Nonlinearities in the response of beam position monitors for the LEP spectrometer

The LEP Spectrometer determines the energy of the beam by measuring the bending angle due to a dipole magnet, using a total of six beam position monitors (BPM). The response to an off-centre beam of a BPM is not strictly linear with displacement and, in addition, depends on the size of the beam. The response also depends on the shape of the aperture of the BPM itself, since the inner surface is conducting and must therefore be an equipotential. For the LEP Spectrometer, BPM nonlinearities may become large enough to affect the beam energy measurement. We calculate the size of the effect for Gaussian beams within circular and elliptical apertures, drawing conclusions regarding the operation of the Spectrometer.