Integral equations collision

Collision Integrals.—The collision integral for Eq. (1-86) is evaluated from the right side of the Boltzmann equation, Eq. (1-39), by multiplying by Y(20) and integrating we obtain ... 

Because non-adiabatic collisions induce transitions between rotational levels, these levels do not participate in the relaxation process independently as in (1.11), but are correlated with each other. The degree of correlation is determined by the kernel of Eq. (1.3). A one-parameter model for such a kernel adopted in Eq. (1.6) meets the requirement formulated in (1.2). Mathematically it is suitable to solve integral equation (1.2) in a general way. The form of the kernel in Eq. (1.6) was first proposed by Keilson and Storer to describe the relaxation of the translational velocity [10]. Later it was employed in a number of other problems [24, 25], including the one under discussion [26, 27]. 

It is necessary to get insight into the kernel of the integral equation (3.26). Since frequency exchange is initiated by non-adiabatic collisions, it is reasonable to use the Keilson-Storer model. However, before employing kernel (1.16) it should be integrated over the angle... 

In the case of weak collisions the change in J is so slight that one may proceed from an integral description of the process to a differential one, just as in Eq. (1.23). However, the kernel of the integral equation (3.26) specified in Eq. (3.28) is different from that in the Feller equation. Thus, the standard procedure described in [20] is more complicated and gives different results (see Appendix 3). The final form of the equation obtained in the limit y — 1, to —> 0 with... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

This integral equation for Q reduces in the limit of instantaneous collisions (t (Aco)-1) to a closed differential equation for 1ITZ ... 

A derivation of an integral equation for Fr will now be presented which closely follows the procedure used by Lindhard et al. and uses the nomenclature of Weisman and Sigmund . A particle, which starts at x — 0, may or may not have suffered a collision after moving a distance 8R = 8x/cos P nevertheless, the final distribution will not have changed. Therefore, Fr based on initial conditions is equated to an expression for Fr based on conditions which prevail after the particle has traveled a distance 5R. 

This is substituted into (6.23) to obtain the integral equation for the collision state... 

In this section we first summarise the meaning of the notation for the channel and collision states with box normalisation and in the continuum limit L —> 00. We then define notation for the limit 6 —> 0-1- and write the corresponding integral equations. 

Introduction of the energy width e enabled us to write an integral equation (6.26) for the collision state. The limiting procedure is represented by... 

The distorted-wave integral equation for the full collision state, corresponding to (6.81), is... 

Lindhard J, Nielsen V, Scharff M, Thomsen PV (1963) Integral equations governing radiation effects (Notes on atomic collisions (III). KDan Vidensk Matematisk-fysiske Meddelelser 33 1-42 Lindhard J, Nielsen V, Scharff M (1968) Approximation method in classical scattering by screened coulomb fields (Notes on atomic colhsiorts (I). K Dan Vidensk Selsk Matematisk-fysiske Meddelelser... 

Since equation (1.47) is valid at the strong collision limit for all possible values of k f, we can perform a simple expansion around this limit. The integral equation (1.42) can be solved by iteration - it... 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

Fxmctions Adj, Bdj, Fdj and Gdj satisfy the linear integral equations with the linearized operators of elastic collisions and inelastic ones with internal energy transitions. 

These integral equations (4.88) and (4.89) for q(u) and F u) may be solved by the same procedure as used in the case of the energy variable. It is easy to show, for example, that the collision density in the first interval F u) is given by... 

These results are entirely consistent with those of our previous analysis of the bare reactor using the Fermi age model (refer to Sec. 6.3). In this formulation, Eq. (8.281a) describes the neutron-energy spectrum and is easily recognized as the integral equation for the collision density in an infinite homogeneous system. If we select, for example [cf. Eq. (4.36)],... 

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