ISSN 10637761, Journal of Experimental and Theoretical Physics, 2009, Vol. 109, No. 4, pp. 590–601. © Pleiades Publishing, Inc., 2009. Original Russian Text © O.V. Anchugov, V.E. Blinov, A.V. Bogomyagkov, A.N. Zhuravlev, S.E. Karnaev, G.V. Karpov, V.A. Kiselev, G.Ya. Kurkin, E.B. Levichev, O.I. Meshkov, S.I. Mishnev, N.Yu. Muchnoi, S.A. Nikitin, I.B. Nikolaev, V.V. Petrov, P.A. Piminov, E.A. Simonov, S.V. Sinyatkin, A.N. Skrinsky, V.V. Smaluk, Yu.A. Tikhonov, G.M. Tumaikin, A.G. Shamov, D.N. Shatilov, D.A. Shvedov, E.I. Shubin, 2009, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2009, Vol. 136, No. 4, pp. 690–702.
NUCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Experiments on the Physics of Charged Particle Beams at the VEPP4M Electron–Positron Collider O. V. Anchugov, V. E. Blinov, A. V. Bogomyagkov, A. N. Zhuravlev, S. E. Karnaev, G. V. Karpov, V. A. Kiselev, G. Ya. Kurkin, E. B. Levichev, O. I. Meshkov, S. I. Mishnev, N. Yu. Muchnoi, S. A. Nikitin, I. B. Nikolaev, V. V. Petrov, P. A. Piminov, E. A. Simonov, S. V. Sinyatkin, A. N. Skrinsky, V. V. Smaluk*, Yu. A. Tikhonov, G. M. Tumaikin, A. G. Shamov, D. N. Shatilov, D. A. Shvedov, and E. I. Shubin Budker Institute of Nuclear Physics, Russian Academy of Sciences, Siberian Branch, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia *email:
[email protected] Received May 26, 2009
Abstract—The VEPP4M accelerator facility with a universal KEDR detector is designed to conduct exper iments with colliding electron–positron beams. Highenergy physics, nuclear physics, and studies using syn chrotron radiation are the main directions of research with this facility. In addition, experiments on poorly explored issues in the physics of beams in the electron–positron storage ring and methodological studies to prepare an experiment aimed at testing corollaries of the CPT theorem for an electron and a positron are reg ularly conducted at the VEPP4 facility. A number of works performed in recent years are described: studies to increase the accuracy of comparing the electron and positron spin precession frequencies by the resonant depolarization method; measurements of the count rate of Touschek electrons as a function of the beam energy in a wide range; comparison of the methods for measuring the beam energy spread; a study of the elec tron beam dynamics when a nonlinear betatron resonance is crossed. PACS numbers: 29.20.c, 29.27.a, 29.27.Fh DOI: 10.1134/S1063776109100057
1. INTRODUCTION The VEPP4 accelerator facility is designed to con duct experiments with colliding electron–positron beams. Experiments on highenergy physics are the main direction of research with this facility. The sys tem of particle energy measurements by the resonant depolarization method makes it possible to measure the masses of elementary particles with a very high accuracy. In 2008, the accumulation of statistics in an experiment on a precise measurement of the τlepton mass at its production threshold was completed [1]. This experiment allowed the limits of applicability of the Standard Model, the theory that currently describes most completely the fundamental properties of matter and elementary particles, to be refined. The J/ψ and ψ(2s) meson masses measured at the VEPP4 facility with the KEDR detector [2] are now among the ten most accurate elementary particle masses mea sured over the entire history of physics. Apart from experiments in the field of highenergy physics, on beams of synchrotron radiation ejected from the VEPP3 and VEPP4M storage rings, exper iments to investigate the properties of materials, nano structures, explosive processes, catalytic reactions, and biological objects are conducted [3]. Experiments
