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Neglecting dissipation and diffusion processes, the distribution function is an invariant. Many different forms of distribution function has been used in accelerator physics calculations. Given a normalized distribution function p(y, y') with f p(y, y')dydy' = 1, we can define an rms beam size by (B15) Apparantly the rms beam size depends the rms emittance, is defined as on the j3 value. An invariant e_'_'- Z(s) associated with the beam size, called " (Bl6) The emittance is an intrinsic invariant of the beam at a constant energy.
With proper magnet placement, the orbit displacements outside the snake are zero and are independent of magnetic excitation. The important feature of the Steffen snake is the antisymmetric configuration of the vertical magnetic field and symmetric radial field. Let the spin rotation angles of H and V magnets, be respectively ¢, and ¢.. (E1), _ °' e -iO'_':'a ei 's:g'a _'*ei *'g'=aa . we obtain the following equations, e cos _ = cos" gaz+ cos 2¢_ sin" _b. = 1 - 2 sin 2 ¢, sin = ¢_. e. (g2) gives the relation sin ¢, sin ¢, = :i:_2.
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