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Scientific Paper
Multiphoton Processes in Gas Mixture: Determination of Rotational Relaxation Parameters†
Mira Terzić,a Dragan D. Markushev,b and Mihajlo Rabasovićb
aFaculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad,
Serbia and Montenegro. mterzic@im.ns.ac.yu
b Institute of Physics, Pregrevica 118, 11080 Beograd-Zemun, Serbia and Montenegro
Received 09-04-2005
†Paper based on a presentation at the 14th International Symposium “Spectroscopy in Theory and Practice”,
Nova Gorica, Slovenia, 2005.
Abstract



We present a new experimental approach to the study of average macroscopic (bulk) (R-»R/T) rotational relaxation parameters of highly vibrationally excited molecules in gas mixtures, where collisions enhance absorption processes. Bulk parameters were derived from measured enhanced cross section, which contains information on photon and collision physics, by means of calibrated multiphoton photoacoustic spectroscopy. All measurements were carried out at room temperature, for the 10P(16) TEA CO2 laser line with fluence range of (1-7)·103 Jm2 and for buffer gas pressures ranging from 1-140 mbar. Particular attention was paid to evaluate the effects of collision partner and buffer pressure dependence on relaxation parameters. Significant buffer pressure effects were observed for all investigated gas mixtures. Two buffer pressure regimes can be distinguished: low and high. It is shown that the data obtained from the proposed method is in good accordance with data obtained from the “slope” and “point” method presented in earlier papers.
Key words: rotational relaxation, multiphoton, photoacoustic spectroscopy.

Introduction
Infrared multiphoton (MP) processes have been the subject of many experimental and theoretical works since the first observation of chemical reactions induced by infrared lasers.1-4 The most important parameters which describe the MP processes are the average number of absorbed photons per one molecule and the cross section. These parameters are in most cases obtained in experiments at different laser fluences and pressures.5–7 The most significant characteristics in understanding the physical and chemical kinetics of MP processes are relaxation mechanisms. Various experimental methods and theoretical models were developed to study vibrational-vibrational (V-V),8 vibrationa-translational (V-T),9–10 vibrational-to-translational/rotational (V^T/R),11 and rotational-to-rotational/translational (R^R/T)12–14 transitions. There are significant changes of rotational relaxation parameters of absorbing molecules with increasing gas pressure or with addition of an absorbing or nonabsorbing gas. A common opinion is that rotational energy transfer (RET) is the dominant process in gas mixtures, resulting in an
increase in number of molecules which are excited effectively by IR laser radiation. Most of experimental investigations related to R-R energy transfer elucidate the specific energy transfer pathways.1516 Investigations of average (bulk) macroscopic R^R/T energy transfer mostly utilize two spectroscopic method: transmission (TR)12 and photoacoustic (PA)1314 spectroscopy. PA experimental studies are restricted to the measurement of quantities averaged over the rotational states, such as the average rotational energy of molecules. Two different approaches to evaluate bulk rotational relaxation parameters were presented till now: the “slope” method13 and the “point” method.14
Determinations of bulk rotational parameters with TR and PA technique are based on the measurements of saturation intensity. In most MP experiments the enhanced absorption cross section is usually measured. In this paper we present a new approach to determine bulk rotational relaxation parameters (R^R/T) from measured enhanced absorption cross sections. The proposed method was tested by a calibrated photoacoustic spectrometer20 in gas mixtures consisting of absorbing SF6 and a nonabsorbing gas.
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Method for determination of rotational relaxation parameters
Experimental studies of bulk rotational relaxation were largely interpreted in the framework of the relaxation equation. The photoacoustic (PA) method determines quantities averaged (bulk) over the rotational states, hence data determined by PA method obeys the relaxation equation
t      = R
. Ei

Eri
d((E„
dt
(1)
where tR is the bulk rotational relaxation time, ({Erot )) is the average rotational energy of the molecule at equilibrium, and ( (Erot y) is the bulk average rotational energy at time t. The bulk cross sections, <JR, are commonly defined as
OR
1
N(v)t
(2)
R
where N is the density of absorbing molecules, and (v) their mean relative velocity. These bulk rotational relaxational parameters, tR, and aR, can be extracted from experimental measurements.
