- Email: [email protected]
Simplified models for the temperature dependence of linewidths at elevated temperatures and applications to CO broadened by Ar and N2
J. Quant. Spectrosc. Radiat. Transfer Vol. 31, No. 1, pp. 23 34, 1984
0022~,073/84 $3.00 + .00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
SIMPLIFIED MODELS FOR THE TEMPERATURE D E P E N D E N C E OF L I N E W I D T H S AT E L E V A T E D T E M P E R A T U R E S A N D A P P L I C A T I O N S TO CO B R O A D E N E D BY Ar A N D N2 J. BONAMY and D. ROBERT Laboratoire de Physique Molrculaire, ERA CNRS no 834, Universit6 de Franche-Comtr---25030 Besancon Cedex, France and C. BOULET D~partement de Physique Atomique et Molrculaire, Universit6 de Rennes, Campus de Beaulieu, 35042, Rennes Cedex, France (Received 15 March 1983)
A~tract--The broadening coefficients for i.r. lines of CO perturbed by Ar are calculated in the temperature range 300-3500 K using the formalism previously developed by two of us (D.R. and J.B.). The results are compared with high-resolution spectroscopicmeasurements of shock-heated CO-Ar gas mixtures. A simplified model is proposed to describe the temperature dependence of the linewidths. The resulting model is applied to CO broadened by N2 and the results are critically discussed. 1. INTRODUCTION Recent studies have dealt with gaseous species concentrations and temperatures in combustion. The line-center absorption coefficients and half-widths of rovibrational lines of gases have been measured over a large range of temperature using high resolution i.r. diode laser spectroscopy. This technique was applied by Hanson ~ to measure the temperature dependence of the P(11) absorption coefficient for C O - A r mixtures in the range 850-3340 K. A similar study for the P(6) line of CO perturbed by N 2 in the range 303-783 K was performed by Sell. 2 Coherent R a m a n spectroscopy is particularly well suited for combustion diagnostics. 3,4 Coherent anti-Stokes R a m a n spectroscopy (CARS) and inverse R a m a n spectroscopy (IRS) have been successfully applied by Rahn et al., 5"6 and by Hall 7 for highly accurate studies on pressure-broadened linewidths of R a m a n Q-branch transitions in nitrogen at room temperature and in methane-air flame at 1730 K. Interpretation of the experimental results 1,2 involved the following theoretical temperature dependence for the linewidth: ~,( T ) = y ( Trer)( T / T~r) - u,
(1)
where Tref is the chosen reference temperature (e.g., 300 K). This simple law has been deduced from the A n d e r s o n - T s a o - C u r n u t t e theory. 8-1° The main assumptions underlying this theory (i.e., truncated power series for differential collision cross sections, a concomitant cutoff procedure, and straightline trajectories) are less valid for close collisions which become more efficient with increasing temperatures. As pointed out by Varanasi, 1~ Eq. (1) also assumes a predominantly anisotropic interaction and nearly resonant collisions, the validity of which are questionable. The experimentally determined isotropic R a m a n Q linewidths for N 2 have been compared with predicted calculated values using more sophisticated approaches. 6,7 In Ref. 6, a quasi-classical theory has been used in which Hamilton's equations are integrated to describe the collision dynamics for given semi-classical initial conditions. The inter23
24
J. BONAMY et al.
molecular potential has been chosen as an atom-atom potential and the resulting collision cross sections were conveniently averaged over initial conditions for the given temperature. In Ref. 7, a semiclassical theory of the width developed by Robert and Bonamy lz has been applied. This theory includes a non-perturbating treatment of the differential collision cross sections and a realistic analytical model of trajectories; the anisotropic potential is represented by the superposition of atom-atom interactions and electrostatic contributions. These two approaches lead to consistent agreement with experiments at all of the temperatures considered but the simple theoretical temperature law given in Eq. (1) is not deduced. Moreover, more complicated molecular systems, for which a large number of rotational levels is populated, require increasing computation time. There is interest in simplified models to predict approximately the temperature dependence of the linebroadening coefficients for a large variety of molecular systems. 2. SIMPLIFIED MODEL FOR LINEWlDTHS CALCULATIONS AT ELEVATED TEMPERATURES If we neglect the imaginary parts of the differential cross section, as is usually found to be convenient for linewidth calculations, ~3,~4 the expression for the line-broadening coefficient (HWHH in cm -l) may be approximated, in the model of Ref. 12, by 7 = (nb/2nc)(v{1
-- [1 -- S~L~,2]e-(S2,,2+s2.,2+6°a2,2)})b,,
,,2 -- ( n b / 2 n c ) ( v a ) v
,
(2)
where nh is the density of perturbing molecules, c the light velocity, v the relative velocity of colliding molecules, b the impact parameter, J2 the rotational quantum number for the perturber. The expressions for S~,L~n,,Sz,f2,2,(c)S2,:2 and S2, t2 are directly related 12to the second order differential collision cross section in the perturbation approach 8'9 and the symbol (...)b,vj2 means a convenient average over these three parameters. In the approach of Leavitt and Korff, ~5 even the c~L) ~-J2,f2i2 contribution is given by an exponential but the weak amount of this contribution, resulting from non-diagonality with respect to the states of the perturber, makes no significant differencd 4 and allows the following simpler form for Eq. (2): 7 = (nb/2nC)(V[1
-- e-(S2:2+s2"2+s'e2'2)])oj2
- (nb/2nc)f(1
-- e-:2(b'~)hj2
(3)
with S 2,J2i2 -_- S ( 20e2i2 C) ¢ ( 2f2t2' L) "~ ~--J
Here, v has been replaced by its mean value f. The explicit expressions for 6e2(t7) are calculated ~2 by using a convenient analytical model of trajectories in which the influence of both the isotropic potential Vlso in energy conservation and fflSO( = -g-r-~ad* Vtso) in the equation of motion are included. This procedure leads to resonance functions in 5:2(tY), which differ from the usual perturbation values obtained for straight-line trajectories. Their argument is now defined by the distance of closest approach r, and the apparent relative velocity v~ (instead of b and ,7) through k = (Dl2,i,2,rc/V/c
with
(4) and
b(rc,=rc{'
Simplified models for the temperature dependence of CO broadened by Ar and N2
25
E and a are the usual Lennard-Jones parameters and the co,2.,~2, are the angular frequencies characterizing induced transitions. This trajectory model leads to an additional dependence of these resonance functions on vc/v'~, viz.
F/o-kn (0-~61 ]1/2
~)c # Vc
-tr-j j
2(-'\r,i i6lJ _lJ
(5)
vc is the relative velocity at closest approach. For distant collisions (r~ ~ b >> a), v~ ~ v< ~ 6 and the resonance functions tend to the corresponding Anderson functions? '9 In order to determine a simple approach for the temperature dependence of the i.r. and Raman linewidths at elevated temperatures, it is necessary to study in detail a model of the 6e2[b (r<), 6] function and to test the reliability of this model with respect to the more general calculation using Eq. (2) and the procedure of Ref. 12. We describe this study for C O - A r and compare the results with experimental data.I The remaining problem is to give a correct description of the interaction potential. Three intermolecular potential energy surfaces are available for CO-Ar. Two of these have been calculated from the electron gas method) 6 Ambiguities in joining short and long range results lead to two potentials. The third potential surface is an empirical a t o m - a t o m potential, which has been fitted to a set of experimental data for the crystal, t7 We discuss simple methods for more complicated systems and use the C O - A r case for several tests in which we adopt an empirical a t o m - a t o m potential ~2but with atomic parameters determined for the gas phase.18 A study of more realistic potential surfaces requires a more accuratd 2 theoretical approach for such diatom-atom systems. ~9'2° The resulting potential is
V = Z [(do/rlZ2J) - (egr6i,2J)] + Vuiuz + VIIIQ2"~ Vll2Qi + VQIQ2'
(6)
i,j
where the electrostatic contributions between the dipole moments (#) and the quadrupole moments (Q) have been added for interactions between linear molecules (see section 3). The numerical values for atomic pair energy parameters (e u and d,j) between the ith atom of molecule 1 and the j t h of molecule 2 are given in Table 1. This potential may be expressed 12in terms of the intermolecular distance r and of the spherical harmonics Y "11Y /2- m tied to each molecule by expanding the inverse powers of the interatomic distances r-n ,.2j (n = 6, 12). The resulting expressions for ~2[b(rc), 6] in Eq. (3) are given in Refs. 12 and 21. This expansion includes the isotropic (in 6-12, 8-14, and 10-16) and anisotropic contributions (in 7-13, 8-14, 9-15, 10-16), as well as the corresponding cross-terms for li = l, 2 and 12= 0, 1, 2. The 10-16 contribution for l~ = 2 and 12 = 0 is not explicitly published in Ref. 21, but is given in Appendix I. The a p r i o r i calculated values from Eq. (2) and Table l, following the procedure of Ref. 12, are compared with the experimental data I for the temperature range 300-3500 K in Fig. 1. This comparison shows good overall agreement in view of the crudeness of the potential surface. The dependence of y on T for several lines is shown in Fig. 2, where results for purely resonant collision (k = 0) are also shown. Such an assumption becomes realistic at elevated temperatures (T > 1000 K). The variation of the differential collision cross section with impact parameter b [see Eq. (3)] is shown in Fig. 3 for various temperatures. The differential cross-section exhibits a sharp decrease with b, which suggests the following simple definition of an effective optical collision diameter b0(rc0): exp { -- 5:2[bo(rco), 6]} = 1/2.
