TRANSVERSE BUNCH-BY-BUNCH FEEDBACK FOR THE VEPP-4M ELECTRON-POSITRON COLLIDER V. Cherepanov, E. Dementev, E. Levichev, A. Medvedko, V. Smaluk, D.Sukhanov, Budker Institute of Nuclear Physics, Novosibirsk, Russia Abstract Transverse mode coupling instability (TMCI or fast head-tail) is the principal beam current limitation of the VEPP-4M electron-positron collider. For the high-energy physics experiments at the 5.5 GeV energy, the VEPP-4M bunch current should exceed much the TMCI threshold. To suppress transverse beam instabilities, a broadband bunch- by-bunch digital feedback system is developed. The feed- back concept is described, the system layout and first beam measurements are presented. HEAD-TAIL EFFECTS For high-energy physics experiments in the 5.2-5.5 GeV energy range, design value of the VEPP-4M beam current is 40 mA per bunch in 2e − × 2e + -bunch operation mode. At the injection energy of E =1.8 GeV, the beam current is limited by the vertical transverse mode coupling insta- bility (TMCI or fast head-tail) [1]. There is an approxi- mate formula for the TMCI threshold current derived using a two-particle model [2]: I tmci = σ z √ 2πR 8π E e ν s ∑ k ℑZ ⊥k β k , (1) where σ z is the r.m.s. bunch length, R is the average ma- chine radius, ν s is the synchrotron tune, and ∑ k ℑZ ⊥k β k is the beta-weighted broad-band reactive impedance of the ring. For the VEPP-4M at the injection energy, the thresh- old current is 10-12 mA. If a machine chromaticity ξ = δν β δp /p is non-zero, the chromatic head-tail effect appears, and some oscillation modes become unstable for any beam current, and the cur- rent threshold can result from radiation damping only. For the chromatic head-tail, an increment/decrement of the co- herent oscillation mode (1/τ + ) and incoherent one (1/τ − ) is expressed as [2]: 1 τ ± = ∓ I b cR 16πν β E e σ z ℑ [Z ⊥ f (2χ)] , (2) where ν β is the betatron tune, f (2χ) is the complex func- tion f (u)=  π 0 e iu sin x dx of the head-tail phase χ = ξ α σ z R , (3) which is a betatron phase advance caused by the chromatic- ity during a half-period of synchrotron oscillation (from head to tail). The coherent mode (center of mass oscilla- tion) is damped if ξ> 0 and anti-damped if ξ< 0 (for a positive momentum compaction α), whereas for the inco- herent modes (beam size) the effect is vice versa. FEEDBACK THEORY As it follows from (2), positive chromaticity suppress the coherent oscillation mode, i.e. makes bunch center of mass stable, but other oscillation modes are unstable, negative chromaticity has the inverse effect. A detailed analysis of a feedback applicability is given in [3]. The main idea is to suppress the coherent oscillation mode using a resistive feedback, while to keep other modes stable due to a negative chromaticity. Because of the beam-environment interaction, each par- ticle in a bunch is perturbed by electro-magnetic fields induced by all other particles. For the model bunch of N macro-particles uniformly distributed over synchrotron phases, a system of differential equations can be written: dy k dz + 1 N ω 0 I b 〈β〉 4π E e N−1  j=0  y j ∞  m=0 W kjm  =0, (4) where W kjm = Z ⊥ [−iω 0 (m − ν s )] exp  iω 0 (m − ξ) z k − z j c  , y k is the complex betatron oscillation amplitude of k−th particle, ω 0 is the revolution frequency, I b is the bunch cur- rent, 〈β〉 is the average beta-function. Z ⊥ is the broad-band transverse coupling impedance, characterizing the short- range beam-environment interaction. The VEPP-4M ver- tical broad-band coupling impedance estimated from the coherent tune shift measurement [1] is about 2 MΩ/m. It is not conveniently to analyze such complicated mo- tion using the system (4), because the number of equations is equal to the number of particles N , which should be big enough to obtain reasonable results. Moreover, since the longitudinal coordinates of the particles z k ,z j are explic- itly time-dependent, the system (4) is a system of differen- tial equations with variable coefficients. As it is shown in [3], an analysis using symmetric mode expansion is much more efficient because only a few of lowest oscillation modes are significant. In addition, this approach allows us to avoid the variable coefficients. Us- ing the continuous medium model and Vlasov equation, the problem of stability can be reduced to a system of algebraic equation: (iλ + in)a nk + I b 〈β〉 4πν s E e ∞  n ′ =−∞ ∞  k ′ =0 A n ′ k ′ nk a n ′ k ′ =0, (5) Proceedings of DIPAC 2007, Venice, Italy WEPC26 Beam Instrumentation and Feedback Feedbacks 367