Betatron tune shifts with current have long been a puzzle. They have opposite signs for the horizontal and vertical and become larger than two synchrotron-side bands at the operating current of 200 mA (Figure 183). When verifying that the effect is not related to a closed-orbit drift and depends only on the average current, the resistive-wall wake field was suspected owing to its long-range nature. In fact, studies have recently been made elsewhere showing that resistive-wall chambers with an asymmetric cross section generate a quadrupolar mean field. A program was developed at the ESRF to compute the transverse wake fields and was applied for the 10 and 15 mm stainless steel ID vacuum chambers that contribute in a major way to the instabilities. Incoherent tunes of each bunch were then computed, taking into account the actual configuration of low-gap chambers in the ring, the beam-filling structure as well as the multi-turn effect.

Fig. 183: Measured current-dependent tune shifts, with and without orbit correction.

The results showed firstly that the single turn effect is much too small to explain the measured tune shifts, suggesting the importance of multi-turn build-up of the wake field, and secondly that the tune shift is much stronger horizontally than vertically due to the different horizontal and vertical lattice functions in the ID straight sections. To verify the findings, head to tail tune shifts in a bunch train were measured experimentally because they are expected to represent the single turn effect. Both horizontal and vertical tune shifts were found to be in good agreement with expectations. Numerical studies also revealed a significant detuning in single bunch due to the strong short-range component of the resistive-wall wake field, which goes well with observations such as a high sensitivity of the single bunch at high current to the horizontal half-integer resonance and an increase of horizontal coherent mode frequency with chromaticity (Figure 184a). In particular, at zero chromaticity, the mode 0 is only slightly detuned, while other head-tail modes are focused by a similar degree (Figure 184b). It may be that the mode zero is under the influence of counteracting inductive impedance and a mean resistive-wall field, while others are only focused by the latter.

Fig. 184: a) Coherent tune versus chromaticity; b) detuning of head-tail modes in the mode-merging regime, measured in the horizontal plane.

Besides the peculiar behaviour of mode frequency shifts, a marked reduction has been recently observed in the single bunch threshold current horizontally (Figure 185). In particular, the horizontal transverse mode coupling instability threshold around 2 mA can no longer be regarded as high compared to the vertical current threshold of 0.8 mA. While the vertical instabilities, which are more critical, were studied extensively within the framework of a PhD thesis presented in 2000, horizontal investigations have been initiated this year. The study is currently focused on quantifying the incoherent tune shift, which should be subtracted from the total tune shifts observed in order to estimate the horizontal impedance.

Fig. 185: Current threshold as a function of the horizontal chromaticity.