whence we find on taking into account of Eq. 4.7.7:
|v'|2 = v2 + b2 + 2vb = v2 + 4(vu + u2 )sin2 (9c/ 2). (4.7.10) Since E - macc it follows from Eq. 4.7.10 that
Similarly, for overtaking collision (v TT u) we obtain (in this case p = 0):
It can be easily seen that if 0c =n then Eq. 4.7.11 and Eq. 4.7.12 turn out to be the corresponding expressions from Sections 4.2 and 4.3 for mirror reflection.
From Fig. 4.7.4 can be seen that for small angle scattering the relation between scattering angle 0c in the cloud coordinate system and scattering angle 0 in the laboratory coordinate system is
Substituting Eq. 4.7.13 in Eq. 4.7.11 and Eq. 4.7.12 we obtain
/Ar./rn\ 2(vu + u2 ). 2 v0 /Arnirn\ 2(- vu + u2). 2 v0
It is easy to see that at v >> u according to Eq. 4.7.13 0c ~0, and instead of Eq. 4.7.14 we obtain
(AE/E)+=^—5-'sin2—; (AE/E)_=^-5-¿sin—. (4.7.15)
4.7.3. Energy change in non-relativistic case for oblique collisions
Consider now an oblique collision (see Fig. 4.7.5). Here, as in Fig. 4.7.4, the cloud velocity u = OA, the particle velocity in the laboratory coordinate system v = OB, the angle between u and v is p = ZAOB.
Using the coordinate system related to the magnetic cloud we find the velocity of particle
In this coordinate system, particles are scattered through angle 9c and, as a result, the vector OC is transformed into
Let us note that actually, the resultant vector circumscribes a cone around axis OC with half-angle 9c of cone apex O. The scattering velocity vector b, which module determined by lb = |bj = |b2| = 2 v - u| sin-9-, (4.7.19)
circumscribes the cone D1CD2 with the half-angle n/ 2 -9c/ 2 of cone apex C (in this case the section of the cone by the plane running through vectors u and vc is determined by the vectors CDj = bj and CD2 = b 2). Using again the laboratory coordinate system we shall find the particle velocity vector after collision with the cloud:
OD'1 = v'1 = v'c1+u = v + b1; |v'1 = (v2 + b2 + 2vb1cos 9 J2, (4.7.20)
01=ZD\BO = -n-V—^-X; 02 = AD\BO = -- + + Z. (4.7.22)
X = ZOCB = arcsin i ■ ^ t \ v sinv I I [2.2 0 -.-¡- ; v - u = lv + u - 2vu cosV.
It may be assumed that approximately
Substituting Eq. 4.7.19 in Eq. 4.7.20-4.7.21 and then in Eq. 4.7.24 and considering
that at non-relativistic energies E ~ macc we shall obtain, after tedious trigonometric transformations, the relation
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