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Self-, N2- and Ar-broadening and line mixing in HCN and C2H2
J. Qumr. Spertrosc.
0022-4073/93$6.00+ 0.00 PergamonPressLtd
Radial. Transfer Vol. SO, No. 2, pp. 149-166, 1993
Printedin Great Britain
SELF-, N,- AND Ar-BROADENING AND LINE MIXING IN HCN AND C,H, A. S.
RNE
Molecular Physics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899. U.S.A. (Received 28 January
1993)
Abstract-Self-, NZ- and Ar-broadening coefficients have been measured for the stretch-bend infrared combination bands v, + vl(4004 cm-‘) of HCN and v, + vi (4091 cm-‘) of C,H, using a tunable difference-frequency laser. At atmospheric pressures, the Q branches of these bands exhibit significant rotational narrowing or line mixing. The broadening coefficients have been fit with empirical rotationally inelastic collision rate laws, which are then used to model the line mixing in the overlapped Q-branch profiles. Simple energy gap fitting laws appear to be
suitable for the shorter-range intermolecular quadrupole-quadrupole and induction forces, whereas an energy-corrected-sudden scaling law works better for the longer-range dipole-dipole and dipole-quadrupole collision partners. In all cases, the line-coupling coefficients are substantially reduced from the rotationally inelastic rates fit to the broadening coefficients, indicating that 35-70% of the broadening may be due to other collisional mechanisms such as cross-relaxation
to the degenerate lI state vibrational
level.
INTRODUCTION Line mixing or rotational narrowing has been observed in pressure-blended infrared Q branches in COz,‘-6 N,O,C,H,,9 and HCN,‘O causing significant deviations from additive Lorentzian components with important consequences for atmospheric monitoring of these species. However, the rotational narrowing of these infrared Q branches is not as strong or dramatic as in Raman Q branches of N, ,“-24 C0,24-27or, particularly, CO2 ,2G30where the 1388 cm-’ Q-branch profile is x30 times narrowe928*29at atmospheric pressure than any isolated S-branch transition. The line coupling responsible for the Q-branch narrowing arises from rotationally inelastic collisions which are believed to dominate both Raman and infrared broadening. Since Raman and infrared broadening coefficients are usually similar, it is somewhat surprising that line coupling in the infrared is only about half that in the Raman. This decoupling of 0.5 for the infrared Q branches was noted by Strow and Gentry’,’ and Hartman et al3 for CO, and used by others to fit other C024*6and N2064 Q branches. Strow and Pine’ suggested that the decoupling may arise from cross-relaxation between the e and f parity sublevels of the degenerate n bending state which would contribute to broadening but not to line mixing between thefparity rotational levels accessed by the Q branch. Other collisional broadening mechanisms in the infrared, including elastic reorientation and vibrational relaxation and dephasing, might also contribute to the decoupling. Since the polarized Raman Q branches are not sensitive to elastic reorientations and occur for singlet C levels with only e parity, there is no comparable cross-relaxation or evident decoupling. Pine and Looney then studied self-broadening of the Q branches of three different infrared bands in nonpolar HCCH9 (written this way to emphasize the molecular symmetry) and polar HCN’O to test for vibrational relaxation and dephasing. For HCCH,9 two of the bands were in close Fermi resonance but were well represented by the broadening and mixing coefficients from the more isolated vl + v: combination band reexamined here with different collision partners. For HCN,‘O two n + Z combination bands, vl + vi and vj + v:, and one C c J7 difference hot band, vl - v& also exhibited identical broadening and mixing characteristics. Because of this vibrational independence for even strongly anharmonically mixed modes or hot bands which can relax from both the upper and lower levels, it was concluded that vibrational relaxation is not a major contributor to line broadening in these molecules. In fact, there appears to be almost no measurable 149
150
A. S. PINE
vibrational dependence to the broadening coefficients for many other atmospheric molecules such as C0,,‘,2*3’*32 Nz0,64 CH,,33 and NH, ,M (with the exception of the v2 tunneling mode ladder in NH3),35 even when comparing degenerate and non-degenerate levels. For self-broadening HCCH9 and HCN,” Pine and Looney found that a factor of 0.5 in the coupling does not fit the pressure-blended Q branches adequately, so they introduced an empirical coupling factor, F, reducing the mixing coefficients from the rotationally inelastic rates inferred from the measured broadening coefficients. Fis then the fraction of total broadening events due to rotationally inelastic collisions within thefparity sublevel. Iff - e cross-relaxation is the dominant decoupling mechanism, then F = 0.5 represents an equal propensity for cross-relaxation and pure rotational relaxation. For self-broadening of HCCH, there was a stronger propensity for cross-relaxation, whereas HCN exhibited a preference for pure rotational relaxation. In the present work we extend the measurements to foreign gas broadening of HCN and HCCH with N, and Ar to study the influence on the decoupling of the polarity, mass and internal rotational structure of the collision partners. Of course, the N,-broadening case is most appropriate to atmospheric conditions. We also examine the utility of various empirical collisional rate laws for the different bimolecular systems and find some trends with the range of the intermolecular potential, but no universally preferred model. EXPERIMENT
HCN v, + vi and HCCH v, + vi band spectra have been recorded with a linear-scan-controlled difference-frequency laser36y37operated as described in the prior self-broadening studies.‘+‘OThe HCN and HCCH gas samples were natural isotopic mixtures with no detectable impurities. Gas pressures were measured with a capacitance manometer with a nominal accuracy of better than 0.5%. Low pressure (133 Pa), near-Doppler-limited spectra were recorded in a 69.5 cm cell, and self-broadened spectra at pressures from 1.3 to 26.7 kPa for HCN in a 5.8 cm cell and 1.3 to 66.7 kPa for HCCH in a 3.0 cm cell were reported previously. 9~‘oThe self-broadening data have been refit here with slightly different constraints and models to compare with the present N,- and Ar-broadening data on a consistent basis. Buffered HCN and HCCH were premixed with N2 or Ar at a nominal concentration of 0.25% in a 6 1 spherical electropolished stainless steel chamber and were spectrally scanned in a 2 m base length stainless steel White cell with Brewster angle CaF, windows. A total path of 8.25 m was used for the Q branches and 32.25 m for the R branches. The White cell alignment was pressure sensitive due to a bellows-mounted micrometer feedthrough and had to be realigned using the visible laser tracer beam through a small aperture after every pressure change. All spectra were normalized to empty cell traces recorded before filling. With care, variations in the 100% transmission baseline were less than 2% over the time necessary (N 3 h) to fill the cell and record the spectra for the pressure sequence of 1.3, 2.7, 6.7, 13.3, 26.7 and 66.7 kPa. These baseline variations were iteratively corrected during the least-squares analysis, but they probably dominate the fitting errors for the broadening coefficients. At the highest pressures, the transmission near the Q branch heads is strongly saturated, so that uncertainties in the 0% transmission baseline can cause disproportionate errors. In our case, the combined zero drift and noise of the lock-in, analog ratio and digitizer is about 0.1% and there is no extraneous light. All spectra were recorded at room temperature, which varied from 295 to 297 K. SPECTROSCOPIC
ANALYSIS
The spectroscopic constant for the HCN v, + vi and HCCH v, + vi Q branches given in Table 1 were obtained from the near-Doppler-limited low pressure spectra previously reported.‘*‘O The transition frequencies are given by v, = v, + (B’ - B”)J(J + 1) - (D’ - D”)Y(J + l)2.
(1)
Recall that infrared Q branches of n c Z bands access the 17 state f-sublevel with rovibrational parity (-l)J+‘, while the P and R branches access the c-sublevel with (- 1)’ parity. The effective rotational constants in Table 1 incorporate the f-doubling constants for the f-sublevel and higher-order contributions from the usual expansion in J(J + 1) - 12.38the measured Q branch absorbance is given by Beer’s law, A(v) =
W(vYWK
(2)
Broadening and line mixing in HCN and C,H, Table 1. Q-branch spectroscopic constants (in cm-‘) and self-,
NN- and Ar-shift and narrowing coefficients for the v, + v1band of HCN and the v, + v5 band of HCCH.
