- Email: [email protected]
Influence of reagent collision energy on the rotational distribution of BaO in the reaction of SO2 with Ba
429
Chemical Physics 104 (1986) 429-434 North-Holland, Amsterdam
INFLUENCE OF REAGENT COLLISION ENERGY ON THE ROTATIONAL OF BaO IN THE REACTION OF SO, WITH Ba Toshiaki
MUNAKATA
and Takahiro
The Institute of Physical and Chemical Research, Received
27 December
DISTRIBUTION
KASUYA Wako, Saitama, 351-01, Japan
1985
The rotational distribution of the product BaO of the crossed molecular beam reaction of Ba with SO* was studied with interest in its dependence on collision energy. The distribution was probed by laser-induced fluorescence at collision energies ranging from 1.2 to 7.2 kcal/mol. The rotational excitation was found to increase very slowly with collision energy. The observed distribution was compared with calculations based on the phase space theory and the transition state theory. As a result, the phase space theory reproduced the observed distribution only at low collision energy, while the transition state theory reproduced it satisfactorily over a wide range of collision energy. This feature was interpreted in terms of the angular momentum restriction involved in the reaction. The present result was consistent with the angular distribution and recoil velocity spectrum study.
1. Introduction Laser-induced fluorescence is a method of common use in detecting nascent reaction products. Another method of probing elementary reaction dynamics is to measure the velocity and angular distribution spectra of the reaction product in a crossed molecular beam arrangement. These two methods are complementary with each other in that the former provides information on the internal state distribution while the latter on the recoil velocity distribution. However, the results of these two are not to be readily combined because they are often provided under far different experimental conditions. The reaction, Ba(‘S) + SO,(‘A)
+ BaO(‘Z+)
AH = - 4.1 kcal/mol,
+ SO(3Z-), (I)
is one of the systems which have been subjected to studies of both laser-induced fluorescence and molecular beam scattering. Smith and Zare [l] reported the laser-induced fluorescence spectrum of BaO(X lx+) from the facile spin-forbidden reaction (1) of 1.1 kcal/mol collision energy. They described that the reaction cross section was about 0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
four times larger than that of the spin-allowed reaction Ba + CO,, and that the internal energy partitioning was far from symmetric. They explained these features in terms of the electron jump mechanism. Freedman et al. [2], on the other hand, observed the angular distribution and recoil velocity spectrum of BaO formed at four different collision energies in the range from 3.2 to 10.6 kcal/mol. In the reaction at the three lower collision energies, the center-of-mass angular distribution very sharply peaked in the forward and backward directions. They concluded that the reaction proceeds through formation of a long-lived BaSO, complex. This conclusion appears to be in contrast to the asymmetric energy distribution presented by the laser-induced fluorescence study. The inconsistency, however, should be carefully argued with consideration of the far different collision energy range between the two studies. The present paper describes the study of the internal energy distribution of BaO produced at a collision energy of more than 1.1 kcal/mol. Being associated with the angular distribution study, this study aims to provide a better insight into the reaction dynamics. In order to make our experimental conditions comparable to those of the anB.V.
430
T. Munakara,
T. Kasuya / Roiaimal
gular distribution study, an effusive Ba beam crossed with a seeded supersonic SO, beam. nascent rotational distribution of the product from reactions of various collision energies probed by the laser-induced fluorescence.
