LHC Schottky Spectrum from Macro-Particle Simulations

We introduce a method for building Schottky spectra from macro-particle simulations performed with the PyHEADTAIL code, applied to LHC beam conditions. In this case, the use of a standard Fast Fourier Transform (FFT) algorithm to recover the spectral content of the beam becomes computationally intractable memory-wise, because of the relatively short bunch length compared to the large revolution period. This would imply having to handle an extremely large amount of data for performing the FFT. To circumvent this difficulty, a semi-analytical method was developed to compute efficiently the Fourier transform. The spectral content of the beam is calculated on the fly along with the macro-particle simulation and stored in a compact manner, independently from the number of particles, thus allowing the processing of one million macro-particles in the LHC, over 10’000 revolutions, in a few hours, on a regular computer. The simulated Schottky spectrum is then compared against theoretical formulas and measurements of Schottky signals previously obtained with lead ion beams in the LHC.