(7)
The ( 1 - e x p {-~2[bo(rco),6]})-F2(bo, f ) function may be replaced by an effective differential cross section equal to unity for b ~< b0 and by zero for b > b0. This procedure
J. ~ ) N A M Y et al.
26
% "5
II
II
II
n
.2.
¢~1
Z
6 o
v 0 II
0
0
~-
II
II
n
H
II
,o
"gt
N L
~
7
o
~
v
0
u
.
o II
II
g
[..
~
II
N ~
~
o
N
N
5
"g d
.~E r~
v ,~
N
n
I.
I
8
Simplified models for the temperature dependence of CO broadened by Ar and N 2 (rnK/otrn) o ii
40
0
20
•
0
o
• 0
10~
I
2000
@0
I
T(K) "
3000
Fig. 1. The argon-broadened haft width for CO P (t l) fine vs temperature; O: experimental data from Ref. 1; O: calculated values.
( r n K / o t rn)
60
z,0
20
T(K) 1000
0
2000
3000
Fig. 2. The argon-broadened halfwidth for various CO lines vs temperature; - - - - V l - - - - : P(I) line; - - O - - : P(I 1) line; A . P(21) line; x: values for k = 0 corresponding to resonant collisions [for T > 500 K, these values have not been reported since no significant difference (<~ 1~o) appears with respect to the P(I) lines].
..~,11~
300K
08
K K
3500 K b/o-
0.2
o # o~B
i
1.2
1~
Fig. 3. { 1 - e x p [ - b ° 2 ( b , 6)]} vs b/tr for the argon-broadened CO P(11) line at different temperatures.
27
28
J. BONAMY
et al.
leads to
(8)
( T) = (nb/2xc )fxbo2(r~o , ,7).
Calculated values from Eq. (8) agree within a few per cent with those calculated from Eq. (2) (see Table 2). In Table 2, the results of calculations are also given when account is taken of the average over the relative velocity. Thus, Eq. (8) permits us to discuss a simplified model for the determination of an approximate temperature dependence. At elevated temperatures, such that k0 = ogii,rco(bo)/V'~o ~<4, the effective optical collision diameter bo(rco) may be calculated for resonant collisions by usingf(k0) = f ( 0 ) ~ 1 (see Fig. 4 of Ref. 12). t ,,~ 0.25 picosec For the Ar-broadened CO P(11) line at 1300 K, ogir ,-, 50 cm -~, r co(bo)/V~o and k0 ~-, 2.4. This assumption is suitable for linear perturbers or for more complicated perturbing molecules since quasi-resonant rotational energy transfers may then occur. Using 21 Eq. (7) and 6ez[bo(r~o), 5], for a predominantly anisotropic potential in r - ' , 5ez[bo(rco), 5]ocrr~ 2" =)v~o-: ~ c".
(9)
Using now Eqs. (8) and (4), a (6) = r~bo2(rc0, 6) = nr~0{ 1 -
[-/(7" "~12
o 6
(10)
,
r 2 t-)Fl)t-2/n-I co - ~ ~ c o
a(f)ocT-(,
F/a\ u_
1)'{1
o" 6
F I/a \ u
= T_(n_l)
I X
l) ~
G(T) x H(T).
(11)
This equation has been obtained under assumptions similar to those introduced by Birnbaum ~°in the Anderson frame 8'9 but it also takes into account a physically meaningful description of the trajectories for close collisions, which are of primary importance at elevated temperatures. The resulting difference appears through the two factors G(T) and H ( T ) on the r.h.s, of Eq. (11). Although the assumption of a predominant interaction potential contribution is questionable, u it becomes more realistic at elevated temperatures
T a b l e 2. C o m p a r i s o n between cross sections (r for the A r - b r o a d e n e d C O P(I 1) line, c a l c u l a t e d from Eqs. (3) a n d (8).
-'l-e ~'2(b'~')>b,[cf. Eg.(B)]
>[i-,21b'Vq>b, v
300
500
800
1300
2000
2700
3500
62.7
55.9
52.7
50.6
49.2
48.6
48.1
62.7
56.6
52.9
50.5
49.1
48.4
47.9
56.4
52.8
50.1
48.2
47.1
46.3
58.3
54.9
51.9
49.6
1 48.3
47.1
(a) 62.3 b2(r , v ) , [ c f . Eq.(8)] 0 CO L
(b) 63.7
1
(a) Calculations taking into account all interactions. (b) Calculations taking into account only the short-range anisotropic contribution in r -13 (cf. Ref. 21).
Simplified models for the temperature dependence of CO broadened by Ar and N 2
29
when short-range anisotropic forces are dominant. For the Ar-broadened P(11) CO line, calculations show (see Table 2) that this assumption is realistic in the whole range of temperature. The dominant anisotropic potential contribution is then proportional to r - ,3 in the atom-atom model. The corresponding values for the G(T) x H(T) function in Eq. (10) have been reported in Table 3. The temperature dependence of this function [G(T) × H(T)otT -M] is also reported. For the lowest reference temperatures, the dependence of M on T is non-negligible. If we choose this reference temperature such that the effective optical parameter b0 lies in the repulsive part of the isotropic potential (b0 < 1.12tr), i.e. T > 1000 K (see Table 3), an almost constant value is obtained for M and consequently for N (see Table 4). The calculated values (see column 2 of Table 4) are consistent with the temperature dependence measured by Hanson (see Fig. 5 of Ref. 1). This result also confirms that, for high temperatures, the hard-sphere kinetic value (N = 0.5) is not obtained. The temperature dependence is related to the collision dynamics through the change of velocity during the collision and through the radial dependence of the anisotropic interaction (see Table 4). As mentioned above, such a simplified model of temperature dependence for the linewidth would be of interest if it is applicable to more complicated systems for which practical applications need a realistic estimation of this dependence. It should avoid performing lengthy calculations required by the nature of such systems and by the number of populated rotational levels of perturbers at elevated temperatures. We will now apply these considerations to CO perturbed by N 2.
Table 3. Calculated values of the G(T), H ( T ) a n d M = In [G(T) x H(T)/G(T,~r) x H(T,~f)]/ln ( T J T ) for the Ar-broadened C O P(11) line. T,K
I in
b0, units of
r a
,
G(T)
H(T)
in uni~ of a
M
Tref
= 300K
Tref
= 800K
Tref
= 1300K
300
1.237
1.095
1.275
0.962
500
1.183
1.096
1.166
0.976
0.147
800
1.148
1.093
1.103
0.983
0.126
1300
1.117
1.084
1.062
0.987
0.108
0.070
2000
1.092
1.072
1.038
0.989
0.094
0.059
0.047
2700
1.077
1.063
1.027
0.990
0.085
0.053
0.041
3500
1.063
1.053
1.019
0.991
0.079
0.048
0.037
Table 4. The temperature exponent N for different reference temperatures calculated from Eqs. (1), (8), and (11) with N = 0 . 5 + 1/(n - I ) + M a n d n = 13.