was The BaO was
2. Experimental The apparatus and experimental procedure were described previously [3,4]. Briefly, a pulsed supersonic beam of SO, crossed an effusive Ba beam at right angles in a reaction chamber of less than 2 X low6 Torr background pressure. The translational energy of the supersonic beam was varied by controlling the seeding ratio of SO, in He carrier gas. The collision energy distribution was evaluated from the velocity distribution of SO, and Ba; the former was given from a time-of-flight measurement, the latter from the oven temperature of 11ClOK as measured by an optical pyrometer. With the equilibration assumption of translation and rotation during the SO2 nozzle expansion, the width of the time-of-flight spectrum corresponds to a rotational temperature of less than 30 K. Under these conditions, the collision energy was variable from 2.6 to 11 kcal/mol with an energy resolution of 2.5 to 3.5 kcal/mol. As a remark, no dimer nor van der Waals complex was detected on a mass spectrometer. The reaction was also studied under “beamgas” condition: the reaction chamber was filled with thermal SO2 of 5 X 10s5 Torr, in which reaction took place at a collision energy of 1.2 kcal/mol. This condition is in contrast to the crossed beam condition in the internal energy of SO,. Taking account of the thermal internal energy of 1.1 kcal/mol, the total reaction energy is close to that of the crossed beam condition of in 2.6 kcal/mol collision energy which is realized by the SO, beam without carrier gas. The light beam of the dye laser (Quanta-Ray DCR-1 and PDL) at = 520 nm wavelength passed the interaction zone of the reactant beams. The induced fluorescence was condensed by a quartz lens onto a photomultiplier which was placed inside of the reaction chamber. The output of the photomultiplier was averaged and then normal-
distrihuiion of BaO
ized to the laser power integrator.
by a two-channel
gated
3. Results Fig. 1 shows the fluorescence excitation spectra of the (A’B+, u’ = 5-X ‘Z+, u” = 0) band system of BaO produced at a collision energy of (a) 2.6 kcal/mol and (b) 7.2 kcal/mol. The spectrum measured under the beam-gas ~ndition was in close resemblance to fig. la. This feature suggests that the influences of the reagent rotational and translational energy on the reaction are not remarkably different from each other. The rotational band head at 521 nm loses intensity with collision energy. The weak lines visible in fig. lb on the left-hand side of the band head are due to the high .J lines of the (6,0) band. The increase of rotational excitation with collision energy is evident in the present observation. On the other hand, the study of the reaction of Ba and SO, dimer [5] describes no rotational heating of BaO. This seems rather unreasonable because the formation of dimer in a high Mach number supersonic jet should be accompanied by translational excitation of SO,. Weak lines due to the (6,l) band are discernible in the wavelength region longer than 527 nm. The (6,l) band lines are weak owing to the small Franck-Condon factor and to the small population in the u” = 1 state. The comparison of figs. la and lb shows that increase of the collision energy brings about no significant vibrational excitation. Since the BaO( u” = 0) is the main product irrespective of the collision energy, we are concerned in this paper only with the rotational distribution of the ground vibrational state. The (5,O) band system was adopted for the determination of the nascent rotational distribution of BaO since the P and R branch lines are resolved. Each rotational line was assigned on the basis of the rotational constants of a previous study [6]. Not taking any perturbed lines into account, we determined the rotational distribution of BaO by iterative least-squares fitting of the peak heights above the fluorescence background to the computer simulation. The populations on
431
T. Munakata, T. Kasuya / Rotational distribution of BaO
BaO(A.v’=5+--X,v”=O)
(b)
524
522 LASER
526
526
530
WAVELENGTH/nm
Fig. 1. Fluorescence excitation spectra of BaO produced in the reaction Ba+ SOs at a collision energy of (a) 2.6 lccal/mol and (b) 7.2 kcal/mol. At the top of the figure is shown the assignment of those lines which are used in the population analysis. The perturbed lines of the (5,O) band and small contributions from the (6,l) band are left unassigned.
the unresolved low J” levels (J” < 9) were assumed to be proportional to 2 J” + 1. Then the simulation arrived at a standard deviation of less than 7% of the peak height of the P(46) line. The residual is due mainly to the fluctuation of fluorescence intensity which caused deviation of the intensity ratio of the P and R lines from J”,/( J” + 1).
BaO (J”)
The rotational populations of BaO determined for the reaction at a collision energy of (a) 1.2 and (b) 7.2 kcal/mol are plotted by the open circles in fig. 2. The mean rotational energy of BaO, as plotted by the open circles in fig. 3, increased slowly with collision energy: only a small fraction of the incremental collision energy appeared as the rotation of the product BaO.
BaO (J” )
Fig. 2. Rotational distribution of BaO produced at a collision energy of (a) 1.2 kcal/mol and (b) 7.2 kcal/mol. Dotted and dashed lines represent calculations by the phase space theory and the transition state theory for the ‘4100se’* complex, respectively.
