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Classical superposition phenomena in H+2 (v = O) - He reactive collisions
Volume 23, numb.
CLASSICAL
CHEMICAL PHYSICS LETTERS
SUPERPOSITION
PHENOMENA G.R. NORTH
Deportment
15 December 1973
IN Hi(v 7 0)-He
REACTIVE
COLLISIONS
and J.J. LEVENTHAL
of Physics. CJniversit.~of Minouri-St.
Louis. Sr. Louis. Missouri 631-71. USA
Received 27 September 1973
A simple classical model is used to analyze reactive processes in Hl-He collisions at relative energies above a few eV. Detailed structure of the cross section occurs and is a consequence of the superposition of probabilities due to distinct binary collision mechanisms which depend on impact parameter and initial orientation. The predicted strticture correlates favorably with available experimental data. Comparison is also made with the (quantum) statistical phase space model.
The short deBroglie wavelengths associated with the translational motion (c-m.) for atomic and molecular systems above a few eV suggests that a classical approach to collision phenomena may be useful for certain systems. For example, in H&He collisions 5 eV corresponds to a wavelength of = l/30 Bohr radii (au). For such studies the @--He system provides the additional simplification that electronic excitation and electron exchange are energetically forbidden below about 13 eV and 9 eV, respectively. Furthermore, the relative simplicity of the species involved has prompted many experimental [l-3] and theoretical [4-61 treatments with which the present work may be compared. We have compiled final state data for about 9000 trajectories using a simplified classical model of H~(LJ = 0) collisions with He atoms. The model potentials are simple and are expected to give accurate results only when the collision-is fairly direct - that is, no complicated (long) trajectories where detailed features might become important. Our results show that above 2 or 3 eV (c.m.) kinetic energy the collisions are always direct so this study should be useful in the understanding of.reaction mechanismi in this energy range. We have chosen to represent the interaction potential as the sum of three pair potentials:
(14
where particle 1 is the helium atom, particle 2 is an H atom and particle 3 is’the proton. The subscript i above refers to the jk pair of particles. The constants in eqs. (1) are given in table 1. No switching [7] was employed so that the sum of the above potentials will be a poor approximation when the three particles are close tcgether. We always consider the target diatomic to be in its classical ground state to avoid the complication of phase averaging. Fig. la shows the initial configuration of the three particles and serves to define our notation *. In order to obtain cross sections one must average over cos0,, and 40. By symmetry Q0 need be considered only in the interval (0”, 180”). To obtain cross sections, probabilities must be multiplied by 2nb and integrated with respect to b. The averaging process is cdnveniently displayed by referring to a cylindrical projection of the unit sphere of initial drientations. We refer to these projections as maps - an example is shown in fig. lb. Equal areas on a map correspond to equal areas on the unit sphere representing initial orientations. For a given impact parameter any final state data, such as scattering angle of particle 1, may be displayed as a * @or.the details
of three-body
kkemaiics.
see ref. [8] _
Volume
23. number
4
CHEMICAL
PHYSICS
Table I Potential parameters [6] used in eq. (1). The equations are written in a form such that when attractive (l-3 and 2-3) ai is the equilibrium internuclear separation and 13; is the binding energy of the classical ground state. The I-2 pair is repulsive, but for simplicity WC arbitrarily chose 02 = oj and B2 = B3
i= 1
i=2
i=3
uj (au)
2.0
Bi (au)
0.1027
1.436 0.074 10
1.436 0.07410
contour representation on the map. In this way final states may be easily studied as a function of initial orientation. Standard (Runge-Kutta) integration methods were used to integrate all trajectories. About 100 equally spaced points were needed on each map to get an accurate picture. An integration over impact parameter required about S-10 maps. Total energy and angular momentum conservation were used as checks on the integrations; several orbits were time reversed to check these criteria. We shall restrict ourselves in this paper to discussion of the 1-3 bound final states (HeH+); we include cpasi-bound final states of HeH’. A quasi-bound state [S] is defined as one where the relative energy of the pair is positive but because of the large rotational barrier they are bound classically. Later we intend to report in detail about map sequences, angular distributions, differential cross sections of energy transfer, and collision induced dissociation (CIDj. Above a few eV, the maps show that production of HeHf occurs in several distinct “islands” on the map. Detailed examination shows that trajectories in the same island are related by collision mechanism. The mechanisms are simple sequences of binary encounters, all of which lead to the production of HeH+. Three distinct sequences occur: (a) knockout of 2 by 1 and subsequent capture of 3 by 1, denoted by (12)(13); (b) grazing of 3 by 1 followed by knockout of 2 by 1 and subsequent binding of 3 and 1 (13)(12)(13);(c) (13)(23)(13). We defme a binary en-. counter as a period of time in which the pair is closer than some preaisigned distance. These separations are 1.30 au for (13), 1.80 au for (23), and 1.70 au for ,(12). Prpcess (a) was studied by Light and Horrocks [9], and (c) was studied by Bates et al. [lo]. A fourth _’
15 December
LETTERS
6eV.