T,K [
N Tre f
= 300K
Tref
= 800K Tref = 1300K
500
0.73
800
0.71
1300
0.69
0.65
2000
0.68
0.64
0.63
2700
0.67
0.64
0.62
3500
0.66
0.63
0.62
J. BONAMY et al.
30 3. A P P L I C A T I O N
TO
N2-BROADENED
CO L I N E S
As a first step, full calculations starting from Eq. (2), Table 1 and the procedure of Ref. 12 were performed for several lines in the temperature range 300-2000 K and were then compared with the available experimental data 2,22 (see Fig. 4, Tables 5 and 6). G o o d agreement is also obtained for the temperature dependence of a given line, as well as for the m-dependence at a given temperature. We note that, for m = 22, the main contribution Ii =/2 = 2 comes from both quadrupole-quadrupole and atom-atom interactions and a semiclassical approach is no longer valid at low temperatures. Indeed, the corresponding energy defects lie between 100 and 200 c m - t so that calculations for 300 and 600 K have to be avoided in the present frame. If we assume a priori an exponential law for the temperature dependence [see Eq. (1)], the resulting exponent N is, as expected, a temperature-dependent quantity (see Table 5) and varies with m over a range of values Table 5. Calculated halfwidth for the P(6) line of CO broadened by N 2 and the temperature exponent N obtained from N = In[(7(T)/7(T~f)]/In(TJT).
(a) T,K
(C) l
(b)
I
(d)
[
N
Y' mK/atm
mK/atm
300
65.8
75.9
67.2
450
48.7
!
55.2
49.6
0.74
600
39.8
!
44.1
40.4
0.73
750
34.0
i
37.1
34.1
0.72
900
30.1
i
32.3
29.6
0.71
1200
24.7
26.0
24.2
1500
21.2
220
I
20.7
2000
17.5
18.0
J i
17.0
I
1
mK/atm
Tref = 300K
i
Tref = 600K
Tre f = 900K
0.71 i
0.69
-
0.71
0.69
0.69
0.70
0.69
0.69
0.70
0.68
0.68
(a) Calculated values from Eq. (2) and following the procedure of Ref. 12 with the samc parameters as in Ref. 12 (cf. Table I ) . (b) Calculated values for the same conditions as (a) but assuming that all collisions are resonant (k = 0). (c) Calculated values for the same conditions as (a) but replacing the J2-average
by a calculation for J2 : 32 ~ INT f ~ J2 P(J2)]" J2=O (d) N : ~n [ ¥(T)/ y(Tref)]/zn (Tref/T) [ cf. Eq. (1)], calculated from colunn(a).
Table 6. Calculated halfwidths (in mK/atm) for several lines of CO perturbed by N 2. The numbers in parentheses are the experimental values of Ref. 22. T,K
300
600
900
1200
1500
2000
3
71.8 (69.6)
41.9 (43.1)
30.9
25.0
21.3
17.6
7
64.5 (60.7)
39.0 (37.3)
29.7
24.5
21.1
17.5
12
59.7 (56.4)
36.6
27.8
23.1
20.1
16.0
(35.2) 25.5
21.7
19.0
15.4
22 (49.8)
(a) m is the i n i t i a l rotational quantum number for the P branch or the final one for the R branch. , the corresponding values have not been calculated due to the breakdown of the semiclassical approximation for such high rotational transitions at these t~mperatures (see the text).
Simplified models for the temperature dependence of CO broadened by Ar and N 2
31
Table 7. The temperature exponent N for several lines of CO perturbed by N 2 (T~f = 300 K) as calculated from the data of Table 6. T,K
m 3
12
22(a)
600
0.78
0.71
900
0.77
0.70
0.60
0.76
0.70
0.60
1500 I 0.76
0.69
0.60
2000 I 0.74
0.69
0.59
1200 l
(a) Because of the lack of a calculated value for T = 30OK, the experimental result (y = 49.8 mKlatm ; cf. Table 6) has been chosen as the reference value.
(see Table 7) that is consistent with experimental studies. 22'23 Moreover, as mentioned by Lowry and Fischer, 22 at elevated temperatures the line-broadening coefficients for various lines tend to converge to a common value (see Fig. 4). We recall that the present calculations take into account anisotropic short-range forces and curved classical trajectories. Thus, these calculations remain physically significant for high temperatures, which is not the case for those based on a perturbative approach using straight-line trajectories described with constant velocity, s,9 If we extend, in this physically significant frame, a single law for the temperature dependence of the line width, as discussed in Section 2 for atomic perturbers [see Eq. (1)], a problem arises from the presence of rotational degrees of freedom for the molecular perturbers. Moreover, the number of populated rotational levels increases with increasing temperatures. A somewhat drastic approximation would consist of neglecting these by considering an intermolecular potential averaged over this angular dependence. Such an approximation does not take into account the possible quasi-resonant rotational transfers between the two colliding molecules. Another approximation avoiding this problem is to consider all collisions as resonant. In this case the k-resonance parameter [see Eq. (4)] is always zero (since co,2.,.2, is set equal to zero) and calculations are oversimplified. The calculated halfwidths with this approximation for CO-N 2 are given in Table 5; they are m-independent (i.e., they vary by less than 29/o between m = l and 20). Comparison with full calculations (see Table 6)
"l'( rnK/at m) "~
~ P(3)
6C
40 ¸
2C
T(K)
tl
T7 s6o
660
960
12'oo
1~o~-
Fig. 4. The N2-broadened line halfwidth for various CO lines vs temperature; I I : experimental data from Ref. 22 for the P(3) line; 0: experimental data from Ref. 2 for the P(6) line; A: experimental data from Ref. 22 for the P(12) line; - - x - - : calculated values from Eq. (2) following the procedure of Ref. (12) with the s a m e parameters as in this Ref. 12 (see Table 1).
32
J. BONAMYet al.
shows an overestimate going from several per cent for the lowest m values to 50~o for the highest m values at 300 K. This discrepancy decreases strongly when the temperature increases and the resonance assumption becomes realistic for T > 1000K (discrepancy ~< 10~). A better approximation, particularly for low B2 rotational constants and for high temperatures, is to describe the rotational states of the perturber molecules by means of some appropriate average angular momentum ~ since the partial (r(j2) cross sections exhibit under these conditions a weak monotonic variation with the J2 rotational quantum number (see Fig. 5). The corresponding calculated halfwidths are reported in Table 8. These values are close to those obtained by performing the j2-average over the collision cross section (see Table 6) but the m-dependence is not accurately reproduced. Nevertheless, we note that, at elevated temperatures, this dependence is weak (see Table 6 and Fig. 4). Within this approximation, the model developed for atomic perturbers (see Section 2) may be extended by introducing an effective optical collision diameter bo,~(rco,72) through a relation that is similar to Eq. (7) with J2 =~. For temperatures and induced transitions such that k0~ = ~o;~.r2,rcoYv'~o,~~<4 and for an assumed potential proportional to r-% Eq. (11) applies, i.e.,
tr(v)otT -('-0-' x G~(T) x H~(T),
(12)
where G~(T) and H~(T) are functions similar to the corresponding functions in Eq. (11) with re0 = rc0.~. The preceding successive assumptions to deduce the simple semianalytical law for the temperature dependence of the cross section [Eq. (12)] must be accepted with caution. Besides thej2-average discussed above, the main problem lies in the assumed predominance of a particular anisotropic potential contribution. Indeed, competition appears in this case between short-range contributions (represented in the present work by an atom-atom
9oi 52 (12~ .300K
80- ~
6
0
0
70:~ . 4 - " ~
6
K
\
/ 900K
~
0
~ 1.500K
-
~
.2000 K
/
50-
J2 0
g
1'2
11}
2'z;
L-
3b
Fig. 5. The collision cross section (%2vs the J2 rotational quantum number of the perturbing molecules for the N2-broadened CO P(6) line at different temperatures.