T. Munakara, T. Kasuya / Rotational distribution of BaO
432
5r .‘Z E 4-
,:.
{3ii c2-
: ..d .:.
&”
:. 0
W 4 dc”,
an induced dipole-induced dipole interaction term to be 740 and 164 in units of 10e6’ erg cm6 for the entrance and exit channel, respectively. The reactant SO, was assumed to be in the ground vibrational and rotational state since it is sufficiently cooled in a supersonic beam. The rotational population a(Mh; Ei) is given
:
:
y.’ 00 &r
. PO0
r
_crRH
by
0.9” ./I0
e(Mt,;
0 s
I
0
Ei) = C C e(Mr,, “f2 Mr2
nf2, Mrz, Ef; Ei),
,
CoIIision5Energy/kcal~~il
Fig. 3. Mean rotational energy of BaO from Ba+SO, plotted as a function of collision energy. Open circles are the experimental points, squares and triangles are the results of the phase space theory and the transition state theory, respectively. Dotted and dashed lines are for the sake of visibility of calculated data.
(2) with o(Mft,
nr2, Mr2r Er; Ei)
= ( nfi2/2~Ei)
Li X cp(Mf,, Lf
Toward a higher collision energy of above 10 kcal/mol, the fluorescence was less intense, and the rotation was more highly excited so that the (5,0) rotational band head was covered by the high J” lines of the (60) and (7,l) bands. Though the quantitative analysis of this type of spectrum was hardly possible, it seemed that the rotational excitation still continued to increase smoothly at this range of collision energy.
4. Discussion A useful aid in understanding the reaction dynamics is to compare the rotational distributions of the present experiment with those derived from purely statistical considerations. The statistical theories may provide an understandable model against the present reaction system. Dagdigian et al. [7] reviewed the phase space theory (PST) [8] and the transition state theory (TST) [9], and presented an extensive application to four-atomic reaction systems. The observed rotational distribution was compared with calculated ones from the two statistical theories on the basis of ref. [7]. The potential functions of the interaction were assumed to be van der Waals type of the form V(r) = - Cc6)rp6 for both the entrance and exit channels. The C@’ constants were estimated from
c (2L, + 1)
Mf2,
nf2,
Ef;
ET,
Li)
(3)
and P(M,,,
nr,,
Mt2, Er; Er,
Li) = l/N(Er,
Li), (4)
where L, M and n are the orbital angular momentum, the rotational angular momentum, and the vibrational quantum number, respectively, and the subscripts i, fl, and f2 stand for SO,, BaO and SO molecules, respectively. Eq. (2) essentially accords to eq. (A3) of ref. [7] except for a slight simplification owing to the low temperature of SO, mentioned above. The total number of accessible states N(E,, Li) is the sum over nR of the number of integer lattice points involved in the space of dimensions (L,, M,, Mn) which satisfies the inequality relationships:
< W:‘3 IL,-M,,I
[ E, (M,, 9 n fz 9 Mf2 j/2]
I M,, - M,, I Q 42
G
2’3,
(5) (6)
W,
+
Mr2.
(7)
As for TST, the energy level densities of the reaction complex are to be calculated for either of four models: the “loose” complex, two forms of the “tight” non-linear complex, and the “tight” linear complex. The last model is not realistic for
T. Munakato,
T. Kasuya / Rotational distribution of BaO
the present system. Among the remaining three models, the “loose” complex model was found to reproduce the experimental results most successfully. Based on the “loose” complex model, the relative rotational populations were calculated by the equations, N(j)=(2j+l)[(E’-
N(j)
W;)‘-$(E’-
IV,)%,
= (2j f l)$?jJ( E’ - 11$1’3,
for
W,> E’-
B,.
