Electron cooling requires small angles between the electrons and the cooled particles. In other words, the electron beam has to be of a cylindrical shape in the cooling section. How to satisfy this requirement for a specific case of a relativistic electron beam transport for the Fermilab electron cooling project? In fact, the requirement splits into two parts: for the beam centroid and its envelope. A straight centroid motion means a good field quality and zero initial conditions; this issue is not a subject of this paper. The cylindrical envelope requires proper initial conditions of the envelope at the entrance of the cooler, it is a problem of matching. A specific complex of measurements and calculations aimed at solving this problem is described here.

This paper presents results of experimental and theoretical investigations of transverse beam stability at injection to Fermilab Booster and discusses a novel scheme for transition crossing allowing to avoid the longitudinal emittance growth related to the transition. At reduced chromaticity a multibunch high order head-tail mode develops with growth time of 12 turns at fractional part of tune close to zero. An estimate of the growth rate based on known sources of impedance results in significantly smaller value and cannot explain observed instability growth rate.

Simulation of beam cooling usually requires performing certain integral transformations every time step or so, which is a significant burden on the CPU. Examples are the dispersion integrals (Hilbert transforms) in the stochastic cooling, wake fields and IBS integrals. An original method is suggested for fast and sufficiently accurate computation of the integrals. This method is applied for the dispersion integral. Some methodical aspects of the IBS analysis are discussed.

The Fermilab's Recycler ring will employ an electron cooler to cool stored 8.9 GeV antiprotons [1]. The cooler is based on an electrostatic accelerator, Pelletron [2], working in an energy-recovery regime. A full-scale prototype of the cooler has been assembled and commissioned in a separate building [3]. The main goal of the experiments with the prototype was to demonstrate stable operation with a 3.5 MeV, 0.5 A DC electron beam while preserving a high beam quality in the cooling section. The quality is characterized, first of all, by a spread of electron velocities in the cooling section, which may be significantly affected by mechanical vibration of the Pelletron elements. This paper describes the results of vibration measurements in the Pelletron terminal and correlates them with the beam motion in the cooling section.

With cooling, beam phase space density increases, which makes the beam motion intrinsically unstable. To suppress instabilities, dampers are required. With a progress of digital technology, digital dampers are getting to be more and more preferable. Conversion of an analog signal into digital one is described by a linear operator with explicit time dependence. Thus, the analog-digital converter (ADC) does not preserve a signal frequency. Instead, a monochromatic input signal is transformed into a mixture of all possible frequencies, combining the input one with multiples of the sampling frequency. Stability analysis has to include a cross-talk between all these combined frequencies. In this paper, we are analyzing a problem of stability for beam transverse microwave oscillations in a presence of digital damper; the impedance and the space charge are taken into account. The developed formalism is applied for antiproton beam in the Recycler Ring (RR) at Fermilab.

Although the luminosity growth for Tevatron Run II was slower than expected, steady growth of luminosity has been demonstrated during the last three years with the peak luminosity of 1.02 x 10{sup 32} cm{sup -2} s{sup -1} achieved in July 2004. Suppression of instabilities has been a valuable contributor to the luminosity growth. This report discusses the transverse instabilities in the Tevatron and Recycler.

Stochastic cooling with bunched beam in a linear bucket has been obtained and implemented operationally in the Fermilab Recycler Ring (RR). This is the first time that linear-rf bunched-beam stochastic cooling has been successfully used operationally in a high-energy facility. In this implementation the particle bunch length is much greater than the cooling system wavelengths, and that property is critical to the cooling success. The simultaneous longitudinal bunching enables cooling to much smaller longitudinal emittances than the coasting beam or barrier bucket system. Characteristics and limitations of bunched beam stochastic cooling are discussed.