2538
J. Phys. Chem. 1982, 86,2538-2549
Vibrational-Rotational-Translational Energy Transfer in Ar 4- OH. Quasiclassical Trajectory State-testate Cross Sections Donald L. Thompson University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received: January 6, 7982)
A study has been made of the energy transfer in Ar + OH using quasiclassical trajectories and a pairwise additive potential. The calculations were carried out for initial vibrational states ui = 0,2, and 4 and initial rotational states J1= 0, 10, and 20 for initial relative translational energies 0.2,0.5,and 1.0 eV. State-to-statecross sections were computed to provide information on pure-rotationalenergy transfer, vibration-rotation (V-R) transitions, and vibrational relaxation. Energy transfer in collisions of Ar with highly excited OH involves multiple-quantum V-R transfers. As the vibrational energy is collisionallydecreased, the rotational energy is increased. Varying the collision energy has a minor influence on V-R transfer. For Ji = 0 there is essentially no vibrational energy effect on pure-rotational energy transfer for 0.2-1.0-eV collision energy. For the larger Jithere is both a vibrational and translational energy effect on the rotational energy transfer. Vibrational relaxation cross sections are strongly dependent on Ji.
Introduction Studies by Robinson and co-workers1p2provide strong evidence that pure-rotational laser emissions from OH (X 2111/z)are the result of vibration-rotation (V-R) energy transfer in argon collisions with the vibrationally excited OH produced in the O('D) H2 reaction. The hydroxyl radical is produced in vibrational levels up to u = 4. It is found that, when the system contains argon as a buffer gas, pure-rotational lasing is observed in the lower vibrational levels, u = 0-3. The collisional energy-transfer mechanism creates or enhances inverted rotational-state populations by the efficient conversion of vibrational energy to rotational energy. The inverted population is maintained by the low probabilities for rotational-to-translational (R-T) energy transfer from high rotational levels. Thus, the energy-transfer pumping mechanism depends on the high probabilities for near-resonant V-R transfers and low probabilities for R-T transfers with large energy gaps. For sufficiently high levels of vibrational and rotational energies, there can exist energy resonances for V-R transfers that greatly enhance the energy We have been carrying out a series of quasiclassical trajectory studies that examine the energy-transfer mechanisms for initial levels of internal excitation appropriate for diatomic molecules produced in exothermic reactions such as those used in the rotational lasing studies. State-to-statecross sections have been reported for He + Hz,3Ar + HC1," and Ar + HF.5 It is found that the result of the V-R energy transfer is such that, as the system cascades down the vibrational ladder, the rotational ladder is ascended. Thus, an inverted rotational population is generated in the lower vibrational levels. Cuellar, Parker, and PimenteP first suggested the possibility of an energy-transfer pumping mechanism for rotational lasing in HF in 1974. From a study of rotational lasing in HF produced by photochemical elimination from CHZF, and CH2CHF, they concluded that the rotation-
al-state population inversion was the result of direct chemical pumping, but they also raised the question of collisional V-R energy-transfer pumping. Recently, Sirkin and Pimentel' reported a more thorough study of the photoelimination/HF rotational lasing and concluded that both chemical and energy-transfer pumping occur. Cuellar and PimenteP had also reached the same conclusion earlier in a study in which the HF was produced by flash photolysis of CIF/Hz/Ar mixtures. Smith and Robinsong found that lasing occurs for lower rotational states of HF than expected on the basis of a collisional energy-transfer mechanism in experiments using flash photolysis of halogenated methane/acetylenic molecules/argon mixtures. The lasing differs from that which occurs when the HF is obtained by difluoroethylene elimination. Thus, it appears that there are two mechanisms, chemical and energy transfer, that can contribute to the inverted rotational populations in the lower vibrational levels of the diatomic molecules. It seems clear that V-R energy transfer plays an important role. The theoretical results*5 imply that the role of the energy-transfer mechanism would be enhanced if the reaction products are produced with inverted rotational populations in the excited vibrational states. No measurements have been made, to our knowledge, of the nascent distributions of vibrationalrotational states in HF for the reactions used in the laser experiments. However, distributions for OH have been reported. Measurements of the nascent product rotational distributions have been made for the O(lD) + Hz OH+ H reaction, however, only for the u = 0 and 1 states.lOJ1 Smith, Butler, and Lin'l studied the rotational distribution in the u = 0 state; the measured distribution was found to be in accord with a prediction based on information theory. Luntz, Schinke, Lester, and GiinthardlO reported results for u = 0 and u = 1; they found that the distributions are in good agreement with classical-trajectory1OJ2 results, but in poor accord with statistical predictions-the
(1)G.D. Downey, D. W. Robinson, and J. H. Smith, J . Chem. Phys., 66,1685 (1977). (2) J. H. Smith and D. W. Robinson, J. Chem. Phys., 68,5474(1978). (3)D. L. Thompson, J. Chem. Phys., 75, 1829 (1981). (4)D. L. Thompson, J. Phys. Chem., 86,630 (1982). (5)D. L.Thompson, J. Chem. Phys., in press. (6)E. Cuellar, J. H. Parker, and G. C. Pimentel, J . Chem. Phys., 61, 422 (1974).