(9)
See refs. [7,9] for the notation. Minor errors in the coefficients of eq. (B8) of ref. f73 were corrected in eqs. (8) and (9). The results of the PST and the TST for the “loose” complex are shown in figs. 2 and 3 by the dotted and dashed lines, respectively. The PST reproduces the experimental distribution only at the low collision energy; it overestimates the rotational excitation at higher collision energy.. While, the TST successfully reproduces the distribution over a wide extension of collision energy and acquires more accuracy at a higher collision energy. The success with the TST is very suggestive of that the reaction of Ba with SO, proceeds via formation of a long-lived complex. A further information on the reaction dynamics could be deduced from the preference of the TST to the PST. The TST for the loose complex is claimed to be equivalent to the PST when angular momentum restrictions are fully taken into account [9,10]. But in the present TST treatment, in contrast to the PST, the initial and final orbital angular momenta L and L’, respectively, are assumed to be equal to the total angular momentum J ]7,91, J=L=L’.
(101
On account of eq. (lo), the TST leads to some lower rotational excitation than the PST. The preference of the TST to the PST suggests that the restriction on the final orbital angular momentum
433
(10) is really satisfied in the present reaction system. When the increase in rotational excitation is not significant, the parameter A’ = L’/( L’ + M’), where M’ denotes the product rotational angular momentum, approaches unity with increasing collision energy. This supports the preference of the TST at a higher collision energy. On the other hand, the PST counts too many combinations of angular momenta to form high rotational BaO when the total angular momentum is large. Under the restriction (lo), the scattering process is rather classical so that the product angular distribution is expected to be sharply peaked in the forward and backward directions [ll]. This is what is pointed out in the angular distribution study by Freedman et al. [2]. The present result of the laser-induced fluorescence is, therefore, consistent with the angular distribution study. It is interesting to estimate the extent of the internal excitation of SO. The fraction of the total energy going to the recoil energy was calculated to be 0.5 it 0.1 from fig. 9 of ref. [2], whereas the fractional energy going to the BaO rotation was found to decrease from 0.23 to 0.17 by the increase of collision energy from 2.6 to 7.2 kcal/mol. Accordingly, the rotational and vibrational energy of SO is estimated to be less than 30% of the total energy. The rotational excitation of products is feasible even under the restriction (10) by antiparallel alignment of rotational momenta of BaO and SO. But since the rotational energy of SO is 40% of the total energy under the perfect antiparallel alignment, it can be concluded that the SO is unable to exceed the BaO in rotational excitation. This is again in good accordance with what is described in ref. [2]. The reaction (1) was reported to acquire the character of a direct reaction at a collision energy of 10.6 kcal/mol [2]. This was not confirmed by the present population analysis, but it is worthwhile noting that no drastic change was observed in the fluorescence excitation spectra for reactions at a collision energy above 10 kcal/mol. In conclusion, the rotational distribution of product BaO in the reaction of Ba + SO, is successfully reproduced by the simple transition state theory. The systematic deviation from the phase space calculation implies that the initial and final
434
T. Munakata,
T. Kasuya / Rotational distribution of BaO
orbital angular momenta are close to the total angular momentum. These results suggest that the reaction proceeds via formation of a long-lived complex. Though the energy partitioning of the present reaction is asymmetric to product modes of freedom, it should be regarded as statistical under the angular momentum restriction (10).
Acknowledgement The authors wish to acknowledge Professor Kajimoto for his suggestions on the phase space calculation.
References [l] G.P. Smith and R.N. Zare, J. Am. Chem. 1985.
Sot. 97 (1975)
[2] A. Freedman, T.P. Parr, R. Behrens Jr. and R.R. Herm, J. Chem. Phys. 70 (1979) 5251. [3] T. Munakata, Y. Matsumi and T. Kasuya, J. Chem. Phys. 79 (1983) 1698. [4] T. Munakata and T. Kasuya, J. Chem. Phys. 81 (1984) 5608. [5] J. Nieman and R. Naaman, Chem. Phys. 90 (1984) 407. [6] K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979). [7] P.J. Dagdigian, H.W. Cruse, A. Schultz and R.N. Zare, J. Chem. Phys. 61 (1974) 4450. [8] P. Pechukas, J.C. Light and C. Rankin, J. Chem. Phys. 44 (1966) 794. [9] S.A. Safron, N.D. Weinstein and D.R. Herschbach, Chem. Phys. Letters 12 (1972) 564. [lo] R.A. Marcus, J. Chem. Phys. 62 (1975) 1372. [ll] W.B. Miller, S.A. Safron and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 108.