on nuclear physics at an internal gas target, a stream of (deuterium or hydrogen) gas with a record intensity injected directly into the vacuum chamber of the VEPP3 storage ring, provide unique information about the proton structure and properties [4]. The physics of charged particle beams in the stor age ring is investigated at the VEPP4 facility. In the next sections, we describe the most interesting works performed in recent years: studies to increase the accuracy of comparing the electron and positron spin precession frequencies in the storage ring by the reso nant depolarization method; measurements of the dependence of the Touschek electron count rate on the beam energy in the range 1.85–4.00 GeV; compar ison of the methods for measuring the beam energy spread; and investigation of the electron beam dynam ics when a nonlinear betatron resonance is crossed. 2. PRECISE COMPARISON OF THE SPIN PRECESSION FREQUENCIES The works on a precise measurement of the spin precession frequency are being carried out in the con text of preparing an experiment to test the CPT theo rem at the electron–positron storage ring [5]. The
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CPT theorem is one of the fundamental principles of modern physics. It states that the physical laws are invariant with respect to the CPT transformation, i.e., they do not change their form if three transformations are made simultaneously: charge conjugation (the replacement of particles by antiparticles), spatial inversion (mirror reflection), and time reversal. It fol lows from the CPT theorem that the particle and anti particle masses, lifetimes, absolute charges, magnetic moments, and other parameters are equal. Experiments to investigate the possibility of testing corollaries of the CPT theorem for an electron and a positron have been conducted at VEPP4M since 2004. The experiments consist in comparing the aver age spin precession frequencies of the electron and positron bunches that simultaneously circulate in the storage ring by the resonant depolarization method with a relative accuracy as high as 10–10. The spin pre cession frequency in the storage ring is determined by a combination of the particle anomalous magnetic moment, mass, and charge. Therefore, the combina tions of simultaneously three fundamental electron and positron parameters can be compared in one mea surement. Previously, such an experiment was carried out at the Institute of Nuclear Physics at the VEPP 2M storage ring [6]. The authors interpreted its results as a comparison of the electron and positron anoma lous magnetic moments with an accuracy of 10–8. At present, the world average accuracy of comparing the electron and positron anomalous magnetic moments, masses, and charges is 3 × 10–9, 8 × 10–9, and 4 × 10–8, respectively. The main goal of the experiments described below is to achieve a minimum statistical error and to study possible sources of the systematic error. A system of absolute energy calibration by the res onant depolarization method [7] is used to measure the average spin precession frequency in a beam. The polarization of an electron or positron beam required for the experiments is produced in the VEPP3 booster storage ring, following which the polarized beam is injected into VEPP4M. The beam injected into VEPP4M is depolarized by a kicker—a pair of matched strip lines inside the vacuum chamber. A traveling TEM wave propagating in the direction opposite to beam motion whose frequency changes linearly with time in a range that includes the expected spin resonance frequency is produced in the strip. The strip connection scheme is implemented in such a way that the standing wave being generated simultaneously acts on the electrons and positrons with the same intensity. A sinusoidal excitation signal is generated by a precision synthesizer developed at the Institute of Nuclear Physics that allows the frequency to be tuned with a step smaller than 10–6 Hz. The intrabunch par ticle scattering effect (Touschek effect) is used to record the beam depolarization. The count rate of scattered particles dependent on their polarization is measured by a pair of movable scintillation counters
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Fig. 1. Superfine scanning at a rate of 5 eV s–1. The measured depolarization frequency is fd = 652896.5518 ± 0.1470 Hz; the beam energy is E = 1851.930 ± 0.008 MeV.
