J. Phys. Chem. 1984, 88, 883-891
883
Rotatlonally Inelastic Collisions Studied by Using Model Potential Energy Surfaces. 1. Polarization and Rainbow Effects and Their Dependence on the Attractive Well Howard R. Mayne Department of Chemistry, Eastern Michigan University, Ypsilanti, Michigan 481 97
and Mark Keil* Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada (Received: February 15, 1983: In Final Form: July 14, 1983) Scattering calculations for rotational energy transfer on model potential energy surfaces have been used to elucidate the direct influence of potential energy surface topography on the polarization of integral cross sections, and on differential cross sections. For purely repulsive surfaces, integral cross section polarization data are sensitive to the potential energy contour eccentricity, just as the differential cross sections are. For scattering on substantially attractive surfaces, it is shown how the two types of data provide complementary information. The polarization data probe the angular orientation of the absolute attractive minimum, and are largely independent of the range of attraction. Rainbow features in the state-to-state differential cross sections should allow the attractive and repulsive parts of the intermolecular potential energy surface to be probed fairly independently. Mechanisms for energy transfer on the model potential energy surfaces are discussed, particularly as to how they differ for surfaces having T-shaped or collinear equilibrium geometries. The qualitative conclusions reached here are shown by way of specific examples to be independent of our use of classical mechanics.
I. Introduction Interconversion of translational and rotational energy is the simplest outcome of an encounter between an atom and a diatomic molecule. Studies of energy transfer processes may broadly be delineated as attempting to explore the atom-diatom potential energy surface (PE surface), which is responsible for the rotational energy transfer process. Collisional studies are usually conducted for atom-diatom systems that are only weakly attractive, certainly in comparison with the collision energy. Much of the recent theoretical work has therefore focussed on purely repulsive PE surfaces.'-I0 Wonderfully detailed rotationally inelastic collisional studies have been carried out both in the molecular beams"-17 and g a ~ - c e l l ' ~environments. J~ Collisional techniques not using (1) R. Schinke and J. M. Bowman in 'Molecular Collision Dynamics", J. M. Bowman, Ed., Springer, Berlin, 1982, and references therein. (2) H. J. Korsch and D. Poppe, Chem. Phys., 69, 99 (1982). (3) R. Schinke, W. Miiller, and W. Meyer, J. Chem. Phys., 76, 895 (1982). (4) H. J. Korsch and R. Schinke, J. Chem. Phys., 75, 3850 (1981). (5) R. Schinke and H. J. Korsch, Chem. Phys. Lett., 74, 449 (1980); P. Eckelt and H. J. Korsch, J . Phys. B, 10, 741 (1977). (6) J. M. Bowman, Chem. Phys. Lett., 62, 309 (1979). (7) L. D. Thomas in YPotential Energy Surfaces and Dynamics Calculations", D. G. Truhlar, Ed., Plenum, New York, 1981, p 737. (8) D. Beck, U. Ross, and W. Schepper, Z . Phys. A , 293, 107 (1979). (9) For earlier studies, see especially (a) R. A. LaBudde and R. B. Bernstein, J . Chem. Phys., 55, 5499 (1971), (b) R. J. Cross and D. R. Herschbach, ibid., 43, 3530 (1965). (1 0) Recent general reviews for rotationally inelastic scattering calculations can be found in the volume by R. B. Bernstein, Ed., "Atom-Molecule Collision Theory", Plenum, New York, 1979, Chapters 8-10. See also A. S . Dickinson, Comp. Phys. Commun., 17, 51 (1979). (11) For a recent review, see H . J. Loesch, Adu. Chem. Phys., 42, 421 (1980), and references therein. (12) P. L. Jones, U. Hefter, A. Mattheus, J. Witt, K. Bergmann, W. Miiller, W. Meyer, and R. Schinke, Phys. Rev. A , 26, 1283 (1982); K. Bergmann, R. Engelhardt, U. Hefter, and J. Witt, Phys. Reu. Lett., 40, 1446 (1978). (13) J. A. Serri, C. H. Becker, M. B. Elbel, J. L. Kinsey, W. P. Moskowitz, and D. E. Pritchard, J . Chem. Phys., 74, 5116 (1981); J. A. Serri, A. Morales, W. Moskowitz, D. E. Pritchard, C. H . Becker, and J. L. Kinsey, J . Chem. Phys., 72, 6304 (1980). (14) P. Andresen, H. Joswig, H. Pauly, and R. Schinke, J . Chem. Phys., 77, 2204 (1982). (15) M. Faubel, K. H . Kohl, J. P. Toennies, K. T. Tang, and Y. Y. Yung, Faroday Discuss. Chem. Soc., 73, 205 (1982); M. Faubel, K. H . Kohl, and J. P. Toennies, J . Chem. Phys., 73, 2506 (1980). (16) U. Buck, F. Huisken, J. Schleusener, and J. Schafer, J . Chem. Phys., 72, 1512 (1980). (17) W. R. Gentry and C. F. Giese, J . Chem. Phys., 67, 5389 (1977). (18) A. J. McCaffery, Gas Kinet. Energy Transfer, 4, 47 (1980), and references therein. (19) T. A. Brunner and D. E. Pritchard, Adv. Chem. Phys., 50,589 (1982). and references therein.
0022-3654/84/2088-0883$01.50/0
rotationally inelastic scattering have also been used to determine anisotropic PE surfaces.20,2' Almost simultaneously, bound-state spectroscopic techniques have come to fruition in the ultracold molecular beams e n v i r o n m e r ~ t . ~ Spectroscopic ~-~~ and collisional techniques have been the main sources for detailed experimental information pertaining to atom-diatom PE surfaces. The application of spectroscopic techniques to determining atom-diatom PE surfaces has been particularly f r ~ i t f u l . ~ ~ , ~ ~ The appeal and utility of spectroscopic studies arises from the ability clearly to relate specific features of the intermolecular potential with the spectroscopic observables. Such a clear connection between the anisotropic PE surface and experimentally observable results is, however, largely absent from scattering studies. Nevertheless, scattering techniques complement spectroscopic ones in that they are uniquely sensitive to the repulsive part of the PE surface.26 However, the sensitivity of the scattering results to particular regions or features of the PE surface has not been mapped out clearly. It is in order to help establish a clearer connection between the PE surface and scattering observables that the present study has been undertaken. Our approach is to conduct a "global" kind of sensitivity analysis. By selectively modifying simple parameterized model PE surfaces, changes in the scattering results are attributed straightforwardly to specific features or regions of the PE surface. The procedure used here is reminiscent of that used in relating features of atom-diatom PE surfaces for reactive collisions to consumption or disposal of internal energy.27 It has also been used in a very recent study of rotational rainbow structure.28 The (20) G. A. Parker, M. Keil, and A. Kuppermann, J. Chem. Phys., 78, 1145 (1983); M. Keil, G. A. Parker, and A. Kuppermann, Chem. Phys. Lett., 59, 443 (1978). (21) R. T Pack, J. J. Valentini, and J. B. Cross, J . Chem. Phys., 77, 5486 (1982). (22) For a recent review, see D. Levy, Annu. Rev. Phys. Chem., 31, 197 (1980). (23) J. M. Hutson and B. J. Howard, Mol. Phys., 43, 493 (1981). (24) H. P. Godfried and I. F. Silvera, Phys. Rev. A , in press. (25) R. J. LeRoy and J . S. Carley, Adu. Chem. Phys., 42, 353 (1980); R. J. LeRoy, J. S . Carley, and J. E. Grabenstetter, Faraday Discuss.Chem. Soc., 62, 1969 (1977). (26) For some favorable diatomic cases, the repulsive wall has also been studied spectroscopically; see, for example, R. E. M. Hedges, D. L. Drummond, and A. Gallagher, Phys. Reu. A , 6, 1519 (1972); W. J. Alford, K. Burnett, and J. Cooper, ibid., 27, 1310 (1983). (27) J. C. Polanyi and J. L. Schreiber in 'Physical Chemistry-an Advanced Treatise", Vol. VIA, H. Eyring, W. Jost, and D. Henderson, Ed., Academic Press, New York, 1978, p 383; J. C. Polanyi, Acc. Chem. Res., 5, 161 (1972), and references therein. (28) R. Schinke, H. J. Korsch, and D. Poppe, J . Chem. Phys., 77, 6005 (1982).