f;: B’ D”/lO-6 D’/lO-6 dv,/dP self MM Ar
HCN uI+ua
HCCHYIW,
4004.16266(2) 1.4782197(X) 1.4753668(16) 2.9061(21) 2.9999(23)
4091.17238(10 1.1766455(12) 1.1743948(14) 1.62382(83) l-6676(12)
(cm.‘/MPa) -0.169(7) -0.049(6) -0.120(10)
-0.048(4) -0.060(9) -0.094(12)
B/P (cm. ’ /MPa) self 4.9 MM 0.9 Ax 0.3
0.42 0.26 0.17
where B(v) is the empty cell background trace and S(v) is the transmission spectrum. The J dependence of the transition intensities for a IT c C Q branch is given by the nuclear spin statistical weight, g,, times the Boltzmann population factor, pJ
=
(2
+
l)exp(-E;)/kT)/ZR,
(3)
where E,” = B”J(J + 1) - D”J*(J + l)* is the lower state rotational energy, k = k,/hc where ka is Boltzmann’s constant and Z, is the rotational partition function. The spin weights are g, = 1 for all J of HCN and even J of HCCH and g, = 3 for odd J of HCCH. We have neglected any Herman-Wallis factor accounting for deviations from rigid-rotor intensities due to vibration-rotation interactions as these appeared to be negligible for these bands. LINE
BROADENING
The foreign gas broadening coefficients for v, + vi of HCN and v, + vi of HCCH given in Table 2 were obtained from our R- and Q-branch spectra at pressures near 13.3 kPa (100 torr), which is high enough to avoid domination by Doppler broadening but low enough to minimize mixing of overlapping lines except for the lowest J Q-branch lines. For the self-broadening of HCCH9 and HCN’O reported previously, lower pressures were used because of the larger broadening coefficients. Selected self-broadening coefficients are also presented in Table 2, where the rows labeled R(J) for HCN/HCN are actually Q(J) from the fully resolved v,-vl hot band.‘O All the lines were individually fit with a Rautian,39 or hard collision, Dicke-narrowed model lineshape given by
R(x, Y, z) = Re{w(x, y +
z)/[l - ,/xzw(x, Y + z)l},
(4)
where w(x, y + z) is the complex probability or error function, fW exp( - t*) dt/[x - t + i(y + z)]. w(x, y + z) = (i/x) (5) s --co Here the variables are given in dimensionless form with the frequencies and collision rates scaled to the Doppler half width at e-’ intensity, (7 = v,(2ks T/MC*)“*,
(6)
where x = (v - vJ)/b, y = y/a and z =/I/a. y is the pressure-broadened half width (HWHM) resulting from state- or phase-changing collisions’ and B is the Dicke-narrowing parameter or velocity-changing collision rate. The Rautian profile reduces to a Voigt profile if Dicke narrowing is neglected (i.e. for z, fi = 0); but in this near-Doppler regime, the fits show large systematic residuals without /?, as has been noted before. 33Another lineshape, known as the Galatry,” or soft
A. S.RNE
152
Table 2.Self-,NNandAr-broadening coefficients (in cm-‘/MPa) for the v,+v2bandofHCN and the v,+vs bandof HCCH. band
HCN vl+vz
HCCH vl+ve NN 297 13.34
Ar 296 13.34
HCCH 295 6.67
NN 297 13.34
Ar 296 13.34
5.609( 43)* 6.800( 35) 7.511( 28) 8.184( 25) 8.738( 24) 9.259( 24) 9.707( 24) 9.682( 24) 9.804( 24)
1.633( 2) 1.550( 2) 1.473( 2) 1.407( 1) 1.354( 2) 1.295( 1) 1.243( 1) 1.202( 1) 1.178( 1) 1.166( 1)
0.860( 1) 0.832( 1) 0.786( 2) 0.759( 1) 0.720( 1) 0.699( 1) 0.674( 1) 0.640( 1) 0.626( 1) 0.601( 1)
2.061(11) 1.913( 4) 1.743( 5) 1.659( 2) 1.648( 7) 1.586( 4) 1.526( 4) 1.492( 3)
1.125( 3) 1.060( 1) 1.024( 5) 0.996( 1) 0.946( 3) 0.929( 1) 0.913( 2) 0.887( 9)
0.829( 3) 0.788( 1) 0.771( 4) 0.716( 1) 0.700( 3) 0.657( 1) 0.641( 2) 0.609( 1)
9.365(123) 8.889( 99) 8.360( 80) 7.818( 69) 7.132( 61) 6.588( 58) 5.958( 57) 5.297( 56) 4.733( 58) 4.428( 62) 3.844( 67) 3.414( 75) 3.248( 86) 3.094(105) 2.697(125) 2.858(165) 2.666(215)
1.170( 5) 1.170( 5) 1.148( 4) 1.133( 4) 1.124( 4) 1.113( 4) 1.104( 5) 1.107( 5) 1.105( 6) 1.102( 7) 1.113( 9) 1.140(11) 1.111(14) 1.105(17) 1.102(22) 1.082(30) 0.985(38) 0.933(49) 0.930(70)
0.627( 1) 0.608( 1) 0.591( 1) 0.575( 1) 0.562( 1) 0.552( 1) 0.539( 2) 0.523( 2) 0.503( 2) 0.495( 2) 0.488( 3) 0.480( 3) 0.460( 4) 0.464( 5) 0.464( 7) 0.458( 9) 0.462(12) 0.433(16) 0.436(21)
buffer HCN T/K 295 P/kPa 2.69 line R(O)t R(l) R(2) R(3) R(4) p(5) R(6) R(7) R(8) R(9) Q(7) Q(8) Q(9) Q(l0)
Q(ll) Q(12) Q(13) Q(14) Q(15) Q(16) Q(17) Q(18) Q(19)
Q(20) Q(21) QW9 ~(23) ~(24) Q(25) Q(26) ~(27) q(28) Q(29)
1.504( 5) 0.874( 4) 0.589( 2) 1.580(10) 0.930( 8)5 0.620( 4) 1.472( 3) 0.853( 2) 0.563( 1) 1.500( 7) 0.868( 5) 0.575( 3) 1.434( 3) 0.832( 2) 0.547( 1) 1.433( 7) 0.836( 5) 0.551( 3) 1.381( 2) 0.806( 2) 0.531( 1) 1.350( 7) 0.805( 5) 0.526( 3) 1.326( 3) 0.794( 2) 0.511( 1) 1.267( 8) 0.796( 6) 0.485( 4) 1.258( 3) 0.778( 2) 0.486( 1) 1.203(10) 0.795( 7)5 0.463( 5) 1.192( 4) 0.762( 3) 0.462( 2) 1.182(13) 0.885( 9)s 0.475( 6) 1.124( 5) 0.730( 4) 0.452( 3) 1.028(18) 0.725(13) 0.423( 9) 1.074( 9) 0.677( 6) 0.419( 4) 0.933(27) 0.635(18) 0.473(14)§ 1.170(12)§0.664(12) 0.453( 7) 0.918(44) 0.570(28) 0.439(22)§ 0.888(18) 0.553(11) 0.402( 9) 0.675(67)§ 0.524(42) 0.537(38)§ 0.892(34) 0.557(20) 0.384(16)
tself-broadeneddata for HCN with J=l to 9 obtained from Q branch of vl-vz *uncertaintiesin parenthesesare one standard deviation in last digits. ireduced or zero weight in rate law fit.
collision, profile fits just as well as the Rautian. We use the Humlicek algorithm4’ for computation of the complex probability function. Since the velocity-changing collision rate is a translational or kinetic effect, we expect it to be independent of J to first order and constrain all the lines in the fitted Q and R branches to have the same /I, which are also given in Table I for the various collision partners. Otherwise we float the frequency, intensity and Lorentzian linewidth, y, of each line in the Q branch with the exception of the lines for J c 7. In this low J region where the broadened lines are strongly overlapped, free-floating fits would not converge; so we constrained the relative integrated intensities for J < 7 to the Boltzmann populations, g,p,, given by Eq. (3) and floated the ys independently. This made the low J Q-branch broadening coefficients unreliable so they are not reported in Table 2. These multiparameter least-squares spectral fits will be shown later in comparison with line-mixing calculations using empirical rate law models. The broadening coefficients in Table 2 are the Lorentzian HWHM, y, scaled by the measurement pressure. The statistical uncertainties for the y given in Table 2 are one standard deviation from the least-squares fit and are generally less than
Broadening and line mixing in HCN and C,H,
153
1% for J < 20 and get progressively larger for the weaker high J lines. These uncertainties do not reflect systematic errors such as pressure calibration, lineshape model inadequacies, baseline uncertainties and underlying weak hot band and isotopic transitions. No correction has been made for self-broadening of the dilute active gas in the mixtures. The self-, Nz- and Ar-broadening coefficients of HCN and HCCH given in Table 2 are plotted versus J in Fig. 1. R-branch lines are plotted on the half integer, J + l/2. All the yJ decrease monotonically with J except for self-broadening in HCN which peaks around the Boltzmann population maximum. These y, generally agree within a few percent with other measurements on different Z t C and II c C vibrational bands of these molecules such as 2~,,~ 4v, ,43 and ground-state microwaveM for self-broadened HCN; v, for N,-broadened HCN;45 v, + v3, 5vcH and 2, 3, 4vca for self-broadening HCCH;46-48 and vi for Nz- or Ar-broadening HCCH.49*S0
0.6
0
HCCH/HCCH
0.4 t
0.4
t
HCN/NN
1
0.4 0.3’
0.2
HCCH/NN
1.0 0.9. 0.8’ 0.7’
2 t
0.6.
-(
0.5.
I ‘;
0.4’
Fig. 1. Self-, N2- and Ar-broadening coefficients for the v, + vi band of HCN and the v, + vi band of HCCH measured at the pressures indicated and fit to empirical MEG (solid lines), PEG (dashed) and ECS (dotted) collision rate laws (see text).
A. S. PINE
154
COLLISION
RATE
LAWS
The measured broadening coefficients provide information about the various collisional mechanisms, such as inelastic rotational and vibrational energy transfer, reorientations and dephasing, needed to calculate the line mixing effects on the blended Q-branch profiles at atmospheric pressures. For longer range forces, semiclassical line broadening theory can provide good estimates of the broadening coefficients 9*‘obut not the individual inelastic rotational state-to-state rates needed for Q branch line-mixing calculations. Therefore, we model the rotational state-to-state collision rates with several empirical laws used with some success in prior studies’-28 and neglect the other collisional processes for the moment. The first empirical fitting model is the hybrid power-exponential-gap (PEG) law,5’
R J-K=
(7)
a,(EKJ/B)-“2exP(-a,EKJ/kT),
where E u = EK - EJ is the rotational energy gap. Here RJ_K is the upward rotationally-inelastic collision rate for K > J; while the downward rate is given by detailed balance, R K-J-
-
(8)
R~+~p~/p~.
The second fitting model is the modified-exponential-gap Koszykowski et a1,‘3*‘s 1 + l.S(E,/a,kT) 1 + l.S(E,/kT)
R J+Kzal
(MEG) law developed
for N2 by
1 ’
‘=P( - W%.d=).
