Journal of Molecular Spectroscopy 251 (2008) 282–292

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Journal of Molecular Spectroscopy

journal homepage: www.elsevier .com/locate / jms

Galatry versus speed-dependent Voigt profiles for millimeter lines of O3

in collision with N2 and O2

F. Rohart a,*, G. Wlodarczak a, J.-M. Colmont a, G. Cazzoli b, L. Dore b, C. Puzzarini b

a Laboratoire de Physique des Lasers, Atomes et Molécules, UMR 8523 CNRS-Lille 1, Centre d’Etudes et Recherches Lasers et Applications, Université de Lille1, F-59655 Villeneuved’Ascq cedex, Franceb Dipartimento di Chimica ‘‘G. Ciamician”, Università di Bologna, Via Selmi 2, I-40126 Bologna, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 February 2008In revised form 4 March 2008Available online 26 March 2008

Keywords:Spectral lineshapeSpeed-dependent effectsDicke narrowingCollisional narrowingMolecular diffusion effectCollisional broadeningPressure broadeningOzoneAbsorption spectroscopyMillimeter wave absorption

0022-2852/$ - see front matter � 2008 Elsevier Inc. Adoi:10.1016/j.jms.2008.03.005

* Corresponding author. Fax: +33 (0) 320 43 40 84.E-mail address: Francois.Rohart@univ-lille1.fr (F. R

Experimental and theoretical investigations of ozone lines broadened by nitrogen as well as oxygen havebeen carried out in the 300–320 GHz frequency range. Lineshape analysis has demonstrated clear depar-tures from the usual Voigt profile, actual lineshapes being narrower and higher than expected. Morerefined models, such as the Galatry and speed-dependent Voigt profiles, have been used. Both of themhave been found to reproduce the experimental lineshapes well. However, while for the latter, the nar-rowing parameter shows a linear behavior with pressure, for the Galatry profile a strong nonlinear behav-ior is observed. Such observations demonstrate that the Dicke narrowing effect, related to moleculardiffusion, cannot be the leading process involved. Experimental results have also been compared tothe theoretical ones. The relaxation rate dependence on molecular speed has been computed employingthe Robert–Bonamy semiclassical theory. These calculations confirm the leading role of molecular speeddependence effects. Finally, it is inferred that optical diffusion rates are much lower than the kinetic dif-fusion rate, a conclusion well supported from the comparison of optical and Lennard–Jones radii.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

A previous paper [1] was devoted to an extensive intercompar-ison, in the millimeter-wave region, of ozone-broadening parame-ters retrieved by laboratories in Lille (Laboratoire de Physique desLasers, Atomes et Molécules, PhLAM) and Bologna (Laboratory ofMillimeter-wave Spectroscopy of Bologna, LMSB). The main goalwas to get information on the reproducibility of pressure broaden-ing parameters and, above all, on the systematic effects affectingthem. In this study, we will essentially reconsider the ozone tran-sitions of Ref. [1] focusing on a detailed analysis of observed line-shapes, namely the departures from the Voigt profiles thatclearly show up as soon as a good signal to noise ratio can be ob-tained. Indeed, nowadays, the spectral resolution and sensitivityachieved by spectrometers allow one to point out deviations fromthe usual Voigt profile, so that actual lineshapes appear narrowerand higher than expected. These discrepancies result from the factthat the Voigt profile does not take into account correlations exist-ing between molecular velocities and collisional processes. Suchcorrelations may be ascribed to different physical effects, eitherto velocity/speed changing collisions (the so-called Dicke-narrow-

ll rights reserved.

ohart).

ing effect that reduces the Doppler broadening) [2–4] or to thedependence of relaxation rates on molecular speeds [5,6], or insome cases to both effects as well [7]. In the literature differentmodels have been introduced for describing the observed line pro-files (for a review, see for instance Ref. [8]).

In the present work we mainly focus on two models: the Gala-try profile [3] and the speed-dependent Voigt (SD-Voigt) profile[5,6]. The former accounts for the Dicke-narrowing effect in a softcollisional model, whereas the second one accounts for the molec-ular speed dependence of relaxation rates. Relaxation coefficientsrelated to either velocity changing effect (Galatry profile) or tospeed dependence effect (SD-Voigt profile) are expected to showa linear behavior versus the gas pressure in the binary collisions re-gime, which should be well fulfilled in our experimental condi-tions. Actually, it will be exactly this expected linear behaviorthat will help us in sorting out the best model to be used.

In fact, in this study it will be shown that, as already seen in theliterature for various molecules [8–18], both the Galatry and SD-Voigt profiles are able to well recover the observed lineshapes aswell as provide reliable and accurate retrieved broadening param-eters. Therefore, lineshape analysis itself and pressure-broadeningparameters do not allow us to discriminate between the two mod-els, although they are based on completely different processes. Onthe other hand, it will be shown that the diffusion rate BG, defined

F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292 283

in the Galatry model, does not have the proper linear behavior inall the pressure range considered and it is even not determinablefor the higher pressure values considered. On the contrary, it willbe shown that the speed-dependence rate CSDV

2 behaves well inthe pressure range considered and it is always determinable.Therefore, it is a deeper investigation of these narrowing parame-ters that can help us to get the right clue.

In this view, we have compared the experimental results ob-tained for some rotational lines of ozone in collision with nitro-gen and oxygen: the 301.8 GHz (J = 140,14 131,13) transitionobserved in Lille in the 235–300 K temperature range; the301.8 GHz, 317.2 GHz (J = 53,3 62,4) and 320.0 GHz(J = 201,19 200,20) transitions observed in Bologna in the 195–300 K temperature range. After a short description of the mainfeatures of the spectrometers employed (Section 2) and of lineprofile models considered (Section 3), Section 4 is devoted to adetailed analysis of the lineshapes observed in both laboratories.For all studied transitions, whichever the temperature and colli-sional partner are, it is shown that the Galatry model leads tonon physical behaviors of relaxation rates in the high pressureregime, in full contrast with the SD-Voigt model. A detailed com-parison of Galatry and SD-Voigt models (Section 5) allows us toexplain these features and to claim that the Galatry profile mustbe disregarded. In Section 6, the retrieved relaxation parametershave been compared to those obtained from semiclassical theo-retical calculations. It is concluded that observed departuresfrom the Voigt profile result mainly from the dependence ofrelaxation rates on molecular speeds and that the optical diffu-sion parameter must be much smaller than the kinetic diffusionone. Finally, Section 7 is devoted to a short conclusion.

2. Experimental details

The spectrometers used have been described in details in previ-ous papers [1,19]; therefore, here, we only report the main detailsrelated to the present study.

