General formula to deduce the space charge tune spread from a quadrupolar pick-up measurement

In 1966, W. Hardt derived the oscillation frequencies obtained in the presence of space charge forces and gradients errors for elliptical beams. Since then, a simple formula is usually used to relate the shift of the quadrupolar mode (obtained from the quadrupolar pick-up) and the space charge tune spread, depending only on the ratio between the two transverse equilibrium beam sizes. However, this formula is not always valid, in particular for machines running close to the coupling resonance Qx = Qy with almost round beams. A new general formula is presented, giving the space charge tune spread as a function of i) the measured shift of the quadrupolar mode, ii) the ratio between the two transverse equilibrium beam sizes and iii) the distance between the two transverse tunes.