It follows from Eq. 1.8.8, in particular, that at rL/r0 < 1 any scattering angle 9 is possible (depending on the impact parameter r), and at rL/r0 > 1 the angle 9 is limited by the value
9max = 2arctg
It can be easily seen that at rL/r0 << 1 the Eq. 1.8.8 gives the classical cross section of scattering by a solid sphere of radius r0 :
1.8.5. Scattering by cylindrical fibers with field of type h = M/rn
The law of conservation of the angular momentum gives for n ^ 2 the following equation for the particle trajectory in the field h = M/rn :
where r, 6 are the cylindrical coordinates of particle motion; ds is the element of the trajectory length; C1 and C2 are constants. Inserting the new unit length we obtain l = (AeM/pc )n-1
where r1 = r/l, s1 = s/l . At n = 3 the Eq. 1.8.12 determines the Stormer unit (see in Dorman et al., M1971). Fig. 1.8.3 shows the dependence of the scattering angle 6 on the impact parameter r1 (in units l) for dipolar field (n = 3).
The dependence of 6 on r1 for a value of 6 smaller than some small 6o may be approximated by the expression 6 = 6o (r1^ /r12), where r1o is the value of the impact parameter at small deflection 6o . From this we get for r1 >> 1:
For other r1 the value of da is determined from Eq. 1.8.6 using the plot shown in Fig. 1.8.3.
1.8.6. Three-dimensional model of scattering by inhomogeneities of the type h = (0, h(x ),0) against the background of general field Ho = (Ho ,0,0)
Parker (1964) considered the scattering of a particle moving along the basic field Ho = (Ho ,0,0) in the x direction from to by an inhomogeneity h = (0,h(x),0), where h(x) was set in the form h(x) = dF/dx (i.e. F(x) is the field flux from to x). The equations of particle motion in such field will be written in the form d2x dz dF d2y dz d2z dx dF dy „ ,
—T =--wl-, —2- = —Wl, —— = — Wl---—Wl , (1.8.15)
dt2 dt dx dt2 dt dt2 dt dx dt where mL = ZeHojMc is the cyclotron frequency. Integration of Eq. 1.8.15 gives
^ = WLZ, ^ = wl (F - y), f+ f iL f + f ^f = v2, (1.8.16)
where v is the particle velocity. It follows from Eq. 1.8.15 and Eq. 1.8.16 that
dt2 dt the solution of which is t y (( ) = œL J F (x )sin(L (t-t))t. (1.8.18)
Assuming that the scattering is inconsiderable, we may set x ~ v//1. Then the scattering angle 6~ v±/v// at t ^^ , where v±=((dy/dt)2 + (dz/dt)2) will be for the three types of inhomogeneities (j = 1, 2, 3 respectively; see Fig. 1.8.4) 1:
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