Landau damping

Abstract Landau damping is the suppression of an instability by a spread of frequencies in the beam. It is treated here from an experimental point of view. To introduce the concept we consider a set of oscillators having a spread in resonant frequencies !r and calculate the response of their there center-of-mass to an external driving force. A pulse excitation gives each oscillator the same initial velocity but, due to their different frequencies, the center-of-mass motion will decay with time. A harmonic excitation with a frequency ! being inside the distribution in !r results in oscillators responding with different phases and only a few of them having !r ! will grow to large amplitudes and absorb energy. The oscillator response to a pulse excitation, called Green function, and the one to a harmonic excitation, called transfer function, serve as a basis to calculate Landau damping which suppresses an instability at infinitesimal level before any large amplitudes are reached. This is illustrated by a negative feed-back system acting on the center-ofmass of the oscillators. In the absence of a frequency spread this leads in an exponential growth, but a sufficiently large spread will provide stability. Landau damping is then applied to the transverse motion of a coasting (un-bunched) beam. The dependence of revolution frequency and betatron tune on momentum establishes a spread of the frequencies contained in the upper and lower betatron side-band. A transverse impedance present in the ring gets excited by the beam and acts back on it, causing an instability. For a given frequency distribution a maximum value of this impedance, being still conform with stability, is calculated and plotted in a stability diagram. Finally, longitudinal instabilities in a coasting beam are investigated. They can be stabilized by a spread of revolution frequencies caused by an energy distribution of the particles in the beam.