Laboratory velocity

Figure B2.3.2. Velocity vector diagram for a crossed-beam experiment, with a beam intersection angle of 90°. The laboratory velocities of the two reagent beams are and while the corresponding velocities in the centre-of-mass coordinate system are and U2, respectively. The laboratory and CM velocities for one of the products (assumed here to be in the plane of the reagent velocities) are denoted if and u, respectively.

The dashed circle denotes the possible laboratory velocities tt for the fidl range of CM scattering angles 9. ... 

Equation (B2.3.10) shows that the scattered intensity observed in the laboratory is distorted from that hr the CM coordinate system. Those products which have a larger laboratory velocity or a smaller CM velocity will be observed in the laboratory with a greater intensity. 

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... 

Actual laboratory velocities or velocities with respect to the tube... 

The equation for q, (Eq. [123]) can be thought of as the velocity of the particle in the laboratory frame. This can be understood more completely by noting that the laboratory velocity, q,- is a sum of ajhermal (or peculiar) velocity P,/ot, and a component due to the velocity field iy y,. At the steady state, the laboratory velocity profile will be linear as in Figure 10, with the slope (the... 

Where yP is defined as the angle between the epr and v. The direction of PR can be either parallel or perpendicular to the axis of the flight path. Only the terms are multiplied by a function of %p, which will influence the shape of the TOF spectra. Empirically, it was found that all of the data could be fitted by considering only the fio(22) parameter, which describes v j correlations. In order to separate the signal dependence on P(Etrans) versus its dependence on S, we first approximated both the beam and laboratory velocity distributions with single velocity vectors. This process allows us to remove the integral of Eq. 38. With this approximation, we can write an equation for / o(22) that is independent of P(Etrans). 

As is true for the primary dissociation step, the secondary fragments recoil in all directions, so there is not a single secondary vector, but a family of them. The tips of the resultant vectors obtained from summing each of these secondary velocities with the molecular beam velocity and with the primary velocity all fall on the large circle in Fig. 2. As is true for the primary steps, the laboratory velocities corresponding to these center-of-mass velocities may be obtained from the intersection of these circles with the lines indicating the molecular beam-to-detector angles. 

The full Newton diagram is much more complicated than what is shown in Fig. 2. First, there is a set of secondary velocity vectors originating at the tip of every primary velocity vector. The circle corresponding to only one such set is shown. Second, although the detected laboratory velocity must be in the plane containing the molecular beam and detector axes (the plane of Fig. 2), the primary and secondary vectors may be out of this plane. For example, the primary step may be such that the fragment velocitj is out of the plane, but tbe secondary velocity may be such that the resultant... 

The first is an obvious choice since it has been studied more than any other reaction. It is not, however, a simple reaction to study. At laboratory energies 2 eV, it proceeds via a complex mechanism and the laboratory velocity of the should be similar to the center-of-mass velocity, ... 

Reactive and nonreactive scattering data are analyzed with the aid of a Newton diagram, illustrated in Fig. 12 for the hypothetical reaction A + BC AB + C. This vector diagram in velocity space facilitates the transformation between the laboratory and center-of-mass (CM) reference frames. Laboratory angles are denoted by 0, with 0 = 0 and 90° defined by the directions of the A and BC beams, respectively. All laboratory velocities are denoted by the variable v, and have their origins in the lower left corner of... 

Corrections for this type of effect are possible, but they require knowledge of angular distributions. We conclude that angular distributions are therefore normally necessary even for an analysis of velocity data. However, with an apparatus providing velocity spectra only, such errors do become small for very-high-energy reactions. Here, the laboratory velocity is so large that the sensitivity of the detector is similar for products emitted in any direction from the center of mass. Fortunately, most of the early uses of... 

Let us assume that the two particles A and B have the same mass and laboratory velocities. What would be the laboratory angle at which the adduct AB is formed if the two beams collide at a right angle The solution is as follows. 

---