At PhLAM (Lille) a video type spectrometer has been employed.Electromagnetic sources are backward wave oscillators (BWO) thatare phase-locked to an emission harmonic of a 1–20 GHz synthe-sizer, locked onto a GPS reference signal. The intermediate fre-quency beat near 320 MHz is compared to the 32nd harmonic ofa 10 MHz signal issued from a second frequency synthesizer thatalso provides the linear frequency scan at a 20 Hz rate. The result-ing electromagnetic power is detected by a liquid-He cooledbolometer, amplified and average about 500� to give the trueabsorption lineshape. The whole spectrometer is managed by acomputer that controls the frequency sweep and stores the corre-sponding absorption signal that consists of about 500–600 datapoints. The gas sample has been set in a 110 cm long cell thatcan be thermo-regulated between 230–350 K with a stability bet-ter than 1 K. Gas pressures have been measured with a MKS Bara-tron capacitance transducer having a 0.1 mTorr resolution and astated accuracy of 0.12% of the reading scale.

At LMSB (Bologna) the spectrometer employed is a frequencymodulated spectrometer whose radiation source is a Gunn-drivenfrequency multiplier. The source is phase-locked to a Rubidiumfrequency standard. The frequency modulation is performed bysine-wave modulating (1.666 kHz) the 90 MHz reference signal ofthe source-synchronizer. The lock-in amplifier is tuned at twicethe modulation frequency so that the recorded signal looks likethe second derivative of the natural lineshape. The cell is a Pyrextube either 148 or 55 cm long and 5 cm in diameter. In both cases,the cell is thermally insulated. The measurements have been per-formed at 195, 240 and 296 K. For measurements at 240 and296 K, the temperature has been kept controlled by a cryostat,

while in the case of measurements at 195 K, the temperature hasbeen maintained by employing an ethyl alcohol—dry ice bath. Inboth cases, the temperature accuracy is ±1 K. The sample pressurehas been measured by a Baratron gauge (MKS type 220 B) with ameasurable pressure range of 10�4 � 1 Torr (�1.33 � 10�2–133 Pa), and with a 0.1 mTorr resolution. The spectrometer isequipped with a liquid-Helium cooled InSb detector.

In both laboratories, ozone was prepared using the silent elec-trical method in a sample of pure dry oxygen (see Ref. [6] of Ref.[1]). After cryogenic pumping, ozone purity was estimated betterthan 95%, a sufficient value for foreign gas broadening experimentssince measurements consisted of about 5–20 lineshape recordingsobtained with the same ozone pressure (about 5–20 mTorr) andvarious buffer gas partial pressures in the 0–500 mTorr pressurerange. Buffer gases were commercial grades of N2 and O2 havingstated purities better than 99.5%.

At PhLAM the true lineshape has been observed and the expo-nential form of the Lambert–Beer law has been used for the lineprofile analysis:

I ¼ I0 � exp½�aðm� m0Þ � L�; ð1Þ

where a(m � m0) is the absorption coefficient at the detuning fre-quency (m � m0) and L the effective cell length.

As accurately explained in Ref. [20], the approximate linearexpression of the absorbed intensity

I ¼ I0 � ½1� aðm� m0Þ � L� ð2Þ

should be used for the modulated line profile analysis performed atLMSB. This expression is valid only if the maximum absorption islower than 6% (amaxL 6 0.06), therefore, this requirement has beenchecked to be fulfilled during all the measurements employingamplitude modulation of the signal [1].

3. Lineshape models

For the fitting of experimental lineshapes, each laboratory usedits own nonlinear least-squares code. They are both based on aFourier transform technique since true lineshapes a(m � m0) can beexpressed as proportional to the real part of the Fourier transformof the molecular correlation function U(t). This function describesthe time domain evolution of the sample polarization after a pulseexcitation occurring at time t = 0 [21]. In the case of frequencymodulation technique with the lock-in amplifier tuned at twicethe modulation frequency, the Fourier transform technique canbe applied by replacing the correlation function U(t) with [22]

J2 m t � sincxmt

2

� �� �� cosðxmtÞ � UðtÞ; ð3Þ

where J2 [x] is the second order Bessel function, with m and xm rep-resenting the modulation depth and frequency, respectively, andsinc (x) = sin(x)/x. Numerical tests on artificial lineshapes showedthat both codes agree within 0.1% on retrieved broadeningparameters.

Lineshapes considered in this work are:(i) the usual Voigt model which considers that molecular dis-

placements and collisions are statistically independent. Its correla-tion function is

UVðtÞ ¼ exp ix0t � Ct � kva0t2

� �2" #

; ð4Þ

where x0 is the line center frequency, C the collisional relaxationrate (in s�1), k = x0/c the wave number and va0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBT=ma

pthe

most probable value of the absorber speed defined by the Boltz-mann constant kB, the temperature T and the molecular mass ma

of the absorber.

1 To avoid confusion, the indexes V, G and SDV will be added to the relaxationarameters according to the line profile considered: Voigt, Galatry and SD-Voigt,spectively.

284 F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292

(ii) the Galatry model which takes into account the moleculardiffusion (Dicke effect). Velocity changing collisions lead to areduction of the Doppler contribution, and hence to a line narrow-ing. As this process influences the Doppler contribution, the Gala-try profile differs significantly from the Voigt profile mainly at lowpressure (Doppler regime). Its correlation function is [3]

UGðtÞ ¼ exp ix0t � Ct þ 12

kva0

B

� �2

� 1� Bt � expð�BtÞf g" #

; ð5Þ

where C is the relaxation rate and B the optical diffusion rate. Forcompleteness, let us mention that we have not considered the Rau-tian profile [4]. Indeed, although this profile considers a hard colli-sion model instead of a soft collision one, it leads to quite similarline fits and conclusions [15].

(iii) the speed-dependent Voigt (SD-Voigt) model which ac-counts for the dependence of relaxation rates on molecular speeds.As fast molecules relax more rapidly than slower ones, the SD-Voi-gt profile is narrower than the Voigt profile. Moreover, it can beeasily understood from collisional theories that this effect occurswhichever the pressure is (see Section 6). In order to have an ana-lytical form of the correlation function, the molecular speed depen-dence of the relaxation rate C (va) has been approximated via thequadratic model [21,11]

CðvaÞ ¼ C0 þ C2va

va0

� �2

� 32

" #ð6Þ

which leads to

USDVðtÞ ¼exp ix0t � ðC0 � 1:5C2Þt½ �

ð1þ C2tÞ3=2 � exp � ðkva0tÞ2

4ð1þ C2tÞ

" #; ð7Þ

where C0 = hC(va)i is the mean relaxation rate over molecularspeeds and C2 describes the speed dependence of the relaxationrate.

