By Peter W. Hawkes (Eds.)
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Extra resources for Aspects of Charged Particle Optics
9. Ion trajectory in a cylindric coordinate system. straight ion main path): + mac,C(ro+ w,)’ - radial equation, I’” =O (14) 18 D. IOANOVICIU e[xY?. ’l”2 1 m,c,C(II, x‘2 + W,l2 - =0 (15) axial equation. The substitution l / p = 0 gives the trajectory equations in Cartesian coordinates. 9. Space Transfer Matrix In the following paragraphs, we describe the ion trajectories through x, a, y, 6, y and 8. We define y as the relative mass difference y = (m, - m)/m, where marm are the rest mass of certain ion and of the reference ion respectively.
Sy = i sinh(-p2)11’Z ’ cY = As results from these notations, the element with crossed fields is radially focusing when k Z > 0, diverging when k 2 < 0 and dispersing when k 2 = 0. s ( c k P 11, P(2a2 + a3P2)s Ca2(k2- 2P2) + a3P41 Sk(k2 - 4 p 2 ) k2-4~2 Y (+) = (2a2 + a 3 p 2 ) ( 1 - c (&) k 2 - 4p2 Y’ - 2s;) + [a3(2p2- k 2 )- 4a, J (2a, a 3 p 2 ) S+ p(k2 - 4 p 2 ) = 2k(k2 - 4 p 2 ) It is easy to see that the above formulae can not be used to calculate /BB) elements when k 2 = 4p2. They must be replaced by: ( /yy), ( / y B ) and ( A c) + -sz, B 4k ($) = 5 (f - cz), B B A with the abbreviations: A = a, - a 3 k 2 / 8 ,B = a, + a3k2/8.
22. Coordinate system to relate ion movement in a cylindric mirror. We write the Lagrangian in cylindric coordinates. The z velocity component is constant always. Therefore we transform the movement equation following Sar-el (1967): r" + (constant)/(m,r) = 0 into a radial trajectory equation: d Z r / d z 2+ l/(p,rsinZcicos2/3i)= 0 where pa = 2U,/[eVJln(b/a)], V, being the potential difference applied between the cylindric plates of a and b radii. ci is the ion trajectory incidence angle defined as for magnets, Pi the angle between the trajectory and the median plane.
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