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Analysis of the Head Tail Damping and Growth Time for the Estimation of the Esrf Machine Impedance P Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 ANALYSIS OF THE HEAD TAIL DAMPING AND GROWTH TIME FOR THE ESTIMATION OF THE ESRF MACHINE IMPEDANCE P. Kernel, J.-L. Revol, R. Nagaoka, ESRF, Grenoble, France G. Besnier, University of Rennes, France Abstract are compared to the tracking simulation in order to dis- cuss the impedance model for single bunch. A theoretical and experimental characterisation of trans- verse damping mechanisms was initiated within the 2 DIFFERENT MECHANISMS REVIEW framework of the single bunch transverse instability The damping time measurement presents many difficul- studies. An attempt is made to estimate the machine im- ties in practice due to the fact that different other damp- pedance through the measurement of the head tail ing mechanisms can occur at the same time and spoil the damping time as a function of the chromaticity. The observed signal. presence of strong sextupole fields used for chromaticity compensation however induces large momentum and 2.1 Transverse Radiation Damping amplitude dependent tune shifts. A systematic study of the different damping mechanisms was therefore per- Radiation damping mechanism presents a characteristic formed in order to quantify the different contributions, and constant decay time of 7 ms for the ESRF storage which for the ESRF storage ring are in the same range. ring. This value is long enough not to perturb the head Numerical simulations and record of the beam response tail measurements. to a kick were performed. The transverse signal decay after the excitation was observed on the tune monitor 2.2 Chromatic Modulation system used in zero span mode. The tune amplitude de- After a transverse kick excitation, the frequency mixing pendence which depends on the kick amplitude is dis- of the individual particles oscillations, at slightly differ- cussed first. Then the modulation induced by chro- ent betatron frequencies depending on the chromaticity maticity overcompensation which could blind the obser- ξ=(dQ/Q)/(dp/p), results in a modulation of the trans- vation is described. The measurement of the head tail verse signal of period Ts, the synchrotron period [Fig.1.a damping and growing time, under precise operating con- and 1.b, Ts=0.5 ms]. The peak widths, which only de- ditions, with positive and negative chromaticity, is pre- pend on the chromaticity, narrow when the chromaticity sented. Finally, a model for the machine impedance is increases. To avoid confusing the chromatic modulation discussed. with the head-tail damping, the head tail damping decay should be much longer than Ts. In that case, from the 1 INTRODUCTION amplitude of a few peaks, the envelope of the modulated The optimisation of the machine operation and the un- signal can be extracted. derstanding of the feedback system efficiency for future operation are the ultimate purpose of the transverse mo- tion study at the ESRF. The modelling of the storage ring transverse impedance is a key point for the study of single bunch instability limitations. For this purpose, a campaign of measurements has been performed, their interpretation is under progress [1,2]. The head tail damping time and the growth rate of the transverse sig- nal resulting from the interaction of the beam modes with the machine impedance are accessible physical quantities. These data are complementary in order to cross-check the impedance model resulting from meas- Fig 1.a (left) Modulated head tail damping (simul.) urements of mode merging observed at low current [2] Fig 1.b (right) Modulated tune spread damping (simul.) and threshold curve associated to higher order head-tail 2.3Amplitude-Dependant Tune Spread Damping instability [1] for the high current regime. Head tail damping time extraction from the measurement requires If the transverse kick is too large, additional tune spread a methodical approach, which is part of this paper. The is induced within the bunch by the non-linearity coming characterisation and the description of the different from the strong sextupoles fields. The corresponding mechanisms occurring at ESRF, the measurement meth- damping signal resulting from this mechanism presents a ods and data are presented. The results (in vertical plane) bell-shape decay (Fig.1.b). The strength of the tune 0-7803-5573-3/99/$10.00@1999 IEEE. 1195 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 1 222 spread damping, which is independent of the beam cur- hmsh(ωω−=−ξξξ ) exp[]() ( ωω − ) + ( ω − ω ) στ × (2) rent, increases with the kick amplitude, Fig.2. For the 2 2 head tail damping time measurement, the