For the PA method two different approaches are reported: (i) the “slope method”,13 based on measurements of saturation intensity dependence on buffer gas pressure, and (ii) the “point method”,14 founded on measurements of saturation intensity, at selected buffer gas pressures.
It would be quite interesting to determine the tR and aR in MP processes from common measurements of enhanced absorption cross section values, ae, without additional measurements. In order to achieve this, a “calibration” procedure should be established, i.e. a relation between the measured quantity, ae, and the saturation intensity I S. In this way, rotational relaxation parameters can be determined directly from credata.
Saturation intensity
Saturation intensity, I S, is the quantity which is defined as the photon irradiant for which the small signal absorption coefficient is reduced the down to half of its small signal value. I S is given by17
S
1
^a teff
(3)
where ?a is the absorption cross section for the specific laser line and teff is the effective lifetime. Numerical computation for an illuminated gas mixture absorber and absorbing gas18,19 shows that rotational relaxation energy transfer is the dominant process. Hence,
the effective lifetime, teff, only determines the bulk rotational relaxation time tR, i.e. teff = tR. This time is an average over all initial and final states. Under this condition, where the other relaxation processes can be neglected, tR can be estimated from IS
t    =
R
1
tfaIS
(4)
The saturation intensity, I(SAB) , in gas mixtures, containing the absorbing (A) and nonabsorbing (B) particles (buffer gas), can be expressed through the
Equation19
I
(AB)
I S A) +
1
1
(AB)
(A)
 +
1
<Vr         <Vr       <Vr
(AB)
(5)
where I(SA) denotes the saturation intensity for pure gas absorber. According to Equation (5), two rotational relaxation times can be distinguished: (i) tR(A) , the bulk rotational relaxation time of the absorbing molecule caused in collision with another absorbing molecule (self collision), and (ii) tR(AB) the bulk rotational relaxation time of absorbing particle A under collision with nonabsorbing B species.
Different experimental methods are used to measure IS. Here we shall mention only the transmission (TR) and the photoacoustic (PA) method. The saturation intensity in gas mixture for optically thick absorber can be deduced from transmission (TR)
measurements19
I
S,TR
4
Ei-ET
ln
ET Ei
(6)
¦ap(A)L
where hv is the laser photon energy, d is the diameter of the laser beam, ti is laser pulse duration, Ei is the laser excitation energy, ET is the laser excitation energy transmitted through the cell of length L, a is the absorption coefficient of absorber gas at pressure p(A). Equation (6) can be adapted for PA measurements by following redefinitions. The photoacoustic method measures the absorbed energy, Eabs, directly by the first peak from microphone signal (photoacoustic signal), Pa (Figure 2), only in the volume in front of the microphone, i.e. along the length lm of the PA detector
Eabs=Ei-ET=S-Pa .                  (7)
In Equation (7), the quantity S = S(p, T, y ) is the sensitivity of the used photoacoustic system (gas-cell-detector). Sensitivity depends on pressure, p, temperature, T, and nature, y, of the gas constituents. Therefore, for quantitative determination of relaxation



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parameters in gas mixtures, the calibration of the PA detector is necessary. In our case for the specified gas mixture, and at room temperature, sensitivity behavior can be described by S = S(p). The detailed calibration procedure was reported in a previous study.20
According to Equation (7), Equation (6) can be adopted for PA parameters13
I
4
SPa
hvd nti
- Ei j + op    ln
(8)
Equation (8) can be applied to determine the saturation intensity in a pure gas absorber, I ( S A ,P )A, as well in a gas mixture, I ( S A ,P B A ) , by the PA method.
“Calibration” procedure
From Equations (2), (4) and (5), the bulk rotational cross sections for a pure gas absorber, aR and for absorbing particle A under collision with nonabsorbing B species, aB)   are related to the saturation intensities,
I S    and I S^
4A)
aa Iff^
N \  v
AA/
4AB)
^a
(IAB)

IA))
N    \    v
(9)
AB/
Thereby, if it is possible to find a relationship between saturation intensity, I S, and measured enhanced cross section, ae, the rotational relaxation parameters can be evaluated without separately measured I S.