Fitted parameters were recorded signal amplitude, central fre-quency, collisional broadening and, when applicable, a line nar-rowing parameter. The Doppler broadening and, when applicable,the modulation depth and frequency were kept fixed at values cor-responding to the experimental conditions. Finally, some experi-mental artifacts, namely the line asymmetry and background,both due to the base line distortion and/or to the gas refractive in-dex through residual standing waves [12,13,22], were taken intoaccount in the line profile analysis. These effects were modeledby introducing an extra term proportional to the imaginary partof the Fourier transform of Eq. (3) and a linear frequency-depen-dent contribution.

4. Analysis of experimental lineshapes

4.1. The 301.8 GHz line observed at PhLAM

The 301.8 GHz line has been studied in Lille by using the videotype spectrometer which allows the observation of the true line-shape. Fig. 1 presents a recording of the lineshape broadened bynitrogen at 232 K (partial pressure PN2 = 80 mTorr). These condi-tions correspond to an intermediate case between Doppler and col-lisional regimes, the collisional broadening being about 1.5 theDoppler width. By using the Voigt profile model (curve b), one ob-tains a rather good agreement with the actual lineshape (curve a).However, residuals (curve c) display a clear deviation characteristicof a line narrowing with a total amplitude of about 0.5% the lineamplitude. This discrepancy is well explained by considering Gala-try or SD-Voigt profiles (residuals displayed in curves d and e,respectively). However, both fits lead to quite similar residuals sothat this single experiment does not allow any discrimination be-tween Galatry and SD-Voigt models.

A better insight is obtained by considering pressure dependen-cies of relaxation rates. They are displayed in Fig. 2 for various N2

pressures ranging from 0 to 400 mTorr. For all the considered lineprofiles, collisional broadening rates, as well as the SD-Voigt nar-rowing rate C2, evolve linearly versus the sample pressure, as ex-pected for a binary collisional regime. As regards the Galatryprofile, the corresponding narrowing rate B, which is related tooptical diffusion, displays a more contrasted behavior: first, itspressure dependence is linear up to about 100 mTorr, i.e., for colli-sional broadening smaller than twice the Doppler one. For higherpressures, its behavior gets nonlinear: this effect is more dramaticas the pressure is larger, with relative uncertainties getting extre-mely large. Finally, no fit at all using the Galatry profile can be per-formed in the high pressure regime (above about 380 mTorr). Thus,the Galatry profile seems inappropriate for reproducing the depar-tures from the Voigt profile, at least in the high pressure regime,whereas the SD-Voigt profile seems to be suitable for all the pres-sure range explored. As a matter of fact, it does not seem reason-able to consider different line profiles according to variouspressure ranges as it has been done by some authors [23,24]. So,these observations suggest that observed departures from the Voi-gt profile result mainly from the speed dependence of relaxationrates and that velocity/speed changing collisions play a minor rolein the case of O3–N2 collisions.

In the binary collisional regime, relaxation rates behave linearlywith the foreign gas pressure Pb. Consequently, for experimentsinvolving a constant absorbing gas pressure Pa and various foreigngas pressures Pb, the broadening parameters c(T) can be defined attemperature T by

C2p¼ Dm0 þ cðTÞ � Pb ð8Þ

where Dm0 is a constant taking account of self collisions or other ef-fects (wall collisions,. . .). Similar expressions have been used fornarrowing parameters c2 and b related to C2 and B, respectively.These parameters have been deduced from a weighted least-squares fit of about 20 data points available at the temperature con-sidered. For the 301.8 GHz line of O3 in collision with N2 at 232 K(see Fig. 2), one obtains1 cV = 3.80(3) MHz/Torr, cG = 3.91(3) MHz/Torr and cSDV

0 ¼ 3:92ð3ÞMHz=Torr, i.e., slightly larger values for thenarrowed Galatry and SD-Voigt models than for the Voigt model.The narrowing parameter cSDV

2 ¼ 0:31ð2ÞMHz=Torr is about 8% thecSDV

0 value, in agreement with previous results published on N2-in-duced relaxation of O3 or of other atmospheric gases (see for exam-ple Refs. [11,13–15,17,18]). Let us note however that nonlinearitiesobserved employing the Galatry profile prevent to apply Eq. (8) tothe optical diffusion rate BG. Thus, a Galatry parameter bG was de-duced from low pressure recordings only. The obtained value,bG = 1.07(11) MHz/Torr, is about 3.5(6) times larger than the cSDV

2 va-lue, a similar ratio was obtained in previous works (see also Section6).

Similar analyses have been performed for all the 3 temperatures(T = 232, 269 and 299 K) considered for this line. As an example,Fig. 3 displays a log–log plot of the temperature dependence ofrelaxation parameters obtained for the 301.8 GHz line of O3 in col-lision with N2. As usually observed, these dependencies are inagreement with the usual empirical power law [25]

cðTÞ ¼ cðT0Þ:TT0

� ��n

; ð9Þ

where T0 is a reference temperature (chosen as T0 = 296 K) and n thetemperature dependence exponent. The parameters obtained for

pre

-3 -2 -1 0 1 2 3

Frequency Detuning (MHz)

ab

cd

e

Fig. 1. Recorded shape of the 301.8 GHz line of O3 in collision with N2: (a) observed signal, (b) (in red) fitted Voigt profile. Residuals (obs. � calc., enlarged by a factor of 10):(c) (in red) Voigt, (d) Galatry, (e) SD-Voigt. O3 pressure = 20 mTorr, N2 pressure = 80 mTorr, cell temperature: 232 K. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this paper.)

0

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200 250 300 350 400

Pressure (mTorr)

Rel

axat

ion

Rat

e (k

Hz)

Fig. 2. Relaxation of the 301.8 GHz line of O3 in collision with N2. For the different considered line profiles, the relaxation rates are displayed versus the total pressure and thestraight lines are derived from weighted least-squares fits: Voigt [CV (e)], Galatry [CG (not shown); BG(d)], SD-Voigt [CSDV

0 (h); 3 CSDV2 (j)]. The dashed line refers to the 1/e

Doppler line width (285 kHz) at cell temperature: 232 K. For 3 CSDV2 and BG, error bars correspond to one standard deviation.

F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292 285

the various considered profiles are collected in Table 1. As far astemperature exponents of narrowing parameters bG and cSDV

2 areconcerned, they look similar to those obtained for broadeningparameters cV, cG and cSDV

0 , but with much larger uncertainties sothat a reliable study of their temperature dependencies does notseem feasible. For the purpose of analysis of relaxation rates, itseems possible to assume in the following that narrowing parame-ters bG and cSDV

2 have the same temperature dependencies as cG andcSDV

0 , respectively.