0 1984 American Chemical Society
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The Journal of Physical Chemistry, Vol. 88, No. 5, 1984
Mayne and Keil
TABLE I: Anisotropy Parameters for the Model Potential Energy Surfacesa lR1/,VC lRIVC 1R2VC
-0.400
0.000 +OS00
9.0 15.0 22.5
18.0 15.0
11.3
1.
The subscripts /I and 1 denote the particular Fixed configurations having y = 0, 71, and 7 = n / 2 , respectively. parameter values are ( d r = 15.0 meV; e = 0 . 4 1 7 ; R m l i = 4.27 A; R,l= 3.88 A; (R,)r= 4.00 A. a Equation
method is independent of the accuracy with which a particular PE surface is known, and correlations inferred between the scattering results and the PE surface should be qualitatively valid. Using model potentials facilitates PE surface adjustments to be made in a more straightforward and systematic way than using more realistic PE surfaces.29 The potentials without attractive wells used in this study are closely akin to hard-shell m o d e 1 ~ . ~ * The ~ ~ ~“no-well” ~ ~ ~ * results ~*~~ serve to establish the role of scattering off the repulsive wall, independently of any attractive component. The role of the attractive well is then established by recalculating the scattering on PE surfaces including the attraction, for otherwise identical R energy transfer can be affected by PE surfaces. Since T the presence of an appreciable attractive ~ e l l , ~ 9 ” documentation ~’ of these effects is necessary before determination of the important attractive region of the PE surface becomes experimentally feasible. In section I1 of this paper, we describe the parameterization used for the model PE surfaces and the calculation of scattering results by the classical trajectory method. Results for integral cross sections are presented in section 111. Differential cross sections and the classical origins of rainbows are discussed in section IV. The polarization of integral cross sections is discussed extensively in section V, showing especially the important role played by the attractive well. A mechanism for the observed polarization behavior is outlined. Finally, we summarize our conclusions in section VI.
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11. Inelastic Scattering Computations A . Model Potential Energy Surfaces. We parameterize the PE surface anisotropy by making the potential minimum position and depth parameters, R , and e , respectively, dependent on the atom-molecule orientation angle y:20,34338
Axial symmetry makes the potential symmetric about the y = 0,a line. In this study, the linear molecule is restricted to be a homonuclear diatom; in addition, we choose the eccentricity e to be less than unity (making the semimajor axis lie along the diatomic axis) and nonzero values for to and t2 only. The orientation-averaged parameter values are then given by (R,)? = ( R , , / e ) sin-’ e, and (E), = eo. Eccentricity of the elliptical V ( R , y ) = 0 contour is termed “radial anisotropy” and is controlled by e. The “energy anisotropy” is controlled by c2: for the attractive well region of the PE surface, c2 controls the anisotropy of the well depth. For the repulsive (29) J. C. Polanyi and N. Sathyamurthy, Chem. Phys., 29, 9 (1978). (30) S.Bosanac, Phys. Reu. A, 22, 2617 (1980). (31) M. H. Alexander and P. J. Dagdigian, J . Chem. Phys., 73, 1233 (1980). (32) J. A. Serri, R. B. Bilotta, and D. E. Pritchard, J . Chem. Phys., 77, 2940 (1982). (33) V. Khare, D. J. Kouri, and D. K. Hoffman, J . Chem. Phys., 74,2275 (1 98 1). (34) M. Keil and H. R. Mayne, Chem. Phys. Lett., 85,456 (1982). (35) J. A. Barnes, M. Keil, R. E. Kutina, and J. C. Polanyi, J. Chem. Phys., 72,6306 (1980); 76, 913 (1982). (36) J. Bentley, J. Chem. Phys., 73, 4708 (1980). (37) G. D. Barg and J. P. Toennies, Chem. Phys. Lett., 51, 23 (1977). (38) R. T Pack, Chem. Phys. Lett., 55, 197 (1978).