mounted symmetrically on both sides of the vacuum chamber in the orbital plane. In working conditions, the counters are inserted into the dynamic aperture and are located at a distance of about 10 mm from the beam without affecting significantly its lifetime. In these conditions, the counters record the Touschek electrons scattered in the bulk of the ring perimeter. A measuring scheme with two electron bunches, one of which is polarized and the other is not polarized and is used to normalize the count rate, is employed to achieve the limiting resolution in comparing the pre cession frequencies. The measurable quantity is · dN 2 /dt (1) S = 1–N 2 ≡ 1 – , · dN 1 /dt N 1
where dN1/dt and dN2/dt are the count rates of the Touschek electrons scattered, respectively, from the polarized and unpolarized bunches. Under “superfine scanning” at a rate of several electronvolts per second (energy units are used, because the spin frequency and the particle energy are proportional), the depolariza tion process has the form of a prolonged jump whose duration depends on the spin line width and on the rel ative spin frequency drift rate determined by the guid ingfield drift. In addition, the existing field pulsations distort the shape of the jump, introducing a systematic error and limiting the accuracy of determining the depolarization frequency. Figure 1 presents one of the depolarization curves measured in 2007 experiments in superfine scanning mode. An accuracy of determin ing the depolarization frequency from 2.5 × 10–9 to 4 × 10–8 was achieved in this series of measurements. The measured depolarization time (jump duration) had a large spread and lay within the range 150–450 s. At the achieved count rate of 150 kHz mA–1, this circum stance limited the resolution in depolarization fre quency. To explain the results of our measurements, we the oretically studied the superfine scanning process, in which the depolarizer line width is much smaller than
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Fig. 2. Modeling the effect of field pulsations on the shape of the prolonged depolarization jump using (a) a kinetic model for the cases without pulsations (1) and in the pres ence of pulsations with a period of 150 s and an amplitude of 2.5 keV (2); the scanning rate is 5 eV s–1, the spin line width is 1 keV; (b) a phenomenological model of the same scanning for pulsations with a period of 200 s and an amplitude of 2 keV.
the spin precession frequency line width (about 5 × 10 ⎯7 rel. units), based on two specially developed mod els [8]. One model is phenomenological and invokes Monte Carlo methods to model the count rate of scat tered Touschek electrons in the scanning process. The other (analytical) model follows from the kinetic the ory of resonant spin diffusion [9] and contains the main physical parameters of the experiment. Both models allow us to describe the depolarization process in agreement with observations and to explain the change in the shape of the jump by the effect of guid ingfield instability. Figure 2 shows the results of mod eling the effect of guidingfield pulsations on the shape of the prolonged depolarization jump (here, Δ is an analogue of the measurable quantity S, see Eq. (1)). It follows from the results obtained that to achieve a higher experimental accuracy, we must improve the guidingfield stabilization and increase the count rate of Touschek particles. For this purpose, a new recording system was set up in 2008. It allowed the count rate of Touschek particles to be raised from 100–150 kHz to 2 MHz at a beam
current of 2 mA. The system consists of three inserts in the vacuum chamber of the storage ring, each contain ing a pair of movable scintillation counters. The sepa ration between the inserts ranges from 50 to 150 m at a machine perimeter of 366 m. According to our numer ical simulations, the count rate of 2 MHz achieved with the new system allows a limiting statistical error of 10–10 to be attained in an experiment compared to the electron and positron spin frequencies. A feedback system based on the signal of a NMR sensor in a calibration magnet has been developed to stabilize the field of the rotating magnets in the VEPP 4M accelerator and to suppress the field pulsations with a period of more than 10 s. The NMR magne tometer measures the magnetic field with an accuracy of at least 0.5 × 10–6. A digitaltoanalog converter generates a signal proportional to the field that is fed to additional turns in the coil of a contactless current sensor. The signal of the current sensor is used in the feedback circuit to stabilize the current source feeding the guidingfield magnets of the VEPP4M accelera tor. The frequency band of the stabilization circuit is 0.1 Hz; the field control range is about ±10–4. The sys tem allowed the constant guidingfield drift compo nent to be reduced approximately by an order of mag nitude—to a level of 1 keV day–1. Thermal stabiliza tion in the magnet water cooling system was achieved with an accuracy as high as 0.1°C. The results obtained are presented in Fig. 3. Once the stabilization system was switched on, it emerged that there was a residual field drift due to instability of the reference frequency in the NMR sen sor unit. To remove this effect and to reduce the sys tematic error related to the possible relative frequency drift of the reference RF generators and the depolar izer, a system of synchronization of these generators using a rubidium frequency standard that provided a frequency stability better than 10–10 as a reference was developed. A series of measurements with the new Touschek particle recording system was performed in 2008 with guidingmagneticfield stabilization and synchroniza tion of the reference generators of an accelerating RF voltage, the depolarizer, and the NMR magnetometer. A record resolution of (1–3) × 10–9 in resonant depo larization frequency was achieved [8]. Figure 4 shows an example of superfine scanning with the new record ing system and with guidingfield stabilization based on the NMR signal. In the 2008 measurements, the depolarization time fitted into a much narrower range than previously, 15– 75 s, indicating that the stability of the experimental conditions on the whole increased. Since a limiting accuracy of ~10–10 corresponds to the achieved maxi mum count rate of about 2 MHz, it should be noted that a further improvement in accuracy requires a higher level of field stability in all magnetic elements of the storage ring and a reduction in the width of the spin frequency distribution determined by the qua
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Fig. 3. Magnetic field stabilization using the feedback sys tem: (a) time dependence of the guiding field measured with the NMR sensor with switchedoff (1) and switched on (2) feedback; (b) longterm field stability with switchedon feedback on a smaller scale (10–3 Oe corre sponds to a 1keV energy of the particles circulating in the storage ring).
