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A kinetic study of infrared multiple photon absorption in SF6-noble gas mixtures during a TEA CO2 laser pulse. Evaluation of V-T relaxation rates
Chemical Physics 142 ( 1990) 473-484 North-Holland
A KINETIC STUDY OF INFRARED MULTIPLE PHOTON ABSORPTION IN SF,-NOBLE GAS MIXTURES DURING A TEA COz LASER PULSE. EVALUATION OF V-T RELAXATION RATES M. LENZI ‘, E. MOLINARI a,c, G. PICIACCHIA b, V. SESSA c and M.L. TERRANOVA c ’ Istituto di Metodologie Avanzate Inorganiche de1 CNR, P.O. Box 10. Monterotondo Scala, 00016 Roma, Italy b Area della Ricerca CNR, Servizio Laser, P.O. Box IO, Monterotondo Scala, 00016 Roma, Italy ’ Dipartimento di Scienze e Tecnologie Chimiche, 2” Universitd di Roma, Tor Vergata, Via 0. Raimondo, 00173 Roma, Italy Received 25 September 1989
The time-resolved attenuation technique described in the preceding paper has been utilized to investigate the kinetics of IR multiple photon absorption by SF, in tbe presence of a large excess of He, Ne, Ar, Kr and Xe. The kinetic model proposed compares favourably with experimental observations provided it includes the possibility of the formation of concentration profiles within the Gaussian laser beam. An evaluation of the net bulk average energy transferred to the noble gas per collision and per SF, molecule, (( AE )) , has been possible and it is shown to decrease regularly from helium to xenon by a factor 19.
1. Introduction
In the preceding paper [ 11, paper I, a light attenuation technique has been described which allows the determination of the number of infrared photons absorbed per SF6 molecule, ( n) , as a function of time and fluence during a TEA CO2 laser pulse. In paper I neat SF, has been investigated at pressures between 0.2 and 1.O Torr. In the present paper this same technique will be applied to a study of the kinetics of IR multiple photon absorption by SF6 in the presence of a large excess of different noble gases: helium, neon, argon, krypton and xenon. In the presence of a buffer gas the total number of photons absorbed per SF6 molecule, ( n), is actually composed of two quantities: the number of photons stored as vibrational energy within the vibrational manifold on the SF, molecule, (n,), and the number of photons ( nT) transferred by V-T relaxation from the vibrationally excited SF, molecule to the atoms of the noble gas. The equality (n>=<&>+
(1)
is valid at any time during the absorption. The quantity which one would like to derive from the absorption kinetics is (( AE )) , the net bulk average energy 0301-0104/90/$03.50 (North-Holland)
Q Elsevier Science Publishers B.V.
transferred per collision and per SF6 molecule to the noble gas bath. This is the ensemble average over the microscopic energy change (A,!?) v [ 2,3 1. This paper is organized as follows: In section 2.1 the IR multiple photon absorption kinetics in the presence of buffers is developed for a hypothetical laser beam of constant radial intensity (rectangular profile with sharp edges) leading to the kinetic model of the IR absorption process in the presence of buffers which allows (n), (n,) (%) and the gas temperature, to be calculated as a function of time and fluence at different laser intensities and different values of (( AE )) . In section 2.2 the problem of the conversion of the experimentally determined number of photons absorbed per molecule ( n ) o ( exp ) into ( n ) and vice versa is discussed at some length because it is a central point for the reduction of the absorption data. The transformation of (n),(exp) into (n), or vice versa, in the presence of significant relaxation of the absorber has been discussed by the authors in ref. [4] to which the reader could refer for a more general understanding of this matter. In section 2.2 this problem is discussed with reference to conditions specific of the present experiments and a numerical procedure is given which allows the transformation
474
M. Lunzi et al. /Infrared
multiple photon absorption
of the values of (n) calculated in section 2.1 into values of ( n ) o. These ( n )o can then directly be compared to the experimental ones. Experimental data are presented in section 3. Section 4 contains a discussion of the data divided into two subsections: in section 4.1 experimental ( n >o( exp) are compared with the corresponding ( n)G calculated from the model of section 2. This comparison brings up features of the process which apparently require for their interpretation the assumption of the presence of radial concentration profiles of excited SF6 of the type discussed in paper I and in rcfs. [ $61 but with an essentially different time dependence. With this assumption it is then possible to evaluate the dependence of (( AK )) on ( n,) . This is done in section 4.2 for He, Ne, Ar, Kr and Xe and leads to a functional dependence of (( U )) / ( n,) on the atomic weight of the noble gas. Section 5 contains the concluding remarks.
2. IR multiple photon absorption kinetics 2.1. The model The time derivative
of eq. ( 1) is:
d(n)/dt=d(n,)ldt+d(n-r)ldt.
(2)
With 0 the excited state absorption cross section (cm’ molecule-’ ) and w the laser intensity (photons s-’ cme2) one has d(n)/dt=aw.
(3)
The vibration-translation written as d
energy transfer rate can be
@ >> ,
(4)
with Z the collision number (cm3 molecule-’ SK’ ) and M the number density (molecule cmm3) of the noble gas atoms. In eq. (4) (( AE)) is expressed in units of (n) (photons molecule-‘): (P(20) photonc944.2 cm-‘). d( n,) /dt, the rate of increase of the vibrational energy content of the SF, molecule can then be written as d(n,)/dt=aw-ZM((
AE))
.
(2’ 1
The rate of increase of the vibrational energy content of the IR absorbing species is therefore regulated by
in St;,noblegas
mixtures
the competition between laser pumping (0~) and VT relaxation (ZM(( AE; )) ). If both 0 and (( U )) are known functions of the vibrational energy content (n,) , w is known as a function of time because it is directly derived from the laser pulse temporal profile, then eq. (2’ ) can be integrated stepwise and yields ( n,) as a function of time and, obviously, of fluence F (J cmw2). In this way 0 will also be known as a function of time and one integrates eq. (3 ) stepwise to obtain ( n) as a function of time and fluence. Eq. ( 1) then yields the corresponding values of ( nT) . A knowledge of ( nT) then allows the determination of the adiabatic temperature of the irradiated volume as a function of time, once the gas composition is known. In order to integrate (2’ ) one should make the proper selection of the o( (n,) ) and of the (( AZ?)) ( ( n, ) ) functions. As already discussed in paper I according to the treatment of ref. [ 7 1, the SF6 molecules present at a given time t can be subdivided into two ensembles: a fraction q( 1) is involved in the process of multiple photon absorption and forms an ensemble of “hot”, i.e. vibrationally excited molecules, while a fraction 1 -q( 1) forms the ensemble of “cold” molecules. q( 1) is related to the relaxation time for rotational hole filling which in SF6 is of the order of 35 ns at 1 Torr [ 8 1. At the buffer pressures utilized in the experiments, which are between 250 and 750 Torr, q( 1) can safely be taken equal to 1 (see also ref. [ 9 ] ). As will become apparent from the presentation of the data, the translational temperature of the gas mixture remains low and in any case much lower than the vibrational temperature of the excited SF,+ One shall therefore select curve (b) of fig. 5 of paper I, which corresponds to a situation of vibrational equilibration but with “cold” translation and rotation, as the a( ( n,) ) function more appropriate to present experimental conditions. The dependence of (( A,??)) on the internal vibrational energy of SF6 has recently been investigated in ref. [ 10 ] for the system SF,-argon and found to be essentially linear: ((M)) =4x 10-4(n,) up to (n,) x20 ( ((M)) photons molecule- ’ ). We shall therefore integrate eqs. (2’ ) and (3) for an argon pressure of 750 Torr, using the mentioned 0 and (( AE )) functions, and a typical experimental w(t) profile. In addition the slope (( A,!? )) / ( n,) will be taken as a parameter variable in the range O-l OX 10e4 to simulate the ef-
M. Lmzi et al. /infrared multiple photon absorptionin SFrnoble gm mixtures
feet of an increasing V-T relaxation rate. The results obtained with this kinetic model are collected in figs. l-3. Curves of fig. 1 give (n) as a function of time, those of fig. 2 (n) as a function of fluence. The corresponding values of (n,) are given in fig. 3 versus time. These results are interesting because they bring out a number of points which should properly be considered: - The total number of photons absorbed per molecule ( n ) at the end of the pulse varies between 20 when (( A,!?)) ~0, i.e. in the absence of V-T relaxation, to approximately 150 when (( AE)) / ( n,) = 10 x 1Om4,showing that the absorption kinetics is particularly sensitive to the V-T relaxation rates. - Values of ( n) and of (n,) are very different and so is their temporal behaviour except when ((AE))=Oand (n)=(n,) ((n,)=O).Inparticular the highest values of ( n) correspond to the lowest ones of ( n,) and ( n,) curves go through definite maxima. High values of ZM (( AE )) yield low values of (n,) which, according to the a( (n,) ) function adopted, correspond to large values of crand, according to eq. ( 3 ), to high rates of absorption. At the end
01” 0
0.2
”
0.4
”
475
0.6
11
0.8
0
11
1.0
F (Jcd)
Fig. 2. Calculated (n) versus fluence for different values of ( (( AE)) /(PI,) ) x 10’ at a buffer pressure of 750 Torr ( 1000 mbar).