The quantity ae contains information on photon and collision physics and can be obtained from the transmission (TR) method21
as where (j>= AEi/nd2 is the laser fluence. Consequently, knowing c7pA the saturation intensities in the pure gas absorber and in gas mixtures can be calculated. Equation (13) can be applied to a gas mixture, as well as to a pure gas absorber. After the “calibration”, i.e. establishing the relation between c7pA and I S,PA, the parameter erR can be determined from Equation 9 (Figure 1). The same consideration can be applied to other experimental methods which determine GjK. For instance, for transmission (TR) measurements, from Equations (6), and (10) it follows
I
S,TRS
Ftr)
1 - exp(- o-
<t>
trNL )
(14)
TR
Figure 1. Schematic representation of the proposed method and the “point” method for evaluation of bulk rotational parameters.
Experimental
ol
1
In
NL  \Ei
E
T
(10)
The enhanced cross sections can also be determined
from PA measurements22
er|
1
PA
---
Nl
In
SPa Ei
(11)
The Equation (11) is a presentation of the Beer-Lambert absorption law appropriate for quantitative PA method. From Equations (11) and (8) the “calibration” procedure can be established. It follows that the relation between measured data (7pA and I S,PA can be expressed
S,PA
k)
0
hvtil ( a
l-exp(-(7pA
Nl
p
(A)
aePAN
(13)
The multiphoton photoacoustic spectrometer has been described previously.20 Briefly, the TEA CO2 laser was operating at 947.75 cm-1 (45ns, FWHM) and in fluence range of 0.1 < ? < 0.5 Jcm-2. The multiphoton parameters were measured by adding the buffer gas in the pressure range of 0–140 mbar to the PA cell containing SF6 under constant pressure of pSF =0.47
6
mbar and pSF =0.13 mbar. All measurements were carried out at 6room temperature (T=300K) .
Results and discussion
Typical signals of PA detector (microphone) in gas mixture SF6–Ar have the waveforms presented in Figure 2. The waveforms (pAr = const) are averaged over five signals to produce a single data point for analysis – the amplitude of the photoacoustic signal, Pa, (the height of the first peak).


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Figure 2. PA detector signals for different pressures of SF6–Ar mixture in the cell. Pa is the first peak from the microphone signal (photoacoustic signal).
The Equation (11) was used to evaluate enhanced cross sections, ?Pe A , from PA measurements. The obtained results are displayed in Figure 3 as a function of buffer gas pressure for SF6–Ar gas mixture for different laser fluences and for pSF =0.13 mbar.
Figure 3. The enhanced absorption cross section versus argon gas pressure for different laser fluences.
The “calibration” relation between
?Pe  and
IS,PA(?PA ) (Equation 13) was used to determine saturation intensities for pure gas and gas mixtures. After that, according to Equation (9), bulk rotational relaxation cross sections, ?P( Aa ) and ?R(AB) were calculated for different laser fluences in the investigated pressure range. The calculated values for ?R(AB) in SF6–Ar gas mixture are shown in Figure 4. The results show that in the investigated pressure and fluence range two pressure regions can be distinguished: (i) the low and (ii) the high-pressure range (Figure 4). Hence, two different sets of rotational relaxation parameters are determined: for low and high pressure ranges. The widths of pressure ranges depend on buffer gas type, i.e.
on the mass ratio of the gas absorber (A) and buffer gas (B), K = mA/mB. For higher K, the low pressure range is narrow, and the high pressure range is broader. For gas mixture SF6–CH4 the low pressure range extends up to pCH = 20 mbar. Opposite to this, for SF6–Ar the low
4
pressure range is two times bigger (pAr = 55 mbar). In SF6–N2 mixture the low pressure range extends up to pN = 40 mbar.