(9)
For both fitting laws, the adjustable parameters, a2 and a3, are dimensionless and a, has units of the broadening coefficients, cm-‘/MPa. The third model we consider is the dynamically based energy-corrected-sudden (ECS) inelastic rate scaling law as discussed by DePristo and collaborators.52-” The effective inelastic rates may be written for K > J in the form, R J-K=
(2K
+
l)exp(-E,/kT)C,(2L
+ 1) (;
;
;>‘RL-o
x [l + (a,EL_)2/BkT]2/[1 + (a4EK-)2/BkTj2,
(10)
in terms of base rates, R L-0
=
a, (EL/B)-a2 exp( - a3E,/kT),
(11)
taken here to be of the PEG form. In Eq. (10) (: : :) is a 3-J symbol, EL_ = EL - EL_, for dipolar HCN or EL - EL _ 2 for quadrupolar HCCH. A fourth dimensionless, parameter, ad, represents a collisional scaling length, &, or collision duration, t,, according to a, = [(n/384)B(8~~c~~~f/h)]‘/~,
I, = fiq,
(12)
where d = (8kB T/~A~)‘/~ and pc is the reduced mass of the collision partners. The efficacy of this ECS model for fitting Raman Q-branch spectra in N, and CO was demonstrated recently by Millet” and in CO2 by Lavorel et al. 28The broadening coefficients, yJ, are taken to be the sum of the inelastic rates over all allowed K # J rotational levels averaged over the upper and lower vibrations (no J = 0 for the n state, otherwise treated as the C state). All K-J # 0 are permitted for HCN, whereas only even K-J # 0 are allowed for HCCH by symmetry (ignoring f -'gcross relaxation in the n state for the time being). Since none of these rate laws explicitly refers to the rotational level of the collision partner, they are inherently rotation-to-translation (R + T) models more characteristic of atom-molecule collisions. However, rotation-to-rotation (R + R) collisions may be significant for molecular buffers, particularly for resonant self-broadening. Hinchen and Hobbs” and Copeland and CrimM measured state-to-state collisions in HF by pump-probe methods and devised a simple R + R energy-gap (REG) law of the form, RJ-K=~I~K~L,P~P$~~P(-~,IEKJ+E$LI/~T),
(13)
where we now sum over all possible initial (~5)and final (M) rotational levels of the collision partner (primed quantities). This REG law has only two adjustable parameters, a, and a3, and was shown
Broadening and
155
line mixing inHCN and C,H,
for self-broadening
in HCN” to be incapable of fitting either the broadening coefficients or the line mixing profiles; so it will not be considered any further here. A number of variants of the PEG and MEG laws using additional m, degeneracy factors have also been tried without any success in improving these fits. The resulting parameters, ai, of the fits of the MEG, PEG and ECS rate laws to the yJ shown in Fig. 1 for each of the collision partners are given in Table 3. A few of the y,, noted in Table 2, appeared to be less reliable so were given reduced or zero weights in the fits. Not all of the parameters could be well determined in the fits. Particularly, the PEG law for self-broadening of HCN fits very poorly with an anomalous negative a* and a highly correlated a,. Similarly, the four parameters ECS law does not rapidly converge for any of the cases except HCN/HCN, with the a3 parameter being small and indeterminate. Therefore, we fixed the ECS a3 to zero in those cases. The MEG and PEG a3 exponential energy gap coefficients are very large for HCN/HCN compared to the other cases here, or to COZ,‘-6 N,G,@ N,,“-24 and C0,24-27which may reflect the long-range dipoledipole forces emphasizing AJ = 1 collisional transitions over higher AJ. For self-broadening in HCN, the MEG and ECS fits to the yJ shown in Fig. 1 are virtually indistinguishable. However, an examination of the corresponding rate matrixes shown in Fig. 2 indicates a much more gradual fall off of rates with increasing AJ for the ECS model than for the MEG law. Here only upward rates are shown and downward rates are given by detailed balance; the y, are the sum along any row. Though we cannot choose between these very dissimilar MEG and ECS laws from this cumulative fit to the y,, the coupling between neighboring lines is crucial for line mixing in the overlapped Q-branch profiles. We will see that this provides a much more discriminating test of the collision dynamics model. For the remaining cases shown in Fig. 1, the rate law fits to the y, are very similar for the MEG and PEG laws throughout, whereas the ECS law tends to decrease faster at high J where, unfortunately, our data are least reliable. Again, the rate law matrixes exhibit more distinction, though not as extreme as for HCN/HCN above, enabling us to choose between them with the line-mixing profiles. In contrast to self-broadening in HCN, the rates for the other collision partners
Table 3. Rate law parameters for self-, NN- and Ar-broadening in the Y, + Y, band of HCN and the Y, + v5 band of HCCH. band
buffer
HCN HCN yr+yz NN
Ar
HCCH HCCH vi+"5 NN
Ar
L fixed.
law
2% cm-l
a2
83
F @ P/kPa 66.7 26.7
13.3
6.67
0.61 0.60 0.65
0.58 0.55 0.64
0.51 0.50 0.64
1.05(12)
0.52 0.52 0.53
0.56 0.52 0.54
0.55 0.51 0.52
0.46 0.42 0.45
0.480(20)
0.38 0.34 0.39
0.39 0.32 0.37
0.32 0.23 0.29
0.29 0.23 0.18
1.09(29)
0.28 0.29 0.31
0.28 0.28 0.32
0.29 0.29 0.34
0.30 0.27 0.34
1.08(21)
0.41 0.40 0.43
0.41 0.38 0.41
0.39 0.34 0.40
0.39 0.36 0.42
0.292(38)
0.46 0.41 0.44
0.44 0.35 0.41
0.43 0.31 0.36
0.32 0.22 0.25
a4
/MPa
MEG 2.608(47) PEG 0.017(21) ECS 0.169(26)
0.2969(90) -1.96(53) 0.303(29)
8.88(21) 9.6(2.2) 0.318(15)
MEG 0.2030(21) PEG 0.612(37) ECS 0.3047(58)
1.584(41) 0.371(20) 0.8948(45)
1.198(21) 1.092(34) 0.t
MEG 0.1144(11) PEG 0.398(15) ECS 0.1643(29)
1.778(49) 0.423(13) 0.9148(34)
1.265(23) 1.191(27) 0.t
MEG 0.5563(68) PEG 1.51(13) ECS 1.184(68)
1.425(42) 0.293(27) 0.9822(88
1.545(37) 1.518(54) ‘)
MEG 0.3074(40) PEG 0.709(70) ECS 0.588(28)
1.330(41) 0.242(31) 0.9496(68
1.506(38) 1.481(55) ) 0.t
MEG 0.2102(34) PEG 1.02(11) ECS 0.449(20)
1.831(83) 0.458(33) 0.9592(83)
1.271(38) 1.190(60) 0.t
0.t
2.18(52)
A. S. PINE
156
0.24
HCN/HCN
0.22
ECS
40
W-Matrix
& 35
0.20 0.18 0.16
0.08 0.06
0
5
15
10
20
25
30
35
40
45
K
HCN/HCN
0.45r
MEG
W-Matrix
10
Y
0.40
-
0.35
-
0.30
-
0.25
2 0.20
0.15
0.00 0
5
10
15
20
25
30
35
40
45
K
Fig. 2. Three-dimensional plots of the upward collision rates (off-diagonal elements of the inelastic rotational relaxation W-matrix) for the ECS and MEG laws fit to the HCN self-broadening coefficients. J is labelled on the K-J = 1 subdiagonal. The IV,, x scale is normalized to unity for the sum over all K # J at the peak of the distribution.
Broadening and line mixing in HCN and C,H,
157
decrease with AJ slowest for the MEG law and fastest for the ECS law. The kinkiness of the model calculations for HCCH at low J arises from the minimum AJ = 2 quadrupolar selection rule. LINE
MIXING
The formalism used for calculating the absorption coefficient, K(V), of a pressure-blended Q branch in the impact approximation was given previously. 9*‘oThe key operation is the diagonalization of a complex matrix, H=v,+iPW,
(14)
where v, is a diagonal matrix of transition frequencies, v,, given by Eq. (2) and W is the so-called relaxation matrix expressing the collision dynamics. The real parts of the diagonal elements of the W-matrix are the broadening coefficients, y,, discussed in the previous section. Here we could use the measured yJ, which are valid independent of the collision model, or, as we have done, use the energy-gap fitting or scaling laws, Eqs. (7-l 1), which nicely fit and smooth the broadening data. This, of course, does not imply that the models are correct, but just that they well represent the broadening data. The imaginary parts of the diagonal elements of the W-matrix are the pressure shifts which we take to be independent of J and ascribe to a band origin shift, Av, , in the following analysis. Further, we neglect the imaginary parts of the off-diagonal elements of the W-matrix. The real parts of the off-diagonal elements of the W-matrix are set proportional to the negative of the rotationally-inelastic collision rates, Re{ W,K} = - FR,,,.
(1%
Here we have introduced an empirical “coupling factor”, F, representing the contribution of the relevant state-to-state collisions to the total line broadening. Greens7 has calculated inelastic rates and line-coupling coefficients for He-CO2 in the 10s approximation, obtaining W,, about half the R We expect full coupling, F = 1, in the Q branch if the total broadening in the Z7 vibrational leii;is due solely to rotationally inelastic collisions within thef-parity sublevel. Thus, we interpret 1 - F primarily as the fraction of f+e cross-relaxation in the Z7 level, although it may have contributions from other extraneous broadening mechanisms such as elastic reorientations and vibrational relaxation and dephasing. f-c cross-relaxation is responsible for coupling Q-branch lines to those in the P and R branches, which may come into evidence at much higher pressures. We note that if the lines are not coupled, say by setting F = 0, then the H matrix is diagonal and the eigenvalues are w, = v, + iy,, yielding a simple sum of Lorentzian lines.9J0 The impact approximation expressing line-mixing for r~(v)~*‘O must be convolved with the Doppler distribution to account for the translational motion evident at subatmospheric pressures. We use the Dicke-narrowed Rautian profile of Eq. (4) for y = 0 to obtain the absorption spectrum, R(V)=I,SR
(;,O,;).,,
-v’)dv’,
(16)
where IO is an overall intensity fitting, scaling or normalization factor. This approach should be valid provided that there is no statistical correlation between velocity- and state-changing collisions.39 The Dicke-narrowing parameters for the various collision partners are given in Table 1. A nonlinear least-squares fitting program using numerical derivatives is used to fit the pressure-broadened absorption spectra using the line-mixing expressions above with the various rate laws determined from the fits to the broadening coefficients in the previous section. These fits refine the overall intensity, IO, the band-origin pressure shift, Av,, and the coupling factor, F, any of which can be fixed at specified values if desired. This is far fewer parameters than those used in the all-lines fit for the broadening coefficients of the previous section. Diagonalization of the complex H matrix, of order J_ + 1, is the time-consuming step in the fits since J,, = 40 is needed to include all thermally populated levels. Observed-calculated residuals for the 2.69 kPa (20 torr) v, + v: Q branch of self-broadened HCN for the various fitting and scaling law models discussed in the previous section are shown in Fig. 3. The multiparameter LSQ fit residuals are within the noise level except near the band origin where the lines severely overlap and were constrained for convergence. The rate model line-mixing fits QSRT
so/z-c
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Wavenumber/cm
4004.0
1
HCN/HCN
-1
I 4002.0
kPa
,,I
--
--
--..-
_
--
Fig. 3. Intermediate and higher pressure observed absorbance spectra of the self-broadened HCN v, + vi Q branch with observed~lculated residuals (offset below zero for clarity) for the multiparameter line-by-line least-squares fit (LSQfit no mix) and MEG, PEG, and ECS empirical rate law models with various coupling factors, F, as labelled.