150 -
100 -
: Y
1.5
0.0
0
1.0
2.0
3.0
i
Fig. 3. Calculated values of (n,) versus time for different values of(((AE))/(n.,,))Xl04atabufferpressureof750Torr(lOOO mbar).
4.0
t (as)
Fig. 1. Calculated (n) versus time for different values of (((AE))/(n,))X104atabufferpressureof750Torr(1000 mbar).
of a laser pulse one can therefore observe a large number of absorbed photons but these will mainly be ( nT) photons transferred to the buffer while (n,) can be small and vibrational excitation low.
h4. Lenzi et al. /Infrared multiplephoton absorption in SI;,-noblegas
476
In fig. 4a one reports the vibrational temperatures TV versus time as derived from the (n,) values of fig. 3 using JANAF tables [ 111. Fig. 4b reports, against time, the translational and rotational temperature T of the various gases which can be calculated from the values of (n=)=(n) -(n,) and the expression: T- 298 = < nr > ~ZWLrr~r
)hvCy'
,
(5)
which is valid for small ratios of the partial pressure ( PSF6) of the IR absorber to the partial pressure of the noble gas buffer ( PbuKer).hv is the energy of the P (20) CO1 photons utilized in the experiments and C, the specific heat at constant volume of the gas bath which, for the small concentrations of SF6 utilized in the experiments, coincides with $R of the noble gas. Fig. 4b shows that the temperature increase of the gas does not exceed 40 K, and this represents an upper limit for present experimental conditions. Values of T calculated from eq. (5) represent the so called “adiabatic” temperatures, i.e. the temperatures reached by the unexpanded irradiated volume. As already pointed out in ref. [ 121 on the basis of calculations carried out for similar systems in refs. [ 13,141, the expansion of the irradiated volume consequent to
1500-
B ‘>
I-
1ooo-
mixtures
the pressure increase will not start before about 5- 10 us so that in our experiments the gas density can be considered constant and the use of the adiabatic temperature correct. Comparison of the temperatures of figs. 4a and 4b clearly shows that the translational degrees of freedom and the rotational ones, remain “cold” (below 340 K) while vibrational temperatures of fig. 4a range from 800 to 2 100 K. In order to compare the results predicted by the kinetic model with the observed kinetics one should remind that the results of the model would correspond to those obtained with a laser beam of uniform radial profile (rectangular with sharp edges). The laser beam utilized in the experiments is essentially smooth and monomodal and the radial profile is Gaussian. The experimentally determined energy absorbed per molecule of SF6 is therefore not ( n) but ( n)o, as in paper I. ( n ) o is a Gaussian average over the spatial intensity profile of the Gaussian beam Z,=Z, exp( -2r’/oi), where Z, is the local intensity at the distance r from the beam axis and Z, the maximum of the intensity at the axis [ 151. The conversion of a measured ( n)c into (n) is therefore preliminary to any comparison between the observations and the predictions of a model. The situation is particularly complex in the presence of effective collisional relaxation, as discussed in ref. [4], and will be examined in some detail. 2.2. Decomposition of a Gaussian averaged nonlinearjkction in the presence of collisional relaxation
50040-a
10
B L
In ref. [ 41 it was pointed out that the “exact” decomposition procedure proposed in ref. [ 15 ] which transforms ( ra)q into (n) according to the following expressions
t6a) Fig. 4. (a) Calculated increase of the vibrational temperature (TV-298) versus time for different values of ( (( AE)) / ( nV) ) x lo4 at a buffer pressure of 750 Torr ( 1000 mbar). (b) Corresponding increase of the temperature of the gas bath (T-298)foraratioP&P-=2.7x10-‘.
(n>=
Fd(n)G dF
(6b)
cannot be applied in the presence of effective collisional relaxation and should be replaced by a more
M. Lenzi et al. /Ini,red
complex numerical procedure which will partly be drafted below with specific reference to the systems under study. The starting point is a proper calculation of the radial profile of the laser beam intensity Z, after absorption of a fraction of the initial energy l-9& where Yo=J, /Jo (see paper I) is the transmission as measured with the initially Gaussian beam. For a linear dependence of ( n ) on fluence and a transmission
60-
(7)
where the radially uniform transmission % has now been replaced by the actual transmission Tat a distance Yfrom the axis. The calculation of the intensity profile becomes therefore the problem of the determination of z The general relationship between (n) and the transmission Scan be written as 1-Y=kvNL(n)/F,
477
multiple photon absorption in SFGnoble gas mixtures
(8)
where v is the frequency of the IR laser line, N is the number density (molecule cm-3) of SF6 and L is the length of the absorption cell (cm ) . .Tcan be derived from ( 8 ) if ( n ) ,/F, is known. Once Z,is known from eqs. (7), (8), the transformation of (n) into ( n)G or vice versa is straightforward as discussed in ref. 141.
In the presence of collisional relaxation of the IR absorber, ( n ) /F curves should be calculated according to the scheme which has been utilized in the derivation of the curves of figs. 2 and 3 (eqs. (2’ ) and (3) ) with the difference that (( AE)) /(n,,) should now be a constant (4 x 10m4) and the parameter becomes the laser intensity. One should in fact utilize at each distance r from the axis the proper intensity according to the Gaussian distribution. Fig. 5 reports typical (n) /F curves calculated in this way together with the curve corresponding to ((M)) =O. This figure shows that ( n) is not a unique function of F but rather (n) versus F functions become dependent on laser intensity. As a consequence each position of the initially Gaussian intensity profile corresponds to a different ( n) JF, function, which is the one that has to be used in eq. (8) at the distance r. No radial mixing of vibrationally excited species oc-
01 0
I
’ 0.2
’
’ 0.4
I
0 0.6
’ 0.E
)
F (JcrC2)
Fig. 5. Calculated (n) vems fluence for ( (( AE)) / ( n,) ) x 104~4.0 and different total laser fluence=s,Fw Curve (1 ), F,,=O.97 J cm-*, Pw6 ~0.2 Torr. The other curves correspond to total fluenccs of 0.5,0.25 and 0.125 of the total fluence of curve ( 1). Also included is the curve for (
M. Lmzi et al. /Infrared multiple
478
photon absorption in SFcnoble
gas mixtures
versus t curve calculated along the beam axis can be compared with the corresponding ( n ) o versus t curve calculated after determination of the radial intensity profiles at different times during the pulse. Calculations also show that the ratio (n)o/ (n) is a function of laser intensity, buffer pressure and 2 (( AE )) [41.
r 0 I0
3. Experimental results 0.2
..I 04
I
3 1.2
r (8.U.)
Fig. 6. Calculated radial relative intensity profiles I,/&, versus distance r (arbitrary units) from the beam axis. ( 1) Gaussian profile corresponding to linear absorption. (2) Calculated from eqs.(7),(8)andvaluesof(n),/F,fromfig.5at3~~.(3)Calculated as curve (2) with inclusion of the concentration profile of fig. 12a. (4) As curve (3) with concentration profile of fig 12b. Dotted curve gives the incident beam profile.
15017
Fig. 7. (n) versus time plots for argon: F,=O.97 J cm-‘. Curve (n) calculatedwith (((AE))/(n,))xl04=4.0andP,,=750 Torr, PsFs = 0.2 Torr for a radially uniform laser beam. Curve (n). calculated for an initially Gaussian laser beam in the absenceofradialconcenttion profiles. Curve (n)& calculatedwith the inclusion of the concentration profile of fig. 12a. Curve (n)& calculated with the inclusion of concentration profile of rig. 12b. Curve (n>o(exp) measured (n>oversus time. Dashed lineis (n),(exp) dividedby2.0.
The experimental technique has been described in paper I. Fig. 8 shows the time dependence of the laser beam effective radius r, ( t) at the exit of the cell filled with neon at three different pressures. This behaviour is very similar to that reported for argon in ref. [ 161 and can be taken as typical of r, ( t) functions in the presence of buffers. These functions differ markedly from those reported in paper I, fig. 2, for neat SF6. Typical experimental plots of ( n ) o versus time and versus fluence have been reported in figs. 9 and 10 respectively, for helium, argon and krypton at pressures of 750 Torr and total pulse fluence close to 1 J cmP2. Fig. 11 reports experimental (n), values determined for neon and xenon again at about 750 Torr but at lower pulse fluences around 0.2 J cme2. The qualitative trends of all these plots compare favourably with those derived from the kinetic scheme proposed and illustrated in figs. 1, 2 and 5. A quantitative comparison is more complex and will form the subject of the discussion to follow.
Fig. 8. Effective radius of the laser beam at the exit of the absorption cell as a function of time. COz laser: 1 W, 10 Hz, Pw6 ~0.2 Tot-r. Neon pressures: (a) 225 Tot-r, (b ) 383 Tot-r, (c) 600 Torr. The initial value r, is 0.16 cm for all runs.
M. Lenzi et al. /Infrared multiple photon absorption in SF,noblegas
479
mixtures
Q
to
100
N.?.
c
r
Xe
0
1
El/L NB
0
I_1.0
..L_... 2.0
tw
i_ 3.0
0
Xe
0.1
0;
6
._
r (J cm-‘)
Fig. 1 I. Experimental plots of (n). versus time (a) and versus fluence (b)forneonandxenon (PNC=765,Ps,=0.1,F,=0.28; Px.=728,PsFb =0.2,F,=O.25). P,isthepartialpressure (Torr), F,,, is the total fluence of the pulse (J cmb2).