kick should be Imsh[]()()ωω−−ξξ ω ωστ chosen small to keep the tune spread damping time long st where Im is the modified Bessel’s function of the 1 kind enough to be negligible (in the order of a few ms). ω ≈ and sh 0 is the shaker frequency, which is null for a Damping time (ms) 0.8 kick. Kicked and shaked modes present exactly the same spectrum. measurement Synchronously with a kick, the transverse signal decay 0.6 calculation (tracking) is measured with a spectrum analyser used in the zero span mode adjusted to the betatron frequency. This 0.4 method improved by averaging was efficient for the de- tection of very low response defined by the following 0.2 operating conditions: • Low intensity per bunch, necessary for a head tail damping to be longer than T =0.5 ms, to be able to ex- 0 s 0 5 10 15 20 tract the envelope from the chromatic modulation. Kick Amplitude (mm) • Small transverse kick to avoid the tune spread Fig 2: Amplitude-dependant tune spread damping time damping. The absence of tune spread can be check by 2.4 Head Tail Damping and Growing time. the linear dependence with current. • Increased average intensity, in some cases, to keep For Hermitian modes that are not interacting in a mode- the transverse signal out of the noise level, (few bunch coupling regime, the head tail characteristic time for a modes, with low single bunch intensity). mode m is given by the formula (cf. [2]): On Fig.3b, the expected shape of the head tail damp- ∑hmp()ωω− ξ ing time after a kick has been computed with a tracking 2m σπ (1) τ = 2 m! τ 4 Ee p code, using a BBR model for the transverse vertical im- m mI− β ∑ ()Zhωωω− ×β Ω ()!!21 Re ( )mp (ξ ) pedance with fr=22 GHz, Q=1, R =13.5 M . p 0.4 ω ωβ ω ω with p= +m s+p 0. −Intensity*growth Intensity*damping ω ω time (ms.mA) time (ms.mA) The mode spectrum hm( p- ξ), is shifted in frequency by 0.3 of spontaneous of kicked ωξ=ω Qβξ/α, proportional to the chromaticity. The 0 pure mode 0 mode measurement of the head tail damping or growing time (tracking) (tracking) τ 0.2 m versus chromaticity, is characteristic of the interaction of the mode spectrum with the real part of the transverse impedance Z(ω). For the head tail mechanism, the trans- 0.1 verse pick-up signal is exponential (fig.1a). The damping τ ξ time m should be inversely proportional to the peak cur- 0 −1.5 −1 −0.5 0 0.5 1 1.5 V rent I/στ. and to the optical beta function β. These pa- Fig3a(left) Head tail growth time simul. (mode 0) rameters can be mastered during the experiments. The Fig3b(right) Head tail damp. time simul. after a kick bunch length which also defines the width of the mode spectrum h (ω -ωξ), should be long in order to scan the m p 3.2 Results impedance in more detail. According to theory, the damping time should increase 3 HEAD TAIL DAMPING TIME (ξ>0) when ξ is close to 0, because of the cancellation due to the anti-symmetric real part of Z(p); and increase at ξ 3.1 Measurement Methods large , due to the decreasing of impedance. The curve should present a minimum at the resonant frequency of At positive chromaticities, the head tail damping time the BBR model. These curves (Fig.4 right, vert. plane) has been scanned versus ξ, in both vertical and horizon- do not present the expected shape and no proportionality tal planes, much below the current threshold. The trans- with the current, which is a hint to say that the observed verse excitation is produced by a kick of the injection damping time was not a pure head tail phenomena. This kicker for the horizontal plane, or by the gated tune discrepancy with the theory might come from a meas- monitor shaker for the vertical plane. The particle distri- urement problem. Despite the efforts employed to avoid bution resulting from the kick is equivalent to a low fre- all the perturbing phenomena and to extract the pure quency shaker mode, and its corresponding envelope head tail damping mechanism, it seems that another spectrum for Gaussian beam is (different of the spectrum mechanism has polluted the measurement. The obtained of the Hermitian modes assumed in Eq.1 when m≠0): results should be clarified by further theoretical studies. 1196 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 4.2 Results 1.2 −Intensity*growth Intensity*damping The growing time which monotonously decreases up 1 time (ms.mA) time (ms.mA) to ξ=-0.5, is in agreement with the 22 GHz BBR imped- 0.8 ance model given by other modelling methods [1] (Fig.4 I=.01 mA left). Nevertheless, the limitation of the measurement at 0.6 I=.2 mA ξ I=.5 mA =-0.5 (corresponding to -13 GHz) does not allow to 0.4 I=.7 mA measure the turning point where the growing time should increase again.
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