Space charge and beam stability of the proposed Fermilab proton driver in Phase I

Issues concerning the beam stability of the proposed Fermilab proton driver are studied in its Phase I stage. Although the betatron tune shifts are dominated by the space charge, these shifts are less than 0.25 and will therefore not drive the symmetric and antisymmetric modes of the beam envelope into instability. The longitudinal space-charge force is large and inductive inserts may be needed to compensate for the distortion of the rf potential. Although the longitudinal impedance is space-charge dominated, it will not drive any microwave instability, unless the real part of the impedance coming from the inductive inserts and wall resistivity of the beam tube are large enough. The design of the beam tube is therefore very important in order to limit the flow of eddy current and keep wall resistivity low. The transverse impedance is also space-charge dominated. With the proton driver operated at an imaginary transition gamma, however, Landau damping will never be canceled and the beam stability can be maintained with negative chromaticities