Table 8. Calculated halfwidths (in mK/atm) for several lines of CO perturbed by N2, without performing the j2-average but taking J2 = ~ =- INT[ E J2P (J2)]j2 = 0
m•
900
1200
1500
2000
3
29.6
24.1
20.5
16.9
7
29.6
24.2
20.7
17.1
12
29.3
24.2
20.7
17.1
22
27.7
23.4
20.1
16.7
Simplified models for the temperature dependence of CO broadened by Ar and N 2
33
Table 9. The temperature exponent N for the N2-broadened CO P(6) line when only the quadrupole-quadrupole interaction is taken into account (all calculated values given in this table have been obtained for J2 =J~). T,K
(a) Y, mK/atm
(b)
N,
r
(c) -
G~(T) xH~(T)(c)
C0,2 Tref = 300K in units~a
N, (d)
Tre f = 300K
1. 209
1.269
0.92
1.164
1.123
0.92
17.5
0.98
1.117
1.060
0.91
.200
13.2
0.98
1.079
1.014
0.91
.500
10.5
0.99
1.048
0.976
0.91
~000
7.8
0.99
3.008
0.924
0.91
300
51.4
600
27.1
1900
[a) Calculated halfwidths (b) Calculated values from column (a) using the definition of Table 5. (c) In this calculation the function l - e x p [ - ~ ( b , 2 , v ) ] - - F ~ ( b ) i s not equal to unity
for b=O. Eq. (7) must then be replacedby F~[bo,~(rco,~),v j : ½ Fz(b:O). (d) Calculated from the GT(T) xHs(T) function using the definition of Table 3 and N = 0.75 + M (cf. TabTe 4). biscrepancies appear with values of column (b) due to the additional assumptions introduced for the effective bo(rco) c o l l i s i o n diameter in the n~de] leading to Eq. (12).
model) and electrostatic long-range contributions with the same angular dependence [particularly for CO-N2 through the Y2m(1)Ym-m(2) terms leading to quasi-resonant energy transfers]. If we only consider the quadrupolar interaction (n = 5), the present model clearly shows (see Table 9) that the value N = 0.5 + 1/(n - 1) = 0.75 may not be obtained due to the kinematical contribution from the G ~ ( T ) x H : ( T ) function in Eq. (12) (i.e., the exponential exponent term called M in the model). This M term is of the same order of magnitude as the 1/(n - 1) contribution. In fact, the anisotropic short-range contributions to the intermolecular potential must not be disregarded, particularly for increasing temperatures since it is mandated by comparison of columns (a) in Table 9 and (c) in Table 5. Therefore, this model is not applicable to the CO-N 2 molecular system since a predominantly anisotropic interaction contribution cannot be determined. The calculated values for N in Table 5, which agree with the experimental results of Varanasi and Sarangi24 and of Sell,2 are not simply related to the radial dependence of the quadrupolar interaction; this point was mentioned previously by Varanasi. H,24 4. C O N C L U S I O N
The theoretical bases underlying the exponential temperature dependence of the linewidth have been discussed in connection with the physical meaning of the temperature exponent frequently used in experiments. For atomic perturbers, the predominance of an anisotropic short-range potential contribution has allowed us to propose a model in which we take account, in a realistic way, of the close collisions which are of primary importance at elevated temperatures. The temperature exponent is then related to the intermolecular potential and to kinematical factors. For diatomic perturbers, the absence of a predominant interaction does not allow us to obtain a simple interpretation of the temperature exponent. A full calculation of linewidths for various temperatures is then needed. Nevertheless, for high temperatures, approximations may be introduced to avoid lengthy calculations that are necessitated by the presence of many populated rotational levels. In a first approximation, we consider all collisions to be resonant; in a better approximation, we introduce an average angular momentum for the perturbing molecule. Our model, ~2 which is reliable at elevated temperatures, may be applied. REFERENCES 1. R.K. Hanson, Shock Tube andShock Wave Research, p. 432. University of Washington Press, Seattle (1978). 2. J. A. Sell, JQSRT 23, 595 (1980). 3. G. L. Eesley, JQSRT 22, 507 (1979). QSRT Vol. 31, No. 1 ~
J. BONAMY et al.
34
4. S. Druet and J. P. Taran, Prog. Quantum Electronics 7, 1 (1981). 5. A. Owyoung and L. A. Rahn, IEEE JQE QE-15, 25D (1979). 6. L. A. Rahn, A. Owyoung, M. E. Coltrin, and M. Kosykowski, Proc. 7th Inter. Raman Conf. (Edited by W. F. Murphy), p. 694. North Holland, Amsterdam (1980). 7. R. J. Hall, Appl. Spectrosc. 34, 700 (1980). 8. P. W. Anderson, Phys. Rev. 76, 647 (1949). 9. C. J. Tsao and B. Curnutte, JQSRT 2, 41 (1962). 10. G. Birnbaum, Adv. Chem. Phys. 12, 487 (1967). II. P. Varanasi, JQSRT 25, 187 (1981). 12. D. Robert and J. Bonamy, J. de Phys. 10, 923 (1979). 13. C. Boulet and D. Robert, Chem. Phys. Left. 60, 162 (1978). 14. N. Lacome, A. Levy and C. Boulet, J. Molec. Spectrosc. (in press). 15. R. P. Leavitt and D. Korff, J. Chem. Phys. 74, 2180 (1981). 16. G. A. Parker and R. T. Pack, 3. Chem. Phys. 69, 3268 (1978). 17. K. Mirsky, Chem. Phys. 46, 445 (1980). 18. M. Oobatake and T. Ooi, Prog. Theor. Phys. 48, 2132 (1972). 19. R. Shafer and R. G. Gordon, J. Chem. Phys. 51, ll (1969); W. Nielsen and R. G. Gordon, J. Chem. Phys. 58, 4231 (1973); Ibid. 58, 4149 (1973). 20. E. W. Smith, M. Giraud and J. Cooper, J. Chem. Phys. 65, 1256 (1976); E. W. Smith and M. Giraud, J. Chem. Phys. 70, 2027 (1979). 21. J. Bonamy and D. Robert, Internal Rep. (1979). 22. H. S. Lowry and C. J. Fisher, JQSRT 27, 585 (1982). 23. P. L. Varghese and R. K. Hanson, JQSRT 26, 339 (1981). 24. P. Varanasi and S. Sarangi, JQSRT 15, 473 (1975). APPENDIX 1: U~s2[rc(b)] An exhaustive list of these functions for l I = 1,2 and l2 = 0, 1, 2 may be found in Appendix C of Ref. 12. However, the 10-16 contribution for l~ = 2, l2 = 0 was not given and is shown here, together with the corresponding cross functions between the 8-14 and 10-16 contributions.
Atom-atom contributions n
2
37
)]2
9125( F~(r4eu+_rLr2 2"°S2[rc(b)l=~12 \hvf.J~ f(L,,j \ ,, ~,e,j X
10 (2) 2,0 10 1 [S~ C(2) J' 2,0f'°(k)+~'C~ f'°(k)+n
r!ScLJ,~
438867 V
J
\(r~'eiY+-Tri,@,~][E3;UL,,J\(r"do+~r~ir~,do)]
4~o
X
'•,o(O)]
20 10
"/10'~'',~.a--J? ;)-
~d~
rio( k ) + D 2'°flo6(0)
/4 7 ~ 2 \12 lZ(r,,4, +~rur~,dU)l Id k /_l x~14~'c1 LrvjC~(2)fi 2"°fll6(k)+ ~C(2)3~Jf 2,0fl6(k)+16 D2'°f,66(0)]1 .
+
4698750771V
~
L
Cross contributions
× ~ [ ~ CJ?' 2"°°°¢l" 8 , - , -~ v[_, e'¢2' --,? 2'°f~°(k) + D r~ L / , ' J? 99198099F
2 qV
/4
+~L~rl,duJL~trl,d~ X
C~2) 2,0f , 416( k )
7 2 2 \3
+~rlir~flu)J
_]_~_.~t.j?~(2) 2.0f [46(k ) ..~ D
' yOlO14(0)
]
Is
1344915[~ r~,eij][~( r4,du+ 7r,ir2 ~,d,y)] • 12288
~.
~.
