By Peter W. Hawkes (Eds.)
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The beam axis is described by the equation r = Re" (Fig. 20). To move along this spiral, D. 20. Coordinate system to describe ion trajectories in spiral electrode deflectors. the main path ion must have the energy U = -eER(l + p2)/2. This is the zeroth order condition that defines also the parameter p. We work with the following small quantities: 5 = (;)e-@ - z 1 and [ = -. R With the help of the zeroth order condition we transscribe the differential equations through the variables 4; and 5: & p + + ( 3 p 2 + 7)@ - 3p5'6 - (1 + p2)d2 + (1 + p 2 ) e - 2 @ [ ' 2 + pe-2'y'"f, ['f- p" = 0 These differential equations can be solved by the conventional successive approximation method.
For nonrelativist ions U << rnc: and q = 1. We consider now the fringing field transfer matrix. In the neighbourhood of the electrode and pole piece ends the fields deviate from the ideal values. They become neglibible at some distance outside the ion optical element. The field distribution part located between a plane where the real fields differ notably from the ideal values and another plane where the fields may be neglected define the fringing field region. For both magnetic and electric fields, effective boundaries may be found.
We give the field integrals later, all together. We substitute the field components and the electric potential in the trajectory Eqs. (14) and (15) in Cartesian coordinates (1/p = 0). After some rearrangements they take the form: 35 To calculate the ion trajectories by successive approximations from the above equations we use the relations: a = a, + x = x1 + :IJ: x” dz, p = B1 + I:, y” dz,y = y, + I:, p dz adz, Here the quantity without index refers to z, while that indexed “1” to zl. The ion trajectory thus obtained must coincide inside the main fields with some trajectory of an ion moving in the ideal fields.
Aspects of Charged Particle Optics by Peter W. Hawkes (Eds.)