-0.7
-0.6
-0.5
-0.2
-0.1
0.0
0.1
0.2
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g II a
0.3
0.4
k
+T
0.5
:
0.6
0.7
HCN/HCN
Broadening and
line mixingin HCN and C,H,
159
show large residuals for F = 0 or 1 corresponding to no mixing and full mixing. When the coupling factor, F, is floated, the ECS law yields the best fit for F = 0.64. At higher pressures, line mixing becomes greatly pronounced as illustrated by the 26.78 kPa (200 torr) trace in Fig. 3. Here the shifts are constrained to those determined from lower pressure data yielding the Av,/P coefficients given in Table 1, since model inadequacies can yield artificial nonlinear pressure-dependent shifts if floated. The F = 0 and 1 calculations respectively under- and over-estimate the peak intensities by 25-30%, compensated by even larger fractional errors in the wings. The best fit is achieved with F = 0.65 for the ECS model, while the MEG and PEG laws still yield residuals > 10%. Note that the floated F factor for the ECS law changes little with pressure, while it drastically increases for the MEG and PEG laws. For a proper physical model, we would expect the coupling factor, F, to be pressure independent since it simply represents a collisional branching ratio. The fitted F factors are given in Table 3 at several measured pressures for the three rate laws for all the collision partners. Thus, based on the smallest residuals and the constancy of F, the ECS law is more appropriate for self-broadened HCN than the MEG or PEG law, the latter not even able to fit the broadening coefficients. In our previous study of self-broadening of HCCH,’ we did not apply the ECS model, so it is of interest to see if a similar improvement results. In Fig. 4, we show intermediate and higher pressure traces of v, + vi of HCCH with a series of observed-calculated residuals for the various models. Note that the increased noise at the peak of the high pressure trace reflects the digitizing step when the transmission is strongly saturated. At all pressures, the MEG, PEG and ECS laws yield comparable fits and relatively constant F factors, with slightly smaller residuals for the MEG law. Here, the most noteworthy feature is the small F x 0.3, indicating that line mixing is weak and closer to F = 0 residuals than to F = 1. If f + e cross-relaxation is the primary decoupling mechanism, then HCCH has a much stronger propensity for cross-relaxation than HCN. We showed previously that the F factor for self-broadened HCCH9 and HCN” was also independent of vibration, even for anharmonically resonant levels and hot bands. The foreign gas broadening spectra and fits are shown in Figs. 5-8. For HCN/NN in Fig. 5, the ECS residuals are marginally smaller than the PEG, with MEG definitely inferior, and the coupling factor, F, is close to 0.5 for all laws. For HCCH/NN in Fig. 6, the MEG law has the smallest residuals and Fis generally about 0.4. For HCN/Ar in Fig. 7, the MEG law has marginally better residuals, though the F factor is not very constant with pressure and the residuals are all worse than for HCN/NN in Fig. 5. Similarly, for HCCH/Ar in Fig. 8, the fits show a slight preference for MEG, but have a somewhat variable F and worse residuals than for HCCH/NN. DISCUSSION We have shown in the previous section that line mixing causes significant deviation of Q branch profiles from an independently additive superposition of Lorentzian lineshapes (i.e. the F = 0 no coupling case). The measured peak intensities are lo-30% higher than if line mixing were ignored, and the wings asymptotically approach 1-F of the additive Lorentzian case. However, the full coupling case, F = 1, predicts far too much narrowing or collapse of the Q branches. Line broadening and line mixing can be modeled successfully with simple empirical fitting or scaling laws for rotationally-inelastic rates using a variable coupling factor, F, though no single law is best for all collision partners studied. The ECS scaling law is clearly preferred for HCN/HCN and is slightly better for HCN/NN, whereas the MEG fitting law works better for the remaining cases. Thus, the longer-range dipole-dipole and dipolequadrupole forces seem better described by the adiabatic energy-correction term, Eqs. (10 and 12), representing a realisticlO collision duration. This contrasts with the Raman line mixing results in nonpolar N2 and COZ, where the ECS law appears to work better than the fitting laws. 24*28 For the infrared HCN and HCCH cases here, the preferred models generally correlate with increased higher AJ collision rates. The shorter-range induction forces in the Ar-broadening case are most difficult to fit with these models, perhaps indicative of still higher AJ collisions. Measurements of line mixing supplement the line broadening information on the collisional dynamics since state-to-state fitting and scaling laws that adequately describe the broadening coefficients strongly overestimate the amount of line coupling when F = 1. Line mixing can also
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10
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Broadening and line mixing in HCN and C,H,
165
discriminate between very different rate laws, such as the MEG and ECS for HCN/HCN, which yield indistinguishable broadening coefficients. The physical interpretation of the coupling factor, F, in terms of possible collisional mechanisms is most problematical with the present set of data. At higher pressures, where line mixing is most evident, the F factors are reasonably independent of rate law model. Association of the decoupling, I-F, with f+ e cross-relaxation implies that self-broadened HCCH (F x 0.3) has a considerably greater propensity for cross-relaxation than self-broadened HCN (F x 0.6). This can be partially rationalized by the quadrupolar selection rules for HCCH which requires AJ odd forf + c collisions and AJ even for f-1. Since the AJ = 1 energy gap is smaller than the AJ = 2, cross-relaxation is favored, but not enough to account for the observed decoupling.’ Also, the difference in coupling is smaller for N,-broadening where F x 0.5 for HCN and F x 0.4 for HCCH, and the order is slightly reversed for Ar broadening where F -N 0.4 for HCN and HCCH, which is difficult to reconcile with cross relaxation. Thus, shorter-range forces tend to reduce coupling in HCN and increase it in HCCH. Interpretation of the empirical F factor in terms of propensity rules and cross-relaxation is not rigorous, but it does provide a simple picture and a convenient means to fit pressure-overlapped infrared spectra. More extensive scattering calculations on polyatomic molecular collision cross sections, particularly for f + e cross-relaxation, would be extremely valuable. Raman Q-branch spectra in HCN and HCCH might provide additional guidance on the interpretation of the F factor, since there the excited vibration is a singlet Z state and reorientational collisions do not contribute; hence we expect F = 1. Direct measurement of state-to-state inelastic rates in these degenerate n vibrations by pulsed pump-probe methods, recently demonstrated for C states in HCCHSs and HCN,59 could elucidate the cross-relaxation mechanism. Acknowledgemenf-This work was supported in part by the Upper Atmosphere Aeronautics and Space Administration.
Research Program of the National
REFERENCES 1. L. L. Strow and B. M. Gentry, J. Chem. Phys. 84, 1149 (1986). 2. B. M. Gentry and L. L. Strow, J. Chem. Phys. 86, 5722 (1987). 3. J. M. Hartmann, J. Taine, L. Bonamy, J. Bonamy, and D. Robert, 10th Congress on High Resolution Molecular Spectroscopy, Dijon, France (1987). 4. M. L. Hoke, S. A. Clough, W. J. Lafferty, and W. B. Olson, Proc. Int. Radiut. Symp., Lille, France (1988). 5. T. Huet, N. Lacome, and A. Levy, J. Mofec. Spectrosc. 138, 141 (1989). 6. M. Margottin-Maclou, A. Henry, and A. Valentine, J. Chem. Phys. 96, 1715 (1992). 7. L. L. Strow and A. S. Pine, J. Chem. Phys. 89, 1427 (1988). 8. M. Margottin-Maclou, A. Henry, and A. Valentine, Appf. Opt. 28, 4920 (1989). 9. A. S. Pine and J. P. Looney, J. Chem. Phys. 93, 6942 (1990). 10. A. S. Pine and J. P. Looney, J. Chem. Phys. %, 1704 (1992). 11. R. J. Hall, J. F. Verdieck, and A. C. Eckbreth, Opt. Commun. 35, 69 (1980). 12. G. J. Rosasco, W. Lempert, W. S. Hurst, and A. Fein, Chem. Phys. Lett. 97, 435 (1983). 13. M. L. Koszykowski, L. A. Rahn, R. E. Palmer, and M. E. Coltrin, J. Phys. Chem. 91, 41 (1987). 14. M. L. Koszykowski, R. L. Farrow, and R. E. Palmer, Opt. Lett. 10, 478 (1985). 15. L. A. Rahn, R. E. Palmer, M. L. Koszykowski, and D. A. Greenhalgh, Chem. Phys. Left. 133,513 (1987). 16. L. A. Rahn and R. E. Palmer, JOSA B3, 1164 (1986). 17. L. A. Rahn and R. L. Farrow, Vibr. Struct. Spectra 17B, 33 (1989). 18. B. Lavorel, G. Millot, R. Saint-Loup, C. Wenger, H. Berger, J. P. Sala, J. Bonamy, and D. Robert, J. Phys. 47, 417 (1986). 19. J. P. Sala, J. Bonamy, D. Robert, B. Lavorel, G. Millet, and H. Berger. Chem. Phys. 106, 427 (1986). 20. B. Lavorel, G. Millot, J. Bonamy, and D. Robert, Chem. Phys. 115, 69 (1987). 21. L. Bonamy, J. Bonamy, D. Robert, B. Lavorel, R. Saint-Loup, R. Chaux, J. Santos, and H. Berger, J. Chem. Phys. 89, 5568 (1988). 22. D. Robert, V&r. Struct. Spectra 17B, 57 (1989). 23. J. Bonamy, D. Robert, J. M. Hartmann, M. L. Gonze, R. Saint-Loup, and H. Berger, J. Chem. Phys. 91, 5916 (1989). 24. G. Millot, J. Chem. Phys. 93, 8001 (1990). 25. G. J. Rosasco, L. A. Rahn, W. S. Hurst, R. E. Palmer, J. P. Looney, and J. W. Hahn, Sot. Photo-Opt. Instrum. Eng. 912, 171 (1988). 26. G. J. Rosasco, L. A. Rahn, W. S. Hurst, R. E. Palmer, and S. M. Dohne, J. Chem. Phys. %,4059 (1989). 27. J. P. Looney, G. J. Rosasco, L. A. Rahn, W. S. Hurst, and J. W. Hahn, Chem. Phys. Left. 161,232 (1989).