- i
b
i   i   r
SF„-Ar
i
i   i   |   i   i   r
p$p  - 0 13 mbar
$ = 0.52 J cm"2 e = 031 J cm"2
J_____L
J_____L
J_____I_____L
50                   100
Pat [mbarl
Figure 4. The pressure dependence of bulk rotational cross sections örAb) in SF6-Ar gas mixture for different laser excitation fluences.
Two pressure ranges for relaxation parameters appear as a consequence of diverse influence of collisional effects in these regions. Laser fluence also has an influence on the rotational relaxation processes. For the low fluence range the hole filling proceses23 have the dominant role in rotational relaxation. They increase the number of molecules interacting with laser radiation. With increasing laser fluence, a greater fraction of rotational states interacts with the laser field. Consequently, the rotational hole filling processes at these fluences do not enhance absorption. Such conditions are met in the experiment described here.
In the low pressure range, collisions effect has the highest influence on molecular absorption efficiency. This influence is followed by rapid growth of the enhanced cross sections, ?Pe a , (Figure 3), i.e. more absorbing molecules inside the irradiated volume are directly coupled with the laser radiation field. Such behavior is due to R–R relaxation, which includes all rovibrational transitions and eliminates the rotational hole filling effect.23
The high pressure range is characterized by the saturation effect of ?Pe a (Figure 3). The collisions achieve their maximum regarding to their influence on MP processes. Hence, all absorbing molecules in the
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irradiated volume are directly coupled with radiation field due to laser fluence and collision inside gas mixture.
It is interesting to compare the results of this calculation for (T^b^ = (T”° with those obtained previously from the “slope" 13, Cr and “point”14/ctr methods that were produced in the same experimenta conditions. In order to perform that, the average values
KAB)
were determined (Figure 5). Typical results for SF6–Ar pSF = 0.47 mbar and pSF = 0.13 mbar and
 66
SF6-N2 (pSF = 0.47 mbar) gas mixtures are presented in Tables 1- 3. The estimated uncertainty of last figures is shown in parentheses.
Table 1. Bulk rotational cross sections in SF6-Ar gas mixtures ( psf = 0.47 mbar).
6
0           (<°)          (or)            ^r
<p [jem"2]     r10-20m2]      [l(T20m2]       [l(T20m2]
P
SE
 0.47 mbar
low pressure range: 0 < p fa < 55 mbar
0.198(7)	1.2(3)	1.3(3)	1.0(8)
0.306(8)	1.53(11)	1.56(12)	1.6(6)
0.4856(8)	1.06(14)	1.09(13)	0.9(9)
high pressure range:		55 < p fa < 140	mbar
0.198(7)	0.48(7)	0.50(7)	0.48(27)
0.306(8)	0.53(7)	0.56(7)	0.5(4)
0.4856(8)	0.53(4)	0.55(4)	0.53(25)
The ratio ör /crp is shown in Figure 6. This is in accordance with the prediction of the four-state kinetic method for laser pulse transmission trought SF6 gas mixure.18 In this method the absorber molecules are divided into four groups. The molecules which can directly interact with laser radiation in (i) ground and (ii) vibrationaly excited states, and the rest of molecules in (iii) ground and in (iv) excited states. The main conclusion of detailed numerical calculation (steady-state rate equations) is that the dominant relaxation processes are the average rotational relaxation transitions.
Table 2. Bulk rotational cross sections in SF6-Ar gas mixtures (pSF = 0.13 mbar).
0  [jcm"2J
<C
[ai)
-22      2
of
~22      2
[10"22m2]     [10"22m 2 ]      [10"" mz]
P
SE
0.13 mbar
low pressure range:     0 < p^ < 50 mbar
Figure 5. Average bulk rotational relaxation cross sections, /<Tr ), in SF6-Ar and SF6-N2 gas mixtures as a function of laser fluence. Average values obtained from the ”point” methods (o^ \
for different gas mixtures are also presented.
In order to determine the efficiency of RET in gas mixtures we compare the bulk rotational cross section, <7r with the enhanced absorption cross section, OrAB), for the same experimental conditions. The results show that for both pressure regions (Tr  > crp .
