*V.Kisele* `kiselev@inp.nsk.su`

INP, Novosibirsk3mm
*V.Smaluk* `smaluk@inp.nsk.su`

3mm

Non-linear dynamics of transverse beam motion have been studied experimentally
at the VEPP-4M electron-positron collider in 1995-1996. The VEPP-4M storage
ring is a 6*GeV* racetrack electron-positron collider with a circumference of
366*m*. The study was performed at the injection energy of 1.8*GeV*. The
relevant parameters of VEPP-4M at this energy are given in Table:

The following aspects of non-linear beam behavior were investigated: phase space topology near the non-linear resonances and the dynamic aperture reduction due to chromatic sextupoles. The results of the observations are compared with the theoretical estimations[1].

To produce the coherent transverse motion, the beam is kicked vertically or
horizontally with the pulse kicker. The oscillation of the beam centroid and
beam intensity are measured turn-by-turn with electrostatic beam position
monitor (BPM)[2]. The measured single turn BPM r.m.s.
resolution in 1 to 5*mA* beam current range is .
The data are sent to the main frame computer for processing including spectral
analysis, phase trajectories plotting, etc.

Beam vibrations in the low frequency range have been studied also[3].

The most fundamental limit for the current at the VEPP-4M is presently the vertical fast head-tail instability. A feedback system for elimination this instability has been developed[4]. The system provides increase the captured current to the value twice more than the threshold.

A new method for observation of non-linear betatron motion on phase space,
using a single BPM, was developed[3]. Applying FFT with
refinement to the array of 1024 coordinates *x*(*n*) measured by BPM and using
few main harmonics to construct the phase space coordinates *X*(*n*), the rules
for harmonics transformation can be found to obtain the array of *X*'(*n*).
This approach was verified with computer tracking of several examples of
non-linear lattice and compared with usual approach when two BPM used with
betatron phase shift between them.

The harmonic decomposition allows to improve significantly the phase trajectories resolution. While the integral resolution of turn-by-turn beam position monitor is , the spectral resolution of betatron harmonics was achieved of for single measurement, and with averaging for 10 measurements. To decrease errors produced by noise of the processing electronics, computer code selects for construction only harmonics with amplitudes exceeded more than 1.5 times the level of r.m.s. spectral noise density.

High spectral resolution permits estimate the parameters of the non-linear system. We have extracted the amplitude for the resonance driving harmonic from the measured data and compare it with that calculated from the non-linear Hamiltonian. The agreement seems not bad: the experimental value is while the theoretical one is of .

Dynamic aperture studies are being carried out with the same turn-by-turn diagnostic technique but now besides the beam displacement, the beam loss measurement is required to define the dynamic or physical aperture limit. We have found that to determine the aperture limit with turn-by-turn loss measurements it is important to consider first 20-40 revolutions of the beam. A long observation includes many other effects and cannot be a figure of merit for the aperture measurement. Fast intensity loss allows us to distinguish between dynamic and physical aperture limitation. The first one displays the intensity reducing during few tens turns because the oscillation amplitude grows at resonance fast (exponentially) but not immediately, while the second one occurs for the very few revolutions (it depends on the fractional tune value).

We determined the aperture boundary as the displacement at which beam lost the half of its initial intensity. With this technique, the following aperture limits were measured with : , .

The beam loss profile for the first few tens turns shows that for the horizontal direction, the aperture is defined by the non-linear processes, while vertical aperture is determined by the physical limitation.

To increase the dynamic aperture we have tried to decrease the strength
of the strong sextupole lenses in interaction region and compensate the
chromaticity with the sextupole correctors distributed around the ring.
The horizontal dynamic aperture is increased up to 14*mm*.

Beam vibrations in the low frequency range (up to 1*kHz*) are produced by
ground vibration or by pulsation of magnet system power supplies. These
vibrations have being studied using the beam diagnostic system provided
turn-by-turn beam position measurement with adding an integrating ADC
connected to low pass filter output. ADC integrating time determines the
bandwidth and spectral resolution. In our case we have 267*Hz* bandwidth
and resolution. This technique is applied now to test the digital
feedback system for closed orbit stabilization, which is under developing.

The most fundamental limit for the current at the VEPP-4M facility is presently the vertical fast head-tail instability. The beam losses is usually observed in a few tens of milliseconds after injection (this corresponds approximately to the time of radiation damping). The threshold current is determined by transverse impedance of vacuum chamber. The impedance value is about evaluated using measured coherent tune shift.

The fast head-tail instability occurs when frequency of the head-tail mode 0 is shifted sufficiently to couple to the -1 mode. In order to increase an instability threshold it is usually suggested introduce the reactive feedback to compensate for the frequency shift of the mode 0. However, as it follows from experiments[5] its turned out that the introducing of the pure active negative feedback increases the threshold current up to substantially higher values.

This effect can be understandable if one can find the eigen modes of particle oscillations in the bunch. As it was studied on simplest two particles model[6], in the vicinity of instability threshold these modes are approximately the same, they have close eigen frequencies and each mode has the approximately equal amplitudes of the dipole and quadrupole components. When switching the negative active feedback an energy extraction occurs from the eigen modes of oscillations excited by the head-tail interaction in a bunch through the dipole degree of freedom, thereby preventing the instability growth. Such an interpretation is additionally supported by the experimental data.

Near instability threshold the center of gravity of each mode performs the oscillations. This oscillations can be detected by pick-ups and used for suppression of instability by feedback, resistive, in particular.

The striplines are used as the pickup of transverse oscillations. The signals from the opposite striplines are applied to the subtracting transformer having the input impedance equal to the wave impedance of striplines that enable us to separate the signals from the electron and positron bunches. The length of striplines was chosen in such a way that the their sensitivity has maximum values in the frequency range .

The suggested system is made selective with the frequency conversion. The preliminary processing of signals is performed at a frequency in the vicinity of the pickup sensitivity maximum and the formation of frequency characteristics and kicker power supply at low frequency .

The differential signal from the transformer output is applied to the selective filter tuned at a frequency of and then to the frequency converter. The heterodyne voltage for the converter is a signal from the accelerating system . In the low frequency part the feedback system has a filter with a range , preamplifier, phase shifter, attenuator, and the power amplifier. The phase is regulated within the range thus enabling the realization of both the active and reactive feedback.

A pair of the diametrically opposite matched striplines of 1*m*
length is used as a kicker thus providing the separate action on the bunches
of electrons and positrons. The power supply of striplines is in series with
the use of the inverter transformer. The inter-lines maximum voltage is
limited by the power of an output amplifier to the value of 400*V*.

The maximum captured current with the feedback is 25*mA*, it exceeds the
thresholds current more than two times. The results give evidence an
efficiency of the active feedback in the fight against the fast head-tail
instability and can be used at other installation for the development of
similar systems.