content_image/350387.jpg content_image/350389.jpg content_image/350388.jpg content_image/350390.jpg content_image/350391.jpg content_image/350392.jpg
## B. Collisions of type (32) n 

The first step in analyzing (32) n or (2)(32) n is to find the relation between the first and second scattering angles. Let the first and second scattering phasors be x = exp( iα ) and y = exp( iβ ) . Suppose the particles 2 and 3 initially have unit velocity. Then following the first collision, the velocities of 2 and 3 are x and ( a + x ) / ( a + 1) . The second collision is between 3 and 1 and we should like to know its scattering angle, φ. If we exponentiate the equation derived above for scattering angle, we get: y = exp( iβ ) = − exp(2 iψ ) / exp(2 iφ ) . In the present case, exp(2 iφ ) = [( a + x ) / ( a + 1)] / [( a + x ) / ( a + 1)] ∗ = x ( a + x ) / ( ax + 1) and exp( iψ ) = x, so that: 

<img src='content_image/350387.jpg'>

Now there is only one unknown variable in the problem, namely x. It is found by insisting that, in the particle-1 frame, 2 have zero velocity finally. To find 2’s final velocity, we need to multiply the string of scattering matrices. Since only the first collision has a unique scattering angle, the latter 2 n − 2 collision matrices may be expressed as a power to which the first two scattering matrices are appended: 

To exponentiate S$_{2}$S$_{3}$, we need to diagonalize it. 

The trace and determinant of the matrix are given by: 

<img src='content_image/350389.jpg'>

<img src='content_image/350388.jpg'>

<img src='content_image/350390.jpg'>

<img src='content_image/350391.jpg'>

Since the determinant is the product of the eigenvalues, the eigenvalues may be expressed as ϵ$_{1}$ = y/g and ϵ$_{2}$ = y · g, where g is to be determined. Furthermore, since the trace is the sum of these, we have: 

<img src='content_image/350392.jpg'>