dratic nonlinearity of the guiding field and its lowfre quency pulsations with a period of less than 0.1 s. The accuracy of comparing the electron and positron spin precession frequencies can be increased if the depolarization frequencies of the electron and positron bunches circulating simultaneously in the storage ring are measured without any separation of their orbits at parasitic collision points. Switching off the electrostatic system of orbit separation removes its contribution to the relative shift of the spin frequen cies. At the same time, this measure leads to a decrease in the currents of the colliding beams due to nonopti mal focusing at a parasitic collision point and, as a result, to a decrease in the count rate of Touschek par ticles depending quadratically on the current approxi mately by a factor of 10. However, even if this factor is taken into account, an accuracy of comparing the electron and positron spin frequencies better than 10 ⎯8 is achievable. This allows the limit of the electron and positron charge difference to be lowered by more than a factor of 4. 3. MEASUREMENT OF THE TOUSCHEK ELECTRON COUNT RATE AS A FUNCTION OF THE BEAM ENERGY To ascertain the limits of applicability of the Tous chek polarimeter [8] as the energy of the beams circu lating in the VEPP4M storage ring increases, we
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Fig. 4. Superfine scanning with the new recording system and with guidingfield stabilization. The scanning rate is 2.5 eV s–1, the duration of the prolonged jump is 14 ± 3 s, the depolarization frequency fd = 652270.68930 ± 0.00646 Hz was determined with a relative accuracy of 1.5 × 10–9. The beam energy is E = 1852.266978 ± 0.000348.
measured the count rate of Touschek electrons as a function of the beam energy in the range from 1.85 to 4.00 GeV. For our measurements, we used the scintil lation counters of the polarimeter; the measurement results are presented in Fig. 5. The energy dependence of the count rate for coin cidence in the pairs of conjugate counters with the uncorrelated background ( N· ) subtracted, normalized to the square of the beam current (I2), and multiplied by the ratio of the current beam volume (VE) to the original one (at energy 1.85 GeV) is described by a power law with an index of –2.2 ± 0.2. Theoretical cal culations in the nonrelativistic approximation, includ ing the geometric factor of the counters and the dis tance from the counters to the beam, give a powerlaw index of –3.5. Thus, the derived experimental depen dence does not fit into the existing theories and requires both more accurate measurements and a more careful theoretical interpretation. Extrapolating the experimental dependence, one might expect the count rate of Touschek electrons at energy 5 GeV to be about 12 kHz at a beam current of 10 mA. A theoretical calculation gives 9 kHz. In any case, the count rate seems sufficient for the Touschek polarimeter to be used in measuring the beam energy by the resonant depolarization method at beam ener gies up to 5 GeV. This result is important in planning physical experiments with the KEDR detector in the region of ypsilon resonances. The experimental dependence of the Touschek electron count rate on the beam energy has been obtained for the first time for a comparatively large energy range at the same facility.
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Fig. 5. Normalized count rate of Touschek electrons versus beam energy: the dots represent the experimental mea surements; the line is a powerlaw fit.