1973
b=1.0 a.”
(b) Fig. 1. (a) Diagram of the initial configuration of particles 1 (He), 2 (H). 3 (Ht). (b) Typical map of final states. Areas lobelled CID represent initial coordinates leading to collision induced dissociation. The area lnbclled H; indicares a final state of 14: and He. The shaded areas represent collisions leading to-tleH+ formation. The shading scheme of lhe latter ‘arc consistent with fig. 3.
Fig. 2. Schematic representation of a collision leading to HeH+ formation (a 13 foal state) indicating the processes which dominarc at different values of impact parameter.
process occurred on a few maps at large impact parameters (~4 A); this last appeared to be a soft peripheral pickup reaction. However, the hard collisions mentioned above accounted for more than 90% of the rearrangement events. A schematic (simpiified to planar) summary of the mechanisms is shown in fig. 2. The island areas (reaction probabilities) have different dependences on energy and impact parameter. Fig. 3 shows two cases at 4 and 6 eV. As the ene& is lowered below about 3 eV, more islands begin to ap- : pear corresponding to Inore complicated processes :such as double rebounds. It is the superposition of :these (classical) probabilities which causes lumps, OK, ,'
Volume 23, number 4
CHEMICAL PHYSICS
15 December
LETTERS
1973
6eV
2
4
KINETIC
m l12)113): [13)(12J(l3): m 1131123)(131: = maps depicting the evolution of island lending to HcHC as a function of impact parameters and energy. Islands are identified as described in the test and illustrated in tig. 2. The ordinates 2nd abscissas of each map are the same as in fig. 1. ix.. cos.00 and 00. respectively. Areas on the maps representing other final states are omirred in this figure for clarily. Fig. 3. Representative
areas for three mechanisms
anomalous str&ture, in the energy dependence of cross sections. Fig_ 4 shows a comparison of our model results with the experimental data of Chupka et al. [I] as well as the statistical model of Rebick and Levine
6 ENERGY
8 (c.ri).
10 eV
Fig. 4. Cross section for formation of HeH+ as a function of relative kinetic energy. The experimental data is that of Chupko et al. [ 1J. The statistictd model result of Rebick and
Levine [5] is also shown.
models (such as impulse approximations). In addition, it is of interest to examine the effects of using even
simpler model potentials duce computer time.
(square wells) in order to re-
The authors would like to thank their colleagues Dr. H.H. Harris and Dr. P.B. James for discussions of this work.
References
PI-
The Rebick and Levine curve does not include quasi-bound states of HeHf since they consider these as a part of the CID probability. Exclusion of these states makes no qualitative difference in the shape of our curve; peaks are not shifted and the maximum added cross section is of order 0.1 AZ. On maps the quasi-bound states tend to occur on the borders of CID islands. Their probability of production then tends to increase with total perimeter of these islands. On the basis of our model, the shoulder in the experimen-
[l]
602
J. Berkowitz
and h1.E. Russel, in: Sixth and Atomic of Papers (MIT Press, Cambridge,
Intern. Conf. on the Physics of Electronic
[2] [3] (41 [S]
tal data is explained by the superposition of distinct collision processes, each leading to HeH+ formation.
A similar shoulder was observed in the rate constant data of Friedman and von Koch [2,4]; however, the H; internal energy was not controlled as in the experiment of Chupka et al. [ 11. Aside from the favorable correlation with experimenta1 data, we believe this simple model will prove useful in devising more refined quantum mechanical
W.A. Chupka,
[9] [IO]
Collisions, Abstracts 1969) p. 71. H. von Koch and L. Friedman, J. Chem. Phys. 38 (1963) 1115. CF. Giese and W.B. hiaier II, J. Chem. Phys. 39 (1963) 739. J.C. Light and J. Lin, J. Chem. Fhys. 43 (1965) 3209. C. Rebick and R.D. Levine, J. Chem. Phys. 58 (1973) 3942, and references therein. D.G. Truhlar, J. Chem. Phys. 56 (1972) 1481; A.F. Wagner and D.G. Truhlar, I. Chem. Phys. 57 (1972) 4063. D.L. Bunker, ic: Proc. of the International School of Physics Enrico Fermi, Course XLIV, ed. Ch. Schlier (Academic Press, New York, 1970) p. 3.55. G.R. North and F.B. James, J.Chem. Phys. 57 (1972) 4415. J.C. Light and D.L. Horrocks, Proc. Phys. Sac. (London) 84 (1964) 527. D-R. Bates,C.J. Cook and F.J. Smith, Proc. Phys. Sot. (London) 83 (1964) 49.
