Analysis of Transverse Resistive Instability in the Recycler

Transverse resistive wall instability is analysed for antiprotons in the Recycler Ring. Damper requirements are formulated. 1 Growth Rate and Landau Damping 1.1 Coasting Beam Coherent instabilities are driven by the real part of impedance. For a coasting beam with insigni…cant frequency spread, the impedance-driven coherent tune shift at the harmonic number n is given by (see e. g. [1]) Z !Z=!0 = i Nr0 4 T0 Z(!b + n!0); (1) where N is the number of particles (pbars), r0 classical radius, = E0=mc relativistic factor, T0 the revolution period, !0 = 2 =T0, the beta-function, and Z( ) the transverse impedance. The beam can be naturally stabilized by the Landau damping, which requires su¢ cient density of particles resonant with the coherent beam motion. Density of the resonant particles is suppressed by the space charge incoherent tune shift, separating coherent and individual betatron frequencies: sc !sc=!0 = Nr0 4 "?B ; (2) with "? as the r. m. s. normalized emittance and B = b=T0 is the bunching factor. Assuming the beam energy spread as a leading source of the single-particle tune spread, via the chromaticity and the slippage factor = 2 t ; the r. m. s. spread of the e¤ective incoherent tunes is given by b(n) !b(n)=!0 = j n j p=p ; the Landau damping rate is calculated as