4. METHODS FOR MEASURING THE BEAM ENERGY SPREAD The experiments on highenergy physics being conducted at collidingbeam facilities impose strin gent requirements on the stability of beam parameters, including the energy spread. In experiments on a pre cise measurement of particle masses [1, 2], the energy spread determines the energy resolution. The best method for measuring the energy spread is scanning in energy of narrow resonances such as ψ(2s), but this procedure requires considerable expenditure of time and cannot be performed frequently. Thus, the prob lem of measuring the energy spread by accelerator methods is very topical. At the VEPP4M storage ring, a technique for par ticle energy calibration by the edge of the spectrum of Compton backscattering of laser radiation from a cir culating electron beam has recently been imple mented for the first time for colliders [10]. The beam energy spread can also be determined using the Comp ton spectrum edge. For this purpose, the spectrum edge can be fitted by a function x–p g ( x, p 0, 1, …, 5 ) = 1 [ p 4 ( x – p 0 ) + p 2 ]erfc ⎛ 0⎞ ⎝ 2p ⎠ 2 1
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where p0 is the edge position corresponding to the average beam energy, p1 is the edge width proportional to the energy spread, p2 is the step height, p3 is the base height, p4 and p5 are the slopes on the curves, respec tively, to the left and the right of the edge. An example of the edge of the measured Compton spectrum with the fitting curve and parameters of the fit is shown in Fig. 6.
The statistical error of this method for measuring the energy spread at a reasonable statistics accumula tion time (~1 h) is 10–15%. The systematic error is related to an uncertainty in the shape of the HPGe detector energy response at photon energies of 4– 6 MeV and is 5%. Assuming the influence of collective effects to be negligible, the longitudinal beam size σz is propor tional to the energy spread σE/E: σ σ z = αc E , (3) ωs E where α is the orbit expansion coefficient, c is the velocity of light, ωs = νsω0 is the synchrotron fre quency, and ω0 is the angular frequency of revolution. Collective effects lead to the dependence of the longi tudinal beam size on the beam current (Fig. 7). Thus, the longitudinal size should be measured at very low beam currents, which, of course, degrades their accu racy. In principle, the energy spread can be determined from measurements of the transverse (usually horizon tal) beam size σ⊥ at points with a large dispersion, where the contribution from the energy spread to the size is dominant. When the condition σ E⎞ 2 ⎛ D ( s ) εβ ( s ), ⎝ E⎠ where D is the dispersion, β is the amplitude function of the betatron oscillations (beta function) at a given azimuths, and ε is the emittance, is satisfied, the energy spread σE/E can be approximately estimated from the simple formula σ σ (4) E ≈ ⊥ . E D However, the accuracy of the measurements may also prove to be insufficient.
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There are also other methods for determining the energy spread from measured beam parameters. A spectral analysis of chromatic synchrobetatron beam oscillation modes underlies one of these methods. Because of chromaticity, satellites at frequencies that are multiples of the synchrotron frequency appear in the spectrum of betatron beam oscillations near the main peak. Their relative amplitude is defined by the expression [11] ∞
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Fig. 8. Spectrum of synchrobetatron oscillations.
Fig. 7. Bunch elongation versus beam current for electrons (䉱) and positrons (䊉); the solid curve represents the calcu lation.