(7)E.R. Sirkin and G. C. Pimentel, J. Chem. Phys., 75,604 (1981). (8)E.Cuellar and G. C. Pimentel, J . Chem. Phys., 71, 1385 (1979). (9)J. H.Smith and D. W. Robinson, J . Chem. Phys., 74,5111 (1981). (10)A. C. Luntz, R. Schinke, W. A. Lester, Jr., and Hs.H. Gunthard, J . Chem. Phys., 70,5908 (1979). (11)G.K.Smith, J. E. Butler, and M. C. Lin, Chem. Phys. Lett., 65, 115 (1979). (12)R.Schinke and W. A. Lester, Jr., J . Chem. Phys., 72,3754 (1980).
+
-
~~~
0022-3654/82/2086-2538$01.25/00 1982 American Chemical Society
~~
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The Journal of Physical Chemistry, Voi. 86, No. 13, 1982 2539
Vibrational-Rotational-Translational Energy Transfer
actual distributions are hotter than statistical. The trajectory results were computed by using an analytical fit to the Howard, McLean, and Lester13ab initio potential energy surface. Sorbie and Murrell14have also reported a trajectory study of this reaction; they used a potential energy surface derived from spectroscopic data.15 Thus, except for the energy transfer (which is the subject of the present paper) the kinetics of the OH rotational lasing system has been studied in considerable detail. The purpose of the present study is to provide some detailed information about the energy transfer between Ar and OH in vibrational and rotational states that are in the region of those produced in the O(lD) + H2 reaction. Initial vibrational states u = 0, 2, and 4 and rotational states J = 0, 10, and 20 were studied at initial relative translational energies 0.2, 0.5, and 1.0 eV.
Method of Computation The quasiclassical trajectory procedures have been described previously (see ref 4 or 5 and references therein). The initial OH internal energy was specified by assigning values for the initial vibrational and rotational quantum numbers ui (0, 2, or 4) and Ji (0, 10, or 20), respectively. The initial relative translational energy Erelwas assigned values of 0.2,0.4, or 1.0 eV. The initial orientation of the 0-H bond and rotational angular momentum vector were determined by selecting orientation angles by Monte Carlo sampling of the appropriate probability density functions. The Porter, Raff, and Miller16 equation was used in the Monte Carlo selection of initial vibrational phases. The impact parameters were obtained from Monte Carlo selected values of the initial orbital angular momentum quantum numbers; impact parameters were averaged over the range 0 Ib I12 au. The trajectories were integrated by the Runge-Kutta-Gill procedure with a fixed size of 2.15 X s. Batches of 2000 trajectories were computed to obtain each set of state-to-state cross section values. The statistical errors reported are for one standard deviation. The final rotational quantum number J f was assigned the value of the closest integer to the solution of
L,2 = J f ( J f+ l ) h 2
(1)
where Lf is final rotational angular momentum of OH. The final vibrational quantum number uf was obtained from Uf
1+2 =-2
h
dr
The inner and outer turning points, rminand r-, respectively, were determined by integrating the trajectory of the final-state OH using a step size of 5.4 X s. In eq 2, r is the 0-H bond length and P, is the momentum along the direction of the bond. The integral in eq 2 was solved by using the trapezoid rule with the Ar values obtained with the integration time step size of 5.4 X and the resulting uf value was rounded to the nearest integer. The intermolecular potential is given by the sum
(13)R.E.Howard, A. D. McLean, and W. A. Lester, Jr., J. Chem. Phys., 71,2412 (1979). (14)K.S.Sorbie and J. N. Murrell, Mol. Phys., 31, 905 (1976). (15) K.S. Sorbie and J. N. Murrell, Mol. Phys., 29, 1387 (1975). (16)R. N.Porter, L. M. Mf, and W. H. Miller, J . Chem. Phys., 63, 2214 (1975).