Chemical Physics 104 (1986) 429-434 North-Holland, Amsterdam
INFLUENCE OF REAGENT COLLISION ENERGY ON THE ROTATIONAL OF BaO IN THE REACTION OF SO, WITH Ba Toshiaki
MUNAKATA
and Takahiro
The Institute of Physical and Chemical Research, Received
27 December
DISTRIBUTION
KASUYA Wako, Saitama, 351-01, Japan
1985
The rotational distribution of the product BaO of the crossed molecular beam reaction of Ba with SO* was studied with interest in its dependence on collision energy. The distribution was probed by laser-induced fluorescence at collision energies ranging from 1.2 to 7.2 kcal/mol. The rotational excitation was found to increase very slowly with collision energy. The observed distribution was compared with calculations based on the phase space theory and the transition state theory. As a result, the phase space theory reproduced the observed distribution only at low collision energy, while the transition state theory reproduced it satisfactorily over a wide range of collision energy. This feature was interpreted in terms of the angular momentum restriction involved in the reaction. The present result was consistent with the angular distribution and recoil velocity spectrum study.
1. Introduction Laser-induced fluorescence is a method of common use in detecting nascent reaction products. Another method of probing elementary reaction dynamics is to measure the velocity and angular distribution spectra of the reaction product in a crossed molecular beam arrangement. These two methods are complementary with each other in that the former provides information on the internal state distribution while the latter on the recoil velocity distribution. However, the results of these two are not to be readily combined because they are often provided under far different experimental conditions. The reaction, Ba(‘S) + SO,(‘A)
+ BaO(‘Z+)
AH = - 4.1 kcal/mol,
+ SO(3Z-), (I)
is one of the systems which have been subjected to studies of both laser-induced fluorescence and molecular beam scattering. Smith and Zare [l] reported the laser-induced fluorescence spectrum of BaO(X lx+) from the facile spin-forbidden reaction (1) of 1.1 kcal/mol collision energy. They described that the reaction cross section was about 0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
four times larger than that of the spin-allowed reaction Ba + CO,, and that the internal energy partitioning was far from symmetric. They explained these features in terms of the electron jump mechanism. Freedman et al. [2], on the other hand, observed the angular distribution and recoil velocity spectrum of BaO formed at four different collision energies in the range from 3.2 to 10.6 kcal/mol. In the reaction at the three lower collision energies, the center-of-mass angular distribution very sharply peaked in the forward and backward directions. They concluded that the reaction proceeds through formation of a long-lived BaSO, complex. This conclusion appears to be in contrast to the asymmetric energy distribution presented by the laser-induced fluorescence study. The inconsistency, however, should be carefully argued with consideration of the far different collision energy range between the two studies. The present paper describes the study of the internal energy distribution of BaO produced at a collision energy of more than 1.1 kcal/mol. Being associated with the angular distribution study, this study aims to provide a better insight into the reaction dynamics. In order to make our experimental conditions comparable to those of the anB.V.
430
T. Munakara,
T. Kasuya / Roiaimal
gular distribution study, an effusive Ba beam crossed with a seeded supersonic SO, beam. nascent rotational distribution of the product from reactions of various collision energies probed by the laser-induced fluorescence.