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[13]. An example of the spectrum is shown in Fig. 8; three pairs of synchrobetatron satellites are clearly seen. The measurements were made for several values of the vertical chromaticity in the range ξy = 5–20. The measured ξy dependence of the amplitude R1 of the first synchrobetatron satellite normalized to the amplitude R0 of the betatron peak is shown in Fig. 9 for three modes of VEPP4M operation differing in energy and energy spread. Fitting the measured data by function (5) yields the energy spread. An alternative method is based on an analysis of the envelope of the betatron oscillations. As was shown in [14], the envelope A(t) of the free betatron oscillations excited by a short kick with an amplitude δa much larger than the transverse beam size σ⊥ is 2
t ⎞ A ( t ) ∝ exp ⎛ – ⎝ 2τ 2⎠
ω β α ω 0 ξ⎞ σ E y = ⎛ + , ⎝ ωs ωs ⎠ E Jm is the Bessel function, m is the synchrobetatron har monic number, ωβ = νβω0 is the betatron frequency, and ξ = Δνβ(Δp/p)–1 is the chromaticity. To eliminate the influence of collective effects on the beam dynamics, the measurements should be made at low beam currents. A multianode photomul tiplier tube (PMT) that ensured measurements of the transverse beam profile with a singleturn time resolu tion was used at VEPP4M [12]. The PMT sensitivity allows the measurements to be made at beam currents of ~10 μA, which makes it possible to get rid of the col lective effects almost completely. The beam oscillations were excited by a short elec tromagnetic pulse in the kicker; the kick amplitude was chosen in such a way that the beam oscillation amplitude exceeded appreciably the vertical bunch size. The oscillation spectrum was calculated by a dis crete Fourier transform (DFT) of the measured set of beam coordinates. The synchrobetatron harmonic frequencies and amplitudes were determined by refin ing the DFT by the method of intermediate harmonics
∂ω σ × exp – ⎛ β E⎞ ( 1 – cos ω s t ) , ⎝ ∂E ω s ⎠
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where ∂ω = 2 2β σ ⊥ δa. ∂a At nonzero chromaticity, the envelope is modulated in amplitude with the synchrotron oscillation frequency; the modulation depth depends on the chromaticity and energy spread. The energy spread can be determined as a parame ter of the fit to the measured envelope of the betatron oscillations by function (6). However, the shape of the envelope deviates from (6) because of the rapid damp ing due to the beam interaction with the transverse impedance of the vacuum chamber. The damping rate is proportional to the beam current. The problem can be solved by two methods: by measuring the damping rate and including the damping in the fit or by per forming measurements of the envelope with a low intensity beam when the damping may be neglected.
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Fig. 10. Application of a discrete comb filter: (a) input measured data without any filter; (b) after filtering (the envelope is shown).
In our experiments at VEPP4M, we performed mea surements with a multianode PMT that had a suffi cient spatial resolution at a beam current of 10–30 μA.
energy spread to be determined with an accuracy of at least 5%.
To further increase the measurement accuracy, we applied a discrete comb filter algorithm. The spatial resolution of the diagnostic system is determined by noise in the output signal band. Being inversely pro portional to the beam intensity, the noise error depends on the band of the sensor frequency response. Various analog and discrete filters are used to reduce the effect of noise and to improve the coordinate reso lution in diagnostic systems. An example of such a fil ter is a discrete Fourier transform, which increases the resolution by a factor of M , where M is the size of the measured data set. If the objective is to determine the shape of oscillations with a complex envelope, then the application of a narrowband discrete comb filter provides optimal noise suppression. Basically, the filter consists in the Fourier synthesis of a set of samples from the small number of Fourier harmonics that determine the useful signal. In this decomposi tion/synthesis operation, the noise harmonics con tained in the full Fourier spectrum of the original sig nal are rejected, thereby improving the signaltonoise ratio in the new set of samples.
5. INVESTIGATION OF THE BEAM DYNAMICS WHEN BETATRON RESONANCES ARE CROSSED The influence of resonances on the beam dynamics is of considerable interest in the physics of circular accelerators. The crossing of a betatron resonance by a beam in a synchrotron or a storage ring and the accompanying particle losses or the perturbation of the beam distribution have been actively studied at various accelerator laboratories [15–17]. The pro cesses under consideration are important, for exam ple, in creating systems of efficient resonant ejection from accelerators. In recent years, interest in investi gating the questions of resonance crossing has increased in connection with the intensive develop ment of FFAG (fixedfield alternatinggradient) syn chrotrons [18] and damping rings for linear colliders, in which the damping of particle oscillations is accom panied by a shift in betatron frequency νβ (to 0.1–0.2) and the crossing of many resonances due to the space charge and strong nonlinearity [19]. The crossing of the betatron resonance 3νy = 23 was studied experimentally at the VEPP4M elec tron–positron collider. The resonance was crossed by an electron beam with various speeds. The strength of the resonance was changed by a separate rotated sex tupole magnet, while the nonlinear betatron fre quency shift was controlled by octupole lenses. During the experiment, we measured a number of parameters, such as the particle loss rate, the beam size, the trans verse distribution, the phase trajectories, the ampli tudedependent frequency shift, etc. The VEPP4 facility is equipped with beam diag nostic tools needed to investigate the nonlinear motion. An electrostatic beam position sensor in turn