CLASSICAL
CHEMICAL PHYSICS LETTERS
SUPERPOSITION
PHENOMENA G.R. NORTH
Deportment
15 December 1973
IN Hi(v 7 0)-He
REACTIVE
COLLISIONS
and J.J. LEVENTHAL
of Physics. CJniversit.~of Minouri-St.
Louis. Sr. Louis. Missouri 631-71. USA
Received 27 September 1973
A simple classical model is used to analyze reactive processes in Hl-He collisions at relative energies above a few eV. Detailed structure of the cross section occurs and is a consequence of the superposition of probabilities due to distinct binary collision mechanisms which depend on impact parameter and initial orientation. The predicted strticture correlates favorably with available experimental data. Comparison is also made with the (quantum) statistical phase space model.
The short deBroglie wavelengths associated with the translational motion (c-m.) for atomic and molecular systems above a few eV suggests that a classical approach to collision phenomena may be useful for certain systems. For example, in H&He collisions 5 eV corresponds to a wavelength of = l/30 Bohr radii (au). For such studies the @--He system provides the additional simplification that electronic excitation and electron exchange are energetically forbidden below about 13 eV and 9 eV, respectively. Furthermore, the relative simplicity of the species involved has prompted many experimental [l-3] and theoretical [4-61 treatments with which the present work may be compared. We have compiled final state data for about 9000 trajectories using a simplified classical model of H~(LJ = 0) collisions with He atoms. The model potentials are simple and are expected to give accurate results only when the collision-is fairly direct - that is, no complicated (long) trajectories where detailed features might become important. Our results show that above 2 or 3 eV (c.m.) kinetic energy the collisions are always direct so this study should be useful in the understanding of.reaction mechanismi in this energy range. We have chosen to represent the interaction potential as the sum of three pair potentials:
(14
where particle 1 is the helium atom, particle 2 is an H atom and particle 3 is’the proton. The subscript i above refers to the jk pair of particles. The constants in eqs. (1) are given in table 1. No switching [7] was employed so that the sum of the above potentials will be a poor approximation when the three particles are close tcgether. We always consider the target diatomic to be in its classical ground state to avoid the complication of phase averaging. Fig. la shows the initial configuration of the three particles and serves to define our notation *. In order to obtain cross sections one must average over cos0,, and 40. By symmetry Q0 need be considered only in the interval (0”, 180”). To obtain cross sections, probabilities must be multiplied by 2nb and integrated with respect to b. The averaging process is cdnveniently displayed by referring to a cylindrical projection of the unit sphere of initial drientations. We refer to these projections as maps - an example is shown in fig. lb. Equal areas on a map correspond to equal areas on the unit sphere representing initial orientations. For a given impact parameter any final state data, such as scattering angle of particle 1, may be displayed as a * @or.the details
of three-body
kkemaiics.
see ref. [8] _
Volume
23. number
4
CHEMICAL
PHYSICS
Table I Potential parameters [6] used in eq. (1). The equations are written in a form such that when attractive (l-3 and 2-3) ai is the equilibrium internuclear separation and 13; is the binding energy of the classical ground state. The I-2 pair is repulsive, but for simplicity WC arbitrarily chose 02 = oj and B2 = B3
i= 1
i=2
i=3
uj (au)
2.0
Bi (au)
0.1027
1.436 0.074 10
1.436 0.07410
contour representation on the map. In this way final states may be easily studied as a function of initial orientation. Standard (Runge-Kutta) integration methods were used to integrate all trajectories. About 100 equally spaced points were needed on each map to get an accurate picture. An integration over impact parameter required about S-10 maps. Total energy and angular momentum conservation were used as checks on the integrations; several orbits were time reversed to check these criteria. We shall restrict ourselves in this paper to discussion of the 1-3 bound final states (HeH+); we include cpasi-bound final states of HeH’. A quasi-bound state [S] is defined as one where the relative energy of the pair is positive but because of the large rotational barrier they are bound classically. Later we intend to report in detail about map sequences, angular distributions, differential cross sections of energy transfer, and collision induced dissociation (CIDj. Above a few eV, the maps show that production of HeHf occurs in several distinct “islands” on the map. Detailed examination shows that trajectories in the same island are related by collision mechanism. The mechanisms are simple sequences of binary encounters, all of which lead to the production of HeH+. Three distinct sequences occur: (a) knockout of 2 by 1 and subsequent capture of 3 by 1, denoted by (12)(13); (b) grazing of 3 by 1 followed by knockout of 2 by 1 and subsequent binding of 3 and 1 (13)(12)(13);(c) (13)(23)(13). We defme a binary en-. counter as a period of time in which the pair is closer than some preaisigned distance. These separations are 1.30 au for (13), 1.80 au for (23), and 1.70 au for ,(12). Prpcess (a) was studied by Light and Horrocks [9], and (c) was studied by Bates et al. [lo]. A fourth _’
15 December
LETTERS
6eV.