4.2. Collisional parameters retrieved at PhLAM

During this work, similar results have been obtained for allexperiments performed on the 301.8 GHz line of O3 in collisionwith N2 and O2 at various temperatures in the 235–300 K range.For all cases, previous conclusions apply, namely a linear pressuredependence of the broadening rates CV, CG, CSDV

0 , and of the SD-Voi-gt narrowing rates CSDV

2 , and a failure of the Galatry optical diffu-sion rate BG in the high pressure regime. These results are similar

to those previously obtained for the 500.4 GHz (342,32–341,33) line[12], except for the fact that the pressure range explored was notsufficiently large to put in evidence a Galatry profile failure.

The results on the 301.8 and 500.4 GHz lines are summarized inTable 1, where the reported relaxation parameters and tempera-ture dependencies have been modeled via the usual empiricallaw of Eq. (9). As indicated before for the 301.8 GHz line, temper-ature dependence exponents of narrowing parameters have beenfixed to values obtained for corresponding broadening parameters.

4.3. The 301.8 GHz line observed at LMSB

For comparison purposes, the 301.8 GHz line has also beenstudied in Bologna by using the frequency modulation technique.Fig. 4 shows, as an example, the spectrum of the line broadenedby oxygen (partial pressure PO2 = 201.8 mTorr) recorded at 240 K:as expected, the observed signal looks like the second derivativeof the true lineshape. By using the Voigt profile model we may no-tice residuals that are characteristic of a line profile narrower and

Table 1Retrieved Ozone relaxation parameters

Buffergases

Transition Absorberfrequency (GHz)

Referencetemperature (K)

Voigt Galatry SD-Voigt

cV cG bG a cSDV0 cSDV

2 cSDV2 =cSDV

0 q b

N2 140,14–131,13 301.8 PhLAM 296 3.136(46) 3.145(31) 0.703(43) 3.148(24) 0.241(19) 0.077(7) 5.6(6)n = 0.82(10) n = 0.90(6) n = 0.90c n = 0.90(4) n = 0.90c

140,14–131,13 301.8 LSMB 296 3.164(4) 3.140(23) 0.937(59) 3.164(16) 0.328(4) 0.104(2) 9.9(6)n = 0.672(22) n = 0.93(18) n = 0.668(19) n = 0.40(5)

53,3–62,4 317.2 LSMB 240d — 3.975(6) 1.389(42) 4.095(13) 0.440(13) 0.107(4) 11(1)

201,19–200,20 320.0 LSMB 296 — 3.050(13) 0.977(25) 3.045(2) 0.278(6) 0.091(2) 7.2(3)n = 0.720(13) n = 0.66(9) n = 0.798(9) n = 0.75(14)

342,32–341,33 500.4 PhLAMe 296 2.887(22) — 0.8(1) 3.00(7) 0.24(4) 0.080(15) 5.9(15)n = 0.87(7) n = 0.71(21)

O2 140,14–131,13 301.8 PhLAM 296 2.745(47) 2.746(30) 0.658(41) 2.765(32) 0.206(12) 0.075(5) 5.0(3)n = 0.63(9) n = 0.63c n = 0.63c n = 0.63c n = 0.63c

140,14–131,13 301.8 LSMB 296 — 2.680(7) 0.918(120) 2.647(13) 0.240(3) 0.091(2) 6.3(2)0.684(10) n = 1.12(36) n = 0.741(17) n = 1.04(4)

53,3–62,4 317.2 LSMB 240d — 3.560(6) 1.339(39) 3.674(9) 0.403(10) 0.110(3) 9.0(7)

201,19–200,20 320.0 LSMB 296 — 2.550(8) 0.836(40) 2.601(45) 0.250(13) 0.096(7) 6.7(8)n = 0.740(15) n = 0.83(19) n = 0.670(57) n = 0.67c

342,32–341,33 500.4 PhLAM e 296 2.394(15) — 0.65(5) 2.45(4) 0.19(3) 0.078(14) 5.2(9)n = 0.91(6) n = 0.84(13)

Relaxation parameters (given in MHz/Torr) result from the least-squares fits performed by using Eq. (9). Quoted errors correspond to 1 standard deviation.a Reported Galatry diffusion parameters results from low pressure experiments only (Doppler regime).b Exponent of the related 1/rq potential.c Value derived with the temperature dependence parameter n fixed at indicated value.d The 317.2 GHz line was observed at 240 K only.

-2

-1

0

1

2

ln (T )

ln (

γ)

5.4 5.5 5.6 5.7

Fig. 3. Relaxation of the 301.8 GHz line of O3 in collision with N2. Log–log plot of temperature dependencies of relaxation parameters: SD-Voigt [cSDV0 (h); cSDV

2 (j)], Galatry [cG

(not shown); bG (d)]. Straight lines are derived from weighted least-squares fits according to Eq. (9).

286 F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292

with peak higher than a true Voigt profile, on the contrary this dis-crepancy is removed when employing the Galatry as well as SD-Voigt profiles. It has to be noted that a systematic study employingthe Voigt profile has been carried out only for the N2-broadening at296 K. For the other temperatures and perturber, only a few seriesof measurements have been analyzed by using the Voigt model astest cases. Since the modulation technique has been employed, foreach recorded spectrum the collisional broadening rate has beendetermined by fitting the observed line profile to the model ofEq. (3) that explicitly accounts for frequency modulation as de-scribed in Refs. [20,22,26].

As already mentioned in the previous section, in the pressureranges considered, the broadening rate C is a linear function ofthe perturber pressure, as expressed by Eq. (8). The pressure broad-

ening parameters c(T) at a given temperature T have then been de-rived by a linear fit of broadening rates against the partial pressurePb of the perturber. For each temperature, some series of measure-ments have been carried out (on the whole 25–35 points), and foreach series of measurements, the pressure-broadening parameterc(T) has been derived by the above mentioned linear fit, in whichthe half-widths have been weighted according to the reciprocalof the squared uncertainties obtained from the profile analysis.Since the series of measurements have been performed usingslightly different quantities of ozone, in order to put all the mea-surements together, we have performed a weighted mean of thevalues of c(T) obtained for each series of measurements.

Fig. 5 shows an example of linear fit of CSDV0 for N2-broadening at

195 K. The behavior of CSDV2 and BG rates is also reported. As already

Fig. 4. Recorded lineshape of the 301.8 GHz line of O3 in collision with O2: (a)observed signal in black and fitted Voigt profile in red. Residuals (obs. � calc., en-larged by a factor of 3): (b) Voigt, (c) SD-Voigt, (d) Galatry. O3 pressure = 5.4 mTorr,O2 pressure = 201.8 mTorr, cell temperature: 240 K. (For interpretation of the ref-erences to color in this figure legend, the reader is referred to the web version ofthis paper.)

