Collision integral elastic

Collisional contributions in the above set of equations due to the elastic Coulomb interactions are the terms Rap and Qap. We now proceed to express more explicitly the source terms Sn[fa] due to atomic and molecular interactions with neutral particles. These result from the collision integrals Can in (2.1). These integrals may be considered separately for each individual collision process, and then summed up. We omit the index labelling a particular type of process in the following discussion. 

For elastic collisions, several different kinetic models have been proposed in order to close the Boltzmann hard-sphere collision term (Eq. 6.9). For inelastic collisions (e < 1), one must correctly account for the dependence of the dissipation of granular energy on the value of e. One method for accomplishing this task is to start from the exact (unclosed) collision integral in Eq. (6.68). From the definition of if given in Eq. (6.60), it can be... 

Here -Bq denotes the charge of the electrons, E(x, t) is the electric field, and C (F) and C[ (F) are the collision integrals for elastic collisions and important conservative inelastic collisions, i.e., the /th excitation or dissociation process of the electrons in collisions with the ground-state atoms or molecules of the gas. For simplicity in the further representation, only the most essential electron... 

The number of neutrons emerging from a collision is automatically determined when the collision is elastic scatter, absorption, (n, n ), ( , 2n), or an (n, 3n) reaction. At a fission, however, the number of secondaries is a function of the incident neutron energy. This function is described in the data library by the mean number of neutrons per fission, v, which is presented in the data library in a similar manner to cross section, except that, generally, fewer points are used and interpolation is linear for v against energy. The number of secondary neutrons released in a fission is an integral random variable whose expectation is equal to v. 

The collision integral Qg g is a ratio giving the deviation of a gas with interactions compared to a gas of rigid, elastic spheres. This value would be 1.0 for a gas with no interactions. Equation (6.2-44) predicts diffusivities with an average deviation of about... 

For elastic collisions (Ae,- = Asj = 0) the only collision integrals required for the complete set of first approximations are the integrals and given by... 

For the elastic-collision case the quantities in Eqs. (3.46) are again available in tables as a function of the reduced temperature T. From the point of view of computation, it is convenient also to have the collision integrals and their ratios expressed as polynomials in T. The appropriate polynomial coefficients for a fourth-order fit of and second-order fits of B, and C over limited ranges of T are given in Table 2. 

For inelastic collisions the Q integrals cannot yet be calculated from first principles. The procedure adopted with polyatomic gases is therefore to treat them as if the collisions were elastic, that is, to use tables of elastic-collision integrals and to derive effective values of eps)ij/kg and (si)y from the temperature variation of the transport properties. The first approximations to the binary diffusion coefficients l ij] 1, as well as and (1.2C — 1), may... 

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... 

The factor 1/2 in front of the double summation is to take care of double counting. All collisions are considered to be completely elastic. To integrate forward to the next position, the computer replaces continuous time with small time steps for integration. 

Eq. (11.30) is strictly applicable only to elastic collisions, in which a = a, and is thus of limited utility. However, it is physically appealing to assume that the cross section o(a, a, / ,/ ) for an inelastic process a = a and ft (i can be written as the integral of the electron scattering cross section oe(J3, f, q) over the velocity distribution of the Rydberg electron in the initial state a. Making this notion explicit, we write19... 

Here. .. depends on the W-kernels introduced above, but in general stands for terms of the form cross-section, appropriately weighted by factors [1 — cosl(0)], l = 1, 2,. .., if the efficiency of exchange of quantity A in a collision depends on the scattering angle 0 in this way (as, e.g., in case of elastic neutral particle - ion collisions, see Sect. 2.2.4). In case of inelastic collision processes a is simply the total cross-section, denoted weighting exponent l = 1, 2,... is possible) or a1. In principle the detector functions qa must be obtained by numerical integration and tabulated for the parameters of the relevant distribution functions fa. 

In the DSMC technique, the probability that a chemical reaction occurs is the ratio of the reaction cross section to the elastic cross section. The most commonly applied chemistry model is the Total Collision Energy (TCE) form employed by Boyd based on a general model proposed by Bird. In this model, the probability of reaction, P, is obtained by integrating the microscopic equilibrium distribution function for the total collision energy, and equating it to a chemical rate coefficient, Kf. Specifically, the mathematical form of the probability is obtained from the following integral ... 

In this section we will briefly review the collision model for binary hard-sphere collisions using the notation in Fox Vedula (2010). The change in the number-density function due to elastic hard-sphere collisions (Boltzmann, 1872 Cercignani, 1988 Chapman Cowling, 1961 Enksog, 1921) obeys an (unclosed) integral expression of the form ... 

---