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Vibrational relaxation of D2 in the range 400-50°K
Volume 27, number
2
CHEMICAL
PHYSICS
VIBRATIONALRELAXATIONOFDt
LETTERS
15 July 1974
INTHERANGE400-50°K
J. LUKASIK and J. DUCUING Laboratoire d’optique Quantique, C.N.R.S. - Ecole Polytechnique, Universitt? de Paris XI, 91405 Orsay, France
Received
18 April 1974
The vibrational relaxation of n-D2 by itself and by n-H2 was studied in the interval 400-50” K. The results are compared to the recent n-H2 studies in the similar temperature range and discussed in terms of various molecular parameters.
The vibrational selfdeexcitation of H, has recently been studied in the low temperature [l] range by the Raman excitation technique [2] . We present here a similar study on the vibrational deexcitation of D2 in D2 -D2 and D2 -II2 collisions which extends previous room temperature work [3] to the interval 400-50°K. The study of the vibrational deexcitation of hydrogen isotopes is of fundamental interest for several reasons. They present large isotopic variations which facilitate the study of the influence of such parameters as the reduced mass or the vibrational quantum. Because of their relative simplicity a priori calculations of the intermolecular potential as a function of the rotational and vibrational coordinates are feasible, as demonstrated in the case of Hz -He [4,5] . This combined with the development of numerical, three-dimensional calculations of the scattering cross sections should allow an interesting confrontation between theory and experiment, particularly in the low temperature domain where the intermediate range features of the potential (such as the well) substantially affect the scattering cross section. The experiment makes use of the tunable Raman excitation scheme in which the vibrational level is populated by a two-photon Raman transition induced by two laser beams one of which is continuously tunable. This scheme has been described in ref. [6] and its application to D2 in ref. [3]. The temperature variation was provided by an arrangement very similar to that used for H, [l] . The gas cell was placed in a vacuum chamber in order to provide good thermal
isolation. It could be heated through electrical resistances up to 500°K or cooled down with circulating around it liquid nitrogen or helium close to the liquid helium temperature. Temperature throughout the cell was determined with platinum resistors (100 52 at 0°C) and was stable and uniform to better than f 1°K above 100°K and + 2°K around 50°K. Relaxation measurements at a particular temperature were made only after the gas in the cell had remained at this temperature for approximately f -1 hour and they were repeated several times within a 3 hours interval. It was found that within this interval the measured rates did not depend on the time at which the experiment was performed. The pressure range of the experiments reported here was from 20 to 60 atm and the real gas corrections given by the compressibility factor [7] were made when necessary. The relaxation rates were measured at 86OK for several densities ranging from 100 to 155 amagat and their values were plotted as a function of the gas density. As in the case of hydrogen [ 1] a linear dependence was found indicating that within experimental accuracy ternary collisions did not contribute, in this density range, to the measured rates. The gases were supplied by Air Liquide. Their impurities are listed in ref. [3]. Two ready mixtures of D2-H2 prepared by Air Liquide (CD, 7 0.746 and c = 0.88 where CD2 is the mole fraction of D2 in D2 . the mrxture) were stu~ed in the experiment. The mole fraction of H, was kept low in order to avoid the 203
Volume
27, number
The experimental uncertainties
2
CHEMICAL
Table 1 results for D2 -Da
AT/T (%)
p7 (atm set)
APr/Pr
400.0 350.0 297.5 296.0 242.0 202.0 152.5 112.5 86.5 86.0 85.0 62.0 61.5 58.0 55.0 54.5
0.1 0.1 0.2 0.2 0.2 0.5 0.7 1.5 1.7 1.7 1.2 2.4 2.8 1.7 2.9 1.8
6.10 x 1.19 x 2.85 x 2.87 x 7.32 x 1.53 x 3.24 x 6.09 x 9.35 x 8.80 x 9.15 x 1.25 x 1.38 x 1.21 x 1.52x 1.39 x
6.9 7.6 7.7 6.6 8.5 4.6 8.0 11.6 13.8 12.8 10.5 22.4 26.1 11.6 8.0 23.8
T (“K)
results
AT/T (%)
Table 2 for Da -Ha
LETTERS
10-4 10-s 1O-3 1O-3 1O-3 10-a 1O-2 1O-2 10-a 10-2 10-a 10-l 10-l 10-i 10-l 10-t
(%)
and the corresponding 19
pr (atm set)
Wr/Pr
.