Fig. 5. Relaxation of the 301.8 GHz line of O3 in collision with N2 at 195 K. For thedifferent considered line profiles, the relaxation rates are displayed versus the totalpressure and the straight lines are derived from weighted least-squares fits: SD-Voigt in blue [CSDV

0 (h); 3 CSDV2 (j)] and Galatry in red [BG (d)]. The dashed line

refers to the limit pressure (see text). Error bars correspond to three times thestandard deviation. (For interpretation of the references to color in this figure leg-end, the reader is referred to the web version of this paper.)

Fig. 7. Relaxation of the 317.2 GHz line of O3 in collision with O2 at 240 K. For thedifferent considered line profiles, the relaxation rates are displayed versus the totalpressure and the straight lines are derived from weighted least-squares fits: SD-Voigt in blue [CSDV

0 (h); 3 CSDV2 (j)] and Galatry in red [BG (d)]. The dashed line

refers to the limit pressure (see text). Error bars correspond to three times thestandard deviation. (For interpretation of the references to color in this figure leg-end, the reader is referred to the web version of this paper.)

Fig. 6. Relaxation of the 301.8 GHz line of O3 in collision with O2: log–log plot of thetemperature dependencies of the SD-Voigt and Galatry profile parameters. Symbolsand colors: cSDV

0 (h) and cSDV2 (j) in blue, cG (s) and bG (d) in red. (For interpretation

of the references to color in this figure legend, the reader is referred to the webversion of this paper.)

F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292 287

noted in Fig. 2, the Galatry rate BG behaves linearly only up to�200 mTorr and it is determinable only up to �350 mTorr. Addi-tionally, the uncertainties increase enormously as soon as the lin-ear behavior is lost. Therefore, the values of bG, obtained fromlinear fits analogous to that of Eq. (8), have been determined con-sidering only the pressure range for which the linear regime is va-lid. Anyway, it should be noted that the uncertainties of the higherpressure values are so large that they would not affect the slope ofthe fit. In contrast, as already observed at PhLAM, the CSDV

2 rate be-haves linearly in all the pressure range considered (0–500 mTorr)and the uncertainties remain small when increasing the pressurevalues.

From the measurements carried out at different temperatures,it is possible to determine the temperature dependence of the

relaxation parameters following the semi-empirical law expressedby Eq. (9). More precisely, a weighted least-squares fit of c(T),based on Eq. (9), has been performed with T0 = 296 K as referencetemperature to retrieve c(T0) and n. Fig. 6 shows such a fit for boththe Galatry and SD-Voigt profiles in the case of O2-relaxation. It isevident that the two models give almost equivalent results for thetemperature dependence of the broadening parameter cG and cSDV

0 .Concerning the narrowing parameters bG and cSDV

2 , a similar behav-ior is also observed.

4.4. Other lines observed at LMSB

In addition to the N2- and O2-induced relaxation of the301.8 GHz line, the 320.0 and 317.2 GHz lines have been investi-gated; the first two lines at three different temperatures (195,240 and 296 K), whereas the last one only at 240 K. As for the

288 F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292

301.8 GHz line, a comparison between the Galatry and SD-Voigtprofiles has been performed, whereas the Voigt model has not beconsidered at all. The spectra recording, the line profile analysisand the retrieving of the collisional parameters have been carriedout as described in the previous subsection for the 301.8 GHz line.

Fig. 7 shows the linear dependence of CSDV0 for O2-broadening of

the 317.2 GHz line at 240 K. The behavior of CSDV2 and BG rates is

also depicted. As already notice in previous figures, the Galatry dif-fusion rate BG behaves linearly only up to 150–200 mTorr and it isnot determinable for Pb > 350 mTorr. Additionally, a few pointsdetermined at higher values than the standard pressure range ofthis work have been considered and reported. It is worthwhile not-ing that the SD-Voigt profile well behaves up to 1 Torr, i.e., the lin-earity of CSDV

0 and CSDV2 and the determinability of corresponding

parameters cSDV0 and cSDV

2 are valid up to such a high pressure value.

4.5. Collisional parameters retrieved at LMSB

The results obtained for the lines investigated at LMSB are sum-marized in Table 1: the pressure broadening and narrowing param-eters and the values of n obtained employing the SD-Voigt modelas well as the Galatry profile are compared. More precisely, the val-ues reported are the c(T0) and n parameters as obtained from theleast-squares fits performed by using Eq. (9). For all the lines con-sidered, Galatry and SD-Voigt profiles lead to equivalent tempera-ture dependencies of the broadening parameters cG and cSDV

0 . Incontrast to the particular case of the O2 relaxation of the301.8 GHz line, narrowing parameters bG and cSDV

2 exhibit differenttemperature dependencies, but their uncertainties are definitelylarger than those of cG and cSDV

0 .A general conclusion that can be drawn is that for all the

transitions and temperatures considered an analogous behavioris observed: while for the SD-Voigt model the linearity withpressure is valid for both the relaxation and its speed depen-dence parameter, in the case of the Galatry profile for the nar-rowing parameter the linearity is observed only up to 100–200 mTorr and often it is no longer determinable for pressureshigher than 300–400 mTorr.

5. Correlation between Galatry and SD-Voigt profiles

Experimental observations show that Galatry and SD-Voigt pro-files are numerically equivalent, which means that their correla-

0

1

2

3

4

5

0 1 2

Reduced P

Β/3

ΓΒ

/3Γ

Β/3

ΓΒ

/3Γ 22 22

Fig. 8. Behavior of BG=ð3CSDV2 Þ versus the reduced pressure CSDV

0 =kva0 for the 301.8 GHz liusing a = 4 and CSDV

2 =CSDV0 ¼ 0:063.

tion functions have similar damping behaviors. This is equivalentto say that they have the same value after a characteristic durationscorr which defines the polarization correlation time, i.e.

