Crossing of an I N C O H E R E N T Integral R E S O N a N C E in the Electron Ring Accelerator*

In one mode of opera t ion o f an electron ring accelerator (ERA), at the end o f compress ion rings are slowly moved th rough the radia l in tegra l be ta t ron resonance Qr = 1. A l though the coherent radial oscillation frequency of the ring as a whole remains below unity, the oscillation frequencies o f individual electron are ( incoherent ly) caused to pass t h rough the resonance because o f the addi t ional focusing f rom ions t rapped in the ring. In this paper the effect o f field errors on ring major and minor radii is e v a l u a t e d t h e o r e t i c a l l y f o r the cases in which the spread in the square o f the electron oscillation frequency (A 2) is (a) m u c h smaller and (b) m u c h larger t h a n the cont r ibu t ion to the square o f the oscillation frequency f rom the ions (A2). It is shown tha t for the ERA, where case (a) applies, the increase in ring minor d imensions , for given field errors and rate o f resonance crossing, is less t h a n in case (b) by a factor o f (A/A) 2. Numerica l examples show that the degradat ion o f ring quality in case (b) should, with suitable a t tent ion to the design and cons t ruc t ion o f the E R A appara tus , be acceptably small.