R (4
Figure 1. Potential energy contours (in meV) for the + W model PE surfaces. The dash-dotted curve is the locus of minimum positions R,(y). The absolute minimum on each PE surface is marked by a cross; it is in the T-shaped configuration on 1R’/2V+W, and in the collinear configuration on lRZV+W. The position of a N atom is noted for reference; the PE surfaces are both symmetric about y = 90’.
region of the PE surface, t2 controls the anisotropy of the repulsive wall steepness. For the purposes of this study, we choose the ratios R , /R,, = 1.10, a moderate radial a n i s o t r ~ p ywith , ~ ~C~~ ~, /~E=~ 0.5b, 1.00, and 2.00. The fixed values chosen for ( R,)? and ( E ) correspond closely to those for the prototypical Ar + N2 systemd (see section IIB). Table I lists the parameter values and our designations for the anisotropies of the various PE surfaces used in this study. Having specified the PE surface orientation angle dependence by eq 1, we must further specify the radial dependence for each angle y. This is accomplished conveniently by choosing a Morse radial function, in terms of which the PE surface is given finally by V ( R , y ) = e(y)e@(l-P)[eB(l-P) - 21 (2) where p 2 R / R , ( y ) , and R is the atom-diatom center-of-mass separation. In addition to modifying the PE surface anisotropy as shown in Table I, the Morse radial function of eq 2 is modified to examine the effects of the long-range attraction and of the attractive well components of the PE surface. In the former case, we remove the large-R attractive part of the PE surface by joining the Morse function smoothly and identically to zero for R / R , ( y ) > 1.25. By using a cubic spline joining function, the repulsive wall and attractive well for R / R , ( y ) 5 1.14 are unaltered. This is designated the “no long-range” (-LR) PE surface. Finally, a “no well” (-W) PE surface is generated by removing the attractive ~~~~
~
(39) M. Keil, J. T. Slankas, and A. Kuppermann, J. Chem. Phys., 70,541 (1979). (40) M. D. Pattengill, R. A. LaBudde, R. B. Bernstein, and C. F. Curtiss, J . Chem. Phys., 55, 5517 (1971); M. D. Pattengill, Chem. Phys. Lett., 36, 25 (1975).
Polarization and Rainbow Effects in T-+R Collisions part of the potential altogether. In this case, the Morse repulsive wall is joined smoothly and identically to zero in such a way that the PE surface is unaltered for R/R,(y) < 0.83. Similar techniques were employed by Polanyi and S a t h y a m ~ r t h y . ~The ~ repulsive walls of the Morse and “no well” and “no long-range” potentials do not differ significantly for V(R,y)5 5 meV (cf. the collision energy of 70 meV, section IIB). The PE surface menagerie used in this study is formed by combining the anisotropies of eq 1 and Table I with the radial Morse, no long-range, and no well functions. The nomenclature used for the PE surfaces reflects this combination; for example, a PE surface with t , , / t l = 0.5 but without the long-range Morse attractive tail is designated lR1/,V-LR. For PE surfaces using the full Morse curve including the entire attractive well, the designation “+W” is added to those of Table I. Contour plots of the lR1/2V+W and lR2V+W model PE surfaces are shown in Figure 1. B. Classical Trajectory Calculations. Scattering calculations were performed by using classical mechanics, since this can provide detailed physical insight as the collision progresses. The question of the qualitative validity of the classical calculations described herein is deferred until a discussion of the results; see sections IV and V.41 The general features of the trajectory program used here have been described p r e v i ~ u s l y . We ~ ~ ~concentrate ~~ here on features specific to this work, where we have confined our studies to excitations from the rotationlessj = 0 state. In this case, only the spherical polar angles 8 and 4 need be selected to fully specify the homonuclear diatomic orientation. Since only even A J transitions are allowed quantum mechanically, the j’ = 0 bin encompasses classical, n ~ n i n t e g r a]’from l ~ ~ 0 to 1, whereas j ’ 1 2 bins are each 2 h wide. To bin m/, we first project I‘onto the