•
370755V L id
The resonance functions 2,0flo,]O2,0f10,16 f16,
ff
2 1-
/ 4
/ L ij \
f14,
']l
/J
fl0 are published in Ref. 21.
0022~,073/84 $3.00 + .00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
SIMPLIFIED MODELS FOR THE TEMPERATURE D E P E N D E N C E OF L I N E W I D T H S AT E L E V A T E D T E M P E R A T U R E S A N D A P P L I C A T I O N S TO CO B R O A D E N E D BY Ar A N D N2 J. BONAMY and D. ROBERT Laboratoire de Physique Molrculaire, ERA CNRS no 834, Universit6 de Franche-Comtr---25030 Besancon Cedex, France and C. BOULET D~partement de Physique Atomique et Molrculaire, Universit6 de Rennes, Campus de Beaulieu, 35042, Rennes Cedex, France (Received 15 March 1983)
A~tract--The broadening coefficients for i.r. lines of CO perturbed by Ar are calculated in the temperature range 300-3500 K using the formalism previously developed by two of us (D.R. and J.B.). The results are compared with high-resolution spectroscopicmeasurements of shock-heated CO-Ar gas mixtures. A simplified model is proposed to describe the temperature dependence of the linewidths. The resulting model is applied to CO broadened by N2 and the results are critically discussed. 1. INTRODUCTION Recent studies have dealt with gaseous species concentrations and temperatures in combustion. The line-center absorption coefficients and half-widths of rovibrational lines of gases have been measured over a large range of temperature using high resolution i.r. diode laser spectroscopy. This technique was applied by Hanson ~ to measure the temperature dependence of the P(11) absorption coefficient for C O - A r mixtures in the range 850-3340 K. A similar study for the P(6) line of CO perturbed by N 2 in the range 303-783 K was performed by Sell. 2 Coherent R a m a n spectroscopy is particularly well suited for combustion diagnostics. 3,4 Coherent anti-Stokes R a m a n spectroscopy (CARS) and inverse R a m a n spectroscopy (IRS) have been successfully applied by Rahn et al., 5"6 and by Hall 7 for highly accurate studies on pressure-broadened linewidths of R a m a n Q-branch transitions in nitrogen at room temperature and in methane-air flame at 1730 K. Interpretation of the experimental results 1,2 involved the following theoretical temperature dependence for the linewidth: ~,( T ) = y ( Trer)( T / T~r) - u,
(1)
where Tref is the chosen reference temperature (e.g., 300 K). This simple law has been deduced from the A n d e r s o n - T s a o - C u r n u t t e theory. 8-1° The main assumptions underlying this theory (i.e., truncated power series for differential collision cross sections, a concomitant cutoff procedure, and straightline trajectories) are less valid for close collisions which become more efficient with increasing temperatures. As pointed out by Varanasi, 1~ Eq. (1) also assumes a predominantly anisotropic interaction and nearly resonant collisions, the validity of which are questionable. The experimentally determined isotropic R a m a n Q linewidths for N 2 have been compared with predicted calculated values using more sophisticated approaches. 6,7 In Ref. 6, a quasi-classical theory has been used in which Hamilton's equations are integrated to describe the collision dynamics for given semi-classical initial conditions. The inter23
24
J. BONAMY et al.
molecular potential has been chosen as an atom-atom potential and the resulting collision cross sections were conveniently averaged over initial conditions for the given temperature. In Ref. 7, a semiclassical theory of the width developed by Robert and Bonamy lz has been applied. This theory includes a non-perturbating treatment of the differential collision cross sections and a realistic analytical model of trajectories; the anisotropic potential is represented by the superposition of atom-atom interactions and electrostatic contributions. These two approaches lead to consistent agreement with experiments at all of the temperatures considered but the simple theoretical temperature law given in Eq. (1) is not deduced. Moreover, more complicated molecular systems, for which a large number of rotational levels is populated, require increasing computation time. There is interest in simplified models to predict approximately the temperature dependence of the linebroadening coefficients for a large variety of molecular systems. 2. SIMPLIFIED MODEL FOR LINEWlDTHS CALCULATIONS AT ELEVATED TEMPERATURES If we neglect the imaginary parts of the differential cross section, as is usually found to be convenient for linewidth calculations, ~3,~4 the expression for the line-broadening coefficient (HWHH in cm -l) may be approximated, in the model of Ref. 12, by 7 = (nb/2nc)(v{1
-- [1 -- S~L~,2]e-(S2,,2+s2.,2+6°a2,2)})b,,
,,2 -- ( n b / 2 n c ) ( v a ) v
,
(2)
where nh is the density of perturbing molecules, c the light velocity, v the relative velocity of colliding molecules, b the impact parameter, J2 the rotational quantum number for the perturber. The expressions for S~,L~n,,Sz,f2,2,(c)S2,:2 and S2, t2 are directly related 12to the second order differential collision cross section in the perturbation approach 8'9 and the symbol (...)b,vj2 means a convenient average over these three parameters. In the approach of Leavitt and Korff, ~5 even the c~L) ~-J2,f2i2 contribution is given by an exponential but the weak amount of this contribution, resulting from non-diagonality with respect to the states of the perturber, makes no significant differencd 4 and allows the following simpler form for Eq. (2): 7 = (nb/2nC)(V[1
-- e-(S2:2+s2"2+s'e2'2)])oj2
- (nb/2nc)f(1
-- e-:2(b'~)hj2
(3)
with S 2,J2i2 -_- S ( 20e2i2 C) ¢ ( 2f2t2' L) "~ ~--J
Here, v has been replaced by its mean value f. The explicit expressions for 6e2(t7) are calculated ~2 by using a convenient analytical model of trajectories in which the influence of both the isotropic potential Vlso in energy conservation and fflSO( = -g-r-~ad* Vtso) in the equation of motion are included. This procedure leads to resonance functions in 5:2(tY), which differ from the usual perturbation values obtained for straight-line trajectories. Their argument is now defined by the distance of closest approach r, and the apparent relative velocity v~ (instead of b and ,7) through k = (Dl2,i,2,rc/V/c
with
(4) and
b(rc,=rc{'
Simplified models for the temperature dependence of CO broadened by Ar and N2
25
E and a are the usual Lennard-Jones parameters and the co,2.,~2, are the angular frequencies characterizing induced transitions. This trajectory model leads to an additional dependence of these resonance functions on vc/v'~, viz.
F/o-kn (0-~61 ]1/2
~)c # Vc
-tr-j j
2(-'\r,i i6lJ _lJ
(5)
vc is the relative velocity at closest approach. For distant collisions (r~ ~ b >> a), v~ ~ v< ~ 6 and the resonance functions tend to the corresponding Anderson functions? '9 In order to determine a simple approach for the temperature dependence of the i.r. and Raman linewidths at elevated temperatures, it is necessary to study in detail a model of the 6e2[b (r<), 6] function and to test the reliability of this model with respect to the more general calculation using Eq. (2) and the procedure of Ref. 12. We describe this study for C O - A r and compare the results with experimental data.I The remaining problem is to give a correct description of the interaction potential. Three intermolecular potential energy surfaces are available for CO-Ar. Two of these have been calculated from the electron gas method) 6 Ambiguities in joining short and long range results lead to two potentials. The third potential surface is an empirical a t o m - a t o m potential, which has been fitted to a set of experimental data for the crystal, t7 We discuss simple methods for more complicated systems and use the C O - A r case for several tests in which we adopt an empirical a t o m - a t o m potential ~2but with atomic parameters determined for the gas phase.18 A study of more realistic potential surfaces requires a more accuratd 2 theoretical approach for such diatom-atom systems. ~9'2° The resulting potential is
V = Z [(do/rlZ2J) - (egr6i,2J)] + Vuiuz + VIIIQ2"~ Vll2Qi + VQIQ2'
(6)
i,j
where the electrostatic contributions between the dipole moments (#) and the quadrupole moments (Q) have been added for interactions between linear molecules (see section 3). The numerical values for atomic pair energy parameters (e u and d,j) between the ith atom of molecule 1 and the j t h of molecule 2 are given in Table 1. This potential may be expressed 12in terms of the intermolecular distance r and of the spherical harmonics Y "11Y /2- m tied to each molecule by expanding the inverse powers of the interatomic distances r-n ,.2j (n = 6, 12). The resulting expressions for ~2[b(rc), 6] in Eq. (3) are given in Refs. 12 and 21. This expansion includes the isotropic (in 6-12, 8-14, and 10-16) and anisotropic contributions (in 7-13, 8-14, 9-15, 10-16), as well as the corresponding cross-terms for li = l, 2 and 12= 0, 1, 2. The 10-16 contribution for l~ = 2 and 12 = 0 is not explicitly published in Ref. 21, but is given in Appendix I. The a p r i o r i calculated values from Eq. (2) and Table l, following the procedure of Ref. 12, are compared with the experimental data I for the temperature range 300-3500 K in Fig. 1. This comparison shows good overall agreement in view of the crudeness of the potential surface. The dependence of y on T for several lines is shown in Fig. 2, where results for purely resonant collision (k = 0) are also shown. Such an assumption becomes realistic at elevated temperatures (T > 1000 K). The variation of the differential collision cross section with impact parameter b [see Eq. (3)] is shown in Fig. 3 for various temperatures. The differential cross-section exhibits a sharp decrease with b, which suggests the following simple definition of an effective optical collision diameter b0(rc0): exp { -- 5:2[bo(rco), 6]} = 1/2.