Fig. 9. Experimental ( n)G versus time plots for He (P,,=750, P,,,=O.OS,F,,=0.99),Ar(P,,=750,P,,,=0.2,F,=0.97),Kr (P,=670, PsF6~0.2, F,,=O.O8). P, is the partial pressure (Torr), Fm is the total fluence of the pulse (J cm-‘).
?
(Jcme2)
Fii. 10. Experimental plot of (n)o and Kr. Conditions as in fig. 9.
versus fluence for He, Ar,
4. Discussion 4.1. Radial concentration profiles A comparison between the observations and the predictions of the model requires a comparison be-
tween experimental ( n ) o values and the ( n)c derived from the model as described in section 3. This is done for the case of argon in fig. 7. The value of (( AE )) / ( n,) selected for argon is that determined in ref. [ 10 ],4 x 10m4, and one notices that the experimental (n), values fall in between the calculated ( n) and ( n ) o values. This situation remains qualitatively similar at different argon pressures and different laser intensities. Therefore, if one takes 4 X 10m4 as the reference value of (( AE )) / ( n,) for argon, calculated (n)o values are about a factor of 1.3 higher than the observed ones at 3 us, this ratio being somewhat dependent on the time. As indicated by fig. 4, the temperatures of the buffers are definitely lower than the vibrational temperatures which correspond to the values of (n,) of fig. 3 so that the adopted selection of the u versus (n,) function appears to be the correct one also in view of the results reported in paper I. There remains however a very important point raised by paper I and which has not been considered, namely the possible formation of a radial concentration profile of SF6 inside the laser beam, accompanied by the increase of the number density of excited SF6 at the beam axis. This dielectric suction toward the region of higher intensity of the electromagnetic field has been discussed in refs. [ 5,6] for equilibrium conditions involving the simultaneous flow of energy, mass and polarisation amplitude (laser induced osmosis ). These radial concentration profiles of SF6 should mimic the radial intensity profile of the laser beam leading, for the present case, to Gaussian or nearly Gaussian concentration profiles with an axial concentration N> No and with N/N,=q( 2) as in paper I.
M. knzi
480
1.0
0
10 r
et al. /Infrared multiple photon absorption in SF6-noblegas
1.0
r,
Fig. 12. Assumed concentration profiles: (a) eq. (9a), (b) eq. (9b).q(2)=2inbothcases.
The influence of these concentration profiles on the relationship between the calculated (n) and (n)o and the experimental ( n ) G is very serious. Let us illustrate this point using two different profiles:
N,/N,=q(2)
exp( -2r2/oi)
(gal
N,/N,,=l+[q(2)-l]exp(-2r*/wg),
(9b)
which are drawn in fig. 12 and for both of which NJ 2) at the axis. The changes of the intensity profile Z, caused by these concentration profiles can again be calculated from eqs. (7) and (8 ) where N becomes however a function of r, N,, given by either (9a) or (9b). The results of the introduction of these concentration profiles are illustrated by curves ( 3 ) and (4) of fig. 6 and by curves (n)& and (n)& of fig. 7. The experimental ( n ) d curve should also be changed because in the presence of these profiles the number of SF6 molecules contained in the irradiated volume w&L is multiplied by q( 2 ) and experimental ( n ) o values should therefore be divided by q(2). In fact, from the definition of the effective radius, r,, which in the model is the same for both the intensity and the concentration profiles, the number of molecules contained in the effective irradiated volume m&L should be calculated as follows: the peak concentration of the distribution is q( 2 )No which, when multiplied by m-h,, yields Tc&q( 2)N, molecules cm -‘. The total number of molecules contained in the optical path L will therefore be rcr&Lq( 2)No. This number should be compared with w.&-LN, which
N,=q(
mixtures
gives the total number of molecules in the irradiated volume utilized for the evaluation of the experimental ( n ) G in the absence of a concentration profile (see papers I, section 3). In the presence of a profile experimental ( n) G values should therefore be divided by q(2). This division by q(2) brings (n)& and experimental (n)&q(2) to a much better agreement when q( 2) =2 as indicated by the dotted line of fig. 7. This is not true for ( n )&. It is appropriate at this point to summarize the discussion of fig. 7. Curve (n) refers to a laser beam of radially constant intensity and has been calculated as discussed in section 2.1 with (( AE)) / (n,) =4x lop4 (argon). Curve (n)d is derived from curve (n) by the numerical procedure outlined in section 2.2 and provides therefore values calculated according to the model for an initially Gaussian intensity profile which is then deformed by nonlinear absorption. These values of ( n)G are those to be compared with the experimental ones reported as (n),(exp). Curves (n)& and (n)& have been calculated as curve ( n)G but with the inclusion of concentration profile (a) and (b) of fig. 12 respectively. Comparison of these curves with ( n)G( exp) requires a division by q( 2 ) of the experimental values. The dashed line is (n),(exp) divided by q(2) =2. All the curves of fig. 7 refer to the same time profile of the laser intensity during the pulse. At each time during the pulse it is therefore possible to test concentration profiles of type (9a) by varying the pre-exponential factor q( 2 ), until ( n) & and the experimental (n)Jq( 2) coincide. This will give q( 2) as a function of time during the laser pulse. This has been done for argon and the result is shown in fig. 13.
Fig. 13. q( 2) = N/N, as a function of time calculated for argon as specified in the text. Point and arrow indicate the value of q(2) at 3 ps utilized in the calculations for the other noble gases.
hf. Lenzi et al. /Infrared multiplephoton absorptionin SF+wble gas mixtures
The result of fig. 13 therefore means that calculated and observed ( n)G can be brought into coincidence if one assumes the existence of concentration profiles of type (9a) whose maxima change with time as indicated in fig. 13. Therefore, if (( AE)) / ( n,) is known independently (4 x 10m4 for Ar ), concentration profiles can be estimated and it should be remarked that it is precisely the suggested existence of these profiles which allows one to bring experiments and model calculations into coincidence. Profiles of type (9b), as well as the absence of a concentration profile, do not allow this matching. The difference between profiles (9a) and (9b) is that in profiles (9b) the wings of the Gaussian distribution approach the bulk value N,, while in profiles (9a) the concentration at the wings can be much lower than NO.Under conditions of the experiments, profiles of type (9a) appear to be the most likely to occur because the characteristic time for dielectric suction of excited SF6 toward the laser axis should not exceed lo-’ s (see fig. 13) while the mass diffusion time necessary to replete, in the presence of a large excess of buffer, the beam wings depleted by this dielectric suction should be of the order of 10m3s. The concentration profile which forms as the laser is switched on successively relaxes because laser intensity falls but should be considered as essentially frozen with respect to relaxation by mass diffusion. The situation is therefore substantially different from that described in ref. [ 61 in connection with the experimental demonstration of the existence of laser induced osmosis and where the time scale of the experiments was of the order of 10d3 s. Creation of a profile of type (9a) has obvious consequences on the (n)q values which are shifted from the (n)o to the ( n ) & curve of fig. 7 because the highly absorbing low intensity wings of the Gaussian distribution have substantially been depleted of the IR absorber. In conclusion the model kinetic scheme of IR multiple photon absorption which one proposes should include the possibility of the creation of concentration profiles within the laser region, a suggestion also coming from paper I and, previously, from refs. [ 5,6 1. 4.2. Evaluation of (( AE )) /( n,) in the presence of concentration profiles The arguments leading to fig. 13 are based on the
481
independent knowledge of (( AE )) / ( n, ) which was available for argon [ lo], In the absence of this information, i.e. for the other noble gases utilized in the experiments, the proposed kinetic scheme obviously lacks the one more equation necessary to solve the problem in the presence of undefined concentration profiles. The procedure which one will tentatively adopt is to assume that at a convenient time, say 3 us, the concentration profile is the same as that determined for argon, N,/N,=2
exp( -2r2/&)
.
This then becomes the additional equation necessary to solve the problem. The procedure now consists in a trial and error selection of (( AE )) / ( &) until the experimental value of (n>J2 at 3 us coincides with the corresponding calculated value ( n ) b. Values of (( AL?)) / ( n,) determined by this procedure for He, Ne, Ar, Kr and Xe have been collected in fig. 14 where the logarithm of (( AE )) / ( n,) has been plotted versus the atomic weight of the noble gas. The least-squares straight line has also been drawn in this figure and the values of ((LIE)) /(n,> derived from this straight line have been collected in table 1 for the five noble gases. This table also gives
Fig. 14. Semilogarithmic plot of ( (( AE)) /(&y> ) x 10’ as a function of the atomic mass MB of the buffers He, Ne, Ar, Kr and Xe. The straight line is the least-squares fit of the experimental points.