TABLE I: Potential Parameters Morse Function ( OH)e
D = 4.62110 eV CY = 1.21391 auz re = 1.83420 au Lennard-Jones Funct,ionb pair ArH ArO
E,
eV
0.004 11 0.076 68
u, au
6.357 5.875
a From B. R. Johnson and N. W. Winter, J. C h e m . Phys. From W. D. Smith and R. T Pack, 66,4116 (1977). C h e m . Phys. Lett., 15, 500 (1972).
Lennard-Jones 6-12 potential. The values of the potential parameters are given in Table I.
Results and Discussion In this study we have used quasiclassical trajectories to investigate the energy in Ar + OH collisions for a wide range of initial conditions. The purpose was to determine the general trends and mechanisms for the energy transfer. The calculations were carried out for fixed values of the initial relative translational energy (0.2, 0.5, and 1.0 eV). Cross sections were computed for state-to-state transitions (ui, Ji J f ) for ui = 0, 2, and 4 and Ji = 0, 10, and 20 at the three values of Erel. The energy transfer can be divided into three types of processes: vibration-rotation transfer (V-R), pure-rotational transitions (R T), and vibrational relaxation (ui, Ji uf, summed Jf). The V-R transfer occurs for internal states that are sufficiently high that the vibrational and rotational motions are in near resonan~e.~.~ The pure-rotational energy transfers are those in which only the rotational quantum number changes (ui, Ji uf = ui, J f ) ;of prime interest here is the influence of J i on the cross section. State-to-State Cross Sections (ui,Ji uf, J f ) . The energy transfer from high vibration-rotation states is dominated by V-R transfers; this has been illustrated previously for He + H2: Ar + HCl? and Ar + HF.5 That this is the case for Ar + OH is shown by the state-to-state cross sections plotted in Figures 1-3 for ui = 4, J i = 20 at Ere, = 0.2, 0.4, and 1.0 eV, respectively. The results are qualitatively the same as we reported for the other syst e m ~ . ~Collisions -~ of highly excited OH with Ar results in V-R transfers; when the vibrational state is lowered in a collision, there is an increase in the rotational state (with the maximum cross section occurring at AJ = 3-4 times Au) and for upward vibrational transitions there is a decrease in rotational energy (see Figures 1-3). As pointed out previously by us3-5and others (most explicity by Smith and Robinson2),the overall effect of V-R transfer is that, as the vibrationally excited molecules relax, the rotational energy increases; the energy-transfer process can thus create (or enhance) an inverted rotational-state population in the lower vibrational levels. The efficiency of this mechanism, of course, depends on the cross sections (or rates) for loss of rotational energy into translation. The inversion process is aided by the decreasing cross section for R-T as the rotational state increases. (Below we present and discuss results for R-T energy transfer as a function of u, J , and Ere,.) Vibration-rotation transfers are also relatively important for ui = 2, J i = 20 as shown by the state-to-state cross section values plotted in Figures 4-6 for E,, = 0.2,0.5, and 1.0 eV, respectively. However, if the initial vibrational state is zero, V-R is relatively unimportant even for Ji =
-
-
-
-
-
Thompson
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982
2540
E,,I =O.zev
ii
' I
)I
16
i
J3
I I\'
14
I
8
16
12
20
24
28
32
36
Jf
Figure 3. Same as Figure 1 except E,,, = 1.0 eV.
4
8
12
16
20
24
28
32
36
-
Flgure 1. Plots of the state-testate cross sections for vi = 4, J , = 20 vf, J f transitions at E, = 0.2 eV as a function o f J,. For an indication of the Monte Carlo statistical error, see the results given in Table 11.
