Integrals over collision angles

We would now like to evaluate explicitly the integrals in the collision terms appearing in Eq. (6.23). We will do this separately for each term involving the collision angles, and then reconstruct the final result at the end. The term V iACv) can be written in the collision frame of reference using xu = L xj2 

Upon applying the multinomial theorem multiple times, we then find 

Considering the first integral in Eq. (6.23) and using the result in Eq. (6.30), after a change of reference into the collision frame, the two integrals over the collision angles are 

Using the fact that n and 2 are even integers, this expression can then be rewritten as 

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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... 

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. 

We assume again that the atoms follow straight line trajectories, and we calculate the transition probability, P(b), from the initial to the final state in a collision with a given impact parameter, b. We then compute the cross section by integrating over impact parameter, and, if necessary, angle of v relative to E to obtain the cross section. The central problem is the calculation of the transition probability P(b). The Schroedinger equation for this problem has the Hamiltonian... 

The reader should not confuse the spherical angles (0i and 0i) (which parameterize v ) with the collision angles (0 and 0) (which parameterize X12). Indeed, the integrals over the collision angles are done with fixed values of the spherical angles (i.e. fixed values of v ). 

Using the analytical expressions for the integrals over the collision angles, we can now rewrite the collision source term in Eq. (6.21) as the sum of two contributions. 

The reader will recognize that the coefficients are related to the integrals over the collision angles discussed in Section 6.1.4. The exact definitions are... 

Numerical cross sections for such reactions can be obtained by calculating the fraction of hindered collisions from plots of ion-dipole orientation angle and then multiplying this fraction by the capture ratio. The product, an eflective reaction ratio, is then integrated over impact parameters to give the reaction cross section. 

Collision termc ksJ j f x,m ) pa m m )dm. It represents the source of photons in the phase space volume dxdm in relation to photons with propagation direction m that are scattered at x in the direction m (see Fig. 8D). c ksj[x,m )dxdm is the rate at which photons within dxdm are scattered, andpsi m m ) is the probabifity density for their scattering direction to be soHd angle accounts for all incoming directions m. ... 

Normally, one assumes that the differential reaction cross-section does not depend on the azimuthal angle. Hence, one takes advantage of the axial symmetry of the collision and works with differential cross-sections integrated over . Thus, one uses the element of sohd angle dm = 27t sin 0 dO ... 

A similar expression can be adopted integrating over the solid angle ii and assuming that the dependence of a, and a, on scattering angles is weak for hyperthermal collisions, to obtain... 

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