(%) 0.0‘
348.5 296.0 231 .O 178.5 141.5 110.0 85.0 53.3
0.1 0.2 0.3 0.6 0.7 1.3 2.4 3.8
1.27 1.85 4.53 8.99 1.68 2.58 3.55 3.40
x x x x x x x x
1O-4 lo4 10-4 lo4 10-s 1O-3 10-s 10-s
17.3 16.2 14.9 13.2 20.5 20.0 19.6 35.0
possible influence of the V-V transfer between D2 and H2 (AL?= -1168 cm-l). The directly measured experimental value was pi, the one-atmosphere relaxation time. The expertiental data and corresponding accuracies are listed in tables 1 and 2. r is related to the deexcitation rate constant klo(T) through the relation 7-l = NklO(r), where N is the number of molecules per cm3. The variation of klo(T) versus T- l/3 is shown in fig. 1 for the deexcitation of n-D2 respectively by itself and by n-H,. The n-H2 results reported in ref. [ 1] are displayed in the same figure for the sake 204
1.5 July 1974
and the corresponding
T(?K)
The experimental uncertainties
PHYSICS
0.10
ma
0.14
y---y 0116
0.u
$5 0.m
0.1-2
0.24
??
0.16
Fig. 1. Log kio(Z) as a function of T-1/a. -e-eD~-J_I~, Da-Ha, present work; -A-AH2-H2, ref. [l]; -m-m-Da-Da, ref. [13]; - - Da-Da, --D2-H2, theory, ref. [12].
of comparison. At high temperature the relaxation rates have a Landau-Teller variation log k =A -BT-1/3. The departure from this behaviour in the low temperature range has been discussed in ref. [ 1 ] . In the following we concentrate on the influence of the isotopic variations on the low temperature rates. Above 300°K the relaxation rates kpteH2 and kFtmD2 are of similar magnitude. When T is decreased, however kDZeD2 decreases faster and falls to much lower vakk:than kzpH 2. This is in sharp contrast with the behaviour of the mixture relaxation rate kzpH2 which is fairly similar to that of klHozmH2. The deexcitation rate klo can be expressed in terms of the total collision cross section ulOQ as:
(1)
Volume 27, number 2
where for a central interaction potential the distorted wave approximation:
CHEMICAL PHYSICS LETTERS
V(r), and in
where 4i. q. are the wave numbers of the incident and outgoing particles, m the reduced mass and Ethe angular momentum quantum number for the relative motion, Vu1 the matrix element of the interaction potential between the initial and final oscillator states and :
VtolJ$lIki) =
je. hR;,(r)cl(r)R&),
(3)
0
the R, being the radial wavefunctions with asymptotic form sin(qr t 6). The matrix element (1,ol V,, lZ,i) is, in general, a rapidly decreasing function of q,-qi = 2mw10/ h(qo+qi). This determines the behaviour of u(E) which decreases with E. In the low temperature range d(qo-qi)/dE z - dqi/dE = - (m/t22E)1/2: the larger the reduced mass the stronger should be the variation of a with E, and as a consequence the variation of k with T. This could account for the more pronounced decrease of the D2-D2 rate and perhaps also for the cross over of the H2-H, and D2-H2 rates around 100°K. As 4i becomes negligible compared to q,, 40-4i tends to the limiting value K = (2molo/A)1/2. Analytical calculations by the distorted wave method of s-wave scattering (I = 0) with either a repulsive exponential [8] or an attractive Morse [9, lo] potential show that in the low energy range the scattering cross section u(O) is proportional to exp(-nrta) where (Yis a characteristic length for the potential. Although s-wave scattering contributes only weakly even in the temperature range considered here, this result suggests that the cross sections for small angular momentum and hence the total cross section will be sensitive functions of the parameter ~a. This is supported by the preliminary results of our numerical distorted wave calculations. The values of KOI are shown in table 3. They are very close for H2-H2 and D2-H2 for which the low temperature rates are of similar magnitude whereas the larger value of KCY for D2-D2 appears to be reflected in a much lower rate. Calvert [ 121 has performed numerical distorted wave calculations of the relaxation rates for the hy-
15 July 1974
Table 3 Various parameters (see text) for the three collision pairs considered. The value o = u-r = 5.77 X 10m9 cm is that used by Bauer [ll] m(10-24g)
%0(eV)
KOL
Hz-Hz
1.66
0.516
9.10
D2(exc)-H2
2.21
0.371
8.91
Da-D2
3.32
0.371
10.92
drogen isotopes between 300 and 2000°K. His results are also shown in fig. 1. They connect very well our low temperature D2 data to the high temperature (1 lOO-3000”K), shock tube data of Kiefer and Lutz [13]. However, as in the case of pure H2 the curvature of the theoretical data around 300°K does not fit the experimental results. The isotopic relative variations of k,, are also correctly predicted in the [ 121. This was already noted in range 300-500°K ref. [3] for the room temperature data. The authors acknowledge fruitful discussions with Dr. C. Joffrin, M.M. Audibert and R. ViIaseca-Alavedra as well as the expert technical assistance of M. Chateau. They appreciate Dr. S. Buhler’s help in designing and building the low temperature apparatus.
