the lines in the 7.8 ,u,8.6 p, and 17 ,u bands. In a later publication describing a cross-correlating spectrometer, GOODY (3) obtained estimates for the ‘optimum’ line widths that provided the best estimates for his measured absorption data on the N,O bands near 3.9p. Reduced to infinite dilution in air, the optimum line widths were 0.083 cm-‘atm-’ for J < 12, 0.078 cm-‘atm-’ for 12 < J < 18, and 0.073 cm-‘atm-’ for J > 18. Comparing the data for pure N,O and 0.1069 mixture of N,O in dry air, G~oDY(~’ found the ratio of the line width for pure N,O to that for N,O broadened by dry air to be 1.25 kO.05, which is substantially the same result obtained originally by GOODY and WORMELL!~) FRALEY et aLc4’ reported a value of 0.05 cm- ‘atm- ’ for some unblended line in the 4.5 p band. GRAY(~)estimated a value of 0.05 cm- ‘atm- ’ by comparing measured spectral absorptance in the parallel bands of N,O with a computation utilizing a random Elsasser band model. OPPENHEIM and GOLDMAN (6) found a linear variation of y” from 0.098 cm- ’ atm- ’ for J = 20 to 0.06 cm- ‘atm- ’ for J = 60, and arrived at an average value of 0.08 cm- ‘atm- ’ for nitrogen-broadened half-width for the 2240-2260 cm- ’ region. ADEL and BARKER(‘) measured a half-width of 0.08 f 0.005 cm- ’ for a line in the long-wave wing of the 17 p band broadened by air. In view of the apparent uncertainty in the reported values for the half widths of N,O, we present here the results of a computation based upon the Anderson-Tsao-Curnutte

* Supported by the National Science Foundation through Grant No. GK-5114. 1659

1660

G. D. T. TEJWANI and P. VARANASI

theory. Since the dependence of half-widths on the rotational quantum number and temperature for this molecule are not well established experimentally, it is hoped that our computed results will prove useful until more careful measurements are performed. COMPUTATION

OF LINE

WIDTHS

It has been shown by YAMAMOTO et al. w that the Anderson-Tsao-Curnutte theory leads to fairly accurate prediction of line widths in C02-CO, and CO,-N, collisions. Since N,O is also a linear triatomic molecule, it may be assumed that the theory will yield line widths in N,O-air collisions with the same degree of accuracy. It should be noted that, in the case of NzO, dipole-quadrupole interactions should also be considered. However, due to its small dipole moment (p = 0.166 x lo- l8 e.s.u.-cm(“), quadrupole-quadrupole interactions dominate the line broadening mechanism in N,O-air collisions. In our calculations, we have included both quadrupole-quadrupole and dipole-quadrupole interactions. A detailed discussion of the theoretical procedure may be found in Refs. (8), (10) and (11). Computations have been carried on an IBM 360 computer, using the following molecular parameters : QNLO= 6.0 x 1O-26 e.s.u.--cm2, QN2= 3.04 x 1O-26 e.s.u.cm’, and Qo2 = 0.78 x lO-26 e.s.u.-cm2, for the quadrupole moments of N,O, N, and 0,) respectively?’

and hm,NzO-02

=

3.7 A2

for the minimum collision diameters in N,O--N, and N20--0, collisions, respectively.!“’ hll of the spectroscopic constants for N,O, N, and 0, were taken from HERZBERG.!‘~-“’ Line-width computations are almost identical for both N,O-N, and N20-0, collisions, except for the differences in the nuclear statistics of the N, and 0, molecules. In deriving line widths for air-broadened N,O, we have assumed standard atmospheric composition of air and used the relation

We have also computed half-widths in resonant N,O-N,O collisions, taking dipoledipole, dipole-quadrupole, quadrupole-dipole and quadrupole-quadrupole interactions into account. The minimum collision diameter in this case is equal to 3.88 ,&:12) RESULTS

AND

DISCUSSION

The results of our computation are shown in Figs. 1 and 2 and in Table 1. In Fig. 1, N,O-0, and N,O-air show the half-width y” (cm-’ atm- ‘) at 300°K in N,O-N,, collisions, as a function of the rotational quantum number m (m = J + 1 for the R-branch of a band, J for the P-branch, and J +$ for the Q-branch lines). Any differences in the computed line widths among various N,O bands would arise from the slightly different rotational constants associated with the corresponding vibrational levels. In the present calculation, we have ignored such slight differences in the line widths as they are indeed very small for all except the first few lines in a band. (U Therefore. the results plotted in ‘we

Theoretical

line widths in N,O-N,O

and N,O-air

collisions

1661

QOS

0.04

Npo-02

_

0.0 I

-I-!

o.o*l II

21

31

41

61

51

m 1.

