Collision angles and the transformation matrix

The integrals over the collision cross section S+ can be computed analytically by defining a linear transformation i(vi2) from the laboratory frame of reference into the collision frame of reference (Fox Vedula, 2010). The linear transformation does not depend on xn (only on V12) and thus it can be taken outside of the integrals. We can define a transformation 

The transformation matrix from the laboratory frame (x) into the collision frame of reference (x ) depends only on the relative velocity vector V12 = vi - V2. By convention, V12 is aligned with the x axis. Thus, the orthonormal transformation matrix defined such that x = Lx can be written as 

Note that the ordering of the first and second rows of L is not important, and that L is only defined for V12 0. The components of L can be written as functions of v, but they could equally well be denoted by T,j(0i, 0i), i.e. as functions of the spherical angles 0i and 0i. 

The following pseudo-code illustrates how the spherical angles can be computed from the vector y = V12  

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 ). 

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