References [l] M.M. Audibert, C. Joffrin and J. Ducuing, Chem. Phys. Letters 25 (1974) 158. [2] J. Ducuing, C. Joffrin and J.P. Coftinet, Opt. Commun. 2 (1970) 245. [3] J. Lukasik and J. Ducuing, J. Chem. Phys. 60 (1974) 331. [4] M. Kraus and F.H. Mies, J. Chem. Phys. 42 (1965) 2703. [S] M.D. Cordon and D. Secrest, J. Chem. Phys. 52 (1970) 120. [6] R. Frey, J. Lukasik and J. Ducuing, Chem. Phys. Letters 14 (1972) 514. [7] D.E. Gray, ed., Handbook of American Institute of Physics, 3rd Ed. (MacGraw-Hill, New York). [8] J.M. Jackson and N.F. Mott, Proc. Roy. Sot. Al37 (1932) 703. [9] S.L. Thompson, J. Chem. Phys. 49 (1968) 3400. [lo] D. Storm and E. Thiele, J. Chem. Phys. 59 (1973) 3313. [ll] E. Bauer, J. Chem. Phys. 23 (1955) 1087. [12] J.B. Calvert, J. Chem. Phys. 56 (1972) 5071. [13] J.H. Kiefer and R.W. Lutz, J. Chem. Phys. 44 (1965) 658.
205
2
CHEMICAL
PHYSICS
VIBRATIONALRELAXATIONOFDt
LETTERS
15 July 1974
INTHERANGE400-50°K
J. LUKASIK and J. DUCUING Laboratoire d’optique Quantique, C.N.R.S. - Ecole Polytechnique, Universitt? de Paris XI, 91405 Orsay, France
Received
18 April 1974
The vibrational relaxation of n-D2 by itself and by n-H2 was studied in the interval 400-50” K. The results are compared to the recent n-H2 studies in the similar temperature range and discussed in terms of various molecular parameters.
The vibrational selfdeexcitation of H, has recently been studied in the low temperature [l] range by the Raman excitation technique [2] . We present here a similar study on the vibrational deexcitation of D2 in D2 -D2 and D2 -II2 collisions which extends previous room temperature work [3] to the interval 400-50°K. The study of the vibrational deexcitation of hydrogen isotopes is of fundamental interest for several reasons. They present large isotopic variations which facilitate the study of the influence of such parameters as the reduced mass or the vibrational quantum. Because of their relative simplicity a priori calculations of the intermolecular potential as a function of the rotational and vibrational coordinates are feasible, as demonstrated in the case of Hz -He [4,5] . This combined with the development of numerical, three-dimensional calculations of the scattering cross sections should allow an interesting confrontation between theory and experiment, particularly in the low temperature domain where the intermediate range features of the potential (such as the well) substantially affect the scattering cross section. The experiment makes use of the tunable Raman excitation scheme in which the vibrational level is populated by a two-photon Raman transition induced by two laser beams one of which is continuously tunable. This scheme has been described in ref. [6] and its application to D2 in ref. [3]. The temperature variation was provided by an arrangement very similar to that used for H, [l] . The gas cell was placed in a vacuum chamber in order to provide good thermal
