Center-of-mass velocity

The collisions that take place at the times x represent the effects of many real collisions in the system.1 These effective collisions are carried out as follows.2 The volume V is divided into Nc cells labeled by cell indices Each cell is assigned at random a rotation operator 6v chosen from a set Q of rotation operators. The center of mass velocity of the particles in cell , is Vj = AT1 JTJj v where is the instantaneous number of particles in the cell. The postcollision velocities of the particles in the cell are then given by... 

Multiparticle collision dynamics can be generalized to treat systems with different species. While there are many different ways to introduce multiparticle collisions that distinguish between the different species [16, 17], all such rules should conserve mass, momentum, and energy. We suppose that the A-particle system contains particles of different species a=A,B,... with masses ma. Different multiparticle collisions can be used to distinguish the interactions among the species. For this purpose we let V 1 denote the center of mass velocity of particles of species a in the cell i ,3... 

Let D(vx, vy, vz) be the product density in the center-of-mass velocity space. The signal under the above condition, i.e. for a given Doppler selection and at a given time bin, is... 

Now consider a Newton sphere with a distribution of the center-of-mass velocity v. The corresponding signal can be expressed as... 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

Fig. 21. The product D-atom velocity-flux contour map, d

Fig. 6. Variation of the center-of-mass velocity with beam crossing angle, 7, for the reaction 0(3P) + C2H2 with the 0(3P) and C2H2 beam velocities of 2798ms-1 and 827ms 1, respectively. The Newton diagrams for 7 = 45°, 90°, and 135° are shown (see text).

In Eqs. (5.1) and (5.2), m is the reduced mass of the colliding system, V is the interaction potential at ion-molecule separation r, 6 is the angle between the direction of r and the center-of-mass velocity, and the dot indicates differentiation with respect to time. Integration of (5.1) just gives the angular momentum L, which is conserved in the collision. Substitution in (5.2) gives... 

Assuming that R, is the same as the center-of-mass velocity R for all i and invoking the preaveraging approximation, we get... 

Figure 13. Cartesian [center-of-mass (CM)] contour diagrams for NH+ produced from reaction of N+ with H2. Numbers indicate relative product intensity corresponding to each contour. Direction of N+ reactant beam is 0° in center-of-mass system. For clarity, beam profiles have been displaced from their true positions (located by dots and 0°). Tip of velocity vector of center of mass with respect to laboratory system is located at origin of coordinate system (+). Scale for production velocities in center-of-mass system is shown at bottom left of each diagram (a) reactant N+ ions formed by impact of 160-eV electrons on N2 two components can be discerned, one approximately symmetric about the center of mass and the other ascribed to N+(IZ3), forward scattered with its maximum intensity near spectator stripping velocity (b) ground-state N+(3/>) reactant ions formed in a microwave discharge in N2. Only one feature is apparent—contours are nearly symmetric about center-of-mass velocity.12 ...

The cartesian velocities of all the molecules were transformed to relative and center of mass velocities. 

The center of mass velocities were multiplied by (TjTr)1/2 and the relative velocities were multiplied by (T/TR)l/2. 

Since the relative speed v appears in the integrand in Eq. (2.18), it will be convenient to change to the center-of-mass velocity V and the relative velocity v (where v = u ). We find... 

That is, the Maxwell-Boltzmann distribution for the two molecules can be written as a product of two terms, where the terms are related to the relative motion and the center-of-mass motion, respectively. After substitution into Eq. (2.18) we can perform the integration over the center-of-mass velocity Vx. This gives the factor y/2iVksTjM (IZo eXP( —ax2)dx = sjnja) and, from the equation above, we obtain the probability distribution for the relative velocity, irrespective of the center-of-mass motion. 

The differential cross-section refers, as in the classical description, to the scattering angle in the center-of-mass coordinate system. In order to relate to experimentally observed differential cross-sections, one has to transform to the appropriate scattering angle. This transformation takes the same form as discussed previously, essentially, because the expectation value of the center-of-mass velocity V is conserved just as in classical mechanics. 

Let us consider the collision between two atoms with masses toa and tob. The center-of-mass coordinate R and center-of-mass velocity V of the two-atom system are given... 

M = to a + to b is the total mass of the system and the superscript 0 on the velocities Vi implies the start velocities before the scattering event. The last identity in Eq. (C.l) follows from the fact that the center-of-mass velocity V is constant, since the action forces only depend on the distance between the atoms, making the total force on the system equal to zero. 

The velocity uA of molecule A with respect to the center-of-mass velocity is given by... 

Let us then go through the derivations leading to the results for non-reactive scattering events and detect where changes are necessary to get a result for a reactive scattering event. We begin by extending the definition of the center-of-mass velocity in Eq. (C.l) with the expression... 

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