The beam quadrupole moment of stored beams can be measured with a four-plate quadrupole pick-up. The frequency spectrum of the quadrupole moment contains not only the usual first-order dipole modes (the betatron tunes) but also the second-order coherent modes, comprising of (1.) (even) normal envelope modes, (2.) odd (skew) envelope modes and (3.) dispersion modes. As a novel diagnostic tool, the measured frequencies and amplitudes provide direct access to transverse space charge strength through the tune shift as well as linear coupling (and mismatch thereof), along with the benefit of a non-invasive beam-based measurement. Technically, quadrupole moment measurements require a pick-up with non-linear position sensitivity function. We discuss recent developments and depict measurements at the GSI SIS18 heavy-ion synchrotron.
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