MULTI MODE APPROACH IN CUMULATIVE BEAM BREAK UP THEORY V.G.Kurakin

Multi mode approach is suggested to evahrate the deflecting gradient of dipole modes. General expression are derived for steady state case foIlowed by numerical calculations. The validity of formulae obtained is discussed. 1. IN-IRODUC~ON Cumulative beam break up (CBBU) in muhi cavity linac manifests itself as the excitation of coherent bet&on osdlations of the accelerated particles under the intluerice of asymmetrical electromagnetic waves, induced by n&lined beam in accelerating cavities [l-4]. Unlike a regenerative beam break up (FU3Bl.J) there is not the effect of field amplification in the cavities through the feed back mechanism, arising in the system due to cavities coupling. CBBU is the instability of convective type and has not any current threshold The latter esults in important consequence, that the single mode approximation, traditionally used in CBBU theories, cannot be justified in non resonance case of cumulative instability. While in RBBU single mode approximation is quite natural in small interval above the instability threshold, where amplifications conditions are satisfied for one mode only, the choice of any particular mode in non resonance case of CBBU becomes completely uncertain. Many of deflecting modes are equivalent o some extent in this case and multi mode approach suggests itself. The attempt of generalisation of CBBU theory to multi mode beam cavity interaction is undertaken in present paper. Appropriate expressions, taking into account the infinite set of deflecting dipole modes, are derived. Steady state deflecting gradients, that are of interest for cw as weIt as for long pulse hnac operation are obtained for some particular cases. 2. THE MAIN EQUATIONS Let us assume, that the linac consists of the cavities electromagnetically decoupled t+om one another, so that RBBU can not take place. tn linear approximation the equation of transverse motion is: d d