initial collision axis, to obtain ji. We label each bin with an integer n, from 0 to j’. Then a trajectory contributes to the nth bin according to45 max (-j’, 2n - j ’ - 1) 5 jz’ < min Q‘, 2n - j ’ + 1) (3) The m i = fj’bins (Le., n = 0 and n = j ? are thus only 1h wide; cross sections into these bins are normalized to compare them with all the other bins, which are 2h wide. Since mi = 0 in this study, we average the cross sections into +m/ and -mi. This was the scheme used to show the (j,mj) Q’,m/) cross sections in our previous p ~ b l i c a t i o n . ~ ~ In our earlier study, we found a preference for conservation of the polarization angle x’, defined by x’ = cosd (mj/j?. This conclusion, also expressed by Alexander46 and observed by Monchick:’ suggests that x’ could be better utilized than mi. For some purposes in this paper, it is more appropriate to define the continuous variable 2’
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2’ = c0s-l ( j ; / j ?
(4)
Only values of 2’ between 0’ and 90’ need be displayed, since the distribution over 2’ must be symmetric about 90’. The last of the final scattering attributes to be analyzed is the scattering angle, 8’. This was binned into units 5’ wide. Integration was carried out with a Hamming predictor-corrector and 10 X routine, with a fixed step size of between 7 X s. The usual tests for trajectory accuracy were made. The method of stratified impact parameter Sam ling was used. Trajectories were started and ended with R > 9 at which point the potential felt was, for all the model surfaces, YR,and “dark“ for 8’ < 8‘R. Trajectory calculations for scattering on the purely repulsive -W PE surfaces also exhibit these orientational rainbows, though the rainbow angles are somewhat larger (e.g., = 80’ (48) W. Schepper, U. Ross, and D. Beck, Z . Phys. A , 290, 131 (1979). (49) R. Schinke, J . Chem. Phys., 73, 6117 (1980). (50) R. B. Gerber, V. Buch, and U. Buck, J . Chem. Phys., 72, 3596 (1980).
on 1R2V-W vs. 8’R = 50’ on 1R2V+W forj’= 10). The DCS becomes progressively more backward scattered as j’ increases, in agreement with quantal scattering from purely repulsive potential~.~ Classical deflection functions for j = 0 j’ scattering are monotonic for largej’,” and the impact parameter is simply related to the scattering angle for fixed j’.2 As the impact parameter is increased to the largest possible value that classically can produce a given j’, there arises a wide range of orientation angles disposed to yield this j’.6 The value of b where this occurs indicates the scattering angle near which the rotational rainbow will arise. Thus, positions of rotational rainbows for eachj = 0 -j’transition yield information on the opacity functions P&b). As sensitive probes of the PE surface anistropy, data on rotational rainbows are complemented by the polarization results discussed in section V. For scattering into lowj’(j’= 2 and 4 on 1R’/2V+W, andj’ = 2, 4 and 6 on 1R2V+W), Figure 4 shows strong forwardscattered peaks, and sharp drops in scattering intensity beyond 8‘ 5 30’. The low-j’ DCS are “bright” for angles below the rainbow and “dark” for angles above it. Investigation of the low-j’ rainbows reported here reveals their origin to be similar to rainbows in atomic s~attering:~’ the deflection functions show a substantial minimum for scattering into low-j’states. The rainbow angle for all the low-j’transitions is largely independent of the rotational i n e l a ~ t i c i t yand ~ ~ depends on the PE surface in the region of its minimum. The low-j’ rainbows are “impact parameter” or “Ctype” rainbows in Thomas’ original classificatior~.~~ Although Thomas has since shown that both types of rainbows are mathematically equi~alent,’~~~ we agree with Bowman in claiming that one of these types can dominate the ~ t h e r . ~The , ~DCS ~ of Figure
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(51) For a review regarding spherical potentials, see H. Pauly in “AtomMolecule Collision Theory”, R. B. Bernstein, Ed., Plenum, New York, 1979, p 111. (52) L. D. Thomas, J . Chem. Phys., 67, 5224 (1977). (53) L. D. Thomas, J . Chem. Phys., 74, 5905 (1980).
