Center-of-mass system

The cross-sections of the various reactions observed are shown as functions of the kinetic energies in the center of mass system and in the laboratory system by Figures 4, 5, and 6. In all cases, the cross-sections for reactions between X+ and H2 or D2, respectively, are equal if one compares them at the same relative kinetic energy. According to the... 

Figure 4. ArH +/ArD + from the reactions of Ar + with HD and HD + with Ar as a function of energy in the center of mass system...

A disadvantage of Langevin thermostats is that they require a (local) reference system. Dissipative particle dynamics (DPD) overcomes this problem by assuming that damping and random forces act on the center-of-mass system of a pair of atoms. The DPD equations of motion read as... 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

The solution of the problem of the proper mass dependence of the relativistic corrections of order (Za) may be found in the effective Hamiltonian framework. In the center of mass system the nonrelativistic Hamiltonian for a system of two particles with Coulomb interaction has the form... 

Figure 10.2 Schematic view of a nuclear reaction in the laboratory and center-of-mass systems. [From Weidner and Sells (1973).]...

This assumes that we are making the measurement of the emitted particles angular distribution in the frame of the moving compound nucleus. In the laboratory frame, there will appear to be more particles emitted in the forward direction (with higher energies) than are emitted in the backward direction due to the motion of the center of mass system. 

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 ...

Figure 17. Rate constants for reactions of Oj"( 4IIU) with N2 as function of kinetic energy in center-of-mass system A and indicate data taken under slightly different experimental conditions.4 ...

Figure 31. Variation of collision-induced dissociation thresholds (in center-of-mass system) for reactions indicated as function of source pressure.1080...

The experiments are carried out at fixed laboratory angles 0tab, and the corresponding center-of-mass system scattering angle and its resolution is... 

Figure 37. Sudden transitions from center-of-mass-system to body-fixed system.

The complex phase shift can be obtained from exact numerical solution of the radial Schrodinger equation.2 The following quantities can immediately be given in terms of 8r The differential elastic cross section in the center-of-mass system... 

Equation (2.39) gives the nuclear Hamiltonian in the center-of-mass system. If wbc( ) represents the vibrational potential of the free BC molecule, the interaction potential Vi vanishes asymptotically [see (2.42)] and the full Hamiltonian becomes... 

The total Hamiltonian including all nuclear degrees of freedom is given in the center-of-mass system by... 

Fig. 11.1. Center-of-mass or Jacobi coordinates R and r used to describe the fragmentation of a triatomic molecule ABC into A and BC. S and s are the centers-of-mass of ABC and BC, respectively, v is the relative velocity of the recoiling fragments in the center-of-mass system. The space-fixed z-axis is parallel to the vector Eo of the electric field, while the body-fixed z -axis is parallel to the scattering vector R at all times. The azimuthal angle ip, which is not indicated in the figure, describes rotation in the plane perpendicular to R.

After a proper Boltzmann averaging of the integral cross section over the kinetic energy in the center of mass system, ECM, one get the rate constants,... 

We consider a microscopic polyatomic system consisting of N nuclei and n electrons (1-4). Let the positions of the nuclei be described by the radius vectors Rx (a = 1,..., N). If the polyatomic system is free of external force, the total linear momentum is conserved and thus its center of mass moves with a constant velocity vector (5). Consequently, a new coordinate system with its origin fixed at the center of mass can be introduced (the center-of-mass coordinate system), where the description of the polyatomic system can be simplified. Since the position of the center of mass of the entire polyatomic system practically coincides with the position of the center of mass of the nuclear subsystem, the number of the degrees of freedom, F, of the nuclei in the center-of-mass system can be reduced by 3 due to the translation of the center of mass, and by 3 connected with the overall rotation about the center of mass (in case of a linear polyatomic system, the reduction due to the overall rotation is only by 2) (5) i.e., the number of independent nuclear coordinates is F = 3N — 6 (3N — 5). The radius vectors Rx can be then expressed in terms of F generalized coordinates Q (5) ... 

The coordinates QJ generate an F-dimensional vector space (or manifold) M-the configuration space of the nuclei. The positions of the nuclei in the configuration space M are given by a single point, the system point Q = QJ. Analogously, let the positions of the electrons in the center-of-mass system be given by the coordinates q = qu, u — 1,..., 3n. 

Worsnop et al., in a study on the exchange of vdW bonds in Xe + At2 collisions, found that the product velocity vectors in the center-of-mass system fall on a narrow drcular band. This is a consequence of the weakness and similarity of the vdW bond strength (D 1 kJ/mol) compared with the collision energy ( = 6-15 kJ/mol). There is a hole in the XeAr distribution for 0 < 45°, with 6 measured from the initial Xe atom direction (Fig. 9). One could perhaps expect to a observe a stripping mechanism in this reaction. 

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