was The BaO was
2. Experimental The apparatus and experimental procedure were described previously [3,4]. Briefly, a pulsed supersonic beam of SO, crossed an effusive Ba beam at right angles in a reaction chamber of less than 2 X low6 Torr background pressure. The translational energy of the supersonic beam was varied by controlling the seeding ratio of SO, in He carrier gas. The collision energy distribution was evaluated from the velocity distribution of SO, and Ba; the former was given from a time-of-flight measurement, the latter from the oven temperature of 11ClOK as measured by an optical pyrometer. With the equilibration assumption of translation and rotation during the SO2 nozzle expansion, the width of the time-of-flight spectrum corresponds to a rotational temperature of less than 30 K. Under these conditions, the collision energy was variable from 2.6 to 11 kcal/mol with an energy resolution of 2.5 to 3.5 kcal/mol. As a remark, no dimer nor van der Waals complex was detected on a mass spectrometer. The reaction was also studied under “beamgas” condition: the reaction chamber was filled with thermal SO2 of 5 X 10s5 Torr, in which reaction took place at a collision energy of 1.2 kcal/mol. This condition is in contrast to the crossed beam condition in the internal energy of SO,. Taking account of the thermal internal energy of 1.1 kcal/mol, the total reaction energy is close to that of the crossed beam condition of in 2.6 kcal/mol collision energy which is realized by the SO, beam without carrier gas. The light beam of the dye laser (Quanta-Ray DCR-1 and PDL) at = 520 nm wavelength passed the interaction zone of the reactant beams. The induced fluorescence was condensed by a quartz lens onto a photomultiplier which was placed inside of the reaction chamber. The output of the photomultiplier was averaged and then normal-
distrihuiion of BaO
ized to the laser power integrator.
by a two-channel
gated
3. Results Fig. 1 shows the fluorescence excitation spectra of the (A’B+, u’ = 5-X ‘Z+, u” = 0) band system of BaO produced at a collision energy of (a) 2.6 kcal/mol and (b) 7.2 kcal/mol. The spectrum measured under the beam-gas ~ndition was in close resemblance to fig. la. This feature suggests that the influences of the reagent rotational and translational energy on the reaction are not remarkably different from each other. The rotational band head at 521 nm loses intensity with collision energy. The weak lines visible in fig. lb on the left-hand side of the band head are due to the high .J lines of the (6,0) band. The increase of rotational excitation with collision energy is evident in the present observation. On the other hand, the study of the reaction of Ba and SO, dimer [5] describes no rotational heating of BaO. This seems rather unreasonable because the formation of dimer in a high Mach number supersonic jet should be accompanied by translational excitation of SO,. Weak lines due to the (6,l) band are discernible in the wavelength region longer than 527 nm. The (6,l) band lines are weak owing to the small Franck-Condon factor and to the small population in the u” = 1 state. The comparison of figs. la and lb shows that increase of the collision energy brings about no significant vibrational excitation. Since the BaO( u” = 0) is the main product irrespective of the collision energy, we are concerned in this paper only with the rotational distribution of the ground vibrational state. The (5,O) band system was adopted for the determination of the nascent rotational distribution of BaO since the P and R branch lines are resolved. Each rotational line was assigned on the basis of the rotational constants of a previous study [6]. Not taking any perturbed lines into account, we determined the rotational distribution of BaO by iterative least-squares fitting of the peak heights above the fluorescence background to the computer simulation. The populations on
431
T. Munakata, T. Kasuya / Rotational distribution of BaO
BaO(A.v’=5+--X,v”=O)
(b)
524
522 LASER
526
526
530
WAVELENGTH/nm
Fig. 1. Fluorescence excitation spectra of BaO produced in the reaction Ba+ SOs at a collision energy of (a) 2.6 lccal/mol and (b) 7.2 kcal/mol. At the top of the figure is shown the assignment of those lines which are used in the population analysis. The perturbed lines of the (5,O) band and small contributions from the (6,l) band are left unassigned.
the unresolved low J” levels (J” < 9) were assumed to be proportional to 2 J” + 1. Then the simulation arrived at a standard deviation of less than 7% of the peak height of the P(46) line. The residual is due mainly to the fluctuation of fluorescence intensity which caused deviation of the intensity ratio of the P and R lines from J”,/( J” + 1).
BaO (J”)
The rotational populations of BaO determined for the reaction at a collision energy of (a) 1.2 and (b) 7.2 kcal/mol are plotted by the open circles in fig. 2. The mean rotational energy of BaO, as plotted by the open circles in fig. 3, increased slowly with collision energy: only a small fraction of the incremental collision energy appeared as the rotation of the product BaO.
BaO (J” )
Fig. 2. Rotational distribution of BaO produced at a collision energy of (a) 1.2 kcal/mol and (b) 7.2 kcal/mol. Dotted and dashed lines represent calculations by the phase space theory and the transition state theory for the ‘4100se’* complex, respectively.