Computed

half-widths

in N,O-N,,

N&O,

and N,O-air

collisions

at 300°K

Figs. 1 and 2 should be equally applicable for a vibration-rotation band of N,O and for its pure rotation spectrum. The relative magnitudes of half-widths for a particular rotational transition in N,O-N, and N,O-0, collisions are determined mostly by the magnitudes of the quadrupole moments of Nz and 02, rather than by the finer details of the rotational levels and their statistical weights. It is interesting to note that the half-widths in N,O-O2 collisions are not too much greater than the kinetic theory value, which is consistent with the very low quadrupole moment of Oz. This fact may provide a partial explanation for the nearly flat y” vs. m curve for oxygen-broadened lines in Fig. 1. Comparison of our computed 0.10,

I

T=220% f % u ;

006

-

0.05 -

“04;

II

21

31

41

I 51

m

FIG. 2. Computed

half-widths

in N,O-air

collisions

at 220”K, 250°K and 280°K.

61

G. D. T.

1662 half-widths

for N,-broadening

with

measurements

shows

low-resolution between

20 and 60 are much

mere coincidence, broadened

N,O

In Table

agree

for

1, we show

derived

reducing

his self-broadened

broadening Goodv

for

infinite

shown

by OPPENHEIM

estimates

than the theoretical be compared

an interesting

by GOODY.(~)

values

derived

that their

larger

should

Goody

comparison

ratio

data

equal

to

values

in air, while

shown

values

between

our

line widths

infinite

dilution

of lines in Fig.

appear

curve

to be for air-

in Fig. were

m

1 and, by 1.

computations which

from with

and

the

obtained

by

in air, utilizing

a self-

to 1.25.

in the last column

dilution

and GOLDMAN@’

for half-widths

with the N,O-air

gives optimum

half-width

to air-broadening

The three

those

and P. VARANASI

m = 60. All of the other published

and, hence,

estimates

TUWANI

of Table

the three

1 are the three estimates

half-widths

in the fourth

given

column

by are

TABLE 1. COMPARIXINBETWEEN OURCOMPUTED HALF-WIDTHS IN N,O-N,O AND N20-~1~ COLLISIONS WITH THE CORRESPONDING ESTIMATFS DERIVEDBY GIDDY; THEVALUESSHOWN IN THE FOURTH COLUMN ARE OBTAINED BY MULTIPLYING THE VALUES IN THE FIFTH COLUMN BY 1.25. THE UNITSOF y” ARE CM-I ATM-’ AT T = 300°K m

1 2 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 55 60

fN20-air

fN*“-N20

(This study)

(This study)

0.0678 0.0648 0.0649 0.0654 0.0661 0.0664 0.0672 0.0676 0.0676 0.0675 0.0670 0.0664 0.0657 0.0649 0.0641 0.0632 0.0624 0.0616 0.0611 0.0604 0.0599 0.0595 0.0593 0.0590 0.0589 0.0586 0.0584 0.0582 0.0581 0.058 1 0.0582 0.0583 0.0579 0.0570 0.0556 0.0539

0.1009 0.098 1 0.0982 0.0992 O.lOCCl 0.1005 0.1005 0.1001 0.0993 0.0985 0.0972 0.0961 0.0951 0.0944 0.0935 0.0927 0.0920 0.0913 0.0907 0.090 1 0.0895 0.0887 0.0879 0.0870 0.0860 0.0849 0.0837 0.0825 0.0811 0.0797 0.0723 0.0646 0.0576 0.0519 0.0478 0.0452

0

Y

N>O-a,r

(Ref. 3)

0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.1038 0.0975 0.0975 0.0975 0.0975 0.0975 0.0975 0.0975 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912 0.0912

0.083 0.083 0.083 0.083 0.083 0.083 0.083 0.083 0.083 0.083 0.083 0.078 0.078 0.078 0.078 0.078 0.078 0.078 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073

