k =-1

10"D 10 s 10 10^ 10 10 10" 101 R/300Ho12

10"D 10 s 10 10^ 10 10 10" 101 R/300Ho12

Fig. 1.9.8. The same as in Fig. 1.9.4, but for j = 3 and X/ A2 = 10-5 .

1.9.8. The transport scattering path including the drift in inhomogeneous fields

Consideration will be given now to the CR propagation as a result of drift in inhomogeneous magnetic fields. In this case a particle trajectory will be a trochoidal with curvature radius rl = cp^/(ZeH) and the center of curvature will drift at the velocity

If the inhomogeneous sectors of the field fill the entire space at a certain distribution N (¿), the particle propagation will be diffusive with the transport path A eff determined by the Eq. 1.9.4 and Eq. 1.9.9; the diffusion coefficient, however, will contain not the particle velocity v, but the velocity of the drift in inhomogeneous sectors of the field vdr:

Such diffusion may be treated as particle propagation at velocity v±, but with transport path

Obviously such modes of propagation will take place only for particles with rL <¿ for which each interaction with inhomogeneity will be effective. If the field inhomogeneities fail to fill the entire space and are spaced apart on the average by a distance l, the particle will traverse the field inhomogeneities within time Atv = ¿¡vdr and pass the space between the inhomogeneities within a time

At2 = l V (¿2v); hence the mean effective velocity of particle motion will be vef =lF%r = ¿¿^. (1.9.38)

¿1 vdr + l I ¿ v ¿3v + l3vdr and the diffusion coefficient will be

where veff is determined by the Eq. 1.9.38.

1.9.9. The transport scattering path in the presence of the regular background field

It should be noted that the particles are scattered in reality by magnetic inhomogeneities practically always against a background of some regular magnetic field (which may be the field of larger inhomogeneities). In this case the scattering, and hence the diffusion coefficient, along the field is practically the same while the diffusion across the field will be significantly hampered. Let a homogeneous magnetic field exist in the space. Then in the absence of inhomogeneities the diffusion coefficient will be zero. An increase in the number of inhomogeneities will result in an increase of the diffusion coefficient, whereas, in the case of absence of the regular field, the same increase would, on the contrary, result in a decrease of the diffusion coefficient (see above the Eq. 1.9.4 and Eq. 1.9.9 which show that Aeff should decrease pronouncedly with decreasing l). Thus the regular field give qualitatively different results for the diffusion coefficient. In the general case of presence of the regular field Ho the particle motion will be an anisotropic diffusion with the diffusion coefficient along the field k// being as a first approximation, the same as that in the absence of the field, whilst the diffusion coefficient across the field will be determined by the expression

where the Larmor frequency is aL = ZeHov/cp , (1.9.41)

and the time t between the 'effective' particle collisions with field inhomogeneities (when the motion direction is significantly changed) is

If mLr2 << 1 the conventional isotropic diffusion takes place. If, however, aLt2 >> 1 the anisotropy in the particle's motion should be taken into account.

Consider now at greater length the mechanism of particle scattering in a plane perpendicular to the regular field which will be assumed, for the sake of simplicity, to be homogeneous with intensity Ho (if the field is inhomogeneous this will additionally give rise to a systematic drift). Let the size of inhomogeneities be A< R±/Ho and the field intensity in them be h. It can be easily seen that, during the time of the particle's motion within an inhomogeneity, the center of curvature will shift by | determined from the relation

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