isolation. It could be heated through electrical resistances up to 500°K or cooled down with circulating around it liquid nitrogen or helium close to the liquid helium temperature. Temperature throughout the cell was determined with platinum resistors (100 52 at 0°C) and was stable and uniform to better than f 1°K above 100°K and + 2°K around 50°K. Relaxation measurements at a particular temperature were made only after the gas in the cell had remained at this temperature for approximately f -1 hour and they were repeated several times within a 3 hours interval. It was found that within this interval the measured rates did not depend on the time at which the experiment was performed. The pressure range of the experiments reported here was from 20 to 60 atm and the real gas corrections given by the compressibility factor [7] were made when necessary. The relaxation rates were measured at 86OK for several densities ranging from 100 to 155 amagat and their values were plotted as a function of the gas density. As in the case of hydrogen [ 1] a linear dependence was found indicating that within experimental accuracy ternary collisions did not contribute, in this density range, to the measured rates. The gases were supplied by Air Liquide. Their impurities are listed in ref. [3]. Two ready mixtures of D2-H2 prepared by Air Liquide (CD, 7 0.746 and c = 0.88 where CD2 is the mole fraction of D2 in D2 . the mrxture) were stu~ed in the experiment. The mole fraction of H, was kept low in order to avoid the 203
Volume
27, number
The experimental uncertainties
2
CHEMICAL
Table 1 results for D2 -Da
AT/T (%)
p7 (atm set)
APr/Pr
400.0 350.0 297.5 296.0 242.0 202.0 152.5 112.5 86.5 86.0 85.0 62.0 61.5 58.0 55.0 54.5
0.1 0.1 0.2 0.2 0.2 0.5 0.7 1.5 1.7 1.7 1.2 2.4 2.8 1.7 2.9 1.8
6.10 x 1.19 x 2.85 x 2.87 x 7.32 x 1.53 x 3.24 x 6.09 x 9.35 x 8.80 x 9.15 x 1.25 x 1.38 x 1.21 x 1.52x 1.39 x
6.9 7.6 7.7 6.6 8.5 4.6 8.0 11.6 13.8 12.8 10.5 22.4 26.1 11.6 8.0 23.8
T (“K)
results
AT/T (%)
Table 2 for Da -Ha
LETTERS
10-4 10-s 1O-3 1O-3 1O-3 10-a 1O-2 1O-2 10-a 10-2 10-a 10-l 10-l 10-i 10-l 10-t
(%)
and the corresponding 19
pr (atm set)
Wr/Pr
.
(%) 0.0‘
348.5 296.0 231 .O 178.5 141.5 110.0 85.0 53.3
0.1 0.2 0.3 0.6 0.7 1.3 2.4 3.8
1.27 1.85 4.53 8.99 1.68 2.58 3.55 3.40
x x x x x x x x
1O-4 lo4 10-4 lo4 10-s 1O-3 10-s 10-s
17.3 16.2 14.9 13.2 20.5 20.0 19.6 35.0
possible influence of the V-V transfer between D2 and H2 (AL?= -1168 cm-l). The directly measured experimental value was pi, the one-atmosphere relaxation time. The expertiental data and corresponding accuracies are listed in tables 1 and 2. r is related to the deexcitation rate constant klo(T) through the relation 7-l = NklO(r), where N is the number of molecules per cm3. The variation of klo(T) versus T- l/3 is shown in fig. 1 for the deexcitation of n-D2 respectively by itself and by n-H,. The n-H2 results reported in ref. [ 1] are displayed in the same figure for the sake 204
1.5 July 1974
and the corresponding
T(?K)
The experimental uncertainties
PHYSICS
0.10
ma
0.14
y---y 0116
0.u
$5 0.m
0.1-2
0.24
??
0.16
Fig. 1. Log kio(Z) as a function of T-1/a. -e-eD~-J_I~, Da-Ha, present work; -A-AH2-H2, ref. [l]; -m-m-Da-Da, ref. [13]; - - Da-Da, --D2-H2, theory, ref. [12].
