Collision vector

Here U is the orientation of vector u before the collision. This only has physical sense for angle S = arccos ( ), which is an even argument of the function ip. This means that, as a result the collision vector u may be oriented inside a cone, whose axis coincides with the position of A characteristic value of conical angle S depends on the width of function ip. 

A further idea introduced in [184] was to use a force decoupling strategy to reduce the cost of computations. The idea is to exploit the observation that impulsive forces ( kicks ) can be supplied without reducing the order of accuracy as long as these have a vanishing component in the direction of the collision vector that is if F = —VUiqc) u = 0. This can be achieved in systems of spheres with pah-potentials (fij only by writing hybrid method that uses only the first part a to define the quadratic Verlet paths for the collision detection scheme, whereas fi is introduced as a standard kick at collision points. 

The first term gives the ideal part of the pressure, Pid, as discussed in [21]. The average of the second term is the non-ideal part of the pressure, Pn. zj, is a vector which indexes collision partners. The first subscript denotes the particle number and the second, /, is the index of the collision vectors ai in Fig. 3 (left panel). The components of Zy are either 0,1, or -1 [55]. Simulation results for P obtained using (62) are in good agreement with the analytical expression, (61). In addition, measurements of the static structure factor S k —> 0,1 = 0) agree with the thermodynamic prediction... 

Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

Figure 6 shows a two-dimensional schematic view of an individual ion s path in the ion implantation process as it comes to rest in a material. The ion does not travel in a straight path to its final position due to elastic collisions with target atoms. The actual integrated distance traveled by the ion is called the range, R The ion s net penetration into the material, measured along the vector of the ion s incident trajectory, which is perpendicular to the... 

Fig. 9.6 Collision of two hard-spheres shown in a reference frame in which sphere 1 is at rest n is a unit vector along the line of the two centers of the colliding spheres.

Some Vector Relations.—Since the relative motion takes place in a plane, one farther parameter must be given in order to describe the collision in three dimensions, namely the orientation of the relative-motion plane. This may be done as in Fig. 1-4, where the collision... 

We have used the transformation of Eq. (1-62) the definition of % in Eq. (1-71) and = (ml2kT)llztya — v] furthermore is the (reduced) velocity vector of the first particle after the collision. Expanding the polynomials, we find noting that the collision... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

Without essential limitation of generality it may be assumed that the orientation of the molecule and its angular momentum are changed by collision independently, therefore F(JU Ji+, gt) = f (Jt, Ji+i)ip(gi). At the same time the functions /(/ , Ji+ ) and xp(gi) have common variables. There are two reasons for this. First, it may be due to the fact that the angle between / and u must be conserved for linear rotators for any transformation. Second, a transformation T includes rotation of the reference system by an angle sufficient to combine axis z with vector /. After substitution of (A7.16) and (A7.14) into (A7.13), one has to integrate over those variables from the set g , which are not common with the arguments of the function / (/ , /j+i). As a result, in the MF operator T becomes the same for all i and depends on the moments of tp as parameters. 

By adopting the basic assumption that the probability density of the direction distribution of particle velocity vector is uniform in the whole space, i.e., 0(0,jS) =sin 0/4tt, the probability of collision between the wall and a particle located S away from the wall (see Fig. 7) can be expressed as the following equation ... 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

Particles are moved along their current velocity vectors without undergoing interactions for a time At which is chosen smaller than the mean collision time. If a particle hits the domain boundary, its velocity vector is modified according to the corresponding boundary condition (for example specular or diffuse reflection if a particle hits a wall) ... 

Based on the molecular collision cross-section, a particle might undergo a collision with another particle in the same cell. In a probabilistic process collision partners are determined and velocity vectors are updated according to the collision cross-section. Typically, simple parametrizations of the cross-section such as the hard-sphere model for monoatomic gases are used. 

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