18
-
/-
-
Vf'4
8
12
20
16
24
28
Jf
Flgure 4. Same as Figure 1 except vi = 2, Ji = 20.
even for the high vibrational state ui = 4;the maximum values of the state-to-state cross sections for Au = -1 transitions are only about 10-15% of the maximum Au = 0 transition. When both ui and Jiare large, the maximum Au = -1 state-to-state cross sections are on the order of 50% of maximum Au = 0 cross section (see Figure 1-3, for example). The state-to-state cross section values for ui = 2, Ji = 10 uf, Jf transitions for Erel= 0.2, 0.4, and 1.0 eV are given in Table 11. The results in Table IT further illustrate the unimportance of V-R transfer in the lower vibration-rotation levels; most of the energy transfer results in a change in rotational state only, that is, R-T. In our previous ~ t u d i e s of ~ -V-R ~ energy transfer the effect of the initial relative translational energy was investigated only for relatively high initial u, J states. In the present study we have carried out a much more extensive investigation of the dependence of the energy transfer on collision energy. That there is a relatively small but nontrivial effect of Ere]on the state-to-state cross sections can be seen by examining the results in Figures 1-12 and
-
Jf Figure 2. Same as Figure 1 except E
= 0.5 eV.
20 as shown by the plots in Figures 7-9 for the three Ere] values studied. Figures 10-12 show the state-to-state cross sections for ui = 4, Ji = 10 at Ere]= 0.2, 0.5, and 1.0 eV. For Ji= 10 V-R transfer becomes relatively unimportant
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2541
Vibrational-Rotational-Translational Energy Transfer
30
"Iz2
2 4 1
Ji=20
r
r 20
16 -
14
12
-
-
-
N .
z
N -
0
b
2 IO -
6
v
b
IO
S-
4
6 -
2
4 -
0
12
8
16
20
24
Jf
2 -
Flgure 7. Same as Figure 1 except vi = 0,J i = 20.
12
8
16
20
24
28
Jf
Flgure 5. Same as Figure 1 except vI = 2, JI = 20, and E, = 0.5 eV. 20[,
,
1
I
,
,
I
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1
,
'
1
3
1
I
'
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14
h
N .
=?
0 IO
v
b 8
12
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Jf Flgure 8. Same as Figure 1 except vi = 0,J , = 20, and E re, = 0.5 eV.
8
12
16
20
24
28
32
Jf
Flgure 6. Same as Figure 1 except vi = 2, J , = 20, and E, eV.
= 1.0
Table 11. First, consider the influence of Erelon V-R transfer for high internal excitation. The effect is illustrated by the results for ui = 4,Ji= 20 shown in Figures 1-3. As Erelincreases from 0.2 to 1.0 eV, the magnitudes of the state-to-state cross sections for the negative Av transition diminish and the cross sections for upward vibrational transitions increase; however, the Au = 0 transitions also occur with smaller cross sections as Erelincreases. The overall effect is a decrease in the cross sections for downward vibrational transitions (which are ac-
companied by upward rotational transitions) and an increase in upward vibrational (and downward rotational) transition cross sections as Ere]increases. Nevertheless, even for the factor of 5 change in Erelthere is not a drastic change in the energy transfer for high u, J. For example, in the case of ui = 4,Ji= 20 the maximum state-to-state cross section for the 4 3 vibrational transition drops by only a factor of 2. The ratio of the maximum Au = 0 cross section to the Au = -1 cross section goes from about 0.6 to about 0.3 as Erelgoes from 0.2 to 1.0 eV. This is in accord with the results reported for Ar + HC1 (ui = 6, Ji = 20) at Erel= 0.3 and 0.8 eV.4 For high u and intermediate J the influence of Erelappears to be even less important as seen by examining the plots in Figures 10-12 for vi = 4,Ji = 10. Note that the peak in the uf = 3 cross sections falls as Erelincreases from
-
2542
The Journal of Physical Chemistry, Voi. 86, No. 13, 1982
Thompson
3 0 1 1 {
20
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16
JrIO
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20
IS
E,,I=0.5eV
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1
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=?
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Y
b
6
2
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Flgure 11. Same as Figure 1 except vi = 4, J , = 10, and E,,, = 0.5 eV.
