Measurements and Simulations of the Damping Effect of the Harmonic Sextupole on Transverse Instabilities L. Tosi # , V. Smaluk, E. Karantzoulis Sincrotrone Trieste, 34012 Trieste, Italy Abstract Measurements at ELETTRA have shown that the harmonic sextupole provides Landau damping capable of suppressing transverse coupled multibunch instabilities. There is strong evidence that the damping is induced by the non-linear tune spread with amplitude among the electrons within the individual bunches together with a change in the electron bunch distribution. Results of measurements are compared to simulations. 1 INTRODUCTION Landau damping is the phenomenon in which a collective motion performed by a certain number of particles is damped by an increasing spread in the oscillation frequencies of the individual particles. As the coherent oscillations build up a centre of mass motion, the latter starts decreasing as particles go out of phase with respect to each other and decohere. Individual particles may be still oscillating but the centre of mass motion is damped. One possibility to generate Landau damping is non-zero chromaticity and another is non- linear elements in a storage ring. A typical example of non-linear elements is the use of octupole magnets in which tune shifts with amplitude are a first order effect. Sextupoles also induce tune shifts with amplitude, although as a second order effect [1], but may nevertheless give rise to the same damping mechanism. In Elettra where a harmonic sextupole S1 is present, strong correlation between the damping of collective motion and the setting of the above sextupole has been noticed in the past [2,3]. It is possible to change the non-linearities of the machine (hence the tune shifts with amplitude) by acting on its settings, maintaining at the same time the constant values of the chromaticities. In this paper results are presented of variations of the harmonic sextupole settings that give rise to changes in the Landau damping. Results are given both of measurements performed using the transverse multibunch feedback system (TMBF) [4] as an acquisition tool, as well as of computer simulations. Due to the relatively simpler mechanism, both measurements and simulations were done in the horizontal plane, thus eliminating the coupling among the two planes due to the sextupoles. In this case the tune shift with amplitude is reduced to ∆ν x = C·2J x , and presents a parabolic behaviour with the settings of S1. The following section deals with damping effects when coupled horizontal multibunch instabilities are present, while section 3 discusses the fast coherent damping when kicking a single bunch. 2 COUPLED MULTIBUNCH INSTABILITIY Coupled multibunch instabilities can be viewed as consisting of two collective motions: a macroscopic one among the centre of mass of the individual bunches and a microscopic one among the single particles within the individual bunches. Thus, in the past, when observations could be not be done with the TMBF, an important question was whether the tune spread induced by the harmonic sextupole was among different bunches or within the particles of single bunch. Simulations done in 1999 using a macroscopic model of the bunches could only reproduce the main features of the phenomena. Although the TMBF allows the visualization in detail of what individual bunches are doing, it is important to realize that it can detect only the centre of mass motion of the bunches. Fortunately the tune shift with amplitude is unidirectional, since amplitudes can be only positive. This means that if there is an increase in tune spread within a bunch during the collective motion, then the centre of mass will result in having a tune shift. 5 10 15 20 25 30 -50 0 50 t, ms beam position, a.u. 5 10 15 20 25 30 0.28 0.285 0.29 0.295 0.3 t, ms ν Figure 1. Horizontal instability at S1 = 50 A Figure 1 shows the centre of mass motion, together with its spectrum (tune resolution is 5·10 -4 ), of one of the bunches undergoing a horizontal coupled multibunch instability (driven by a known higher order dipole mode of one of the RF cavities) with S1 set near the minimum value for the tune shift with amplitude. The damping of the centre of mass motion is not due to a simple detuning of the mode. In fact, observing figure 2 where the same analysis is shown for the same bunch but with the sextupole set to a stronger tune shift with amplitude, one can note that the damping occurs at a much smaller tune shift of the centre of mass. _____________________ #tosi@elettra.trieste.it Proceedings of EPAC 2002, Paris, France 1571