mac c2 c2
The result obtained coincides with Eq. 4.3.2 for non-relativistic case.
Let us determine now the frequencies of the head-on, v+, and overtaking, v_, collisions in the relativistic case. We need to take into account the relativistic summation of velocities of particle and cloud (see Eq. 4.3.7 and Eq. 4.3.8), and the relativistic transformation of the transport path:
Therefore the frequencies of the head-on, v+, and overtaking, v_, collisions in the relativistic case will be v ± u v ± u /„ -, , v±=-i-—, (4.3.11)
i.e. the same as was obtained for non-relativistic case (see Eq. 4.3.3). Considering Eq. 4.3.9 and Eq. 4.3.11, we obtain that the mean variation of energy with time in the relativistic case is dE/dt = (AE)+ v+ + (AE)- v_=aE , (4.3.12)
i.e. the parameter of acceleration a is two times larger than was obtained in Fermi (1949): compare with Eq. 4.2.4.
4.4. Development of the Fermi model: inclusion of oblique collisions
In this case the particle velocity components perpendicular and parallel to the front of cloud will be vL = uv/u and v// = v(l- (uv/uv)2 ) . In the coordinate system related to the cloud, we shall obtain v'± = v± — u; v'// = v//. After reflection in the coordinate system of the cloud v'^2 = —v'x ; v'//2 = v'//. When, after that, the laboratory coordinate system is again used, we shall obtain for the particle velocity after collision that v^2 = v'^2 +u = —v± + 2u = — uv/u + 2u , (4.4.1)
v//2 = v//2 = v whence vf = v22 + v//2 = v2 — 4uv + 4u2 . (4.4.3)
It follows from Eq. 4.4.3 that
Ek2 = Ek - 2uvmac + 2u2mac, (4.4.4) whence the energy change in a single collision is
The relative change of energy in a single collision can be found to be
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