Polarization and Rainbow Effects in T-R
........ lR’/*V-W
-1 R l V - W
The Journal of Physical Chemistry, Vol. 88, No. 5, 1984 887
Collisions
p - p ’ COS 8’
a = cos-’
----- 1 RPV-W
[p2
1
1 1 Figure 5. Mean polarization angles (averaged over all initial trajectory variables) as functions of the final rotational state j‘for scattering on the “no well” (-W) PE surfaces. For the panel on the right,j’is normalized by the maximum observed inelasticity, j’,,,,, = 8, 12, and 14 for the lR’/,V-W, IRIV-W, and lR2V-W PE surfaces, respectively.
4 clearly show the transition from impact parameter rainbows to orientational rainbows, as the rotational inelasticity is increased. The classification of rainbow types as impact parameter or orientational retains validity even with the strongly anisotropic potential well present in our model +W PE surfaces. Nor is the rainbow classification an artifact of our reliance on classical mechanics.28 For infinite-order sudden approximation (IOSA) calculations, generously performed for us by Parker using the lR’/,V+W and 1R2V+W PE surfaces, the distinction between the two rainbow types persists.55 Parker’s IOSA calculations show that the high-frequency quantum oscillations diminish rapidly for angles much beyond the classical impact parameter rainbows found in the present work. This behavior again is typical of atomic scattering. The type of rainbow for a particular j’is easily determined since the impact parameter rainbow is ”bright ... dark”, whereas the orientational rainbow is “dark ... bright”. We see that the type of rainbow (impact parameter or orientational) allows a clear distinction to be drawn between rotational transitions dominated by repulsion and those dominated by attraction. We shall see in section V that strong enhancement of polarization appears for just those j’ whose DCS simultaneously exhibit impact parameter rainbows: domination by repulsion or attraction can also be distinguished from the polarization of integral cross sections. V. Polarization of the Integral Cross Sections A . Surfaces without Attractive Wells. For each particular (j=0) ( j ’ , ~ ’inelastic ) transition on the “no well” (-W) PE surfaces, we have calculated the transition probabilities, defined as P(j’,x’) 3 u ( j ’ , x ’ ) / ~ , ~ u ( j ’ , x ’In ) . Figure 5a are collected (x’)(j’),the mean of the polarization angle distributions for each j’. The figure shows that ( x ’ ) ( j ? tends toward 90’ (Le., m: = 0) asj’increases, and that ( x ’ ) ( j ? decreases toward a limit of -57O as j’decreases. This qualitative trend is observed regardless of which “no well” PE surface is considered. We also find that low-j’ levels are completely unpolarized: P(j’,x’) is largely independent of x’ for j‘ I 4 on lR’/,V-W, or for j’ I 6 on 1R2V-W. In order to interpret the above data, we consider polarization aspects of inelastic scattering from a hard shell. Summarizing Schinke and Korsch’s results for j = 0,5 the maximum mi classically allowed for a given j‘and center of mass (cm) scattering angle 8‘ is my 5 j’sin a, where a is the angle between the initial momentum p and the momentum change vector Ap = p - p’
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(54) J. M . Bowman and K. T. Lee, J . Chem. Phys., 73, 2664 (1981). (55) G. A. Parker, unpublished results.