(7)
The ( 1 - e x p {-~2[bo(rco),6]})-F2(bo, f ) function may be replaced by an effective differential cross section equal to unity for b ~< b0 and by zero for b > b0. This procedure
J. ~ ) N A M Y et al.
26
% "5
II
II
II
n
.2.
¢~1
Z
6 o
v 0 II
0
0
~-
II
II
n
H
II
,o
"gt
N L
~
7
o
~
v
0
u
.
o II
II
g
[..
~
II
N ~
~
o
N
N
5
"g d
.~E r~
v ,~
N
n
I.
I
8
Simplified models for the temperature dependence of CO broadened by Ar and N 2 (rnK/otrn) o ii
40
0
20
•
0
o
• 0
10~
I
2000
@0
I
T(K) "
3000
Fig. 1. The argon-broadened haft width for CO P (t l) fine vs temperature; O: experimental data from Ref. 1; O: calculated values.
( r n K / o t rn)
60
z,0
20
T(K) 1000
0
2000
3000
Fig. 2. The argon-broadened halfwidth for various CO lines vs temperature; - - - - V l - - - - : P(I) line; - - O - - : P(I 1) line; A . P(21) line; x: values for k = 0 corresponding to resonant collisions [for T > 500 K, these values have not been reported since no significant difference (<~ 1~o) appears with respect to the P(I) lines].
..~,11~
300K
08
K K
3500 K b/o-
0.2
o # o~B
i
1.2
1~
Fig. 3. { 1 - e x p [ - b ° 2 ( b , 6)]} vs b/tr for the argon-broadened CO P(11) line at different temperatures.
27
28
J. BONAMY
et al.
leads to
(8)
( T) = (nb/2xc )fxbo2(r~o , ,7).
Calculated values from Eq. (8) agree within a few per cent with those calculated from Eq. (2) (see Table 2). In Table 2, the results of calculations are also given when account is taken of the average over the relative velocity. Thus, Eq. (8) permits us to discuss a simplified model for the determination of an approximate temperature dependence. At elevated temperatures, such that k0 = ogii,rco(bo)/V'~o ~<4, the effective optical collision diameter bo(rco) may be calculated for resonant collisions by usingf(k0) = f ( 0 ) ~ 1 (see Fig. 4 of Ref. 12). t ,,~ 0.25 picosec For the Ar-broadened CO P(11) line at 1300 K, ogir ,-, 50 cm -~, r co(bo)/V~o and k0 ~-, 2.4. This assumption is suitable for linear perturbers or for more complicated perturbing molecules since quasi-resonant rotational energy transfers may then occur. Using 21 Eq. (7) and 6ez[bo(r~o), 5], for a predominantly anisotropic potential in r - ' , 5ez[bo(rco), 5]ocrr~ 2" =)v~o-: ~ c".
(9)
Using now Eqs. (8) and (4), a (6) = r~bo2(rc0, 6) = nr~0{ 1 -
[-/(7" "~12
o 6
(10)
,
r 2 t-)Fl)t-2/n-I co - ~ ~ c o
a(f)ocT-(,
F/a\ u_
1)'{1
o" 6
F I/a \ u
= T_(n_l)
I X
l) ~
G(T) x H(T).
(11)
This equation has been obtained under assumptions similar to those introduced by Birnbaum ~°in the Anderson frame 8'9 but it also takes into account a physically meaningful description of the trajectories for close collisions, which are of primary importance at elevated temperatures. The resulting difference appears through the two factors G(T) and H ( T ) on the r.h.s, of Eq. (11). Although the assumption of a predominant interaction potential contribution is questionable, u it becomes more realistic at elevated temperatures
T a b l e 2. C o m p a r i s o n between cross sections (r for the A r - b r o a d e n e d C O P(I 1) line, c a l c u l a t e d from Eqs. (3) a n d (8).
-'l-e ~'2(b'~')>b,[cf. Eg.(B)]
>[i-,21b'Vq>b, v
300
500
800
1300
2000
2700
3500
62.7
55.9
52.7
50.6
49.2
48.6
48.1
62.7
56.6
52.9
50.5
49.1
48.4
47.9
56.4
52.8
50.1
48.2
47.1
46.3
58.3
54.9
51.9
49.6
1 48.3
47.1
(a) 62.3 b2(r , v ) , [ c f . Eq.(8)] 0 CO L
(b) 63.7
1
(a) Calculations taking into account all interactions. (b) Calculations taking into account only the short-range anisotropic contribution in r -13 (cf. Ref. 21).
Simplified models for the temperature dependence of CO broadened by Ar and N 2
29
when short-range anisotropic forces are dominant. For the Ar-broadened P(11) CO line, calculations show (see Table 2) that this assumption is realistic in the whole range of temperature. The dominant anisotropic potential contribution is then proportional to r - ,3 in the atom-atom model. The corresponding values for the G(T) x H(T) function in Eq. (10) have been reported in Table 3. The temperature dependence of this function [G(T) × H(T)otT -M] is also reported. For the lowest reference temperatures, the dependence of M on T is non-negligible. If we choose this reference temperature such that the effective optical parameter b0 lies in the repulsive part of the isotropic potential (b0 < 1.12tr), i.e. T > 1000 K (see Table 3), an almost constant value is obtained for M and consequently for N (see Table 4). The calculated values (see column 2 of Table 4) are consistent with the temperature dependence measured by Hanson (see Fig. 5 of Ref. 1). This result also confirms that, for high temperatures, the hard-sphere kinetic value (N = 0.5) is not obtained. The temperature dependence is related to the collision dynamics through the change of velocity during the collision and through the radial dependence of the anisotropic interaction (see Table 4). As mentioned above, such a simplified model of temperature dependence for the linewidth would be of interest if it is applicable to more complicated systems for which practical applications need a realistic estimation of this dependence. It should avoid performing lengthy calculations required by the nature of such systems and by the number of populated rotational levels of perturbers at elevated temperatures. We will now apply these considerations to CO perturbed by N 2.
Table 3. Calculated values of the G(T), H ( T ) a n d M = In [G(T) x H(T)/G(T,~r) x H(T,~f)]/ln ( T J T ) for the Ar-broadened C O P(11) line. T,K
I in
b0, units of
r a
,
G(T)
H(T)
in uni~ of a
M
Tref
= 300K
Tref
= 800K
Tref
= 1300K
300
1.237
1.095
1.275
0.962
500
1.183
1.096
1.166
0.976
0.147
800
1.148
1.093
1.103
0.983
0.126
1300
1.117
1.084
1.062
0.987
0.108
0.070
2000
1.092
1.072
1.038
0.989
0.094
0.059
0.047
2700
1.077
1.063
1.027
0.990
0.085
0.053
0.041
3500
1.063
1.053
1.019
0.991
0.079
0.048
0.037
Table 4. The temperature exponent N for different reference temperatures calculated from Eqs. (1), (8), and (11) with N = 0 . 5 + 1/(n - I ) + M a n d n = 13.