482
34. Lunzi et al. /Infrared
multiple photon absorption in SF,-noble
gas mixtures
Table I Valuesof(((~))/(n,))x104andofZx10’0(cm3molecule-‘s-‘)forthenoblegases
(((~))/
He
Ne
Ar
Kr
Xe
9.5 5.3
6.5 3.0
4.1 3.2
1.5 2.8
0.5 2.8
the Lennard-Jones collision numbers Z utilized in the calculations and which have been evaluated according to ref. [ 17 1. For each noble gas the scattering of the points is remarkable. These points refer, for each buffer, to variable conditions of laser intensity, SF6 and buffer pressure, initial beam radius. If one takes the value of (( AE )) / ( n,) for krypton from table 1 ( 1.5 x 10w4) and reverses the calculation procedure one obtains values of q( 2 ) which fluctuate randomly with an average value of 2.05 and a standard deviation of 0.35. The presence of these concentration profiles appears therefore to be a severely perturbing factor in the kinetics of IR multiple photon absorption and could jeopardize the efforts aimed at an accurate determination of (( AE )) /(a,) . The apparently random nature of the values of q(2) and the linear relationships of In (( AE)) /(n,) versus the atomic weight of the noble gas of fig. 14, derived by inclusion of the experimental points for all buffers in the regression procedure, do give however reasonable confidence to the (( AE )) / ( n,) values reported in table 1. These values then confirm the expected [ 18 ] negative influence of the mass of the collider on the efficiency of V-T energy transfer from vibrationally excited SF6 to particles with translational degrees of freedom only.
5. Conclusions The kinetic analysis of IR multiple photon absorption in the presence of an excess of a noble gas buffer has been based on the following assumptions: ( 1) (( LIE )) is a linear function of ( n,) . This has been shown experimentally in ref. [ lo] for the system SF,-argon at least above ( n,) = 3-4 and in ref. [ 191 for systems SF6+C6H6 or fluorobenzenes. Below (n,) = 3-4 the situation can be more complex [ 10,201 but this assumption remains acceptable for
the sake of present calculations: in particular the value of (( AE )) / ( n,) 4 X 10P4 for SFs-Ar which has been utilized as the basis of our treatment. This value of (( AE)) / ( n,) is different from that reported in our previous work on the SF6-Ar system [ 16 1. This discrepancy was pointed out by Beck and Gordon [ lo] and formed the subject of our comment on their work [ 2 1 ] and of the author’s reply [ 22 1. Reasons for this discrepancy appear now somewhat clearer. There are in fact two causes of error in our data of ref. [ 161, the most important one being the decomposition procedure. In ref. [ 161 ( n ) was calculated from ( n ) d according to expressions (6a), or (6b) which cannot be applied in the presence of effective collisional relaxation, as discussed above and in ref. [4]. Values of (( AZ?)) derived from these erroneous (n) turn out to be much larger. The second cause of error is represented by the presence of concentration profiles which were also not considered in ref. [ 16 1. (2) The dependence of the IR multiple photon absorption cross section 0 on ( n,) is that of curve (b) of fig. 5 of paper I. This curve has been calculated in ref. [ 231 from the experimental results of ref. [ 241, which refer to complete Boltzmann equilibrium in the system, for conditions in which translational and rotational degrees of freedom remain “cold” and the system is equilibrated with respect to vibration only. This is in fact the situation illustrated by the model calculations reported in fig. 4. One should remind, from the discussion of paper I, that there is experimental evidence also favouring the selection of this particular a( (ra,) ) function. With these assumptions and a knowledge of the temporal laser pulse shape, model calculations allow us to determine (n), ( a,) and (nT> under various conditions of w, A4, and (( AE)) with results which are qualitatively acceptable when compared to the experimental ones. The quantitative comparison between model cal-
h-f Lenzi et al. /Infrared multiple photon absorption in SF=noblegas mixtures
culations and the experiments is somewhat more involved in that it requires a decomposition procedure which transforms the calculated values of ( n ), which refer to a radially uniform laser beam, into ( n ) d the number of photons per SF6 molecule expected for a beam with Gaussian radial intensity profile. The decomposition procedure adopted is that described in ref. [ 41. The result of the comparison between model calculations and experiments indicates the existence of radial concentration profiles accumulating excited SF6 molecules towards the beam axis according to former predictions [ $61. The suggestion is conform to the kinetic analysis of paper I which also suggested the existence of this type of concentration profile in neat SF6 at pressures between 0.2 and 1.O Torr. The temporal evolution of these concentration profiles is different for neat and for buffered SF6 (fig. 7 of paper I and fig. 13 of this work respectively). The differences can be traced back to the fact that IR multiple photon absorption in neat SF6 is initially a slow process limited by bottleneck effects which is progressively accelerated leading to a temporal increase of ( n,) . In the presence of a buffer, q( 1) = 1 at the beginning (no bottleneck) with a rapid initial absorption and with a temporal evolution of (n,) (fig. 3) which depends on ZA4 (( AE )) and is substantially different from that observed in neat SF6. When laser intensity drops at the end of the pulse, the concentration profiles relax more rapidly in neat SF, as one might expect. It is worth noting that the values of q(2) derived from entirely different kinetic treatments such as those of paper I and of paper II are close to 2. This is also the corresponding value given in ref. [ 6 ] for very different experimental conditions. With proper consideration of these concentration profiles, it is possible to assign values of (( AE)) / ( n,) to helium, neon, argon, krypton, and xenon (table 1). They have been determined by a least-squares fitting to an equation of type ln( ((u))
/(nv))=a-b%,
where MB is the atomic weight of the noble gas buffer. These results show helium to be 19 times more active than xenon in relaxing vibrational energy from excited SF+ They are definitely consistent with Ehrenfest adiabatic principle [ 18 ] but cannot easily be in-
483
terpreted on the basis of available treatments of V-T transfer mechanisms [ 2,3]. Measurements are in progress with diatomic and polyatomic buffers to investigate the contribution of V-R and V-V processes. Preliminarly data obtained with CO2 as a buffer yield values of (( AE )) / ( n,) which are about one order of magnitude larger than the value expected for a particle of mass 44 with translational degrees of freedom only. The most severe limitation of the present kinetic approach to the accuracy of <( AE )) / ( n,) is represented by the presence of radial concentration profiles, which become an additional unknown of the problem. This unknown can reasonably be estimated, as in the present case, but requires independent information on ((A.!?)) /(n,), or should be measured directly and this requires independent experiments. These experiments are now being planned and are concerned with the attempt of identifying these hypothetical concentration profiles and their dependence on laser parameters including spatial intensity distributions other than Gaussian.
References [ 1] M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and M.L. Terranova, Chem. Phys. 142 ( 1990) 463. [2] R.J. Gordon, CommentsAt. Mol. Phys. 21 (1988) 123. [3] W. Forst, Chem. Phys. 131 (1989) 209. [ 41 M. Lenzi, E. Molinari, V. Sessa and M.L. Terranova, Opt. Commun. 73 (1989) 67. [51X. de Hemptinne, Phys. Rept. 122 (1985) 1; IEEE J. Quantum Electron. QE-21 (1985) 755. [ 61 X. de Hemptinne, Spectrochim. Acta 43 A ( 1987) 155; J. Chem. Phys. 86 (1987) 1824. [ 71 V.N. Bagratashvili, VS. Letokov, A.A. Makrov and E.A. Ryabov, MultiplePhoton Infrared Laser Photophysics and Photochemistry (Hanvood,NewYork, 1985). [8] M. Dubs, D. Harradine, E. Schweitzer and J.I. Steinfeld, J. Chem. Phys. 77 (1982) 3824. [9] M. Koshi, Y.P. Vlahoyannis and R.J. Gordon, J. Chem. Phys. 86 (1987) 1311. [IO] K.M. Beck and R.J. Gordon, J. Chem. Phys. 87 (1987) 5681. [ 1 I ] D.R. Stull and H. Prophet eds., JANAF Thermochemical Tables, 2nd Ed., NSRDS-NBS 37 ( 197 1). [ 121 M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and ML. Terranova, Chem. Phys. 108 (1986) 167. [ 131 D.F. McMillen, R.E. Lewis, G.P. Smith and D.M. Golden, J. Phys. Chem. 86 ( 1982) 709.
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[ 141 B. Herzog, H. Hippler, L. Kong and J. Troe, Chem. Phys. Letters 120 (1985) 124. [ 151 P. Kolodner, H.S. Kwok, J.G. Black and E. Yablonovitch, Opt. Letters 4 ( 1979) 38. [ 161 M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and M.L. Terranova, Spectrochim. Acta 43 A ( 1987) 137. [ 171 H. Hippler, J. Troe and H.J. Wendeiken, J. Chem. Phys. 78 ( 1983) 6709. [ 181 J.D. Lambert, Vibrational and Rotational Relaxation in Gases ( Clarendon Press, Oxford, 1977 ) .
[ 191 P. Dagant, T.J. Wallington and W. Braun, J. Photochem. Photobiol. A 45 ( 1988) 15 I. [20] W. Braun, M.D. Sheer and R.J. Cvetanovic, J. Chem. Phys. 88 (1988) 3715. [ 211 M. Lenzi, E. Mohnari, G. Piciacchia, V. Sessa and M.L. Terranova, J. Chem. Phys. 89 (1988) 3398. [22] K.M. Beck and R.J. Gordon, J. Chem. Phys. 89 (1988) 3399. [23] U. Schmailzl, Chem. Phys. Letters 68 (1979) 443. [ 241 A.V. Nowak and J.L. Lyman, J. Quant. Spectry. Radiative Transfer 15 (1975) 945.