+ (Pq2 - ~ ~ P ’ C O 8’]1/2 S
)
(5)
Averaging Schinke and Korsch’s probability distribution for particular projections [eq 12 of ref 51, we obtain the To obtain the equivalent ( x ’ ) , “depolarization” quantity (m,.). we use the approximation that (mi)N j’cos ( x ’ ) , and obtain
(x’)(j’,8’) = cos-] (2n-1 sin a )
(6)
This average polarization angle (x’)(j’,8’) is a function of 8’ through a,and of j’ through p ’ in eq 5. We note that in the limit of sudden collisions, Ap is small: for large collision energy and small Aj, aSUd = 1 / 2 ( ~- 8’). We then have the very simple results
( x ’ ) ~ I”8~’ / 2
( x ’ ” ” ~= cos-’ [2& cos ( 8 ’ / 2 ) ]
(7)
In this limit, we see that ( x ’ ) no longer depends explicitly on j’, but rather has a simple correspondence with the scattering angle 8’. The classical hard-shell model used above illustrates that the x’ distribution is related directly to the distribution over scattering angles: once the angular dependence of the cross section is known, the polarization data provide no new i n f ~ r m a t i o n . ~ ~ - ~ ~ The route from polarization data to angular distributions may be mapped explicitly by recalling that 8’ is strongly correlated with the impact parameter b, even for rotationally inelastic collisions.” Our data show a typically linear relation, with 8’ = K - 7rb/b*, where b* is just the semimajor axis of the V ( R , y )= 0 contour. From eq 7 we then have56
( ~ ’ ) ” ~ ( ( e=’ )( ~ ’ ) ” ~ (=b cos-I[ ) 2r-I sin
($)I
(8)
To compare the predictions of eq 8 with the observed first moment polarization data of Figure 5a we integrate eq 8 over the impact parameter. This is easily done by convoluting eq 8 with the inelastic opacity functions (Figure 3). Since large j’arise from small-6 collisions, their ( x ’ ) tend to 90’. Conversely, small j’arise predominantly from large4 collisions, so ( x ’ ) for small j’ tends to the 6 b* limiting value of eq 8, namely, ( x ’ ) ( b )= 50.46’. It is interesting to note that a completely random distribution over my states yields ( x ’ ) = 57.30°, very close to the observed small-j’ limit displayed in Figure 5a. The larger eccentricity of the 1R2V-W PE surface means that a given j’ transition can occur for larger 6 . Thus, b/b* is larger, so ( x ’ ) ( j ’ ) for a given j’ is smaller on 1R2V-W than on 1R’/2V-W. Scaling the rotational inelasticityj’by j’,,,, to account for the eccentricity of the different “no well” PE surfaces is illuminating. The utility of such a scaling is recognized by noting that as j ’ / j h a xapproaches 0, the impact parameter increases to its maximum value b*, and ( x ’ ) tends to -50’. Conversely, as j’increases to the maximum inelasticity, b approaches 0 and ( x ’ ) increases to 90’. These considerations are independent of the PE surface eccentricity. Indeed, the plot of ( x ’ ) ( b )as a function of j’/jrmaX in Figure 5b shows the “universality” of the hard-shell polarization results: the scaling of j’ by j’,,, collapses the three curves of Figure 5a into a single curve. In this sense, ICs polarization data for scattering on a purely repulsive PE surface would not add any new information, since it could be predicted We will see in section VB how simply from the observed j’max. different a conclusion is obtained for scattering on an attractive PE surface. The kinematic nature of the polarization behavior is confirmed by a detailed examination of the actual trajectories calculated on two of the “no well” PE surfaces. These are shown in Figure 6 as plots of 2’ as functions of the initial orientation angles 8 and 4, for fixed impact parameters on the lR’/,V-W and lR2V-W PE surfaces. It is evident that the detailed polarization behavior
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(56) D. Richards, J . Phys. B, 15, 3025 (1982). (57) V. Khare, D. J. Kouri, and D. K . Hoffman, J . Chem. Phys., 76,4493 (1982). ( 5 8 ) D. Richards, J . Phys. E , 14, 4799 (1981).
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1 R%V-W
Mayne and Keil
1 RZV-W
.........IR‘/sV 3
