Description of Coherent Beam Instabilities in the Presence of Space Charge Forces using the Circulant Matrix Model

Low energy hadron synchrotrons face significant limitations in the number of particles per bunch because of the direct electromagnetic interaction of beam particles, known as space charge. This report aimed to improve previous models showing how space charge affects the limitations of transverse mode coupling instabilities due to electromagnetic wake fields by using a Gaussian charge distribution in the code BimBim which implements the circulant matrix model. The space charge tune shift modes were compared for the airbag in a linear potential and Gaussian distributions. We used the airbag in a square-well distribution as a benchmark. The main result was the comparison of the transverse mode coupling instabilities (TMCI) generated due to space charge and wake fields for the airbag and Gaussian distributions. We confirmed with theory that sufficiently strong space charge can suppress the instabilities caused by wake fields alone. For the airbag distribution, positive mode coupling was found to be a result of our approximation to a finite ring width. The Gaussian distribution showed that when considering multiple radial modes, increasing space charge induced new types of instabilities that cannot be attributed to numerical artefacts. We thus find that space charge suppresses the negative TMCI due to wake fields and causes positive mode coupling.