A KINETIC STUDY OF INFRARED MULTIPLE PHOTON ABSORPTION IN SF,-NOBLE GAS MIXTURES DURING A TEA COz LASER PULSE. EVALUATION OF V-T RELAXATION RATES M. LENZI ‘, E. MOLINARI a,c, G. PICIACCHIA b, V. SESSA c and M.L. TERRANOVA c ’ Istituto di Metodologie Avanzate Inorganiche de1 CNR, P.O. Box 10. Monterotondo Scala, 00016 Roma, Italy b Area della Ricerca CNR, Servizio Laser, P.O. Box IO, Monterotondo Scala, 00016 Roma, Italy ’ Dipartimento di Scienze e Tecnologie Chimiche, 2” Universitd di Roma, Tor Vergata, Via 0. Raimondo, 00173 Roma, Italy Received 25 September 1989
The time-resolved attenuation technique described in the preceding paper has been utilized to investigate the kinetics of IR multiple photon absorption by SF, in tbe presence of a large excess of He, Ne, Ar, Kr and Xe. The kinetic model proposed compares favourably with experimental observations provided it includes the possibility of the formation of concentration profiles within the Gaussian laser beam. An evaluation of the net bulk average energy transferred to the noble gas per collision and per SF, molecule, (( AE )) , has been possible and it is shown to decrease regularly from helium to xenon by a factor 19.
1. Introduction
In the preceding paper [ 11, paper I, a light attenuation technique has been described which allows the determination of the number of infrared photons absorbed per SF6 molecule, ( n) , as a function of time and fluence during a TEA CO2 laser pulse. In paper I neat SF, has been investigated at pressures between 0.2 and 1.O Torr. In the present paper this same technique will be applied to a study of the kinetics of IR multiple photon absorption by SF6 in the presence of a large excess of different noble gases: helium, neon, argon, krypton and xenon. In the presence of a buffer gas the total number of photons absorbed per SF6 molecule, ( n), is actually composed of two quantities: the number of photons stored as vibrational energy within the vibrational manifold on the SF, molecule, (n,), and the number of photons ( nT) transferred by V-T relaxation from the vibrationally excited SF, molecule to the atoms of the noble gas. The equality (n>=<&>+
(1)
is valid at any time during the absorption. The quantity which one would like to derive from the absorption kinetics is (( AE )) , the net bulk average energy 0301-0104/90/$03.50 (North-Holland)
Q Elsevier Science Publishers B.V.
transferred per collision and per SF6 molecule to the noble gas bath. This is the ensemble average over the microscopic energy change (A,!?) v [ 2,3 1. This paper is organized as follows: In section 2.1 the IR multiple photon absorption kinetics in the presence of buffers is developed for a hypothetical laser beam of constant radial intensity (rectangular profile with sharp edges) leading to the kinetic model of the IR absorption process in the presence of buffers which allows (n), (n,) (%) and the gas temperature, to be calculated as a function of time and fluence at different laser intensities and different values of (( AE )) . In section 2.2 the problem of the conversion of the experimentally determined number of photons absorbed per molecule ( n ) o ( exp ) into ( n ) and vice versa is discussed at some length because it is a central point for the reduction of the absorption data. The transformation of (n),(exp) into (n), or vice versa, in the presence of significant relaxation of the absorber has been discussed by the authors in ref. [4] to which the reader could refer for a more general understanding of this matter. In section 2.2 this problem is discussed with reference to conditions specific of the present experiments and a numerical procedure is given which allows the transformation
474
M. Lunzi et al. /Infrared
multiple photon absorption
of the values of (n) calculated in section 2.1 into values of ( n ) o. These ( n )o can then directly be compared to the experimental ones. Experimental data are presented in section 3. Section 4 contains a discussion of the data divided into two subsections: in section 4.1 experimental ( n >o( exp) are compared with the corresponding ( n)G calculated from the model of section 2. This comparison brings up features of the process which apparently require for their interpretation the assumption of the presence of radial concentration profiles of excited SF6 of the type discussed in paper I and in rcfs. [ $61 but with an essentially different time dependence. With this assumption it is then possible to evaluate the dependence of (( AK )) on ( n,) . This is done in section 4.2 for He, Ne, Ar, Kr and Xe and leads to a functional dependence of (( U )) / ( n,) on the atomic weight of the noble gas. Section 5 contains the concluding remarks.
2. IR multiple photon absorption kinetics 2.1. The model The time derivative
of eq. ( 1) is:
d(n)/dt=d(n,)ldt+d(n-r)ldt.
(2)
With 0 the excited state absorption cross section (cm’ molecule-’ ) and w the laser intensity (photons s-’ cme2) one has d(n)/dt=aw.
(3)
The vibration-translation written as d
energy transfer rate can be
@ >> ,
(4)
with Z the collision number (cm3 molecule-’ SK’ ) and M the number density (molecule cmm3) of the noble gas atoms. In eq. (4) (( AE)) is expressed in units of (n) (photons molecule-‘): (P(20) photonc944.2 cm-‘). d( n,) /dt, the rate of increase of the vibrational energy content of the SF, molecule can then be written as d(n,)/dt=aw-ZM((
AE))
.
(2’ 1
The rate of increase of the vibrational energy content of the IR absorbing species is therefore regulated by
in St;,noblegas
mixtures
the competition between laser pumping (0~) and VT relaxation (ZM(( AE; )) ). If both 0 and (( U )) are known functions of the vibrational energy content (n,) , w is known as a function of time because it is directly derived from the laser pulse temporal profile, then eq. (2’ ) can be integrated stepwise and yields ( n,) as a function of time and, obviously, of fluence F (J cmw2). In this way 0 will also be known as a function of time and one integrates eq. (3 ) stepwise to obtain ( n) as a function of time and fluence. Eq. ( 1) then yields the corresponding values of ( nT) . A knowledge of ( nT) then allows the determination of the adiabatic temperature of the irradiated volume as a function of time, once the gas composition is known. In order to integrate (2’ ) one should make the proper selection of the o( (n,) ) and of the (( AZ?)) ( ( n, ) ) functions. As already discussed in paper I according to the treatment of ref. [ 7 1, the SF6 molecules present at a given time t can be subdivided into two ensembles: a fraction q( 1) is involved in the process of multiple photon absorption and forms an ensemble of “hot”, i.e. vibrationally excited molecules, while a fraction 1 -q( 1) forms the ensemble of “cold” molecules. q( 1) is related to the relaxation time for rotational hole filling which in SF6 is of the order of 35 ns at 1 Torr [ 8 1. At the buffer pressures utilized in the experiments, which are between 250 and 750 Torr, q( 1) can safely be taken equal to 1 (see also ref. [ 9 ] ). As will become apparent from the presentation of the data, the translational temperature of the gas mixture remains low and in any case much lower than the vibrational temperature of the excited SF,+ One shall therefore select curve (b) of fig. 5 of paper I, which corresponds to a situation of vibrational equilibration but with “cold” translation and rotation, as the a( ( n,) ) function more appropriate to present experimental conditions. The dependence of (( A,??)) on the internal vibrational energy of SF6 has recently been investigated in ref. [ 10 ] for the system SF,-argon and found to be essentially linear: ((M)) =4x 10-4(n,) up to (n,) x20 ( ((M)) photons molecule- ’ ). We shall therefore integrate eqs. (2’ ) and (3) for an argon pressure of 750 Torr, using the mentioned 0 and (( AE )) functions, and a typical experimental w(t) profile. In addition the slope (( A,!? )) / ( n,) will be taken as a parameter variable in the range O-l OX 10e4 to simulate the ef-
M. Lmzi et al. /infrared multiple photon absorptionin SFrnoble gm mixtures
feet of an increasing V-T relaxation rate. The results obtained with this kinetic model are collected in figs. l-3. Curves of fig. 1 give (n) as a function of time, those of fig. 2 (n) as a function of fluence. The corresponding values of (n,) are given in fig. 3 versus time. These results are interesting because they bring out a number of points which should properly be considered: - The total number of photons absorbed per molecule ( n ) at the end of the pulse varies between 20 when (( A,!?)) ~0, i.e. in the absence of V-T relaxation, to approximately 150 when (( AE)) / ( n,) = 10 x 1Om4,showing that the absorption kinetics is particularly sensitive to the V-T relaxation rates. - Values of ( n) and of (n,) are very different and so is their temporal behaviour except when ((AE))=Oand (n)=(n,) ((n,)=O).Inparticular the highest values of ( n) correspond to the lowest ones of ( n,) and ( n,) curves go through definite maxima. High values of ZM (( AE )) yield low values of (n,) which, according to the a( (n,) ) function adopted, correspond to large values of crand, according to eq. ( 3 ), to high rates of absorption. At the end
01” 0
0.2
”
0.4
”
475
0.6
11
0.8
0
11
1.0
F (Jcd)
Fig. 2. Calculated (n) versus fluence for different values of ( (( AE)) /(PI,) ) x 10’ at a buffer pressure of 750 Torr ( 1000 mbar).
150 -
100 -
: Y
1.5
0.0
0
1.0
2.0
3.0
i
Fig. 3. Calculated values of (n,) versus time for different values of(((AE))/(n.,,))Xl04atabufferpressureof750Torr(lOOO mbar).