T,K [
N Tre f
= 300K
Tref
= 800K Tref = 1300K
500
0.73
800
0.71
1300
0.69
0.65
2000
0.68
0.64
0.63
2700
0.67
0.64
0.62
3500
0.66
0.63
0.62
J. BONAMY et al.
30 3. A P P L I C A T I O N
TO
N2-BROADENED
CO L I N E S
As a first step, full calculations starting from Eq. (2), Table 1 and the procedure of Ref. 12 were performed for several lines in the temperature range 300-2000 K and were then compared with the available experimental data 2,22 (see Fig. 4, Tables 5 and 6). G o o d agreement is also obtained for the temperature dependence of a given line, as well as for the m-dependence at a given temperature. We note that, for m = 22, the main contribution Ii =/2 = 2 comes from both quadrupole-quadrupole and atom-atom interactions and a semiclassical approach is no longer valid at low temperatures. Indeed, the corresponding energy defects lie between 100 and 200 c m - t so that calculations for 300 and 600 K have to be avoided in the present frame. If we assume a priori an exponential law for the temperature dependence [see Eq. (1)], the resulting exponent N is, as expected, a temperature-dependent quantity (see Table 5) and varies with m over a range of values Table 5. Calculated halfwidth for the P(6) line of CO broadened by N 2 and the temperature exponent N obtained from N = In[(7(T)/7(T~f)]/In(TJT).
(a) T,K
(C) l
(b)
I
(d)
[
N
Y' mK/atm
mK/atm
300
65.8
75.9
67.2
450
48.7
!
55.2
49.6
0.74
600
39.8
!
44.1
40.4
0.73
750
34.0
i
37.1
34.1
0.72
900
30.1
i
32.3
29.6
0.71
1200
24.7
26.0
24.2
1500
21.2
220
I
20.7
2000
17.5
18.0
J i
17.0
I
1
mK/atm
Tref = 300K
i
Tref = 600K
Tre f = 900K
0.71 i
0.69
-
0.71
0.69
0.69
0.70
0.69
0.69
0.70
0.68
0.68
(a) Calculated values from Eq. (2) and following the procedure of Ref. 12 with the samc parameters as in Ref. 12 (cf. Table I ) . (b) Calculated values for the same conditions as (a) but assuming that all collisions are resonant (k = 0). (c) Calculated values for the same conditions as (a) but replacing the J2-average
by a calculation for J2 : 32 ~ INT f ~ J2 P(J2)]" J2=O (d) N : ~n [ ¥(T)/ y(Tref)]/zn (Tref/T) [ cf. Eq. (1)], calculated from colunn(a).
Table 6. Calculated halfwidths (in mK/atm) for several lines of CO perturbed by N 2. The numbers in parentheses are the experimental values of Ref. 22. T,K
300
600
900
1200
1500
2000
3
71.8 (69.6)
41.9 (43.1)
30.9
25.0
21.3
17.6
7
64.5 (60.7)
39.0 (37.3)
29.7
24.5
21.1
17.5
12
59.7 (56.4)
36.6
27.8
23.1
20.1
16.0
(35.2) 25.5
21.7
19.0
15.4
22 (49.8)
(a) m is the i n i t i a l rotational quantum number for the P branch or the final one for the R branch. , the corresponding values have not been calculated due to the breakdown of the semiclassical approximation for such high rotational transitions at these t~mperatures (see the text).
Simplified models for the temperature dependence of CO broadened by Ar and N 2
31
Table 7. The temperature exponent N for several lines of CO perturbed by N 2 (T~f = 300 K) as calculated from the data of Table 6. T,K
m 3
12
22(a)
600
0.78
0.71
900
0.77
0.70
0.60
0.76
0.70
0.60
1500 I 0.76
0.69
0.60
2000 I 0.74
0.69
0.59
1200 l
(a) Because of the lack of a calculated value for T = 30OK, the experimental result (y = 49.8 mKlatm ; cf. Table 6) has been chosen as the reference value.
(see Table 7) that is consistent with experimental studies. 22'23 Moreover, as mentioned by Lowry and Fischer, 22 at elevated temperatures the line-broadening coefficients for various lines tend to converge to a common value (see Fig. 4). We recall that the present calculations take into account anisotropic short-range forces and curved classical trajectories. Thus, these calculations remain physically significant for high temperatures, which is not the case for those based on a perturbative approach using straight-line trajectories described with constant velocity, s,9 If we extend, in this physically significant frame, a single law for the temperature dependence of the line width, as discussed in Section 2 for atomic perturbers [see Eq. (1)], a problem arises from the presence of rotational degrees of freedom for the molecular perturbers. Moreover, the number of populated rotational levels increases with increasing temperatures. A somewhat drastic approximation would consist of neglecting these by considering an intermolecular potential averaged over this angular dependence. Such an approximation does not take into account the possible quasi-resonant rotational transfers between the two colliding molecules. Another approximation avoiding this problem is to consider all collisions as resonant. In this case the k-resonance parameter [see Eq. (4)] is always zero (since co,2.,.2, is set equal to zero) and calculations are oversimplified. The calculated halfwidths with this approximation for CO-N 2 are given in Table 5; they are m-independent (i.e., they vary by less than 29/o between m = l and 20). Comparison with full calculations (see Table 6)
"l'( rnK/at m) "~
~ P(3)
6C
40 ¸
2C
T(K)
tl
T7 s6o
660
960
12'oo
1~o~-
Fig. 4. The N2-broadened line halfwidth for various CO lines vs temperature; I I : experimental data from Ref. 22 for the P(3) line; 0: experimental data from Ref. 2 for the P(6) line; A: experimental data from Ref. 22 for the P(12) line; - - x - - : calculated values from Eq. (2) following the procedure of Ref. (12) with the s a m e parameters as in this Ref. 12 (see Table 1).
32
J. BONAMYet al.
shows an overestimate going from several per cent for the lowest m values to 50~o for the highest m values at 300 K. This discrepancy decreases strongly when the temperature increases and the resonance assumption becomes realistic for T > 1000K (discrepancy ~< 10~). A better approximation, particularly for low B2 rotational constants and for high temperatures, is to describe the rotational states of the perturber molecules by means of some appropriate average angular momentum ~ since the partial (r(j2) cross sections exhibit under these conditions a weak monotonic variation with the J2 rotational quantum number (see Fig. 5). The corresponding calculated halfwidths are reported in Table 8. These values are close to those obtained by performing the j2-average over the collision cross section (see Table 6) but the m-dependence is not accurately reproduced. Nevertheless, we note that, at elevated temperatures, this dependence is weak (see Table 6 and Fig. 4). Within this approximation, the model developed for atomic perturbers (see Section 2) may be extended by introducing an effective optical collision diameter bo,~(rco,72) through a relation that is similar to Eq. (7) with J2 =~. For temperatures and induced transitions such that k0~ = ~o;~.r2,rcoYv'~o,~~<4 and for an assumed potential proportional to r-% Eq. (11) applies, i.e.,
tr(v)otT -('-0-' x G~(T) x H~(T),
(12)
where G~(T) and H~(T) are functions similar to the corresponding functions in Eq. (11) with re0 = rc0.~. The preceding successive assumptions to deduce the simple semianalytical law for the temperature dependence of the cross section [Eq. (12)] must be accepted with caution. Besides thej2-average discussed above, the main problem lies in the assumed predominance of a particular anisotropic potential contribution. Indeed, competition appears in this case between short-range contributions (represented in the present work by an atom-atom
9oi 52 (12~ .300K
80- ~
6
0
0
70:~ . 4 - " ~
6
K
\
/ 900K
~
0
~ 1.500K
-
~
.2000 K
/
50-
J2 0
g
1'2
11}
2'z;
L-
3b
Fig. 5. The collision cross section (%2vs the J2 rotational quantum number of the perturbing molecules for the N2-broadened CO P(6) line at different temperatures.