UGðscorrÞ ¼ USDVðscorrÞ: ð10Þ

As the broadening rates CG and CSDV0 are similar, D’Eu et al. [15] have

shown that this correlation time can be defined as

scorr ffia

CG ffia

CSDV0

; ð11Þ

where a is an adjustable scaling constant that allows to choose thesignificant time scale of correlation functions. Then, Eqs. (5), (7),(10) and (11) lead to [15]

1

1þ aCSDV2

CSDV0

� 6CSDV

0

akva0

� �2aCSDV

2

CSDV0

� ln 1þ aCSDV2

CSDV0

!" #

¼ 2CG

aBG

� �2aBG

CG � 1þ exp � aBG

CG

!" #ð12Þ

where the ratio CSDV0 =kva0 of the broadening rate to the Doppler

width can be seen as a reduced pressure. Recalling that reducednarrowing rates BG/CG and CSDV

2 =CSDV0 must be pressure independent

in the binary collisional regime, one gets a contradiction: the lefthand side of Eq. (12) is pressure dependent, whereas the right handside is not. As a result, this leads to a pressure dependent relationbetween CSDV

2 and BG.As inferred from present experimental results on O3, only the

ratio CSDV2 =CSDV

0 related to the SD-Voigt profile seems pressure inde-pendent, whereas the BG/CG one related to the Galatry profile is not(see Figs. 2, 5 and 7). Thus, BG appears only as an ad hoc pressuredependent narrowing rate, Eq. (12) allowing for a determinationof its pressure dependence.

In the low pressure regime (CSDV0 =kva0 � 1), one gets

BG

3CSDV2

ffi 1þ 1a

3CSDV2

CSDV0

CSDV0

kva0

� �2

: ð13Þ

In the zero pressure limit, SD-Voigt and Galatry profiles are identicalby setting BG = 3 CSDV

2 , in agreement with results reported inTable 1 and Figs. 2, 5 and 7. For increasing pressures, the ratioBG=CSDV

2 exhibits a quadratic behavior versus the reduced pressureCSDV

0 =kva0, in agreement with our experimental observations. In thehigh pressure regime (CSDV

0 =kva0 1), whereas the right hand sideof Eq. (12) remains positive, the left hand side is decreasing and gets

3 4 5

ressure

ne broadened by N2 at 232 K. Full line is the theoretical behavior (Eq. (12)) obtained

F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292 289

negative beyond a limiting value of the pressure. This limit, obtainedfrom Eq. (12), takes a simple form in case of a weak narrowing [15]

CSDV0

kva0

� �lim¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6 1þ aCSDV2

CSDV0

� �aCSDV

2CSDV

0� ln 1þ aCSDV

2CSDV

0

� �h ir ffi CSDV0

CSDV2

ffiffiffi3p : ð14Þ

As a consequence, no fit can be performed by using the Galatry pro-file beyond the limiting pressure. This result is displayed in Figs. 8and 9 where BG/(3 CSDV

2 ) ratios have been plotted versus the reducedpressure CSDV

0 =kva0 for experimental results obtained in Lille (Fig. 8:301.8 GHz line of O3 in collision with N2 at 232 K) and in Bologna(Fig. 9: 301.8, 317.2 and 320.0 GHz lines of O3 in collision with O2

at 240 K): a good agreement is obtained with the theoretical behav-ior of these ratios that are deduced from Eq. (12) by using the the-oretical value for the cSDV

2 =cSDV0 ratio (see Section 6) and setting the

scaling constant at a = 4. Furthermore, the observed limiting pres-sures are in good agreement with Eq. (14) (see also Figs. 5 and 7).These results are in full agreement with similar observations previ-ously done in the infrared domain on HCN [15] or in the millimeterrange on N2O [14].

6. Theoretical calculations

6.1. General considerations

The theoretical dependence of collisional relaxation rates onmolecular speeds can be derived from usual molecular collisiontheories, such as the formalism of Robert and Bonamy [27] whichaccounts for electrostatic and atom–atom interactions along withparabolic collisional trajectories. Since we are interested in thespeed dependence of relaxation rates, collisional cross sectionsr (vr, J2) have been computed for various classes of the absorbinggas corresponding to different relative speeds vr and to collisionpartners in the different J2 states [12,28]. For a given temperatureT, the relaxation rate C(vr) is then written as

CðvrÞ ¼ n2vr

XJ2

qðJ2Þrðvr; J2Þ

¼ n2vr

XJ2

qðJ2ÞZ 1

0Sðb; vr; J2Þ2pbdb; ð15Þ

where n2 and q(J2) are the number density and the density operatorof perturbers at the temperature T. As pressure induced frequency

Fig. 9. Behavior of BG=ð3CSDV2 Þ versus the reduced pressure CSDV

0 =kva0 for the 301.8,317.2 and 320.0 GHz lines broadened by O2 at 240 K. Full line is the theoreticalbehavior (Eq. (12)) obtained using a = 4 and CSDV

2 =CSDV0 ¼ 0:063. Dashed lines refer to

the limit pressures (see text). Symbols and colors as specified in the legend.

shifts are neglected, S(b,vr, J2) is the real part of the collisional effi-ciency [27]. The integration over the impact parameter b has beenactually performed via an integration over the closest approachparameter rc and care has been taken to discard the contributionof orbiting collisions corresponding to bound translational statesfor low vr velocities. As an example, the result obtained for theO3–N2 system at 300 K is reported in Fig. 10. The relaxation rate in-creases with the relative velocity vr and can be approximated by thepower law C(vr) / (vr)l which results from an empirical 1/rq inter-action potential, l and q being connected by q = (l � 3)/(l � 1)[5,6]. The relaxation rate C(va) of the absorber molecular class va

is then obtained from the conditional average of Eq. (15) over rela-tive speeds. Using the approximated (vr)l model, one obtains

CðvaÞ ¼Z 1

0CðvrÞf ðvrjvaÞdvr ffi

C0

ð1þ kÞl=2 M � l2

;32

;�kva

va0

� �2" #

;

ð16Þ

where f (vrjva) is the conditional distribution of relative speeds [6], kthe ratio of buffer to absorbing partner masses and M[a; b; x] thehypergeometric confluent function [29]. Quite similar results areobtained either from a direct numerical integration of C(vr) valuesby using Eq. (16) or from the hypergeometric function using theapproximated l value. Finally, the C0 and C2 rates of the quadraticrelaxation model of Eq. (6) have been deduced from C(va) by using aleast-squares procedure weighted by the Maxwell distribution of va.The corresponding results shown in Fig. 10 demonstrate that thequadratic model is a quite good representation of the actual behav-ior of C(va), at least for the most significant part of the Maxwelldistribution.

6.2. Comparison with retrieved parameters

Table 2 summarizes the theoretical results obtained for the SD-Voigt profile parameters of the 4 studied lines of ozone in collisionwith N2 and O2 [12,28]. As expected from the strong quadrupolemoment of nitrogen, N2-induced cSDV

0 broadening parameters arelarger than O2-induced ones. All broadening temperature depen-dencies are similar with temperature dependence exponents aboutn0 = 0.80.