T. Munakara, T. Kasuya / Rotational distribution of BaO
432
5r .‘Z E 4-
,:.
{3ii c2-
: ..d .:.
&”
:. 0
W 4 dc”,
an induced dipole-induced dipole interaction term to be 740 and 164 in units of 10e6’ erg cm6 for the entrance and exit channel, respectively. The reactant SO, was assumed to be in the ground vibrational and rotational state since it is sufficiently cooled in a supersonic beam. The rotational population a(Mh; Ei) is given
:
:
y.’ 00 &r
. PO0
r
_crRH
by
0.9” ./I0
e(Mt,;
0 s
I
0
Ei) = C C e(Mr,, “f2 Mr2
nf2, Mrz, Ef; Ei),
,
CoIIision5Energy/kcal~~il
Fig. 3. Mean rotational energy of BaO from Ba+SO, plotted as a function of collision energy. Open circles are the experimental points, squares and triangles are the results of the phase space theory and the transition state theory, respectively. Dotted and dashed lines are for the sake of visibility of calculated data.
(2) with o(Mft,
nr2, Mr2r Er; Ei)
= ( nfi2/2~Ei)
Li X cp(Mf,, Lf
Toward a higher collision energy of above 10 kcal/mol, the fluorescence was less intense, and the rotation was more highly excited so that the (5,0) rotational band head was covered by the high J” lines of the (60) and (7,l) bands. Though the quantitative analysis of this type of spectrum was hardly possible, it seemed that the rotational excitation still continued to increase smoothly at this range of collision energy.
4. Discussion A useful aid in understanding the reaction dynamics is to compare the rotational distributions of the present experiment with those derived from purely statistical considerations. The statistical theories may provide an understandable model against the present reaction system. Dagdigian et al. [7] reviewed the phase space theory (PST) [8] and the transition state theory (TST) [9], and presented an extensive application to four-atomic reaction systems. The observed rotational distribution was compared with calculated ones from the two statistical theories on the basis of ref. [7]. The potential functions of the interaction were assumed to be van der Waals type of the form V(r) = - Cc6)rp6 for both the entrance and exit channels. The C@’ constants were estimated from
c (2L, + 1)
Mf2,
nf2,
Ef;
ET,
Li)
(3)
and P(M,,,
nr,,
Mt2, Er; Er,
Li) = l/N(Er,
Li), (4)
where L, M and n are the orbital angular momentum, the rotational angular momentum, and the vibrational quantum number, respectively, and the subscripts i, fl, and f2 stand for SO,, BaO and SO molecules, respectively. Eq. (2) essentially accords to eq. (A3) of ref. [7] except for a slight simplification owing to the low temperature of SO, mentioned above. The total number of accessible states N(E,, Li) is the sum over nR of the number of integer lattice points involved in the space of dimensions (L,, M,, Mn) which satisfies the inequality relationships:
< W:‘3 IL,-M,,I
[ E, (M,, 9 n fz 9 Mf2 j/2]
I M,, - M,, I Q 42
G
2’3,
(5) (6)
W,
+
Mr2.
(7)
As for TST, the energy level densities of the reaction complex are to be calculated for either of four models: the “loose” complex, two forms of the “tight” non-linear complex, and the “tight” linear complex. The last model is not realistic for
T. Munakato,
T. Kasuya / Rotational distribution of BaO
the present system. Among the remaining three models, the “loose” complex model was found to reproduce the experimental results most successfully. Based on the “loose” complex model, the relative rotational populations were calculated by the equations, N(j)=(2j+l)[(E’-
N(j)
W;)‘-$(E’-
IV,)%,
= (2j f l)$?jJ( E’ - 11$1’3,
for
W,> E’-
B,.