1973
b=1.0 a.”
(b) Fig. 1. (a) Diagram of the initial configuration of particles 1 (He), 2 (H). 3 (Ht). (b) Typical map of final states. Areas lobelled CID represent initial coordinates leading to collision induced dissociation. The area lnbclled H; indicares a final state of 14: and He. The shaded areas represent collisions leading to-tleH+ formation. The shading scheme of lhe latter ‘arc consistent with fig. 3.
Fig. 2. Schematic representation of a collision leading to HeH+ formation (a 13 foal state) indicating the processes which dominarc at different values of impact parameter.
process occurred on a few maps at large impact parameters (~4 A); this last appeared to be a soft peripheral pickup reaction. However, the hard collisions mentioned above accounted for more than 90% of the rearrangement events. A schematic (simpiified to planar) summary of the mechanisms is shown in fig. 2. The island areas (reaction probabilities) have different dependences on energy and impact parameter. Fig. 3 shows two cases at 4 and 6 eV. As the ene& is lowered below about 3 eV, more islands begin to ap- : pear corresponding to Inore complicated processes :such as double rebounds. It is the superposition of :these (classical) probabilities which causes lumps, OK, ,'
Volume 23, number 4
CHEMICAL PHYSICS
15 December
LETTERS
1973
6eV
2
4
KINETIC
m l12)113): [13)(12J(l3): m 1131123)(131: = maps depicting the evolution of island lending to HcHC as a function of impact parameters and energy. Islands are identified as described in the test and illustrated in tig. 2. The ordinates 2nd abscissas of each map are the same as in fig. 1. ix.. cos.00 and 00. respectively. Areas on the maps representing other final states are omirred in this figure for clarily. Fig. 3. Representative
areas for three mechanisms
anomalous str&ture, in the energy dependence of cross sections. Fig_ 4 shows a comparison of our model results with the experimental data of Chupka et al. [I] as well as the statistical model of Rebick and Levine
6 ENERGY
8 (c.ri).
10 eV
Fig. 4. Cross section for formation of HeH+ as a function of relative kinetic energy. The experimental data is that of Chupko et al. [ 1J. The statistictd model result of Rebick and
Levine [5] is also shown.
models (such as impulse approximations). In addition, it is of interest to examine the effects of using even
simpler model potentials duce computer time.
(square wells) in order to re-
The authors would like to thank their colleagues Dr. H.H. Harris and Dr. P.B. James for discussions of this work.
References
PI-
The Rebick and Levine curve does not include quasi-bound states of HeHf since they consider these as a part of the CID probability. Exclusion of these states makes no qualitative difference in the shape of our curve; peaks are not shifted and the maximum added cross section is of order 0.1 AZ. On maps the quasi-bound states tend to occur on the borders of CID islands. Their probability of production then tends to increase with total perimeter of these islands. On the basis of our model, the shoulder in the experimen-
[l]
602
J. Berkowitz
and h1.E. Russel, in: Sixth and Atomic of Papers (MIT Press, Cambridge,
Intern. Conf. on the Physics of Electronic
[2] [3] (41 [S]
tal data is explained by the superposition of distinct collision processes, each leading to HeH+ formation.
A similar shoulder was observed in the rate constant data of Friedman and von Koch [2,4]; however, the H; internal energy was not controlled as in the experiment of Chupka et al. [ 11. Aside from the favorable correlation with experimenta1 data, we believe this simple model will prove useful in devising more refined quantum mechanical
W.A. Chupka,
[9] [IO]
Collisions, Abstracts 1969) p. 71. H. von Koch and L. Friedman, J. Chem. Phys. 38 (1963) 1115. CF. Giese and W.B. hiaier II, J. Chem. Phys. 39 (1963) 739. J.C. Light and J. Lin, J. Chem. Fhys. 43 (1965) 3209. C. Rebick and R.D. Levine, J. Chem. Phys. 58 (1973) 3942, and references therein. D.G. Truhlar, J. Chem. Phys. 56 (1972) 1481; A.F. Wagner and D.G. Truhlar, I. Chem. Phys. 57 (1972) 4063. D.L. Bunker, ic: Proc. of the International School of Physics Enrico Fermi, Course XLIV, ed. Ch. Schlier (Academic Press, New York, 1970) p. 3.55. G.R. North and F.B. James, J.Chem. Phys. 57 (1972) 4415. J.C. Light and D.L. Horrocks, Proc. Phys. Sac. (London) 84 (1964) 527. D-R. Bates,C.J. Cook and F.J. Smith, Proc. Phys. Sot. (London) 83 (1964) 49.