Optimization of radial matching section

Optimization approach to define geometry (parameters) of radial matching section in RFQ accelerator is suggested. This approach gives wider opportunities to design of radial matching section because it does not have certain prescribed laws of variation of focusing strength along the section. INTRODUCTION Acceptance at the entrance of RFQ accelerator, as it is known, depends on time and rotates with frequency of RF field. On the other hand, the input beam has constant emittance, not changing in time. In this connection there is a problem of the beam matching with an accelerating RFQ channel. In work [1] it was suggested to provide transverse matching of beam with the accelerating channel with the help of radial matching sections, and concrete laws of change of focusing strength in these sections have been considered. The next years this problem was considered also by different authors [2-4]. In the paper the new approach for the solution of the given problem is suggested. Parameters of radial matching section are defined as a result of the solution of some optimization problem. The work is done under support of the National Taiwan University of Science and Technology and Saint-Petersburg State University Joint Research Project No. RP07-05 1 PROBLEM STATEMENT For radial matching section of accelerator charged particles dynamics in ) , ( y x plane which is perpendicular to longitudinal axis in the case of microcanonical charge distribution can be described by the following equations [5]: ( ) ( )x r r Q x r r r r r vr W eI x a W eU d x d y x x y x y x y x , , , 1 2 cos 0 0 0 0 2 0 2 2 φ τ πε φ θτ τ =         + − − − + = , (1) ( ) ( )y r r Q x r r r r r vr W eI x a W eU d y d y x y y x y x y x , , , 1 2 cos 0 0 0 0 2 0 2 2 φ τ πε φ θτ τ =         + − + − + − = , (2) where сt = τ , c πω θ 2 = , U is intervane voltage, 0 W is charged particle rest energy, ω is accelerating field frequency, 0 φ is initial phase, c is light velocity, a is radius of channel, z v & = is a longitudinal velocity of particle which is constant along matching section, x r and y r are normalized beam envelopes, I is beam current. The equations (1), (2) can be transformed in system of the equations ξ τ ξ x A d d = , η τ η y A d d = , (3) where ( ) 2 1 ,ξ ξ ξ = , x = 1 ξ , τ ξ d dx = 2 , ( ) 2 1 ,η η η = , y = 1 η , τ η d dy = 2 , and matrices x A and y A