Concerning narrowing parameters, all cSDV2 have similar values

for the reference temperature T = 296 K, but a better insight in nar-rowing processes can be obtained from the consideration of therelation [18]

cSDV2

cSDV0

¼ kl

3ð1þ kÞl=2 M 1� l2

;52

;�3k2

� �; ð17Þ

which links the cSDV2 =cSDV

0 ratio to the empirical 1/rq interaction po-tential with q = (l � 3)/(l � 1). As reported in Table 2, all q-valuesare similar, in the 4.4–4.7 range, either for N2- and O2-inducedrelaxation. Since these values are significantly lower than theq = 6 value corresponding to a Van der Waals interaction potential,it can be inferred that O3–N2 and O3–O2 collisions are not very hard.Finally, one observes that the cSDV

2 =cSDV0 ratios are a little bit larger in

the O2 relaxation case because of its larger molecular mass. By con-trast, n2 theoretical temperature dependencies of narrowing param-eters are smaller than those related to broadening and exhibitstrong variations between the various lines as well as for N2- andO2-induced relaxation. More particularly, it seems that n2 valuesget smaller as ozone rotational quantum numbers J get larger,which means that speed dependent effects are more efficient atlow J-values.

These theoretical results are in rather good agreement withexperimental values retrieved at PhLAM and LMSB. For the 301.8and 320.0 GHz ozone lines, the SD-Voigt broadening parameters

0

1

2

3

4

5

0 250 500 750 1000

Molecular Speed (m/s)

Rel

axat

ion

Par

amet

er (

MH

z/T

orr)

e

d

b a

c

Fig. 10. Theoretical speed dependence of collisional relaxation for the 301.8 GHz line of O3 in collision with N2 at 300 K. The relaxation parameters c(vr) (curve a) and c(va)(curve c) are plotted versus the relative speed vr and the absorber speed va, respectively. Curve b (in red) results from a fit of the c(vr) curve to the (vr)l model. The parabola(curve d, in red) results from a fit of the c(va) curve to the quadratic model of Eq. (6). The Maxwell absorber speed distribution is shown for comparison (curve e, in blue). (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Table 2Theoretical Ozone relaxation parameters at the reference temperature T0 = 296 K

Buffer gases Transition Absorber frequency (GHz) SD-Voigt

cSDV0 (MHz/Torr) n0 cSDV

2 (MHz/Torr) n2 cSDV2 =cSDV

0 q

N2 140,14–131,13 301.8 3.05 0.80 0.19 0.65 0.063 4.753,3–62,4 317.2 3.24 0.79 0.20 0.73 0.061 4.6201,19–200,20 320.0 2.92 0.81 0.19 0.50 0.065 4.8342,32–341,33 500.4 2.83 0.82 0.19 0.43 0.067 4.9

O2 140,14–131,13 301.8 2.52 0.78 0.18 0.49 0.071 4.853,3–62,4 317.2 2.86 0.79 0.18 0.68 0.063 4.4201,19–200,20 320.0 2.39 0.79 0.17 0.28 0.070 4.7342,32–341,33 500.4 2.20 0.75 0.18 0.44 0.082 5.4

290 F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292

cSDV0 are similar, theoretical values being smaller than experimental

ones by 3–4% for N2-broadening and 6–8% for O2-broadening. Asregards the temperature dependencies, the agreements seem notso good, but it must be recalled that the actual uncertainties ontemperature dependence exponents n0 are generally much largerthan the retrieved standard deviations. In the case of experimentsperformed in the 200–300 K range a conservative relative uncer-tainty is about 15% [30,1].

6.3. Analysis of narrowing parameters

As far as speed dependence parameters are concerned, theagreement is not so good and experimental cSDV

2 parameters appearmuch larger than the theoretical ones. If one considers cSDV

2 =cSDV0 ra-

tios, the disagreement gets as large as 30% for the 320.0 GHz line, adiscrepancy incompatible with the retrieved standard deviation.Similarly, retrieved l- or q-values are much larger than theoreticalones. This would mean that the observed speed dependence ofrelaxation rates is much larger than theoretically predicted. As amatter of fact, the Robert–Bonamy formalism [27] used for theo-retical calculations is one of the most sophisticated semi-classicalcollision theory presently available for molecular systems. Sincethis theory has revealed quite accurate for the analysis of a lot ofexperimental situations, it can be inferred that observed discrepan-cies from the Voigt profile do not result exclusively from the speeddependence of relaxation rates. Then, although the SD-Voigt profileleads to satisfactory fits of observed lineshapes, namely linearpressure dependencies of relaxation rates, it seems that the sole

consideration of a speed dependence of relaxation rates is insuffi-cient for an adequate interpretation of obtained relaxation param-eters. In such conditions, Dicke effect, i.e., optical diffusion viavelocity changing collisions, should be taken into considerationsimultaneously by using a more sophisticated line profile. In thispurpose, the simplest available model is the SD-Galatry profile pro-posed by Ciuryło and Szudy [7]. As relaxation speed dependenceand optical diffusion parameters are numerically correlated, ithas been shown in a previous paper dealing with HCN lineshapes[18] that similar insights can be obtained either from line fits ofexperimental signals by using the SD-Galatry profile,—relaxationspeed dependence effects being fixed at their theoretical value,—or from an analysis of collisional relaxation rates derived fromthe Galatry and SD-Voigt profiles in the low pressure regime. Sinceour signal to noise ratios are insufficient for reliable SD-Galatry linefits, we have retained the second method. Following Ref. [18], theoptical diffusion parameter bopt can be obtained from

bopt

cSDV0

þ 3c2

c0

� �th

¼ bG

cSDV0

or 3c2

c0

� �SDV

; ð18Þ

where cSDV0 and cSDV

2 are retrieved SD-Voigt profile parameters, (c2/c0)th is their theoretical ratio, and bG is the narrowing Galatryparameter retrieved from the low pressure regime. Using the exper-imental and theoretical values reported in Tables 1 and 2, resultshave been obtained for optical diffusion parameters and collectedin Table 3 along with kinetic diffusion parameters. Although ob-tained bopt-values exhibit quite large uncertainties, it is clear thatthese values are significantly smaller than the bkin-values related

Table 3Relaxation and diffusion parameters of Ozone

Buffergases

Transition Absorberfrequency(GHz)

Referencetemperature(K)

SD-Voigt Galatry Kinetic diffusion

cSDV0

(MHz/Torr)b0

(Å)(c2/c0)SDV (c2/c0)th bG a

(MHz/Torr)bopt

(MHz/Torr)bkin

(MHz/Torr)rLJ

(Å)

