Stability Diagrams for Landau Damping with Two-Dimensional Betatron Tune Spread from both Octupoles and Nonlinear Space Charge Applied to the LHC at Injection

The joint effect of space-charge nonlinearities and octupole lenses is discussed for the case of a quasi-parabolic distribution function in the two transverse planes, considering a monochromatic beam and neglecting the longitudinal variation of the transverse space-charge forces. The self-consistent nonlinear space-charge tune shift corresponding to the above distribution function is first derived analytically. Noting that a reasonable approximation of the space-charge tune shift is given considering only linear terms in the betatron action variables, the dispersion relation is solved analytically in this approximate case. As expected, in the absence of external (octupolar) nonlinearities, the result of Möhl and Schönauer is recovered: there is no stability region. In the absence of space charge, the stability diagrams of Berg and Ruggiero are recovered. The new result is applied to the LHC at injection.