Impedance Analysis of the Pep-i1 Vacuum Chamber*

The PEP-I1 high energy ring (HER) vacuum chamber consists of a copper tube with periodically spaced p u m p ing slots. The impedance of the vacuum chamber due to the slots is analyzed. Both narrow-band and broadband impedances are considered as well as longitudinal and transverse components thereof. It is found that although the broad-band impedance is tolerable, the narrow-band impedance may exceed the instability limit given by the natural damping with no feedback system on. Traveling wave modes in the chamber are the major source of this high value narrow-band impedance. We also study the dependences of the impedance on the slot length and the geometrical cross section. I. ABOUT SMALL IMPEDANCES In oder to create a "feeling" about the magnitude of small impedances, we shall start with a general note of caution showing that even a tiny little cavity in a beam pipe can easily create kS1 of impedance. Thus one such small cavity by itself might be capable of driving the beam unstable. Let us consider for simplicity a circular symmetric beam pipe with a small pill-box cavity in it. Such a cavity (see Fig. 1) has one trapped mode [l] for each rn, where m is the azimuthal mode number. We restrict our consideration to the TMolo mode here for simplicity and obtain by numerical analysis with MAFIA the trapped mode as shown in Fig. 2. The trapped mode parameters are listed in Table 1. It can be seen that a tiny cavity with only 3 mm dent at the outside of the beam pipe can exceed the tolerable impedance limit in a B-factory machine without feedback by a significant factor. A deeper dent of 10 mm over the length of 20 mm is even 12 times higher and becomes relevant even if there is a powerful feedback. Figure 1. Three dimensional cutaway view of two very small cavities in a circular beam pipe. The cavity is 20 mm long. Their inner radius is 35 mm, and the outer radii are 38 mm and 45 mm for the two cases. Work supported by the Department of Energy, contract DEAC03tpermanent address: Univenity of Technology, FB18, Schlon76SF00515. gortenstr.8, D64289, Darmstadt, Germany. z Im Figure 2. Longitudinal component of the electric field of the trapped mode TMolo in a small "cavity" (router=45 mm) in a circular beam pipe. Note the exponential decay over a long distance at both sides. Gap length (mm) rinner (mm)