4.0
t (as)
Fig. 1. Calculated (n) versus time for different values of (((AE))/(n,))X104atabufferpressureof750Torr(1000 mbar).
of a laser pulse one can therefore observe a large number of absorbed photons but these will mainly be ( nT) photons transferred to the buffer while (n,) can be small and vibrational excitation low.
h4. Lenzi et al. /Infrared multiplephoton absorption in SI;,-noblegas
476
In fig. 4a one reports the vibrational temperatures TV versus time as derived from the (n,) values of fig. 3 using JANAF tables [ 111. Fig. 4b reports, against time, the translational and rotational temperature T of the various gases which can be calculated from the values of (n=)=(n) -(n,) and the expression: T- 298 = < nr > ~ZWLrr~r
)hvCy'
,
(5)
which is valid for small ratios of the partial pressure ( PSF6) of the IR absorber to the partial pressure of the noble gas buffer ( PbuKer).hv is the energy of the P (20) CO1 photons utilized in the experiments and C, the specific heat at constant volume of the gas bath which, for the small concentrations of SF6 utilized in the experiments, coincides with $R of the noble gas. Fig. 4b shows that the temperature increase of the gas does not exceed 40 K, and this represents an upper limit for present experimental conditions. Values of T calculated from eq. (5) represent the so called “adiabatic” temperatures, i.e. the temperatures reached by the unexpanded irradiated volume. As already pointed out in ref. [ 121 on the basis of calculations carried out for similar systems in refs. [ 13,141, the expansion of the irradiated volume consequent to
1500-
B ‘>
I-
1ooo-
mixtures
the pressure increase will not start before about 5- 10 us so that in our experiments the gas density can be considered constant and the use of the adiabatic temperature correct. Comparison of the temperatures of figs. 4a and 4b clearly shows that the translational degrees of freedom and the rotational ones, remain “cold” (below 340 K) while vibrational temperatures of fig. 4a range from 800 to 2 100 K. In order to compare the results predicted by the kinetic model with the observed kinetics one should remind that the results of the model would correspond to those obtained with a laser beam of uniform radial profile (rectangular with sharp edges). The laser beam utilized in the experiments is essentially smooth and monomodal and the radial profile is Gaussian. The experimentally determined energy absorbed per molecule of SF6 is therefore not ( n) but ( n)o, as in paper I. ( n ) o is a Gaussian average over the spatial intensity profile of the Gaussian beam Z,=Z, exp( -2r’/oi), where Z, is the local intensity at the distance r from the beam axis and Z, the maximum of the intensity at the axis [ 151. The conversion of a measured ( n)c into (n) is therefore preliminary to any comparison between the observations and the predictions of a model. The situation is particularly complex in the presence of effective collisional relaxation, as discussed in ref. [4], and will be examined in some detail. 2.2. Decomposition of a Gaussian averaged nonlinearjkction in the presence of collisional relaxation
50040-a
10
B L
In ref. [ 41 it was pointed out that the “exact” decomposition procedure proposed in ref. [ 15 ] which transforms ( ra)q into (n) according to the following expressions
t6a) Fig. 4. (a) Calculated increase of the vibrational temperature (TV-298) versus time for different values of ( (( AE)) / ( nV) ) x lo4 at a buffer pressure of 750 Torr ( 1000 mbar). (b) Corresponding increase of the temperature of the gas bath (T-298)foraratioP&P-=2.7x10-‘.
(n>=
Fd(n)G dF
(6b)
cannot be applied in the presence of effective collisional relaxation and should be replaced by a more
M. Lenzi et al. /Ini,red
complex numerical procedure which will partly be drafted below with specific reference to the systems under study. The starting point is a proper calculation of the radial profile of the laser beam intensity Z, after absorption of a fraction of the initial energy l-9& where Yo=J, /Jo (see paper I) is the transmission as measured with the initially Gaussian beam. For a linear dependence of ( n ) on fluence and a transmission
60-
(7)
where the radially uniform transmission % has now been replaced by the actual transmission Tat a distance Yfrom the axis. The calculation of the intensity profile becomes therefore the problem of the determination of z The general relationship between (n) and the transmission Scan be written as 1-Y=kvNL(n)/F,
477
multiple photon absorption in SFGnoble gas mixtures
(8)
where v is the frequency of the IR laser line, N is the number density (molecule cm-3) of SF6 and L is the length of the absorption cell (cm ) . .Tcan be derived from ( 8 ) if ( n ) ,/F, is known. Once Z,is known from eqs. (7), (8), the transformation of (n) into ( n)G or vice versa is straightforward as discussed in ref. 141.
In the presence of collisional relaxation of the IR absorber, ( n ) /F curves should be calculated according to the scheme which has been utilized in the derivation of the curves of figs. 2 and 3 (eqs. (2’ ) and (3) ) with the difference that (( AE)) /(n,,) should now be a constant (4 x 10m4) and the parameter becomes the laser intensity. One should in fact utilize at each distance r from the axis the proper intensity according to the Gaussian distribution. Fig. 5 reports typical (n) /F curves calculated in this way together with the curve corresponding to ((M)) =O. This figure shows that ( n) is not a unique function of F but rather (n) versus F functions become dependent on laser intensity. As a consequence each position of the initially Gaussian intensity profile corresponds to a different ( n) JF, function, which is the one that has to be used in eq. (8) at the distance r. No radial mixing of vibrationally excited species oc-
01 0
I
’ 0.2
’
’ 0.4
I
0 0.6
’ 0.E
)
F (JcrC2)
Fig. 5. Calculated (n) vems fluence for ( (( AE)) / ( n,) ) x 104~4.0 and different total laser fluence=s,Fw Curve (1 ), F,,=O.97 J cm-*, Pw6 ~0.2 Torr. The other curves correspond to total fluenccs of 0.5,0.25 and 0.125 of the total fluence of curve ( 1). Also included is the curve for (
M. Lmzi et al. /Infrared multiple
478
photon absorption in SFcnoble
gas mixtures
versus t curve calculated along the beam axis can be compared with the corresponding ( n ) o versus t curve calculated after determination of the radial intensity profiles at different times during the pulse. Calculations also show that the ratio (n)o/ (n) is a function of laser intensity, buffer pressure and 2 (( AE )) [41.
r 0 I0
3. Experimental results 0.2
..I 04
I
3 1.2
r (8.U.)
Fig. 6. Calculated radial relative intensity profiles I,/&, versus distance r (arbitrary units) from the beam axis. ( 1) Gaussian profile corresponding to linear absorption. (2) Calculated from eqs.(7),(8)andvaluesof(n),/F,fromfig.5at3~~.(3)Calculated as curve (2) with inclusion of the concentration profile of fig. 12a. (4) As curve (3) with concentration profile of fig 12b. Dotted curve gives the incident beam profile.
15017
Fig. 7. (n) versus time plots for argon: F,=O.97 J cm-‘. Curve (n) calculatedwith (((AE))/(n,))xl04=4.0andP,,=750 Torr, PsFs = 0.2 Torr for a radially uniform laser beam. Curve (n). calculated for an initially Gaussian laser beam in the absenceofradialconcenttion profiles. Curve (n)& calculatedwith the inclusion of the concentration profile of fig. 12a. Curve (n)& calculated with the inclusion of concentration profile of rig. 12b. Curve (n>o(exp) measured (n>oversus time. Dashed lineis (n),(exp) dividedby2.0.
The experimental technique has been described in paper I. Fig. 8 shows the time dependence of the laser beam effective radius r, ( t) at the exit of the cell filled with neon at three different pressures. This behaviour is very similar to that reported for argon in ref. [ 161 and can be taken as typical of r, ( t) functions in the presence of buffers. These functions differ markedly from those reported in paper I, fig. 2, for neat SF6. Typical experimental plots of ( n ) o versus time and versus fluence have been reported in figs. 9 and 10 respectively, for helium, argon and krypton at pressures of 750 Torr and total pulse fluence close to 1 J cmP2. Fig. 11 reports experimental (n), values determined for neon and xenon again at about 750 Torr but at lower pulse fluences around 0.2 J cme2. The qualitative trends of all these plots compare favourably with those derived from the kinetic scheme proposed and illustrated in figs. 1, 2 and 5. A quantitative comparison is more complex and will form the subject of the discussion to follow.
Fig. 8. Effective radius of the laser beam at the exit of the absorption cell as a function of time. COz laser: 1 W, 10 Hz, Pw6 ~0.2 Tot-r. Neon pressures: (a) 225 Tot-r, (b ) 383 Tot-r, (c) 600 Torr. The initial value r, is 0.16 cm for all runs.
M. Lenzi et al. /Infrared multiple photon absorption in SF,noblegas
479
mixtures
Q
to
100
N.?.
c
r
Xe
0
1
El/L NB
0
I_1.0
..L_... 2.0
tw
i_ 3.0
0
Xe
0.1
0;
6
._
r (J cm-‘)
Fig. 1 I. Experimental plots of (n). versus time (a) and versus fluence (b)forneonandxenon (PNC=765,Ps,=0.1,F,=0.28; Px.=728,PsFb =0.2,F,=O.25). P,isthepartialpressure (Torr), F,,, is the total fluence of the pulse (J cmb2).