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28. B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, J. Chem. Phys. 93, 2176, 2185 (1990). 29. J. Baran, A. Grofcsik, and W. J. Jones, Molec. Phys. 45, 1291 (1982). 30. R. Roy, D. S. Elliot, D. Meschede, F. M. Pipkin, and S. J. Smith, Chem. Phys. Z.ett. 93, 603 (1982). 31. J. W. C. Johns, J. Molec. Spectrosc. 125, 442 (1987); J. W. C. Johns and J. Vander Auwera, J. Molec. Spectrosc. 140, 71 (1990). 32. V. M. Devi, D. C. Benner, C. P. Rinsland, and M. A. H. Smith, JQSRT 48, 581 (1992). 33. A. S. Pine, J. Chem. Phys. 97, 773 (1992). 34. V. N. Markov, A. S. Pine, G. Buffa, and 0. Tarrini, JQSRT SO, 167 (1993). 35. G. Baldicchini, G. BuIfa, and 0. Tarrini, Nouuo Cimento D13, 719 (1991). 36. A. S. Pine, JOSA 64, 1683 (1974); 66, 97 (1976). 37. M. J. Coulombe and A. S. Pine, Appl. Opt. 18, 1505 (1979). 38. G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, Princeton, NJ (1945). 39. S. G. Rautian and I. I. Sobel’man, Sou. Phys. Usp. 9, 701 (1967). 40. L. Galatry, Phys. Rev. 122, 1218 (1961). 41. J. Humlicek, JQSRT 21, 309 (1979). 42. M. T. Pigott and D. H. Rank, J. Chem. Phys. 26, 384 (1957). 43. E. Lindholm, Z. Phys. 109, 223 (1938). 44. A. G. Smith, W. Gordy, J. W. Simmons, and W. V. Smith, Phys. Reu. 75, 260 (1949). 45. M. A. H. Smith, G. A. Harvey, G. L. Pellet& A. Goldman, and D. J. Richardson, J. Molec. Spectrosc. 105, 105 (1984). 46. P. Varanasi and B. R. P. Bangaru, JQSRT 15, 267 (1975). 47. J. S. Wong, J. Molec. Spectrosc. 82, 449 (1980). 48. A. L. Utz, J. D. Tobiason, E. Carrasquillo, M. D. Fritz, and F. F. Crim, J. Chem. Phys. 97,389 (1992). 49. J. P. Bounich, D. Lambot, G. Blanquet, and J. Waldran, J. Molec. Spectrosc. 140, 195 (1990). 50. P. Varanasi, JQSRT 47, 263 (1992). 51. T. A. Brunner and D. Pritchard, Ado. Chem. Phys. 50, 589 (1982). 52. A. E. DePristo, J. J. BelBruno, J. Gelfand, and H. Rabitz, J. Chem. Phys. 74, 5031 (1981). 53. A. E. DePristo, S. D. Augustin, R. Ramaswamy, and H. Rabitz, J. Chem. Phys. 71, 850 (1979). 54. A. E. DePristo, J. Chem. Phys. 73, 2145 (1980). 55. J. J. Hinchen and R. H. Hobbs, J. Chem. Phys. 65, 2732 (1976). 56. R. A. Copeland and F. F. Crim, J. Chem. Phys. 78, 5551 (1983). 57. S. Green, J. Chem. Phys. 90, 3603 (1989). 58. J. D. Tobiason, A. L. Utz, and F. F. Crim, J. Chem. Phys. 97, 7437 (1992). 59. R. Huang, J. Wu, M. Gong, and E. Carrasquillo, submitted to J. Chem. Phys. (1993).
0022-4073/93$6.00+ 0.00 PergamonPressLtd
Radial. Transfer Vol. SO, No. 2, pp. 149-166, 1993
Printedin Great Britain
SELF-, N,- AND Ar-BROADENING AND LINE MIXING IN HCN AND C,H, A. S.
RNE
Molecular Physics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899. U.S.A. (Received 28 January
1993)
Abstract-Self-, NZ- and Ar-broadening coefficients have been measured for the stretch-bend infrared combination bands v, + vl(4004 cm-‘) of HCN and v, + vi (4091 cm-‘) of C,H, using a tunable difference-frequency laser. At atmospheric pressures, the Q branches of these bands exhibit significant rotational narrowing or line mixing. The broadening coefficients have been fit with empirical rotationally inelastic collision rate laws, which are then used to model the line mixing in the overlapped Q-branch profiles. Simple energy gap fitting laws appear to be
suitable for the shorter-range intermolecular quadrupole-quadrupole and induction forces, whereas an energy-corrected-sudden scaling law works better for the longer-range dipole-dipole and dipole-quadrupole collision partners. In all cases, the line-coupling coefficients are substantially reduced from the rotationally inelastic rates fit to the broadening coefficients, indicating that 35-70% of the broadening may be due to other collisional mechanisms such as cross-relaxation
to the degenerate lI state vibrational
level.
INTRODUCTION Line mixing or rotational narrowing has been observed in pressure-blended infrared Q branches in COz,‘-6 N,O,C,H,,9 and HCN,‘O causing significant deviations from additive Lorentzian components with important consequences for atmospheric monitoring of these species. However, the rotational narrowing of these infrared Q branches is not as strong or dramatic as in Raman Q branches of N, ,“-24 C0,24-27or, particularly, CO2 ,2G30where the 1388 cm-’ Q-branch profile is x30 times narrowe928*29at atmospheric pressure than any isolated S-branch transition. The line coupling responsible for the Q-branch narrowing arises from rotationally inelastic collisions which are believed to dominate both Raman and infrared broadening. Since Raman and infrared broadening coefficients are usually similar, it is somewhat surprising that line coupling in the infrared is only about half that in the Raman. This decoupling of 0.5 for the infrared Q branches was noted by Strow and Gentry’,’ and Hartman et al3 for CO, and used by others to fit other C024*6and N2064 Q branches. Strow and Pine’ suggested that the decoupling may arise from cross-relaxation between the e and f parity sublevels of the degenerate n bending state which would contribute to broadening but not to line mixing between thefparity rotational levels accessed by the Q branch. Other collisional broadening mechanisms in the infrared, including elastic reorientation and vibrational relaxation and dephasing, might also contribute to the decoupling. Since the polarized Raman Q branches are not sensitive to elastic reorientations and occur for singlet C levels with only e parity, there is no comparable cross-relaxation or evident decoupling. Pine and Looney then studied self-broadening of the Q branches of three different infrared bands in nonpolar HCCH9 (written this way to emphasize the molecular symmetry) and polar HCN’O to test for vibrational relaxation and dephasing. For HCCH,9 two of the bands were in close Fermi resonance but were well represented by the broadening and mixing coefficients from the more isolated vl + v: combination band reexamined here with different collision partners. For HCN,‘O two n + Z combination bands, vl + vi and vj + v:, and one C c J7 difference hot band, vl - v& also exhibited identical broadening and mixing characteristics. Because of this vibrational independence for even strongly anharmonically mixed modes or hot bands which can relax from both the upper and lower levels, it was concluded that vibrational relaxation is not a major contributor to line broadening in these molecules. In fact, there appears to be almost no measurable 149
150
A. S. PINE
vibrational dependence to the broadening coefficients for many other atmospheric molecules such as C0,,‘,2*3’*32 Nz0,64 CH,,33 and NH, ,M (with the exception of the v2 tunneling mode ladder in NH3),35 even when comparing degenerate and non-degenerate levels. For self-broadening HCCH9 and HCN,” Pine and Looney found that a factor of 0.5 in the coupling does not fit the pressure-blended Q branches adequately, so they introduced an empirical coupling factor, F, reducing the mixing coefficients from the rotationally inelastic rates inferred from the measured broadening coefficients. Fis then the fraction of total broadening events due to rotationally inelastic collisions within thefparity sublevel. Iff - e cross-relaxation is the dominant decoupling mechanism, then F = 0.5 represents an equal propensity for cross-relaxation and pure rotational relaxation. For self-broadening of HCCH, there was a stronger propensity for cross-relaxation, whereas HCN exhibited a preference for pure rotational relaxation. In the present work we extend the measurements to foreign gas broadening of HCN and HCCH with N, and Ar to study the influence on the decoupling of the polarity, mass and internal rotational structure of the collision partners. Of course, the N,-broadening case is most appropriate to atmospheric conditions. We also examine the utility of various empirical collisional rate laws for the different bimolecular systems and find some trends with the range of the intermolecular potential, but no universally preferred model. EXPERIMENT
HCN v, + vi and HCCH v, + vi band spectra have been recorded with a linear-scan-controlled difference-frequency laser36y37operated as described in the prior self-broadening studies.‘+‘OThe HCN and HCCH gas samples were natural isotopic mixtures with no detectable impurities. Gas pressures were measured with a capacitance manometer with a nominal accuracy of better than 0.5%. Low pressure (133 Pa), near-Doppler-limited spectra were recorded in a 69.5 cm cell, and self-broadened spectra at pressures from 1.3 to 26.7 kPa for HCN in a 5.8 cm cell and 1.3 to 66.7 kPa for HCCH in a 3.0 cm cell were reported previously. 9~‘oThe self-broadening data have been refit here with slightly different constraints and models to compare with the present N,- and Ar-broadening data on a consistent basis. Buffered HCN and HCCH were premixed with N2 or Ar at a nominal concentration of 0.25% in a 6 1 spherical electropolished stainless steel chamber and were spectrally scanned in a 2 m base length stainless steel White cell with Brewster angle CaF, windows. A total path of 8.25 m was used for the Q branches and 32.25 m for the R branches. The White cell alignment was pressure sensitive due to a bellows-mounted micrometer feedthrough and had to be realigned using the visible laser tracer beam through a small aperture after every pressure change. All spectra were normalized to empty cell traces recorded before filling. With care, variations in the 100% transmission baseline were less than 2% over the time necessary (N 3 h) to fill the cell and record the spectra for the pressure sequence of 1.3, 2.7, 6.7, 13.3, 26.7 and 66.7 kPa. These baseline variations were iteratively corrected during the least-squares analysis, but they probably dominate the fitting errors for the broadening coefficients. At the highest pressures, the transmission near the Q branch heads is strongly saturated, so that uncertainties in the 0% transmission baseline can cause disproportionate errors. In our case, the combined zero drift and noise of the lock-in, analog ratio and digitizer is about 0.1% and there is no extraneous light. All spectra were recorded at room temperature, which varied from 295 to 297 K. SPECTROSCOPIC
ANALYSIS
The spectroscopic constant for the HCN v, + vi and HCCH v, + vi Q branches given in Table 1 were obtained from the near-Doppler-limited low pressure spectra previously reported.‘*‘O The transition frequencies are given by v, = v, + (B’ - B”)J(J + 1) - (D’ - D”)Y(J + l)2.
