Trajectories particles moving along

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

When we study the effect of charged particles on a substance, we often need to estimate the probabilities of excitation or ionization as functions of the distance from the axis of the track (i.e., of the impact parameter b). This is done using the quasi-classical approach, within which we assume that the charged particle moves along a definite trajectory. In the... 

Explanation of the Theorem. The idea of the theorem is following. Let us choose some instant t = to and consider a chain of the particles termed the to chain. These particles are characterized by the same phase point T and lifetime tv but have phases 0 differing by 2nn, where n is integer. The first particle of this chain starts interaction with field E(t) after a strong collision occurring just at the instant t0 the second particle started its interaction at a period 2n/(t> of the field E(t) earlier the third particles started at two such periods earlier, and so on, until the last particle of this chain (before it had exerted a strong collision). Because of a periodicity of a.c. field, each particle moves along the same trajectory. 

The power radiated by a particle moving along a circular trajectory with radius p is given by the relation ... 

Displacement of particles along the near part is similar to displacement along the circumference so that inertia forces appear as centrifugal forces inhibiting deposition. The primary effect of inertia forces on the near part of the trajectory at St < St was determined by Dukhin (1983). A displacement of the particle trajectory with respect to stream-line 1 which is the grazing trajectory is shown in Fig. 10.14. After displacement, particles move along the stream-line 1 away from the bubble and do not touch its surface. 

In another limiting case of highly inertial particles, it is possible to assume that particles move along straight trajectories. Then the cross section of collision can be found from simple geometrical arguments ... 

The particle s motion from the point x ti) to the point x(ii) may, in principle, occur along one of an infinite number of trajectories. In reality the particle moves along the tra-... 

In order to study the peripheral classification we injected a particle with zero velocity through the upper orifice near the chamber axis. For all our computations we used Discrete Random Walk Tracking (DRWT) option of Fluent code to take into account the influence of air velocity random component, caused by turbulence (k-e model), on particle trajectories. In Fig. 4, for example, one can see calculated trajectories for particles with various sizes. Large (200 pm ) particles move along the complex polygonal trajectory (Fig. 4-... 

As a result, the particle moves along a curved path (Fig. 2, path 2 and 3). Obviously, the smaller the mass and the greater viscosity of the gas, the closer it will be to the trajectory of the circle (Fig. 2, path 3). 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

Because particles have wavelike properties, we cannot expect them to behave like pointlike objects moving along precise trajectories. Schrodinger s approach was to replace the precise trajectory of a particle by a wavefunction, i]i (the Greek letter psi), a mathematical function with values that vary with position. Some wavefunctions are very simple shortly we shall meet one that is simply sin x when we get to the hydrogen atom, we shall meet one that is like e x. 

If the angle (3 is much less than 1, then, in accord with Figs. 7 and 9, the most part of the rotators move freely under effect of a constant potential U0, since their trajectories do not intersect the conical cavity. A small part of the rotators moves along a trajectory of the type 1 shown in Fig. 10. However, at d > (3—that is, in the most part of such a trajectory—they are affected by the same constant potential U0- Therefore, for this second group of the particles the law of motion is also rather close to the law of free rotation. For the latter the dielectric response is described by Eq. (77). We shall represent this formula as a particular case of the general expression (51), in which the contributions to the spectral function due to longitudinal A) and transverse KL components are determined, respectively, by the first and second terms under summation sign. Free rotators present a medium isotropic in a local-order scale. Therefore, we set = K . Then the second term... 

Noting that dr/dt v, the angle between v and VR is 8, where tand = (dr/dt)/v. As the particle moves inward towards 0, the component of T along the spiral trajectory is responsible for increasing its speed. 

Particles with initial velocities in different directions will move along different trajectories, and so need different times for falling down to a certain height. The residence time distribution of particles in this stage can be determined with the motion equation, which is obtained from the force balance, while neglecting the buoyant force, as follows ... 

Equation (30) gives the energy that a charged particle moving with constant velocity along an arbitrary trajectory loses due to electronic excitations in an electron system that is translationally invariant in two directions, as occurs in the case of a simple metal surface modeled by jellium. 

Now picture your particle moving toward the surface of the liquid. If it continues along the same trajectory when it gets to the surface, its momentum (mass times... 

The liquid flow envelops the bubble surface, and the particles are entrained to a greater or a lesser extent by the liquid. The smaller the particles and the less different their density relative to that of the medium, the weaker are the inertia forces acting upon them and the more closely the particle trajectory coincides with the liquid streamlines. Thus, at the same target distance fairly large particles move almost linearly (Fig. 10.1, line 1), while fairly small particles move essentially along the corresponding liquid flow line (line 2). The trajectories of particles of intermediate size are distributed within lines 1 and 2 as the size of particles decreases, the trajectories shift from line 1 to line 2 and the probability of collision decreases. 

Strictly speaking, the size of the scale Tq [appearing in (5.34) and (5.35)] must exceed the threshold value T,h PmD. Here m denotes the mass of the diffusing particle and / = l/hgT with kg the Boltzmann constant. For intervals of time shorter than r., the particle motions are not controlled by collisions with other particles of the medium. Instead, the particles move freely along ballistic trajectories, R(t) = R(0) + (p/m)t, as they do in the example of a dilute gas to which we now turn. Specifically, we consider the van Hove intermediate scattering function for a single thermalized particle. 

Let us show how one can And the limiting trajectory and collision frequency of particles with the cylinder without solving these equations. The limiting trajectory (see Fig. 10.9) ends at the back point Op =n, y = y. To determine y, one should proceed as follows. When the particle is moving along the critical trajectory near the back stagnation point, the velocity is Ur = = 0 and since the particle mo-... 

Cells undergoing positive dielectrophoresis move in the radial direction of the merging of the curved rear electrode and the flat front electrode, whereas cells experiencing negative dielectrophoresis move in the opposite direction. Meanwhile, the bulk carrier flow perpendicular to this electrical force carries the cells forward toward the carrier exit. The cells or particles unaffected by the nonuniform electrical field continue to move along the line of original introduction. Thus different cells trace out different trajectories and therefore can be withdrawn at different locations from the stream near the liquid outlet. 

Since r and z are the coordinates of a particle of radius tp and density as it moves along its trajectory, equation (7.3.93) defines the smallest value of the radial location r of the particle at z = 0, where it must be located if it is to hit the wall (r = Tq) and be captured when z = L. If the particle is located at a value of r greater than that defined by equation (7.3.93) at z = 0, then it will surely hit the wall (r = Tq) before z = L. However, if the particle at z = 0 is located at an r smaller than that defined by equation (7.3.93), it will not hit the wall (r = to) by the time z = L then the particle escapes with the overflow. A critical radius r is thus identified with each particle size rp via equation (7.3.93). 

---