N2 140,14–131,13 301.8 PhLAM 296 3.148(24) 5.73 0.077(7) 0.063 0.703(43) 0.13(8) 0.67 3.97140,14–131,13 301.8 LSMB 296 3.164(16) 5.74 0.104(2) 0.063 0.937(59) 0.39(4) 0.67 3.9753,3–62,4 317.2 LSMB 240 4.095(13) 6.20 0.107(4) 0.061 1.389(42) 0.58(5) 0.78 3.97201,19-200,20 320.0 LSMB 296 3.045(2) 5.64 0.091(2) 0.065 0.977(25) 0.25(3) 0.67 3.97342,32–341,33 500.4 PhLAM 296 3.00(7) 5.59 0.080(15) 0.067 0.8(1) <0.35 b 0.67 3.97

O2 140,14–131,13 301.8 PhLAM 296 2.765(32) 5.49 0.075(5) 0.071 0.658(41) <0.10 b 0.68 3.89140,14–131,13 301.8 LSMB 296 2.647(13) 5.37 0.091(2) 0.071 0.918(120) 0.16(10) 0.68 3.8953,3–62,4 317.2 LSMB 240 3.674(9) 5.99 0.110(3) 0.063 1.339(39) 0.52(4) 0.80 3.89201,19–200,20 320.0 LSMB 296 2.601(45) 5.32 0.096(7) 0.070 0.836(40) 0.21(10) 0.68 3.89342,32–341,33 500.4 PhLAM 296 2.45(4) 5.16 0.078(14) 0.082 0.65(5) <0.15 b 0.68 3.89

Quoted errors correspond to 1 standard deviation.a Reported Galatry diffusion parameters results from low pressure experiments only (Doppler regime).b As bopt cannot be negative, only an upper bound can be obtained from Eq. (18).

F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292 291

to kinetic diffusion processes. This means that, in the case of O3–N2

and O3–O2 pairs, the influence of velocity changing collisions ismuch lower than expected from kinetic diffusion theory.

This situation can be easily understood. Although no reliabletheoretical determination of the optical diffusion parameter bopt

seems presently available, velocity changing collisions can leadto effective consequences on lineshapes only if molecular coher-ences are preserved or at least weakly perturbed during the colli-sion process [31]. Let us now consider optical radii b0, related toline broadenings by cSDV

0 ¼ n�vrpb20, and Lennard–Jones radii rLJ, re-

lated to molecular diffusion. These radii give the order of magni-tude of impact parameters b leading to significant relaxation orvelocity changes, respectively. For collision partners consideredin this work, b0 is about 1.5 times larger than rLJ, which means thata molecule exhibiting a strong collision-induced velocity changesuffers a strong phase change as well, and then cannot contributesignificantly to optical diffusion. These results are similar to thosepreviously obtained for other molecules of atmospheric interestsuch as HCN [15,18], N2O [14] and CO [17] in collision with N2

and O2. In these cases, collisional broadenings are rather large sothat optical diffusion effects play nearly no role and departuresfrom the usual Voigt profile result nearly exclusively from thespeed dependence of relaxation rates.

By contrast, let us consider previous experimental observationsof real Dicke line narrowing effects performed, as an example, onH2O [32–34] or H2 [35]. In these cases, one observes that, as thepressure increases, the total linewidth, equal to the Doppler widthin the zero pressure limit, first actually decreases and then, beyondsome higher pressure, increases. These observations are in agree-ment with our previous arguments, that is collisional line broaden-ing parameters are much smaller than kinetic diffusion ones,allowing for collisions inducing efficient velocity changes but smallphase changes.

7. Conclusion

The pressure dependence of the lineshapes and relaxation ratesof ozone lines (in the 300–320 GHz frequency region) broadenedby nitrogen as well as oxygen has been experimentally and theo-retically investigated. As generally observed when sensitive spec-trometers are employed, in the millimeter-wave region as well asin the infrared domain, absorption lines exhibit clear departuresfrom the usual Voigt profile, the actual profiles being narrowed.Two lineshape models have been used: the Galatry profile whichconsiders the molecular diffusion (Dicke effect) due to velocitychanging collisions, and the SD-Voigt profile which takes into

account the speed dependence of relaxation rates. The pressuredependence of the narrowing rates is expected to be linear as forcollisional linewidth since only binary collisions should take placein our experimental conditions. This investigation allows us topoint out that the B diffusion rate of the Galatry profile does notbehave linearly in the high pressure regime. Upon the definitionof a reduced pressure by the ratio of collisional and Doppler line-width, it is noted that the non linear behavior occurs for reducedpressures larger than 2–3 and BG gets undeterminable for reducedpressures larger than 5–6. On the contrary, the SD-Voigt profile re-sults to be adequate whichever the pressure considered is and evenfor the largest reduced pressure used (about 13). The specific SD-Voigt profile employed is based on a quadratic model of the relax-ation rates speed dependence, thus involving a speed dependencerate CSDV

2 , which allows to critically test the line shape model via itspressure dependence. As a result, it can be concluded that the SD-Voigt profile leads to a realistic modeling of ozone lines and thatthe Galatry profile must be disregarded.

This work has been completed by using the Robert–Bonamy for-malism for theoretical calculations of relaxation rates and of theirspeed dependences. Since observed speed dependence relaxationrates CSDV

2 are significantly larger than theoretical ones, it is inferredthat observed departures from the Voigt profile could also resultfrom a small contribution due to molecular diffusion, in additionto speed dependence effects. From a careful analysis of the correla-tion existing between retrieved experimental values of the BG andCSDV

2 rates, a rough estimate of the molecular diffusion parameterbopt

has been proposed. It is concluded that, in the case of ozone lines inthe millimeter-wave region, the optical diffusion parameter bopt ismuch smaller than the kinetic diffusion one bkin. This conclusion isin agreement with similar studies already performed on other mol-ecules of atmospheric interest, and it confirms the idea that the spec-troscopic consequences of molecular diffusion cannot be observed incase of absorption lines exhibiting too large relaxation rates.

Acknowledgments

Professor R.-R. Gamache is gratefully acknowledged for numer-ical results of theoretical calculations. In Lille, this work has bene-fited of grants from Programme National de Planétologie andProgramme National de Chimie Atmosphérique from CNRS-INSU.The Centre d’Etudes et Recherches Lasers et Applications (CERLA) issupported by the Ministère chargé de la Recherche, the RégionNord-Pas de Calais, and the Fonds Européen de DéveloppementEconomique des Régions. In Bologna, this work has been supportedby ’PRIN 2005’ funds (project ‘‘Trasferimenti di energia e di carica alivello molecolare”), and by University of Bologna (RFO funds).

292 F. Rohart et al. / Journal of Molecular Spectroscopy 251 (2008) 282–292

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