Fig. 9. Experimental ( n)G versus time plots for He (P,,=750, P,,,=O.OS,F,,=0.99),Ar(P,,=750,P,,,=0.2,F,=0.97),Kr (P,=670, PsF6~0.2, F,,=O.O8). P, is the partial pressure (Torr), Fm is the total fluence of the pulse (J cm-‘).
?
(Jcme2)
Fii. 10. Experimental plot of (n)o and Kr. Conditions as in fig. 9.
versus fluence for He, Ar,
4. Discussion 4.1. Radial concentration profiles A comparison between the observations and the predictions of the model requires a comparison be-
tween experimental ( n ) o values and the ( n)c derived from the model as described in section 3. This is done for the case of argon in fig. 7. The value of (( AE )) / ( n,) selected for argon is that determined in ref. [ 10 ],4 x 10m4, and one notices that the experimental (n), values fall in between the calculated ( n) and ( n ) o values. This situation remains qualitatively similar at different argon pressures and different laser intensities. Therefore, if one takes 4 X 10m4 as the reference value of (( AE )) / ( n,) for argon, calculated (n)o values are about a factor of 1.3 higher than the observed ones at 3 us, this ratio being somewhat dependent on the time. As indicated by fig. 4, the temperatures of the buffers are definitely lower than the vibrational temperatures which correspond to the values of (n,) of fig. 3 so that the adopted selection of the u versus (n,) function appears to be the correct one also in view of the results reported in paper I. There remains however a very important point raised by paper I and which has not been considered, namely the possible formation of a radial concentration profile of SF6 inside the laser beam, accompanied by the increase of the number density of excited SF6 at the beam axis. This dielectric suction toward the region of higher intensity of the electromagnetic field has been discussed in refs. [ 5,6] for equilibrium conditions involving the simultaneous flow of energy, mass and polarisation amplitude (laser induced osmosis ). These radial concentration profiles of SF6 should mimic the radial intensity profile of the laser beam leading, for the present case, to Gaussian or nearly Gaussian concentration profiles with an axial concentration N> No and with N/N,=q( 2) as in paper I.
M. knzi
480
1.0
0
10 r
et al. /Infrared multiple photon absorption in SF6-noblegas
1.0
r,
Fig. 12. Assumed concentration profiles: (a) eq. (9a), (b) eq. (9b).q(2)=2inbothcases.
The influence of these concentration profiles on the relationship between the calculated (n) and (n)o and the experimental ( n ) G is very serious. Let us illustrate this point using two different profiles:
N,/N,=q(2)
exp( -2r2/oi)
(gal
N,/N,,=l+[q(2)-l]exp(-2r*/wg),
(9b)
which are drawn in fig. 12 and for both of which NJ 2) at the axis. The changes of the intensity profile Z, caused by these concentration profiles can again be calculated from eqs. (7) and (8 ) where N becomes however a function of r, N,, given by either (9a) or (9b). The results of the introduction of these concentration profiles are illustrated by curves ( 3 ) and (4) of fig. 6 and by curves (n)& and (n)& of fig. 7. The experimental ( n ) d curve should also be changed because in the presence of these profiles the number of SF6 molecules contained in the irradiated volume w&L is multiplied by q( 2 ) and experimental ( n ) o values should therefore be divided by q(2). In fact, from the definition of the effective radius, r,, which in the model is the same for both the intensity and the concentration profiles, the number of molecules contained in the effective irradiated volume m&L should be calculated as follows: the peak concentration of the distribution is q( 2 )No which, when multiplied by m-h,, yields Tc&q( 2)N, molecules cm -‘. The total number of molecules contained in the optical path L will therefore be rcr&Lq( 2)No. This number should be compared with w.&-LN, which
N,=q(
mixtures
gives the total number of molecules in the irradiated volume utilized for the evaluation of the experimental ( n ) G in the absence of a concentration profile (see papers I, section 3). In the presence of a profile experimental ( n) G values should therefore be divided by q(2). This division by q(2) brings (n)& and experimental (n)&q(2) to a much better agreement when q( 2) =2 as indicated by the dotted line of fig. 7. This is not true for ( n )&. It is appropriate at this point to summarize the discussion of fig. 7. Curve (n) refers to a laser beam of radially constant intensity and has been calculated as discussed in section 2.1 with (( AE)) / (n,) =4x lop4 (argon). Curve (n)d is derived from curve (n) by the numerical procedure outlined in section 2.2 and provides therefore values calculated according to the model for an initially Gaussian intensity profile which is then deformed by nonlinear absorption. These values of ( n)G are those to be compared with the experimental ones reported as (n),(exp). Curves (n)& and (n)& have been calculated as curve ( n)G but with the inclusion of concentration profile (a) and (b) of fig. 12 respectively. Comparison of these curves with ( n)G( exp) requires a division by q( 2 ) of the experimental values. The dashed line is (n),(exp) divided by q(2) =2. All the curves of fig. 7 refer to the same time profile of the laser intensity during the pulse. At each time during the pulse it is therefore possible to test concentration profiles of type (9a) by varying the pre-exponential factor q( 2 ), until ( n) & and the experimental (n)Jq( 2) coincide. This will give q( 2) as a function of time during the laser pulse. This has been done for argon and the result is shown in fig. 13.
Fig. 13. q( 2) = N/N, as a function of time calculated for argon as specified in the text. Point and arrow indicate the value of q(2) at 3 ps utilized in the calculations for the other noble gases.
hf. Lenzi et al. /Infrared multiplephoton absorptionin SF+wble gas mixtures
The result of fig. 13 therefore means that calculated and observed ( n)G can be brought into coincidence if one assumes the existence of concentration profiles of type (9a) whose maxima change with time as indicated in fig. 13. Therefore, if (( AE)) / ( n,) is known independently (4 x 10m4 for Ar ), concentration profiles can be estimated and it should be remarked that it is precisely the suggested existence of these profiles which allows one to bring experiments and model calculations into coincidence. Profiles of type (9b), as well as the absence of a concentration profile, do not allow this matching. The difference between profiles (9a) and (9b) is that in profiles (9b) the wings of the Gaussian distribution approach the bulk value N,, while in profiles (9a) the concentration at the wings can be much lower than NO.Under conditions of the experiments, profiles of type (9a) appear to be the most likely to occur because the characteristic time for dielectric suction of excited SF6 toward the laser axis should not exceed lo-’ s (see fig. 13) while the mass diffusion time necessary to replete, in the presence of a large excess of buffer, the beam wings depleted by this dielectric suction should be of the order of 10m3s. The concentration profile which forms as the laser is switched on successively relaxes because laser intensity falls but should be considered as essentially frozen with respect to relaxation by mass diffusion. The situation is therefore substantially different from that described in ref. [ 61 in connection with the experimental demonstration of the existence of laser induced osmosis and where the time scale of the experiments was of the order of 10d3 s. Creation of a profile of type (9a) has obvious consequences on the (n)q values which are shifted from the (n)o to the ( n ) & curve of fig. 7 because the highly absorbing low intensity wings of the Gaussian distribution have substantially been depleted of the IR absorber. In conclusion the model kinetic scheme of IR multiple photon absorption which one proposes should include the possibility of the creation of concentration profiles within the laser region, a suggestion also coming from paper I and, previously, from refs. [ 5,6 1. 4.2. Evaluation of (( AE )) /( n,) in the presence of concentration profiles The arguments leading to fig. 13 are based on the
481
independent knowledge of (( AE )) / ( n, ) which was available for argon [ lo], In the absence of this information, i.e. for the other noble gases utilized in the experiments, the proposed kinetic scheme obviously lacks the one more equation necessary to solve the problem in the presence of undefined concentration profiles. The procedure which one will tentatively adopt is to assume that at a convenient time, say 3 us, the concentration profile is the same as that determined for argon, N,/N,=2
exp( -2r2/&)
.
This then becomes the additional equation necessary to solve the problem. The procedure now consists in a trial and error selection of (( AE )) / ( &) until the experimental value of (n>J2 at 3 us coincides with the corresponding calculated value ( n ) b. Values of (( AL?)) / ( n,) determined by this procedure for He, Ne, Ar, Kr and Xe have been collected in fig. 14 where the logarithm of (( AE )) / ( n,) has been plotted versus the atomic weight of the noble gas. The least-squares straight line has also been drawn in this figure and the values of ((LIE)) /(n,> derived from this straight line have been collected in table 1 for the five noble gases. This table also gives
Fig. 14. Semilogarithmic plot of ( (( AE)) /(&y> ) x 10’ as a function of the atomic mass MB of the buffers He, Ne, Ar, Kr and Xe. The straight line is the least-squares fit of the experimental points.