S
12
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Flgure 9. Same as Figure 1 except vi = 0, J , = 20, and E,,, = 1.0 SV. 301
I
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4 -
2 -
0 4
s
12
16
20
Jf
Flgure 10. Same as Figure 1 except vi = 4, J , = 10.
0.2 to 0.5 to 1.0 eV. This is clearly illustrated by the plots shown in Figure 13 for u2 = 4, J1 = 10. In Figure 13 the cross sections for ui = 4, Ji= 10 uf, Jffor E,1 = 0.2 (solid circles), 0.5 (solid squares), and 1.0 (open circles) eV are superimposed to illustrate the effect of changing E,,,; the Au = -1, +1,0, and +2 transitions are shown in the a-d frames, respectively, of Figure 13. For the Au = -1 transitions (Figure 13a) the main feature is the drop in the cross sections as E,, is increased. For Au = +1 (Figure 13b) there is a large change as Erelis changed from 0.2 to 0.5 eV, but there are relatively minor changes in the cross
-
sections (for example, the Jfdistribution becomes broader) as E,, is increased from 0.5 to 1.0 eV. As shown by Figure 13c, E,, has essentially no effect on the Au = 0 transitions (pure-rotational energy-transfer results are discussed in some detail below). For Au = -2 transitions (Figure 13d) there is an increase in the magnitude of the cross sections as Erelis increased. There is a shift in the locations of the peaks of the Jf distributions in the negative Au cases. The shifts that occur for the case Au = -1 as E,, is changed are minor, but there is a significant shift in the case of Au = -2 as Erelis increased to 1.0 eV. The cross section curves for Erel= 0.2 and 0.5 eV peak at Jf= 18 or 19, but the maximum cross section is for the transition to Jf= 21 at Erel= 1.0 eV. It is interesting, though perhaps not significant, that the cross sections for the E,, = 1.0 eV shown in Figure 13d display a “shoulder” at the location (Jf values) of the peaks in the Erel= 0.2 and 0.5 eV cross sections curves; similar structure is also present in the cross sections for the Au = -1 transitions shown in Figure 13a.
The Journal
Vibrational-Rotational-Translational Energy Transfer
PhysicalChemistry, Vol. 86,No. 13, 1982 2543
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The Journal of Physical Chemistry, Vol. 86, No. 13, 7982
TABLE IV: State-to-State Cross Sections (in au2)for Pure-Rotational Transitions vi = 0 and 4, E,, = 0.2, 0.5, and 1.0 eVa = 0.2 eV
E,,
v = o
Jf
E,1= 0.18
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 22
0.09
0.09
i
1.22 i 0.54 9.69 i 1.38 3 3 . 4 0 i 2.36
-
24.74 i: 1.98 3.40 c 0.75 0.44 i 0.23 0.23 i 0.14
t
0.62 i 1.72 i 2.80 i 8.28 f 9.16 i 15.16 i 20.13 i
23.89 i. 2.09 8.89 i 1.23 4.07 i. 0.85 1.61 i 0.57 0.06 i 0.06
23.48 i 1.98 12.24 * 1.36 5.04 i 0.84 1.01 i 0.39 0.56 i. 0.29 0.26 t 0.16
-
0.01
i
-
v f = vi, Jf for
1.0 eV
v=4
v=o
v=4
0.28 i 0.28 0.48 i 0.34 1.16 i 0.50 3.10 i 0.75 4.26 i 0.91 11.13 i 1.49 10.74 i 1.37 24.85 i 2.22
0.24 c 0.19 0.46 i 0.26 1.30 i 0.48 2.20 i. 0.62 2.11 i: 0.59 5.55 i 1.00 7.07 f 1.12 14.78 i 1.64 2 2 . 9 8 i 2.02
0.39 i 0.27 0.68 i. 0.34 1.41 i. 0.52 1.43 i 0.50 3.30 i. 0.80 6.96 i: 1.16 7.57 i 1.23 11.49 i. 1.50 21.77 i 2.05
21.87 i. 1.94 14.62 i 1.58 8.89 i 1.15 3.49 i 0.68 4.18 i 0.74 1.51 t 0.51 0.63 i 0.33 0.13 i 0.13
2 6 . 6 5 i 2.24 11.95 i. 1.47 5.33 i 0.97 2.96 i 0.76 2.34 i 0.68 0.80 i 0.38 0.22 i 0.22 0.07 i 0.05 0.15 t 0.15
0.18
0.29 t 0.29 0.36 i 0.26 1.06 t 0.48 1.80 i. 0.60 5.18 i 1.03 10.92 i. 1.43 29.79 i 2.48
-
10
E,,=
0.5 eV
v= 0
v=4
Ji =
Thompson
0.26 0.51 0.69 1.23 1.21 1.64 1.88
-
20.36 10.89 6.28 2.30 0.74
0.01
* 1.94 f
i i i
1.43 1.07 0.63 0.36
-
-