Table 8. Calculated halfwidths (in mK/atm) for several lines of CO perturbed by N2, without performing the j2-average but taking J2 = ~ =- INT[ E J2P (J2)]j2 = 0
m•
900
1200
1500
2000
3
29.6
24.1
20.5
16.9
7
29.6
24.2
20.7
17.1
12
29.3
24.2
20.7
17.1
22
27.7
23.4
20.1
16.7
Simplified models for the temperature dependence of CO broadened by Ar and N 2
33
Table 9. The temperature exponent N for the N2-broadened CO P(6) line when only the quadrupole-quadrupole interaction is taken into account (all calculated values given in this table have been obtained for J2 =J~). T,K
(a) Y, mK/atm
(b)
N,
r
(c) -
G~(T) xH~(T)(c)
C0,2 Tref = 300K in units~a
N, (d)
Tre f = 300K
1. 209
1.269
0.92
1.164
1.123
0.92
17.5
0.98
1.117
1.060
0.91
.200
13.2
0.98
1.079
1.014
0.91
.500
10.5
0.99
1.048
0.976
0.91
~000
7.8
0.99
3.008
0.924
0.91
300
51.4
600
27.1
1900
[a) Calculated halfwidths (b) Calculated values from column (a) using the definition of Table 5. (c) In this calculation the function l - e x p [ - ~ ( b , 2 , v ) ] - - F ~ ( b ) i s not equal to unity
for b=O. Eq. (7) must then be replacedby F~[bo,~(rco,~),v j : ½ Fz(b:O). (d) Calculated from the GT(T) xHs(T) function using the definition of Table 3 and N = 0.75 + M (cf. TabTe 4). biscrepancies appear with values of column (b) due to the additional assumptions introduced for the effective bo(rco) c o l l i s i o n diameter in the n~de] leading to Eq. (12).
model) and electrostatic long-range contributions with the same angular dependence [particularly for CO-N2 through the Y2m(1)Ym-m(2) terms leading to quasi-resonant energy transfers]. If we only consider the quadrupolar interaction (n = 5), the present model clearly shows (see Table 9) that the value N = 0.5 + 1/(n - 1) = 0.75 may not be obtained due to the kinematical contribution from the G ~ ( T ) x H : ( T ) function in Eq. (12) (i.e., the exponential exponent term called M in the model). This M term is of the same order of magnitude as the 1/(n - 1) contribution. In fact, the anisotropic short-range contributions to the intermolecular potential must not be disregarded, particularly for increasing temperatures since it is mandated by comparison of columns (a) in Table 9 and (c) in Table 5. Therefore, this model is not applicable to the CO-N 2 molecular system since a predominantly anisotropic interaction contribution cannot be determined. The calculated values for N in Table 5, which agree with the experimental results of Varanasi and Sarangi24 and of Sell,2 are not simply related to the radial dependence of the quadrupolar interaction; this point was mentioned previously by Varanasi. H,24 4. C O N C L U S I O N
The theoretical bases underlying the exponential temperature dependence of the linewidth have been discussed in connection with the physical meaning of the temperature exponent frequently used in experiments. For atomic perturbers, the predominance of an anisotropic short-range potential contribution has allowed us to propose a model in which we take account, in a realistic way, of the close collisions which are of primary importance at elevated temperatures. The temperature exponent is then related to the intermolecular potential and to kinematical factors. For diatomic perturbers, the absence of a predominant interaction does not allow us to obtain a simple interpretation of the temperature exponent. A full calculation of linewidths for various temperatures is then needed. Nevertheless, for high temperatures, approximations may be introduced to avoid lengthy calculations that are necessitated by the presence of many populated rotational levels. In a first approximation, we consider all collisions to be resonant; in a better approximation, we introduce an average angular momentum for the perturbing molecule. Our model, ~2 which is reliable at elevated temperatures, may be applied. REFERENCES 1. R.K. Hanson, Shock Tube andShock Wave Research, p. 432. University of Washington Press, Seattle (1978). 2. J. A. Sell, JQSRT 23, 595 (1980). 3. G. L. Eesley, JQSRT 22, 507 (1979). QSRT Vol. 31, No. 1 ~
J. BONAMY et al.
34
4. S. Druet and J. P. Taran, Prog. Quantum Electronics 7, 1 (1981). 5. A. Owyoung and L. A. Rahn, IEEE JQE QE-15, 25D (1979). 6. L. A. Rahn, A. Owyoung, M. E. Coltrin, and M. Kosykowski, Proc. 7th Inter. Raman Conf. (Edited by W. F. Murphy), p. 694. North Holland, Amsterdam (1980). 7. R. J. Hall, Appl. Spectrosc. 34, 700 (1980). 8. P. W. Anderson, Phys. Rev. 76, 647 (1949). 9. C. J. Tsao and B. Curnutte, JQSRT 2, 41 (1962). 10. G. Birnbaum, Adv. Chem. Phys. 12, 487 (1967). II. P. Varanasi, JQSRT 25, 187 (1981). 12. D. Robert and J. Bonamy, J. de Phys. 10, 923 (1979). 13. C. Boulet and D. Robert, Chem. Phys. Left. 60, 162 (1978). 14. N. Lacome, A. Levy and C. Boulet, J. Molec. Spectrosc. (in press). 15. R. P. Leavitt and D. Korff, J. Chem. Phys. 74, 2180 (1981). 16. G. A. Parker and R. T. Pack, 3. Chem. Phys. 69, 3268 (1978). 17. K. Mirsky, Chem. Phys. 46, 445 (1980). 18. M. Oobatake and T. Ooi, Prog. Theor. Phys. 48, 2132 (1972). 19. R. Shafer and R. G. Gordon, J. Chem. Phys. 51, ll (1969); W. Nielsen and R. G. Gordon, J. Chem. Phys. 58, 4231 (1973); Ibid. 58, 4149 (1973). 20. E. W. Smith, M. Giraud and J. Cooper, J. Chem. Phys. 65, 1256 (1976); E. W. Smith and M. Giraud, J. Chem. Phys. 70, 2027 (1979). 21. J. Bonamy and D. Robert, Internal Rep. (1979). 22. H. S. Lowry and C. J. Fisher, JQSRT 27, 585 (1982). 23. P. L. Varghese and R. K. Hanson, JQSRT 26, 339 (1981). 24. P. Varanasi and S. Sarangi, JQSRT 15, 473 (1975). APPENDIX 1: U~s2[rc(b)] An exhaustive list of these functions for l I = 1,2 and l2 = 0, 1, 2 may be found in Appendix C of Ref. 12. However, the 10-16 contribution for l~ = 2, l2 = 0 was not given and is shown here, together with the corresponding cross functions between the 8-14 and 10-16 contributions.
Atom-atom contributions n
2
37
)]2
9125( F~(r4eu+_rLr2 2"°S2[rc(b)l=~12 \hvf.J~ f(L,,j \ ,, ~,e,j X
10 (2) 2,0 10 1 [S~ C(2) J' 2,0f'°(k)+~'C~ f'°(k)+n
r!ScLJ,~
438867 V
J
\(r~'eiY+-Tri,@,~][E3;UL,,J\(r"do+~r~ir~,do)]
4~o
X
'•,o(O)]
20 10
"/10'~'',~.a--J? ;)-
~d~
rio( k ) + D 2'°flo6(0)
/4 7 ~ 2 \12 lZ(r,,4, +~rur~,dU)l Id k /_l x~14~'c1 LrvjC~(2)fi 2"°fll6(k)+ ~C(2)3~Jf 2,0fl6(k)+16 D2'°f,66(0)]1 .
+
4698750771V
~
L
Cross contributions
× ~ [ ~ CJ?' 2"°°°¢l" 8 , - , -~ v[_, e'¢2' --,? 2'°f~°(k) + D r~ L / , ' J? 99198099F
2 qV
/4
+~L~rl,duJL~trl,d~ X
C~2) 2,0f , 416( k )
7 2 2 \3
+~rlir~flu)J
_]_~_.~t.j?~(2) 2.0f [46(k ) ..~ D
' yOlO14(0)
]
Is
1344915[~ r~,eij][~( r4,du+ 7r,ir2 ~,d,y)] • 12288
~.
~.
•
370755V L id
The resonance functions 2,0flo,]O2,0f10,16 f16,
ff
2 1-
/ 4
/ L ij \
f14,
']l
/J
fl0 are published in Ref. 21.