of comparison. At high temperature the relaxation rates have a Landau-Teller variation log k =A -BT-1/3. The departure from this behaviour in the low temperature range has been discussed in ref. [ 1 ] . In the following we concentrate on the influence of the isotopic variations on the low temperature rates. Above 300°K the relaxation rates kpteH2 and kFtmD2 are of similar magnitude. When T is decreased, however kDZeD2 decreases faster and falls to much lower vakk:than kzpH 2. This is in sharp contrast with the behaviour of the mixture relaxation rate kzpH2 which is fairly similar to that of klHozmH2. The deexcitation rate klo can be expressed in terms of the total collision cross section ulOQ as:
(1)
Volume 27, number 2
where for a central interaction potential the distorted wave approximation:
CHEMICAL PHYSICS LETTERS
V(r), and in
where 4i. q. are the wave numbers of the incident and outgoing particles, m the reduced mass and Ethe angular momentum quantum number for the relative motion, Vu1 the matrix element of the interaction potential between the initial and final oscillator states and :
VtolJ$lIki) =
je. hR;,(r)cl(r)R&),
(3)
0
the R, being the radial wavefunctions with asymptotic form sin(qr t 6). The matrix element (1,ol V,, lZ,i) is, in general, a rapidly decreasing function of q,-qi = 2mw10/ h(qo+qi). This determines the behaviour of u(E) which decreases with E. In the low temperature range d(qo-qi)/dE z - dqi/dE = - (m/t22E)1/2: the larger the reduced mass the stronger should be the variation of a with E, and as a consequence the variation of k with T. This could account for the more pronounced decrease of the D2-D2 rate and perhaps also for the cross over of the H2-H, and D2-H2 rates around 100°K. As 4i becomes negligible compared to q,, 40-4i tends to the limiting value K = (2molo/A)1/2. Analytical calculations by the distorted wave method of s-wave scattering (I = 0) with either a repulsive exponential [8] or an attractive Morse [9, lo] potential show that in the low energy range the scattering cross section u(O) is proportional to exp(-nrta) where (Yis a characteristic length for the potential. Although s-wave scattering contributes only weakly even in the temperature range considered here, this result suggests that the cross sections for small angular momentum and hence the total cross section will be sensitive functions of the parameter ~a. This is supported by the preliminary results of our numerical distorted wave calculations. The values of KOI are shown in table 3. They are very close for H2-H2 and D2-H2 for which the low temperature rates are of similar magnitude whereas the larger value of KCY for D2-D2 appears to be reflected in a much lower rate. Calvert [ 121 has performed numerical distorted wave calculations of the relaxation rates for the hy-
15 July 1974
Table 3 Various parameters (see text) for the three collision pairs considered. The value o = u-r = 5.77 X 10m9 cm is that used by Bauer [ll] m(10-24g)
%0(eV)
KOL
Hz-Hz
1.66
0.516
9.10
D2(exc)-H2
2.21
0.371
8.91
Da-D2
3.32
0.371
10.92
drogen isotopes between 300 and 2000°K. His results are also shown in fig. 1. They connect very well our low temperature D2 data to the high temperature (1 lOO-3000”K), shock tube data of Kiefer and Lutz [13]. However, as in the case of pure H2 the curvature of the theoretical data around 300°K does not fit the experimental results. The isotopic relative variations of k,, are also correctly predicted in the [ 121. This was already noted in range 300-500°K ref. [3] for the room temperature data. The authors acknowledge fruitful discussions with Dr. C. Joffrin, M.M. Audibert and R. ViIaseca-Alavedra as well as the expert technical assistance of M. Chateau. They appreciate Dr. S. Buhler’s help in designing and building the low temperature apparatus.
References [l] M.M. Audibert, C. Joffrin and J. Ducuing, Chem. Phys. Letters 25 (1974) 158. [2] J. Ducuing, C. Joffrin and J.P. Coftinet, Opt. Commun. 2 (1970) 245. [3] J. Lukasik and J. Ducuing, J. Chem. Phys. 60 (1974) 331. [4] M. Kraus and F.H. Mies, J. Chem. Phys. 42 (1965) 2703. [S] M.D. Cordon and D. Secrest, J. Chem. Phys. 52 (1970) 120. [6] R. Frey, J. Lukasik and J. Ducuing, Chem. Phys. Letters 14 (1972) 514. [7] D.E. Gray, ed., Handbook of American Institute of Physics, 3rd Ed. (MacGraw-Hill, New York). [8] J.M. Jackson and N.F. Mott, Proc. Roy. Sot. Al37 (1932) 703. [9] S.L. Thompson, J. Chem. Phys. 49 (1968) 3400. [lo] D. Storm and E. Thiele, J. Chem. Phys. 59 (1973) 3313. [ll] E. Bauer, J. Chem. Phys. 23 (1955) 1087. [12] J.B. Calvert, J. Chem. Phys. 56 (1972) 5071. [13] J.H. Kiefer and R.W. Lutz, J. Chem. Phys. 44 (1965) 658.
205