(1)
Recall that infrared Q branches of n c Z bands access the 17 state f-sublevel with rovibrational parity (-l)J+‘, while the P and R branches access the c-sublevel with (- 1)’ parity. The effective rotational constants in Table 1 incorporate the f-doubling constants for the f-sublevel and higher-order contributions from the usual expansion in J(J + 1) - 12.38the measured Q branch absorbance is given by Beer’s law, A(v) =
W(vYWK
(2)
Broadening and line mixing in HCN and C,H, Table 1. Q-branch spectroscopic constants (in cm-‘) and self-,
NN- and Ar-shift and narrowing coefficients for the v, + v1band of HCN and the v, + v5 band of HCCH.
f;: B’ D”/lO-6 D’/lO-6 dv,/dP self MM Ar
HCN uI+ua
HCCHYIW,
4004.16266(2) 1.4782197(X) 1.4753668(16) 2.9061(21) 2.9999(23)
4091.17238(10 1.1766455(12) 1.1743948(14) 1.62382(83) l-6676(12)
(cm.‘/MPa) -0.169(7) -0.049(6) -0.120(10)
-0.048(4) -0.060(9) -0.094(12)
B/P (cm. ’ /MPa) self 4.9 MM 0.9 Ax 0.3
0.42 0.26 0.17
where B(v) is the empty cell background trace and S(v) is the transmission spectrum. The J dependence of the transition intensities for a IT c C Q branch is given by the nuclear spin statistical weight, g,, times the Boltzmann population factor, pJ
=
(2
+
l)exp(-E;)/kT)/ZR,
(3)
where E,” = B”J(J + 1) - D”J*(J + l)* is the lower state rotational energy, k = k,/hc where ka is Boltzmann’s constant and Z, is the rotational partition function. The spin weights are g, = 1 for all J of HCN and even J of HCCH and g, = 3 for odd J of HCCH. We have neglected any Herman-Wallis factor accounting for deviations from rigid-rotor intensities due to vibration-rotation interactions as these appeared to be negligible for these bands. LINE
BROADENING
The foreign gas broadening coefficients for v, + vi of HCN and v, + vi of HCCH given in Table 2 were obtained from our R- and Q-branch spectra at pressures near 13.3 kPa (100 torr), which is high enough to avoid domination by Doppler broadening but low enough to minimize mixing of overlapping lines except for the lowest J Q-branch lines. For the self-broadening of HCCH9 and HCN’O reported previously, lower pressures were used because of the larger broadening coefficients. Selected self-broadening coefficients are also presented in Table 2, where the rows labeled R(J) for HCN/HCN are actually Q(J) from the fully resolved v,-vl hot band.‘O All the lines were individually fit with a Rautian,39 or hard collision, Dicke-narrowed model lineshape given by
R(x, Y, z) = Re{w(x, y +
z)/[l - ,/xzw(x, Y + z)l},
(4)
where w(x, y + z) is the complex probability or error function, fW exp( - t*) dt/[x - t + i(y + z)]. w(x, y + z) = (i/x) (5) s --co Here the variables are given in dimensionless form with the frequencies and collision rates scaled to the Doppler half width at e-’ intensity, (7 = v,(2ks T/MC*)“*,
(6)
where x = (v - vJ)/b, y = y/a and z =/I/a. y is the pressure-broadened half width (HWHM) resulting from state- or phase-changing collisions’ and B is the Dicke-narrowing parameter or velocity-changing collision rate. The Rautian profile reduces to a Voigt profile if Dicke narrowing is neglected (i.e. for z, fi = 0); but in this near-Doppler regime, the fits show large systematic residuals without /?, as has been noted before. 33Another lineshape, known as the Galatry,” or soft
A. S.RNE
152
Table 2.Self-,NNandAr-broadening coefficients (in cm-‘/MPa) for the v,+v2bandofHCN and the v,+vs bandof HCCH. band
HCN vl+vz
HCCH vl+ve NN 297 13.34
Ar 296 13.34
HCCH 295 6.67
NN 297 13.34
Ar 296 13.34
5.609( 43)* 6.800( 35) 7.511( 28) 8.184( 25) 8.738( 24) 9.259( 24) 9.707( 24) 9.682( 24) 9.804( 24)
1.633( 2) 1.550( 2) 1.473( 2) 1.407( 1) 1.354( 2) 1.295( 1) 1.243( 1) 1.202( 1) 1.178( 1) 1.166( 1)
0.860( 1) 0.832( 1) 0.786( 2) 0.759( 1) 0.720( 1) 0.699( 1) 0.674( 1) 0.640( 1) 0.626( 1) 0.601( 1)
2.061(11) 1.913( 4) 1.743( 5) 1.659( 2) 1.648( 7) 1.586( 4) 1.526( 4) 1.492( 3)
1.125( 3) 1.060( 1) 1.024( 5) 0.996( 1) 0.946( 3) 0.929( 1) 0.913( 2) 0.887( 9)
0.829( 3) 0.788( 1) 0.771( 4) 0.716( 1) 0.700( 3) 0.657( 1) 0.641( 2) 0.609( 1)
9.365(123) 8.889( 99) 8.360( 80) 7.818( 69) 7.132( 61) 6.588( 58) 5.958( 57) 5.297( 56) 4.733( 58) 4.428( 62) 3.844( 67) 3.414( 75) 3.248( 86) 3.094(105) 2.697(125) 2.858(165) 2.666(215)
1.170( 5) 1.170( 5) 1.148( 4) 1.133( 4) 1.124( 4) 1.113( 4) 1.104( 5) 1.107( 5) 1.105( 6) 1.102( 7) 1.113( 9) 1.140(11) 1.111(14) 1.105(17) 1.102(22) 1.082(30) 0.985(38) 0.933(49) 0.930(70)
0.627( 1) 0.608( 1) 0.591( 1) 0.575( 1) 0.562( 1) 0.552( 1) 0.539( 2) 0.523( 2) 0.503( 2) 0.495( 2) 0.488( 3) 0.480( 3) 0.460( 4) 0.464( 5) 0.464( 7) 0.458( 9) 0.462(12) 0.433(16) 0.436(21)
buffer HCN T/K 295 P/kPa 2.69 line R(O)t R(l) R(2) R(3) R(4) p(5) R(6) R(7) R(8) R(9) Q(7) Q(8) Q(9) Q(l0)
Q(ll) Q(12) Q(13) Q(14) Q(15) Q(16) Q(17) Q(18) Q(19)
Q(20) Q(21) QW9 ~(23) ~(24) Q(25) Q(26) ~(27) q(28) Q(29)
1.504( 5) 0.874( 4) 0.589( 2) 1.580(10) 0.930( 8)5 0.620( 4) 1.472( 3) 0.853( 2) 0.563( 1) 1.500( 7) 0.868( 5) 0.575( 3) 1.434( 3) 0.832( 2) 0.547( 1) 1.433( 7) 0.836( 5) 0.551( 3) 1.381( 2) 0.806( 2) 0.531( 1) 1.350( 7) 0.805( 5) 0.526( 3) 1.326( 3) 0.794( 2) 0.511( 1) 1.267( 8) 0.796( 6) 0.485( 4) 1.258( 3) 0.778( 2) 0.486( 1) 1.203(10) 0.795( 7)5 0.463( 5) 1.192( 4) 0.762( 3) 0.462( 2) 1.182(13) 0.885( 9)s 0.475( 6) 1.124( 5) 0.730( 4) 0.452( 3) 1.028(18) 0.725(13) 0.423( 9) 1.074( 9) 0.677( 6) 0.419( 4) 0.933(27) 0.635(18) 0.473(14)§ 1.170(12)§0.664(12) 0.453( 7) 0.918(44) 0.570(28) 0.439(22)§ 0.888(18) 0.553(11) 0.402( 9) 0.675(67)§ 0.524(42) 0.537(38)§ 0.892(34) 0.557(20) 0.384(16)
tself-broadeneddata for HCN with J=l to 9 obtained from Q branch of vl-vz *uncertaintiesin parenthesesare one standard deviation in last digits. ireduced or zero weight in rate law fit.
collision, profile fits just as well as the Rautian. We use the Humlicek algorithm4’ for computation of the complex probability function. Since the velocity-changing collision rate is a translational or kinetic effect, we expect it to be independent of J to first order and constrain all the lines in the fitted Q and R branches to have the same /I, which are also given in Table I for the various collision partners. Otherwise we float the frequency, intensity and Lorentzian linewidth, y, of each line in the Q branch with the exception of the lines for J c 7. In this low J region where the broadened lines are strongly overlapped, free-floating fits would not converge; so we constrained the relative integrated intensities for J < 7 to the Boltzmann populations, g,p,, given by Eq. (3) and floated the ys independently. This made the low J Q-branch broadening coefficients unreliable so they are not reported in Table 2. These multiparameter least-squares spectral fits will be shown later in comparison with line-mixing calculations using empirical rate law models. The broadening coefficients in Table 2 are the Lorentzian HWHM, y, scaled by the measurement pressure. The statistical uncertainties for the y given in Table 2 are one standard deviation from the least-squares fit and are generally less than
Broadening and line mixing in HCN and C,H,
153
1% for J < 20 and get progressively larger for the weaker high J lines. These uncertainties do not reflect systematic errors such as pressure calibration, lineshape model inadequacies, baseline uncertainties and underlying weak hot band and isotopic transitions. No correction has been made for self-broadening of the dilute active gas in the mixtures. The self-, Nz- and Ar-broadening coefficients of HCN and HCCH given in Table 2 are plotted versus J in Fig. 1. R-branch lines are plotted on the half integer, J + l/2. All the yJ decrease monotonically with J except for self-broadening in HCN which peaks around the Boltzmann population maximum. These y, generally agree within a few percent with other measurements on different Z t C and II c C vibrational bands of these molecules such as 2~,,~ 4v, ,43 and ground-state microwaveM for self-broadened HCN; v, for N,-broadened HCN;45 v, + v3, 5vcH and 2, 3, 4vca for self-broadening HCCH;46-48 and vi for Nz- or Ar-broadening HCCH.49*S0
0.6
0
HCCH/HCCH
0.4 t
0.4
t
HCN/NN
1
0.4 0.3’
0.2
HCCH/NN
1.0 0.9. 0.8’ 0.7’
2 t
0.6.
