Storage ring saw-tooth and microwave instability and free electron laser

A number of theoretical and experimental investigations have suggested that the Free Electron Laser operating with Storage Rings produce effects which inhibit various type of beam instabilities. We exploit simple arguments, based on well-known heuristic criteria, to der ive the critical intracavity power necessary to switch off longitudinal ins tabilities of the microwave and saw-tooth type. The dynamics of the instabi lity is described by exploiting a couple of equations which account for the evolution of the instability growth rate and for the consequent e-beam ene rgy spread enhancement. These equations, implicitly containing the Boussard criterion, are combined to the Free electron laser rate equations model, t o study the dynamical interplay between the laser growth and the mechanisms which lead to the instability suppression and to precise the conditions un der which this effect occurs.