482
34. Lunzi et al. /Infrared
multiple photon absorption in SF,-noble
gas mixtures
Table I Valuesof(((~))/(n,))x104andofZx10’0(cm3molecule-‘s-‘)forthenoblegases
(((~))/
He
Ne
Ar
Kr
Xe
9.5 5.3
6.5 3.0
4.1 3.2
1.5 2.8
0.5 2.8
the Lennard-Jones collision numbers Z utilized in the calculations and which have been evaluated according to ref. [ 17 1. For each noble gas the scattering of the points is remarkable. These points refer, for each buffer, to variable conditions of laser intensity, SF6 and buffer pressure, initial beam radius. If one takes the value of (( AE )) / ( n,) for krypton from table 1 ( 1.5 x 10w4) and reverses the calculation procedure one obtains values of q( 2 ) which fluctuate randomly with an average value of 2.05 and a standard deviation of 0.35. The presence of these concentration profiles appears therefore to be a severely perturbing factor in the kinetics of IR multiple photon absorption and could jeopardize the efforts aimed at an accurate determination of (( AE )) /(a,) . The apparently random nature of the values of q(2) and the linear relationships of In (( AE)) /(n,) versus the atomic weight of the noble gas of fig. 14, derived by inclusion of the experimental points for all buffers in the regression procedure, do give however reasonable confidence to the (( AE )) / ( n,) values reported in table 1. These values then confirm the expected [ 18 ] negative influence of the mass of the collider on the efficiency of V-T energy transfer from vibrationally excited SF6 to particles with translational degrees of freedom only.
5. Conclusions The kinetic analysis of IR multiple photon absorption in the presence of an excess of a noble gas buffer has been based on the following assumptions: ( 1) (( LIE )) is a linear function of ( n,) . This has been shown experimentally in ref. [ lo] for the system SF,-argon at least above ( n,) = 3-4 and in ref. [ 191 for systems SF6+C6H6 or fluorobenzenes. Below (n,) = 3-4 the situation can be more complex [ 10,201 but this assumption remains acceptable for
the sake of present calculations: in particular the value of (( AE )) / ( n,) 4 X 10P4 for SFs-Ar which has been utilized as the basis of our treatment. This value of (( AE)) / ( n,) is different from that reported in our previous work on the SF6-Ar system [ 16 1. This discrepancy was pointed out by Beck and Gordon [ lo] and formed the subject of our comment on their work [ 2 1 ] and of the author’s reply [ 22 1. Reasons for this discrepancy appear now somewhat clearer. There are in fact two causes of error in our data of ref. [ 161, the most important one being the decomposition procedure. In ref. [ 161 ( n ) was calculated from ( n ) d according to expressions (6a), or (6b) which cannot be applied in the presence of effective collisional relaxation, as discussed above and in ref. [4]. Values of (( AZ?)) derived from these erroneous (n) turn out to be much larger. The second cause of error is represented by the presence of concentration profiles which were also not considered in ref. [ 16 1. (2) The dependence of the IR multiple photon absorption cross section 0 on ( n,) is that of curve (b) of fig. 5 of paper I. This curve has been calculated in ref. [ 231 from the experimental results of ref. [ 241, which refer to complete Boltzmann equilibrium in the system, for conditions in which translational and rotational degrees of freedom remain “cold” and the system is equilibrated with respect to vibration only. This is in fact the situation illustrated by the model calculations reported in fig. 4. One should remind, from the discussion of paper I, that there is experimental evidence also favouring the selection of this particular a( (ra,) ) function. With these assumptions and a knowledge of the temporal laser pulse shape, model calculations allow us to determine (n), ( a,) and (nT> under various conditions of w, A4, and (( AE)) with results which are qualitatively acceptable when compared to the experimental ones. The quantitative comparison between model cal-
h-f Lenzi et al. /Infrared multiple photon absorption in SF=noblegas mixtures
culations and the experiments is somewhat more involved in that it requires a decomposition procedure which transforms the calculated values of ( n ), which refer to a radially uniform laser beam, into ( n ) d the number of photons per SF6 molecule expected for a beam with Gaussian radial intensity profile. The decomposition procedure adopted is that described in ref. [ 41. The result of the comparison between model calculations and experiments indicates the existence of radial concentration profiles accumulating excited SF6 molecules towards the beam axis according to former predictions [ $61. The suggestion is conform to the kinetic analysis of paper I which also suggested the existence of this type of concentration profile in neat SF6 at pressures between 0.2 and 1.O Torr. The temporal evolution of these concentration profiles is different for neat and for buffered SF6 (fig. 7 of paper I and fig. 13 of this work respectively). The differences can be traced back to the fact that IR multiple photon absorption in neat SF6 is initially a slow process limited by bottleneck effects which is progressively accelerated leading to a temporal increase of ( n,) . In the presence of a buffer, q( 1) = 1 at the beginning (no bottleneck) with a rapid initial absorption and with a temporal evolution of (n,) (fig. 3) which depends on ZA4 (( AE )) and is substantially different from that observed in neat SF6. When laser intensity drops at the end of the pulse, the concentration profiles relax more rapidly in neat SF, as one might expect. It is worth noting that the values of q(2) derived from entirely different kinetic treatments such as those of paper I and of paper II are close to 2. This is also the corresponding value given in ref. [ 6 ] for very different experimental conditions. With proper consideration of these concentration profiles, it is possible to assign values of (( AE)) / ( n,) to helium, neon, argon, krypton, and xenon (table 1). They have been determined by a least-squares fitting to an equation of type ln( ((u))
/(nv))=a-b%,
where MB is the atomic weight of the noble gas buffer. These results show helium to be 19 times more active than xenon in relaxing vibrational energy from excited SF+ They are definitely consistent with Ehrenfest adiabatic principle [ 18 ] but cannot easily be in-
483
terpreted on the basis of available treatments of V-T transfer mechanisms [ 2,3]. Measurements are in progress with diatomic and polyatomic buffers to investigate the contribution of V-R and V-V processes. Preliminarly data obtained with CO2 as a buffer yield values of (( AE )) / ( n,) which are about one order of magnitude larger than the value expected for a particle of mass 44 with translational degrees of freedom only. The most severe limitation of the present kinetic approach to the accuracy of <( AE )) / ( n,) is represented by the presence of radial concentration profiles, which become an additional unknown of the problem. This unknown can reasonably be estimated, as in the present case, but requires independent information on ((A.!?)) /(n,), or should be measured directly and this requires independent experiments. These experiments are now being planned and are concerned with the attempt of identifying these hypothetical concentration profiles and their dependence on laser parameters including spatial intensity distributions other than Gaussian.
References [ 1] M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and M.L. Terranova, Chem. Phys. 142 ( 1990) 463. [2] R.J. Gordon, CommentsAt. Mol. Phys. 21 (1988) 123. [3] W. Forst, Chem. Phys. 131 (1989) 209. [ 41 M. Lenzi, E. Molinari, V. Sessa and M.L. Terranova, Opt. Commun. 73 (1989) 67. [51X. de Hemptinne, Phys. Rept. 122 (1985) 1; IEEE J. Quantum Electron. QE-21 (1985) 755. [ 61 X. de Hemptinne, Spectrochim. Acta 43 A ( 1987) 155; J. Chem. Phys. 86 (1987) 1824. [ 71 V.N. Bagratashvili, VS. Letokov, A.A. Makrov and E.A. Ryabov, MultiplePhoton Infrared Laser Photophysics and Photochemistry (Hanvood,NewYork, 1985). [8] M. Dubs, D. Harradine, E. Schweitzer and J.I. Steinfeld, J. Chem. Phys. 77 (1982) 3824. [9] M. Koshi, Y.P. Vlahoyannis and R.J. Gordon, J. Chem. Phys. 86 (1987) 1311. [IO] K.M. Beck and R.J. Gordon, J. Chem. Phys. 87 (1987) 5681. [ 1 I ] D.R. Stull and H. Prophet eds., JANAF Thermochemical Tables, 2nd Ed., NSRDS-NBS 37 ( 197 1). [ 121 M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and ML. Terranova, Chem. Phys. 108 (1986) 167. [ 131 D.F. McMillen, R.E. Lewis, G.P. Smith and D.M. Golden, J. Phys. Chem. 86 ( 1982) 709.
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IM.Lenzi et al. /Infrared multiplephoton absorption in SI;,-noblegas mixtures
[ 141 B. Herzog, H. Hippler, L. Kong and J. Troe, Chem. Phys. Letters 120 (1985) 124. [ 151 P. Kolodner, H.S. Kwok, J.G. Black and E. Yablonovitch, Opt. Letters 4 ( 1979) 38. [ 161 M. Lenzi, E. Molinari, G. Piciacchia, V. Sessa and M.L. Terranova, Spectrochim. Acta 43 A ( 1987) 137. [ 171 H. Hippler, J. Troe and H.J. Wendeiken, J. Chem. Phys. 78 ( 1983) 6709. [ 181 J.D. Lambert, Vibrational and Rotational Relaxation in Gases ( Clarendon Press, Oxford, 1977 ) .
[ 191 P. Dagant, T.J. Wallington and W. Braun, J. Photochem. Photobiol. A 45 ( 1988) 15 I. [20] W. Braun, M.D. Sheer and R.J. Cvetanovic, J. Chem. Phys. 88 (1988) 3715. [ 211 M. Lenzi, E. Mohnari, G. Piciacchia, V. Sessa and M.L. Terranova, J. Chem. Phys. 89 (1988) 3398. [22] K.M. Beck and R.J. Gordon, J. Chem. Phys. 89 (1988) 3399. [23] U. Schmailzl, Chem. Phys. Letters 68 (1979) 443. [ 241 A.V. Nowak and J.L. Lyman, J. Quant. Spectry. Radiative Transfer 15 (1975) 945.