-(
0.5.
I ‘;
0.4’
Fig. 1. Self-, N2- and Ar-broadening coefficients for the v, + vi band of HCN and the v, + vi band of HCCH measured at the pressures indicated and fit to empirical MEG (solid lines), PEG (dashed) and ECS (dotted) collision rate laws (see text).
A. S. PINE
154
COLLISION
RATE
LAWS
The measured broadening coefficients provide information about the various collisional mechanisms, such as inelastic rotational and vibrational energy transfer, reorientations and dephasing, needed to calculate the line mixing effects on the blended Q-branch profiles at atmospheric pressures. For longer range forces, semiclassical line broadening theory can provide good estimates of the broadening coefficients 9*‘obut not the individual inelastic rotational state-to-state rates needed for Q branch line-mixing calculations. Therefore, we model the rotational state-to-state collision rates with several empirical laws used with some success in prior studies’-28 and neglect the other collisional processes for the moment. The first empirical fitting model is the hybrid power-exponential-gap (PEG) law,5’
R J-K=
(7)
a,(EKJ/B)-“2exP(-a,EKJ/kT),
where E u = EK - EJ is the rotational energy gap. Here RJ_K is the upward rotationally-inelastic collision rate for K > J; while the downward rate is given by detailed balance, R K-J-
-
(8)
R~+~p~/p~.
The second fitting model is the modified-exponential-gap Koszykowski et a1,‘3*‘s 1 + l.S(E,/a,kT) 1 + l.S(E,/kT)
R J+Kzal
(MEG) law developed
for N2 by
1 ’
‘=P( - W%.d=).
(9)
For both fitting laws, the adjustable parameters, a2 and a3, are dimensionless and a, has units of the broadening coefficients, cm-‘/MPa. The third model we consider is the dynamically based energy-corrected-sudden (ECS) inelastic rate scaling law as discussed by DePristo and collaborators.52-” The effective inelastic rates may be written for K > J in the form, R J-K=
(2K
+
l)exp(-E,/kT)C,(2L
+ 1) (;
;
;>‘RL-o
x [l + (a,EL_)2/BkT]2/[1 + (a4EK-)2/BkTj2,
(10)
in terms of base rates, R L-0
=
a, (EL/B)-a2 exp( - a3E,/kT),
(11)
taken here to be of the PEG form. In Eq. (10) (: : :) is a 3-J symbol, EL_ = EL - EL_, for dipolar HCN or EL - EL _ 2 for quadrupolar HCCH. A fourth dimensionless, parameter, ad, represents a collisional scaling length, &, or collision duration, t,, according to a, = [(n/384)B(8~~c~~~f/h)]‘/~,
I, = fiq,
(12)
where d = (8kB T/~A~)‘/~ and pc is the reduced mass of the collision partners. The efficacy of this ECS model for fitting Raman Q-branch spectra in N, and CO was demonstrated recently by Millet” and in CO2 by Lavorel et al. 28The broadening coefficients, yJ, are taken to be the sum of the inelastic rates over all allowed K # J rotational levels averaged over the upper and lower vibrations (no J = 0 for the n state, otherwise treated as the C state). All K-J # 0 are permitted for HCN, whereas only even K-J # 0 are allowed for HCCH by symmetry (ignoring f -'gcross relaxation in the n state for the time being). Since none of these rate laws explicitly refers to the rotational level of the collision partner, they are inherently rotation-to-translation (R + T) models more characteristic of atom-molecule collisions. However, rotation-to-rotation (R + R) collisions may be significant for molecular buffers, particularly for resonant self-broadening. Hinchen and Hobbs” and Copeland and CrimM measured state-to-state collisions in HF by pump-probe methods and devised a simple R + R energy-gap (REG) law of the form, RJ-K=~I~K~L,P~P$~~P(-~,IEKJ+E$LI/~T),
(13)
where we now sum over all possible initial (~5)and final (M) rotational levels of the collision partner (primed quantities). This REG law has only two adjustable parameters, a, and a3, and was shown
Broadening and
155
line mixing inHCN and C,H,
for self-broadening
in HCN” to be incapable of fitting either the broadening coefficients or the line mixing profiles; so it will not be considered any further here. A number of variants of the PEG and MEG laws using additional m, degeneracy factors have also been tried without any success in improving these fits. The resulting parameters, ai, of the fits of the MEG, PEG and ECS rate laws to the yJ shown in Fig. 1 for each of the collision partners are given in Table 3. A few of the y,, noted in Table 2, appeared to be less reliable so were given reduced or zero weights in the fits. Not all of the parameters could be well determined in the fits. Particularly, the PEG law for self-broadening of HCN fits very poorly with an anomalous negative a* and a highly correlated a,. Similarly, the four parameters ECS law does not rapidly converge for any of the cases except HCN/HCN, with the a3 parameter being small and indeterminate. Therefore, we fixed the ECS a3 to zero in those cases. The MEG and PEG a3 exponential energy gap coefficients are very large for HCN/HCN compared to the other cases here, or to COZ,‘-6 N,G,@ N,,“-24 and C0,24-27which may reflect the long-range dipoledipole forces emphasizing AJ = 1 collisional transitions over higher AJ. For self-broadening in HCN, the MEG and ECS fits to the yJ shown in Fig. 1 are virtually indistinguishable. However, an examination of the corresponding rate matrixes shown in Fig. 2 indicates a much more gradual fall off of rates with increasing AJ for the ECS model than for the MEG law. Here only upward rates are shown and downward rates are given by detailed balance; the y, are the sum along any row. Though we cannot choose between these very dissimilar MEG and ECS laws from this cumulative fit to the y,, the coupling between neighboring lines is crucial for line mixing in the overlapped Q-branch profiles. We will see that this provides a much more discriminating test of the collision dynamics model. For the remaining cases shown in Fig. 1, the rate law fits to the y, are very similar for the MEG and PEG laws throughout, whereas the ECS law tends to decrease faster at high J where, unfortunately, our data are least reliable. Again, the rate law matrixes exhibit more distinction, though not as extreme as for HCN/HCN above, enabling us to choose between them with the line-mixing profiles. In contrast to self-broadening in HCN, the rates for the other collision partners
Table 3. Rate law parameters for self-, NN- and Ar-broadening in the Y, + Y, band of HCN and the Y, + v5 band of HCCH. band
buffer
HCN HCN yr+yz NN
Ar
HCCH HCCH vi+"5 NN
Ar
L fixed.
law
2% cm-l
a2
83
F @ P/kPa 66.7 26.7
13.3
6.67
0.61 0.60 0.65
0.58 0.55 0.64
0.51 0.50 0.64
1.05(12)
0.52 0.52 0.53
0.56 0.52 0.54
0.55 0.51 0.52
0.46 0.42 0.45
0.480(20)
0.38 0.34 0.39
0.39 0.32 0.37
0.32 0.23 0.29
0.29 0.23 0.18
1.09(29)
0.28 0.29 0.31
0.28 0.28 0.32
0.29 0.29 0.34
0.30 0.27 0.34
1.08(21)
0.41 0.40 0.43
0.41 0.38 0.41
0.39 0.34 0.40
0.39 0.36 0.42
0.292(38)
0.46 0.41 0.44
0.44 0.35 0.41
0.43 0.31 0.36
0.32 0.22 0.25
a4
/MPa
MEG 2.608(47) PEG 0.017(21) ECS 0.169(26)
0.2969(90) -1.96(53) 0.303(29)
8.88(21) 9.6(2.2) 0.318(15)
MEG 0.2030(21) PEG 0.612(37) ECS 0.3047(58)
1.584(41) 0.371(20) 0.8948(45)
1.198(21) 1.092(34) 0.t
MEG 0.1144(11) PEG 0.398(15) ECS 0.1643(29)
1.778(49) 0.423(13) 0.9148(34)
1.265(23) 1.191(27) 0.t
MEG 0.5563(68) PEG 1.51(13) ECS 1.184(68)
1.425(42) 0.293(27) 0.9822(88
1.545(37) 1.518(54) ‘)
MEG 0.3074(40) PEG 0.709(70) ECS 0.588(28)
1.330(41) 0.242(31) 0.9496(68
1.506(38) 1.481(55) ) 0.t
MEG 0.2102(34) PEG 1.02(11) ECS 0.449(20)
1.831(83) 0.458(33) 0.9592(83)
1.271(38) 1.190(60) 0.t
0.t
2.18(52)
A. S. PINE
156
0.24
HCN/HCN
0.22
ECS
40
W-Matrix
& 35
0.20 0.18 0.16
0.08 0.06
0
5
15
10
20
25
30
35
40
45
K
HCN/HCN
0.45r
MEG
W-Matrix
10
Y
0.40
-
0.35
-
0.30
-
0.25
2 0.20
0.15
0.00 0
5
10
15
20
25
30
35
40
45
K
Fig. 2. Three-dimensional plots of the upward collision rates (off-diagonal elements of the inelastic rotational relaxation W-matrix) for the ECS and MEG laws fit to the HCN self-broadening coefficients. J is labelled on the K-J = 1 subdiagonal. The IV,, x scale is normalized to unity for the sum over all K # J at the peak of the distribution.
