Estimation of longitudinal bunch characteristics in the LHC using Schottky-based diagnostics

The Large Hadron Collider (LHC) Schottky monitors have been designed to measure various parameters of relevance to beam quality, namely tune, momentum spread, and chromaticity. In this work, we present how this instrument can be used to estimate longitudinal bunch characteristics, such as longitudinal bunch profile or synchrotron frequency distribution. Under the assumption of bunched beams with no intrabunch coherent motion, we start by deriving the relation between the distribution of synchrotron amplitudes within the bunch population and the longitudinal bunch profile from probabilistic principles. Subsequently, we fit the cumulative power density of acquired Schottky spectra with the underlying distribution of synchrotron amplitudes. Finally, the result of this fit is used to reconstruct the bunch profile using the derived model. The results obtained with this method are verified by comparison with longitudinal profile measurements from the LHC wall current monitors.