0.210(21)	8.5(11)	8.9(11)	8.5(12)
0.310(4)	10.1(16)	10.1(17)	10.2(18)
0.516(11)	9.40(13)	9.42(16)	9.5(17)
high pressure range:		50 < pfa < 140	mbar
0.210(21)	3.5(3)	3.5(3)	3.2(9)
0.310(4)	2.4(10)	3.0(4)	2.6(4)
0.516(11)	3.37(20)	3.39(22)	3.1(9)
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Table 3. Rotational cross sections in SF6–N2 gas mixtures (pSF = 0.13 mbar).
6
<p [jcm"2J
[10_   m ]
'°l)
oî
[10-20m2]
[l(T20m2]
PSF = 0.47 mbar
low pressure range; 0 < pm < 40 mbar
0.510(3)	1.0(5)	1.1(5)	0.59(21)
0.365(10)	1.0(4)	1.0(5)	0.57(21)
0.304(4)	1.1(6)	0.91(17)	0.51(17)
0.253(7)	0.9(5)	0.9(5)	0.43(15)
0.224(6)	0.9(5)	1.0(6)	0.39(14)
high	pressure range:	40 < pN2 < 140 mbar	
0.510(3)	0.45(8)	0.45(8)	0.50(15)
0.365(10)	0.48(4)	0.49(4)	0.50(15)
0.304(4)	0.44(9)	0.47(15)	0.47(15)
0.253(7)	0.33(3)	0.34(3)	0.33(15)
0.224(6)	0.30(6)	0.31(6)	0.34(12)
Figure 6. The ratio of bulk rotational cross-section a^ and enhanced absorption cross-section, aR , as a function of buffer gas pressure for different laser fluences.
Conclusions
In this work we have presented a new approach to determine bulk rotational relaxation cross sections, <7R. It is shown that significant information for crR can be determined from measured values of enhanced cross sections, ae, by using a relationship between ae and saturation intensity values, I S. The method was tested in gas mixture SF6-Ar for two different absorber gas pressures, pSF = (0.47 mbar and 0.13 mbar) and in SF6-N2 (0.47 mbar). Enhanced cross sections involved in
MP processes were measured by calibrated pulsed PAS technique. It was shown that the obtained results for crR are in good accordance with the data obtained from the “slope” and the “point” method. This method offers the number advantages. First of all, it gives the possibility to determine bulk data from the usual measured crevalue in MP excitation processes. This method also avoids the need for additional measurements of saturation intensities in gas mixtures as in case of “point” and “slope” methods. The proposed method was tested in relatively large pressure range of buffer gas (0 - 140 mbar), and hence it is not restricted to any pressure range. The same “calibration” enables to determine bulk rotational in both pressure ranges. In this work the “calibration” for TR determination of aR was also established. The accuracy of this method is limited by the accuracy of credata.
Acknowledgements
This work was supported by the Ministry of Science and Environmental Protection of the Republic of Serbia under Project No 141011.
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Povzetek
Predstavljamo nov eksperimentalni pristop k raziskavam povprečnih makroskopskih (R->R/T) rotacijskih relaksacijskih parametrov visoko vibracijsko vzbujenih molekul v plinski zmesi, kjer trki ojačajo absorpcijske
) procese. Z več fotonsko umerjeno optoakustično spektroskopijo smo želene parametre izračunali iz izmerjenega ojačanega absorpcijskega preseka, ki vsebuje podatke o fotonih in fiziki trkov. Vse meritve smo opravili pri sobni temperaturi z uporabo 10P(16) emisijske črte laserja TEA CO2 s površinsko gostoto energije v območju 1 - 7 × 103 Jm2 in za tlake od 1 do 140 milibarov. Posebno pozornost smo posvetili oceni vplivov kolizijskih parametrov in tlaka na relaksacijske parametre. V vseh zmeseh smo ugotovili pomemben vpliv tlaka na osnovi katerega lahko razlikujemo visoko in nizkotlačno območje meritev. Podatki dobljeni s predlagano in enostavnejšo metodo se dobro ujemajo s podatki, ki jih dajeta že poznani točkovna metoda in metoda naklona.
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