Fig. 4.7.4. A scheme of determination of the particle velocity change during non-mirror interactions for head-on collisions.
Let the cloud velocity be u, the particle velocity v, and u Ti v . Fig. 4.7.4 shows the vectors OA = u, OB = v, and BC = -u . In a coordinate system related to the cloud the particle velocity will be
OC = vc = v - u; |vc| = (v2 + u2 - 2vucospj ; (p = n. (4.7.5)
In this coordinate system the particle energy fails to vary, and as a result of scattering, the particle velocity vector will turn through angle 9 so that
is the particle velocity vector after scattering in the coordinate system related to the magnetic cloud. In Eq. 4.7.6
where 9c = ZCOD, i.e. the angle between vectors vc and v'c (scattering angle in the coordinate system related to the magnetic cloud). Since the magnetic field in the cloud may be arbitrarily oriented, the scattering vector b rotates around the OC axis and circumscribes a cone surface. After scattering in the coordinate system of the cloud the velocity vector OD will circumscribe a similar cone around the OC axis. Now we shall again use the laboratory coordinate system, i.e. we shall add the vector OA to the vector OD; as a result we shall obtain for the particle velocity vector OD' = v' after scattering in the laboratory system:
It can be seen from the scheme in Fig. 4.7.4 that the angle between the vectors v and b is n 9
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