Trajectory straight-line

This nomially refers to the use of the straight-line trajectory/ (t) = (b +v t ), 0(t) = arctan(b/vt) within the classical path treatment. See Bates [18,19] for examples and fiirtlier discussion. 

Figure S-IS. Representation of a reaction coordinate diagram by straight line trajectories determined by the condition of minimum time or distance. The reaction coordinate is specibed as the bond order of a bond formed in the reaction.

The kinetic theory of gases is described in any physical chemistry text, for example, Refs. [14,22], The theory assumes that molecules in a gas consist of rigid, hard spheres of mass m and diameter d in continuous, randomly directed translational motion. Collisions between molecules are instantaneous, and the molecules travel in straight-line trajectories between collisions until randomly encountering another collision partner. 

Niels Bohr suggested that the energy loss rate could be estimated in a very simple picture as the series of impulses delivered to individual electrons by the ion. Imagine an ion moving on a straight-line trajectory past an electron (see Fig. 17.2). A net impulse to the electron will occur in the direction perpendicular to the trajectory... 

If we again assume that the atoms collide with an impact parameter b and velocity v, following straight line trajectories, we can again assume that V is only non zero for r b and a time interval r = b/v. Requiring that Vr = 1 for a significant transition probability leads to... 

We assume again that the atoms follow straight line trajectories, and we calculate the transition probability, P(b), from the initial to the final state in a collision with a given impact parameter, b. We then compute the cross section by integrating over impact parameter, and, if necessary, angle of v relative to E to obtain the cross section. The central problem is the calculation of the transition probability P(b). The Schroedinger equation for this problem has the Hamiltonian... 

The dynamics of the two-particle problem can be separated into center-of-mass motion and relative motion with the reduced mass /i = morn s/(rnp + me), of the two particles. The kinetic energy of the relative motion is a conserved quantity. The outcome of the elastic collision is described by the deflection angle of the trajectory, and this is the main quantity to be determined in the following. The deflection angle, X, gives the deviation from the incident straight line trajectory due to attractive and repulsive forces. Thus, x is the angle between the final and initial directions of the relative velocity vector for the two particles. 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

Figure 11. Schematic view of a TS (thick black line), with the same type of view as in Fig. 10. The equilibrium point is in the middle with its stable manifold and unstable manifold extending as straight lines. Trajectories in dot-dashed lines are reactive (inside the tubes) and cross TS trajectories in dashed lines are not reactive. The whole gray surface is the energy level. For a linear motion, it takes the form of a parabolic hyperboloid.

Figure 8 plots the relative velocity dependence of Na -h O excitation cross sections of the 3 lowest channels shown in Fig. 7. The cross sections are calculated using simple classical straight line trajectories in which...

Figure 5. Schematic representation of O + H2 and O + HO IDCI) collisions. The heavy line repesents a part of the critical dividing surface, lighter lines represent equipotential contours. Ellipsoidal approximations are used for both kind of surfaces as described in the text, v is the relative velocity of the collision partners, R is their center-of-mass separation vector, y is te Jacobi angle, n is the normal to the equipotential energy surfeces. The coordinate origin is at the center of mass of the molecule axis x coincides with its longitudinal axis. In the present model calculations, straight line trajectories up to the critical dividing surface are assumed...

Example 5.1.2 provides a clue about how to proceed. Recall that the x and y axes played a crucial geometric role. They determined the direction of the trajectories as z —> 1. They also contained special straight-line trajectories a trajectory starting on one of the coordinate axes stayed on that axis forever, and exhibited simple exponential growth or decay along it. 

For the general case, we would like to find the analog of these straight-line trajectories. That is, we seek trajectories of the form... 

Combining Figures 6.4.1-6.4.4, we get Figure 6.4.5, which already conveys a good sense of the entire phase portrait. Furthermore, notice that the x and y axes contain straight-line trajectories, since x = 0 when x = 0, and y = 0 when y = 0. 

We first state that as a particle approaches the center of force, its orbit will deviate from the incident straight line trajectory. After passing the center of force, the force acting on the particle will eventually diminish so that the orbit once again approaches a straight line (as sketched in Fig. 2.3). In general the final direction of motion is not the same as the incident direction, and the particle is said to be scattered. 

Having in mind this idea and assuming a straight-line trajectory of the projectile, the energy loss in ion-atom collision as a function of the impact parameter b can be written as... 

H arises mainly from the straight-line trajectory approximation (see Figure... 

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