The cross sections for vi = 2, Ji = 10 -+ vf = 2, Jf are given in Table 11. The Monte Carlo statistical errors are given for one standard deviation. 6
4
"'; 3 sb
-
2
I
0
0
-
changing Ere]from 0.2 to 0.5 eV is large, as can be seen by comparing the results in Figures 7 and 8. Changing Erel from 0.5 to 1.0 eV for these initial states (ui = 0, Ji = 20) however produces a much less significant change-compare the results in Figures 8 and 9. The change in the magnitude of the maximum cross section for the ui = 0, Ji = 20 uf, J f transitions is statistically insignificant as Ere, goes from 0.5 to 1.0 eV; however, the spread of the Jf distribution in uf = 1is much greater at 1.0 eV. There is a definite drop in the values of the cross sections for the Au = +1 transitions from ui = 4,Ji = 20 as Erelis varied (see Figures 1-3). Also, the J f distributions appear to broaden as Erelis increased, although only slightly. Thus, while there are some discernible trends in the changes in the state-to-state cross sections when the relative transtional energy is varied, they are large only for certain initial states. Pure-Rotational Energy Transfer. The pure-rotational energy-transfer cross sections (for transitions in which the vibrational state does not change) are given in Table I11 for Ji = 0, Table I1 (see columns 6-8) for ui = 2, Table IV for Ji = 10, and Table V for Ji = 20. The cross sections for initial relative translational energies 0.2, 0.5, and 1.0 eV and vibrational states 0, 2, and 4 are given in these tables. Consider first the energy transfer for J i = 0. The influence of vibrational excitation on the rotational excitation cross section is essentially negligible as shown by the results in Table 111. At Ere]= 0.2 eV there appears to be a slight increase in the cross section for the AJ = +1transition as u goes from 0 to 4; however, the difference falls within the Monte Carlo error limits for one standard deviation. The conclusion that must be drawn from the results in Table I11 is that changing the vibrational state from 0 to 4 has no effect on the energy transfer from Ji= 0 for the collision energy range 0.2-1.0 eV. This is in accord with the results for Ar HF for rotational states less than 10 and vibrational states up to 6 at 0.6505-eV collision energy.16 The experimental results of Barnes, Keil, Kutina, and Polanyil7 for Ar + HF also show no effect of vibrational excitation
4
8
12 Jf
16
20
Jf
Flgure 13. Plots of the state-testate cross sections for vi = 4, J i = 0 v,, Jf at E , = 0.2 (sdid circles), 0.5 (did squares), and 1.0 (open circles) eV as functions of JG (a) Av = -1, (b) Av = +1, (c) Av = 0, and (d) A v = -2. For an indication of the Monte Carlo statistical errors, see the results given in Table 11.
The effects due to changes in Ere]are also seen in the results for other initial conditions; for example, for ui = 4,J i = 20 (see Figures 1-3) the influence of E,,] is seen in the results for the transitions to uf = 3, but to a lesser extent than for ui = 4,J i = 10 (see Figures 10-12). However, as illustrated by the results for ui = 0, Ji = 20 and ui = 4,Ji = 20 the influence of Ere]on the state-to-state cross sections for the upward vibrational (and downward rotational) transitions is somewhat greater than for the negative Au transitions (which are accompanied by upward rotational-state changes). For ui = 0, Ji= 20 the effect of
+
(17) D. L. Thompson, Chem. Phys. Lett., 84, 397 (1981). (18) J. A. Barnes, M. Keil, R. E. Kutina, and J. C. Polanyi, J. Chem. Phys., 72, 6306 (1980); 76, 913 (1982).
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2545
Vibrational-Rotational-Translational Energy Transfer
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