# Info

where

ctg0

Applying Eq. 4.15.36 it is possible to obtain the differential equation which determines the relation of pitch angle and phase along a trajectory. The solution of this equation is sin2 0[2(n-ç -ç)+ (1 - cos Y)sin2ç + sin Y sinçctgO

where C is a constant of integration. Similarly to the Eq. 4.15.27, the Eq. 4.15.38 implies the conservation of the magnetic flux O which is enclosed by a particle during one revolution. A particle's behavior is essentially dependent on Y. At Y = n/ 2 the particles moving from the top (x > 0) cross the plane of discontinuity, and conservation of O means that 6k (the pitch angle of the particles coming down to x < 0) is equal to the initial pitch-angle. For moving particles upward, conservation of Ok does not require any restriction of the final pitch angle. For nl2<Y<n a portion of the particles of the upper region with pitch angle 0 <do < Y - n/2 will cross the discontinuity with conservation of their pitch angle, and another part of them will be reflected returning to the top region with the pitch angle = Y-9o. If Y-n/2<6o <n/2 a particle will cross the discontinuity plane and Eq. 4.15.38 results in Ok be equal either to Oo or to n - Oo. The particles moving from below with pitch-angle Y < 0o <n will cross the discontinuity (Ok = Oo) but if n/2 <do < Y a partial reflection of the particles into lower half space (Ok = Y-Oo) will occur and a fraction of particles will cross the discontinuity plane (Ok = Oo ).

Using the integrals of motion in the field of a rotational discontinuity

h = Py +- | z + |x|ctg«sm— |; I2 = pz +-— I y - xctg«cos—