(9)
See refs. [7,9] for the notation. Minor errors in the coefficients of eq. (B8) of ref. f73 were corrected in eqs. (8) and (9). The results of the PST and the TST for the “loose” complex are shown in figs. 2 and 3 by the dotted and dashed lines, respectively. The PST reproduces the experimental distribution only at the low collision energy; it overestimates the rotational excitation at higher collision energy.. While, the TST successfully reproduces the distribution over a wide extension of collision energy and acquires more accuracy at a higher collision energy. The success with the TST is very suggestive of that the reaction of Ba with SO, proceeds via formation of a long-lived complex. A further information on the reaction dynamics could be deduced from the preference of the TST to the PST. The TST for the loose complex is claimed to be equivalent to the PST when angular momentum restrictions are fully taken into account [9,10]. But in the present TST treatment, in contrast to the PST, the initial and final orbital angular momenta L and L’, respectively, are assumed to be equal to the total angular momentum J ]7,91, J=L=L’.
(101
On account of eq. (lo), the TST leads to some lower rotational excitation than the PST. The preference of the TST to the PST suggests that the restriction on the final orbital angular momentum
433
(10) is really satisfied in the present reaction system. When the increase in rotational excitation is not significant, the parameter A’ = L’/( L’ + M’), where M’ denotes the product rotational angular momentum, approaches unity with increasing collision energy. This supports the preference of the TST at a higher collision energy. On the other hand, the PST counts too many combinations of angular momenta to form high rotational BaO when the total angular momentum is large. Under the restriction (lo), the scattering process is rather classical so that the product angular distribution is expected to be sharply peaked in the forward and backward directions [ll]. This is what is pointed out in the angular distribution study by Freedman et al. [2]. The present result of the laser-induced fluorescence is, therefore, consistent with the angular distribution study. It is interesting to estimate the extent of the internal excitation of SO. The fraction of the total energy going to the recoil energy was calculated to be 0.5 it 0.1 from fig. 9 of ref. [2], whereas the fractional energy going to the BaO rotation was found to decrease from 0.23 to 0.17 by the increase of collision energy from 2.6 to 7.2 kcal/mol. Accordingly, the rotational and vibrational energy of SO is estimated to be less than 30% of the total energy. The rotational excitation of products is feasible even under the restriction (10) by antiparallel alignment of rotational momenta of BaO and SO. But since the rotational energy of SO is 40% of the total energy under the perfect antiparallel alignment, it can be concluded that the SO is unable to exceed the BaO in rotational excitation. This is again in good accordance with what is described in ref. [2]. The reaction (1) was reported to acquire the character of a direct reaction at a collision energy of 10.6 kcal/mol [2]. This was not confirmed by the present population analysis, but it is worthwhile noting that no drastic change was observed in the fluorescence excitation spectra for reactions at a collision energy above 10 kcal/mol. In conclusion, the rotational distribution of product BaO in the reaction of Ba + SO, is successfully reproduced by the simple transition state theory. The systematic deviation from the phase space calculation implies that the initial and final
434
T. Munakata,
T. Kasuya / Rotational distribution of BaO
orbital angular momenta are close to the total angular momentum. These results suggest that the reaction proceeds via formation of a long-lived complex. Though the energy partitioning of the present reaction is asymmetric to product modes of freedom, it should be regarded as statistical under the angular momentum restriction (10).
Acknowledgement The authors wish to acknowledge Professor Kajimoto for his suggestions on the phase space calculation.
References [l] G.P. Smith and R.N. Zare, J. Am. Chem. 1985.
Sot. 97 (1975)
[2] A. Freedman, T.P. Parr, R. Behrens Jr. and R.R. Herm, J. Chem. Phys. 70 (1979) 5251. [3] T. Munakata, Y. Matsumi and T. Kasuya, J. Chem. Phys. 79 (1983) 1698. [4] T. Munakata and T. Kasuya, J. Chem. Phys. 81 (1984) 5608. [5] J. Nieman and R. Naaman, Chem. Phys. 90 (1984) 407. [6] K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979). [7] P.J. Dagdigian, H.W. Cruse, A. Schultz and R.N. Zare, J. Chem. Phys. 61 (1974) 4450. [8] P. Pechukas, J.C. Light and C. Rankin, J. Chem. Phys. 44 (1966) 794. [9] S.A. Safron, N.D. Weinstein and D.R. Herschbach, Chem. Phys. Letters 12 (1972) 564. [lo] R.A. Marcus, J. Chem. Phys. 62 (1975) 1372. [ll] W.B. Miller, S.A. Safron and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 108.