Broadening and line mixing in HCN and C,H,
157
decrease with AJ slowest for the MEG law and fastest for the ECS law. The kinkiness of the model calculations for HCCH at low J arises from the minimum AJ = 2 quadrupolar selection rule. LINE
MIXING
The formalism used for calculating the absorption coefficient, K(V), of a pressure-blended Q branch in the impact approximation was given previously. 9*‘oThe key operation is the diagonalization of a complex matrix, H=v,+iPW,
(14)
where v, is a diagonal matrix of transition frequencies, v,, given by Eq. (2) and W is the so-called relaxation matrix expressing the collision dynamics. The real parts of the diagonal elements of the W-matrix are the broadening coefficients, y,, discussed in the previous section. Here we could use the measured yJ, which are valid independent of the collision model, or, as we have done, use the energy-gap fitting or scaling laws, Eqs. (7-l 1), which nicely fit and smooth the broadening data. This, of course, does not imply that the models are correct, but just that they well represent the broadening data. The imaginary parts of the diagonal elements of the W-matrix are the pressure shifts which we take to be independent of J and ascribe to a band origin shift, Av, , in the following analysis. Further, we neglect the imaginary parts of the off-diagonal elements of the W-matrix. The real parts of the off-diagonal elements of the W-matrix are set proportional to the negative of the rotationally-inelastic collision rates, Re{ W,K} = - FR,,,.
(1%
Here we have introduced an empirical “coupling factor”, F, representing the contribution of the relevant state-to-state collisions to the total line broadening. Greens7 has calculated inelastic rates and line-coupling coefficients for He-CO2 in the 10s approximation, obtaining W,, about half the R We expect full coupling, F = 1, in the Q branch if the total broadening in the Z7 vibrational leii;is due solely to rotationally inelastic collisions within thef-parity sublevel. Thus, we interpret 1 - F primarily as the fraction of f+e cross-relaxation in the Z7 level, although it may have contributions from other extraneous broadening mechanisms such as elastic reorientations and vibrational relaxation and dephasing. f-c cross-relaxation is responsible for coupling Q-branch lines to those in the P and R branches, which may come into evidence at much higher pressures. We note that if the lines are not coupled, say by setting F = 0, then the H matrix is diagonal and the eigenvalues are w, = v, + iy,, yielding a simple sum of Lorentzian lines.9J0 The impact approximation expressing line-mixing for r~(v)~*‘O must be convolved with the Doppler distribution to account for the translational motion evident at subatmospheric pressures. We use the Dicke-narrowed Rautian profile of Eq. (4) for y = 0 to obtain the absorption spectrum, R(V)=I,SR
(;,O,;).,,
-v’)dv’,
(16)
where IO is an overall intensity fitting, scaling or normalization factor. This approach should be valid provided that there is no statistical correlation between velocity- and state-changing collisions.39 The Dicke-narrowing parameters for the various collision partners are given in Table 1. A nonlinear least-squares fitting program using numerical derivatives is used to fit the pressure-broadened absorption spectra using the line-mixing expressions above with the various rate laws determined from the fits to the broadening coefficients in the previous section. These fits refine the overall intensity, IO, the band-origin pressure shift, Av,, and the coupling factor, F, any of which can be fixed at specified values if desired. This is far fewer parameters than those used in the all-lines fit for the broadening coefficients of the previous section. Diagonalization of the complex H matrix, of order J_ + 1, is the time-consuming step in the fits since J,, = 40 is needed to include all thermally populated levels. Observed-calculated residuals for the 2.69 kPa (20 torr) v, + v: Q branch of self-broadened HCN for the various fitting and scaling law models discussed in the previous section are shown in Fig. 3. The multiparameter LSQ fit residuals are within the noise level except near the band origin where the lines severely overlap and were constrained for convergence. The rate model line-mixing fits QSRT
so/z-c
-0.3
-0.4
Y in
g
I1
I1
I
I
---
I
I
I
I
ECS 4003.0
I
&I....
ECS
PEG
MEG
----.
no
I
-1
I
I
-
-
4002.0
I
mix
kPa
L...___
LSOfit
2.69
Wavenumber/cm
4004.0
h.
A
YVM-
u,+v2
I1
F=l.
F=0.64
F=O.44
F=O.47
F=O.
Au
I 1
z
g
Y ul
0 z
:
-e 0
EJ
-6.0
-5.0
-4.0
-3.0
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Fig. 3. Intermediate and higher pressure observed absorbance spectra of the self-broadened HCN v, + vi Q branch with observed~lculated residuals (offset below zero for clarity) for the multiparameter line-by-line least-squares fit (LSQfit no mix) and MEG, PEG, and ECS empirical rate law models with various coupling factors, F, as labelled.
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Broadening and
line mixingin HCN and C,H,
159
show large residuals for F = 0 or 1 corresponding to no mixing and full mixing. When the coupling factor, F, is floated, the ECS law yields the best fit for F = 0.64. At higher pressures, line mixing becomes greatly pronounced as illustrated by the 26.78 kPa (200 torr) trace in Fig. 3. Here the shifts are constrained to those determined from lower pressure data yielding the Av,/P coefficients given in Table 1, since model inadequacies can yield artificial nonlinear pressure-dependent shifts if floated. The F = 0 and 1 calculations respectively under- and over-estimate the peak intensities by 25-30%, compensated by even larger fractional errors in the wings. The best fit is achieved with F = 0.65 for the ECS model, while the MEG and PEG laws still yield residuals > 10%. Note that the floated F factor for the ECS law changes little with pressure, while it drastically increases for the MEG and PEG laws. For a proper physical model, we would expect the coupling factor, F, to be pressure independent since it simply represents a collisional branching ratio. The fitted F factors are given in Table 3 at several measured pressures for the three rate laws for all the collision partners. Thus, based on the smallest residuals and the constancy of F, the ECS law is more appropriate for self-broadened HCN than the MEG or PEG law, the latter not even able to fit the broadening coefficients. In our previous study of self-broadening of HCCH,’ we did not apply the ECS model, so it is of interest to see if a similar improvement results. In Fig. 4, we show intermediate and higher pressure traces of v, + vi of HCCH with a series of observed-calculated residuals for the various models. Note that the increased noise at the peak of the high pressure trace reflects the digitizing step when the transmission is strongly saturated. At all pressures, the MEG, PEG and ECS laws yield comparable fits and relatively constant F factors, with slightly smaller residuals for the MEG law. Here, the most noteworthy feature is the small F x 0.3, indicating that line mixing is weak and closer to F = 0 residuals than to F = 1. If f + e cross-relaxation is the primary decoupling mechanism, then HCCH has a much stronger propensity for cross-relaxation than HCN. We showed previously that the F factor for self-broadened HCCH9 and HCN” was also independent of vibration, even for anharmonically resonant levels and hot bands. The foreign gas broadening spectra and fits are shown in Figs. 5-8. For HCN/NN in Fig. 5, the ECS residuals are marginally smaller than the PEG, with MEG definitely inferior, and the coupling factor, F, is close to 0.5 for all laws. For HCCH/NN in Fig. 6, the MEG law has the smallest residuals and Fis generally about 0.4. For HCN/Ar in Fig. 7, the MEG law has marginally better residuals, though the F factor is not very constant with pressure and the residuals are all worse than for HCN/NN in Fig. 5. Similarly, for HCCH/Ar in Fig. 8, the fits show a slight preference for MEG, but have a somewhat variable F and worse residuals than for HCCH/NN. DISCUSSION We have shown in the previous section that line mixing causes significant deviation of Q branch profiles from an independently additive superposition of Lorentzian lineshapes (i.e. the F = 0 no coupling case). The measured peak intensities are lo-30% higher than if line mixing were ignored, and the wings asymptotically approach 1-F of the additive Lorentzian case. However, the full coupling case, F = 1, predicts far too much narrowing or collapse of the Q branches. Line broadening and line mixing can be modeled successfully with simple empirical fitting or scaling laws for rotationally-inelastic rates using a variable coupling factor, F, though no single law is best for all collision partners studied. The ECS scaling law is clearly preferred for HCN/HCN and is slightly better for HCN/NN, whereas the MEG fitting law works better for the remaining cases. Thus, the longer-range dipole-dipole and dipolequadrupole forces seem better described by the adiabatic energy-correction term, Eqs. (10 and 12), representing a realisticlO collision duration. This contrasts with the Raman line mixing results in nonpolar N2 and COZ, where the ECS law appears to work better than the fitting laws. 24*28 For the infrared HCN and HCCH cases here, the preferred models generally correlate with increased higher AJ collision rates. The shorter-range induction forces in the Ar-broadening case are most difficult to fit with these models, perhaps indicative of still higher AJ collisions. Measurements of line mixing supplement the line broadening information on the collisional dynamics since state-to-state fitting and scaling laws that adequately describe the broadening coefficients strongly overestimate the amount of line coupling when F = 1. Line mixing can also
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Broadening and line mixing in HCN and C,H,
165
discriminate between very different rate laws, such as the MEG and ECS for HCN/HCN, which yield indistinguishable broadening coefficients. The physical interpretation of the coupling factor, F, in terms of possible collisional mechanisms is most problematical with the present set of data. At higher pressures, where line mixing is most evident, the F factors are reasonably independent of rate law model. Association of the decoupling, I-F, with f+ e cross-relaxation implies that self-broadened HCCH (F x 0.3) has a considerably greater propensity for cross-relaxation than self-broadened HCN (F x 0.6). This can be partially rationalized by the quadrupolar selection rules for HCCH which requires AJ odd forf + c collisions and AJ even for f-1. Since the AJ = 1 energy gap is smaller than the AJ = 2, cross-relaxation is favored, but not enough to account for the observed decoupling.’ Also, the difference in coupling is smaller for N,-broadening where F x 0.5 for HCN and F x 0.4 for HCCH, and the order is slightly reversed for Ar broadening where F -N 0.4 for HCN and HCCH, which is difficult to reconcile with cross relaxation. Thus, shorter-range forces tend to reduce coupling in HCN and increase it in HCCH. Interpretation of the empirical F factor in terms of propensity rules and cross-relaxation is not rigorous, but it does provide a simple picture and a convenient means to fit pressure-overlapped infrared spectra. More extensive scattering calculations on polyatomic molecular collision cross sections, particularly for f + e cross-relaxation, would be extremely valuable. Raman Q-branch spectra in HCN and HCCH might provide additional guidance on the interpretation of the F factor, since there the excited vibration is a singlet Z state and reorientational collisions do not contribute; hence we expect F = 1. Direct measurement of state-to-state inelastic rates in these degenerate n vibrations by pulsed pump-probe methods, recently demonstrated for C states in HCCHSs and HCN,59 could elucidate the cross-relaxation mechanism. Acknowledgemenf-This work was supported in part by the Upper Atmosphere Aeronautics and Space Administration.
Research Program of the National
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