Simple analytical computations of intensity dependent emittance transfer

In high-intensity proton synchrotrons with larger horizontal than vertical emittance the fourth-order coupling resonance driven by space charge only, known as the Montague resonance, may lead to a sharing of the emittances and possibly beam loss due to vertical acceptance limitation. The influence of space charge on the transverse emittances is discussed by solving the Kapchinskij and Vladimirskij envelope equations for small perturbations on top of equilibrium beam sizes. This model works therefore only for beams, which are not too asymmetric in their transverse dimensions. Formulae for the transverse emittances in the presence of space charge are derived that include the uncoupled emittances, the space-charge coupling strength, and the tune distance from the resonance. A stop band is also defined.