Theoretical line widths in N20-N20

and N,O-air collisions

1663

obtained by multiplying the N,O-air values of Goody by the factor 1.25. The agreement between our calculated half-widths and the experimentally-derived estimates of Goody is acceptable for self-broadening of N,O lines, for the range of quantum numbers considered in his study. On the other hand, the discrepancy between his values and the calculated half-widths in N,O-air collisions is puzzling. While he reports a single value of 1.25 for the ratio of self-broadened half-widths to air-broadened half-widths, we find theoretically that it varies from 1.46 form = 1 to a value less than unity for m = 60. Using the theoretical ratios to reduce Goody’s results for pure N,O to infinite dilution in air would naturally bring about agreement with our computed half-widths in air-broadening. Hence, the disagreement between his experimental data for air-broadened half-widths and our computations may be attributed either to the factor 1.25 used by him or to the uncertainties in the values of the molecular parameters we have used. The fact that the theoretical halfwidths underestimate all of the reported experimental data on foreign-gas broadening is not due to neglect of such other interactions as dispersion interactions, as we have found these to be quite unimportant. It appears that the value of 0.05 cm- ‘atm- I, used as an average half-width by GRAY(~) and by GOLDMAN et al. (I)to yield the best correlation of spectral transmission measurements, is too low.* Of course, any confidence that one may place in the computed halfwidths rests heavily on (i) the remarkable success with which YAMAMOTOet al.@' have been able to apply the theory in the case of CO2 (which is a ‘non-identical twin’ of N,O), and on (ii) the precision with which the quadrupole moments of N,O, N, and O2 have been measured by Buckingham and his co-workers using the induced birefringence method (see Ref. 9). Because of the much closer spacing of lines in NzO bands as compared to CO, bands, it may be necessary to use a tunable N,O laser or a spectrometer with a resolution of 0.05 cm- 1 or better to make accurate direct line-width measurements. Until such measured data are made available, it is hoped that the results presented here will be of some use. On the other hand, reference to Fig. 1 and, in particular, to the nearly flat portions of the N,O-air curve for 1 < m < 12 and for m > 25, clearly suggests that three values such as those proposed by Goody may provide all of the necessary line-width information in an atmospheric transmission computation. We present in Fig. 2, air-broadened line widths of N20 computed at 220”K, 250°K and 280°K corresponding to the atmospheric temperatures at higher altitudes. Acknowledgements-We wish to thank Dr. L. D. GRAYYOUNGfor suggesting this investigation. The many helpfuldiscussions with Dr. GRAYYOUNG,Professor R. M. GOODY,Professor S. S. PENNER~~~Mr. J. E. LOWDER, which gave our results a fuller meaning, are gratefully acknowledged.

REFERENCES 1. A. GOLDMAN,D. G. MURCRAY,F. H. MURCRAY,W. J. WILLIAMS,T. G. KYLEand J. N. BROOKS,J. opr.

SOC.

Am. 60, 1466 (1970). 2. R. M. GIDDY and T. W. WORMELL, Proc. R. Sot. (Lond.) A209,178 (1951). 3. R. M. GOODY,J. opt. Sot. Am. 58,904 (1968).

* L. D. Gray Young, in a private communication, shares this view with transmission computations of R. McClatchey at AFCRL, which cover a pressures and the amount of nitrous oxide, indicate that the value 0.05 cm-’ It is not clear, therefore, how one may account for the agreement with observed by GOLDMANel al!” using this low value for y”.

us. It appears that the atmospheric wide range of values for the total atm - 1is too low for the line width. atmospheric transmission obtained

1664

G. D. T. TEIWAN~ and P. VARANASI

4. P. E. FRALEY, W. W. BRIM and K. N. RAO, J. molec. Spectrosc. 9,487 (1962). 5. L. D. GRAY, Appl. Opt. 4, 1494 (1965). 6. U. P. OPPENHEIMand A. GOLDMAN, J. opt. Sot. Am. 56,675 (1966). 7. A. ADEL and E. F. BARKER, Rev. Mod. Phys. 16,236 (1944). 8. G. YAMAMOTO, M. TANAKA and T. AOKI, JQSRT9,371 (1969). 9. D. E. STOGRYN and A. P. STOGRYN, Mol. Ph.ys. 11, 371 (1966). IO. W. S. BENEDICTand R. S. HERMAN, JQSRTJ, 265 (1963). I I. P. VARANASI and G. D. T. TEJWANI. JQSRT 11,255 (1971). 12. J. 0. HIRSCHFELDER,C. F. CURXSS and R. B. BIRD. Molecular Theory q/ Gases and Liquids. John Wiley, N.Y. (1954). 13. G. HERZBERG, Infrared and Raman Spectra, p. 396. Van Nostrand, Princeton, N.J. (1945). 14. G. HERZBERG, Spectra qf Diatomic Molecules. Van Nostrand, Princeton, N.J. (1950).