An example of using a discrete comb filter is shown in Fig. 10; the oscillations with envelope (6) were syn thesized from the set of harmonics νβ ± mνs, where νβ and νs are, respectively, the betatron and synchrotron frequencies in units of the frequency of revolution, m = 0, 1, 2, 3. The harmonic frequencies and ampli tudes were calculated with a high accuracy using a dis crete Fourier transform refined by the method of intermediate harmonics [13]. An example of the fit to the measured envelope of the betatron oscillations by function (6) is shown in Fig. 11. It is believed that an optimal fit allows the
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tion is used for a vertical kick to the beam. A special thyratron generator was developed for a horizontal kick. Two pairs of plates that are not matched in impedance are used as the kicker. In addition, they are arranged with a gap, since another pair of plates placed between them is used for an electrostatic separation of the electron and positron beams at a parasitic collision point. These circumstances required making a gener ator with rectangular pulses that would not be sub jected to significant distortion at such a large inhomo geneity. The beam was acted upon only by the mag netic field through the current flowing in the plates. The shape of the kicker pulse used to excite coherent oscillations of the VEPP4M beam is shown in Fig. 13.
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Fig. 11. Fit to the measured envelope of the betatron oscil lations (solid line) by function (6) for the energy spreads σE/E = 6.0 × 10–4 (1), 6.6 × 10–4 (2), and 7.4 × 10–4 (3).
The behavior of the system near a onedimensional nonlinear resonance of the form mνβ = n is described by the Hamiltonian (in action I–phase ϕ variables)
byturn mode with a spatial resolution of about 50 μm was used to record coherent transverse beam oscilla tions. The particle losses and the tails of the beam dis tribution were measured by the scintillation counters inserted into the vacuum chamber. The counters can be displaced by stepping motors with an accuracy of at least 0.1 mm. CCD cameras that could be activated by an external start pulse were used to measure the trans verse sizes and position of the beam; the time error was about 100 μs. A unique device [12] based on a multi anode R5900U00L16 Hamamatsu photomultiplier tube was employed for turnbyturn measurements of the transverse beam distribution during tens of thou sands of turns. This device records the transverse beam profile at each turn at 16 points during 217 bean turns. Figure 12 shows an example of the beam profile mea surements over 120 turns. To investigate the phase trajectories, coherent beta tron oscillations were excited by an impulsive electro magnetic kicker. A positron inflector operating at a lower voltage than that in the mode of positron injec
2
H = δ ( θ )I + α 0 I + A n I
n/2
cos mϕ,
(7)
where the nonlinear amplitude dependence of the fre quency is determined by the coefficient α0, m is the resonance order, n is the perturbation harmonic num ber, An describes the strength of the resonance pertur bation, and the detuning from the exact resonance fre quency δ(θ) depends on the “time”—the azimuthal angle θ. Two cases can be distinguished, depending on the parameter α0. At α0 ≠ 0, three more additional regions (separa trices) are formed around the central stable resonance region. When the resonance is crossed slowly (adiabat ically), the particles are captured into these resonance “islands” and are transported together with the islands to large amplitudes, where they can be lost at aperture restrictions. The adiabatic crossing condition relates the action I (amplitude) of a particle that can be cap rel. units
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Fig. 13. Oscillogram of the pulse from the electromagnetic kicker.
tured into resonance to the crossing speed ν 'β = dνβ/dθ as [20] ν β ⎞ 1/n I > ⎛ . ⎝ 2mα 0 A n⎠
(8)
Indeed, fine tuning the nonlinearity α0 (with the octupole lenses) allowed three stability regions of the resonance 3νy = 23 to be obtained. Figure 14 shows the phase trajectory within the resonance islands con structed by the method described in [21] from turn byturn measurements of coherent beam oscillations. Figures 15 and 16 show the time evolutions of the transverse particle density distribution in the beam when a nonlinear resonance is crossed. The horizontal and vertical axes correspond to the time and vertical coordinate, respectively; different shades of gray cor respond to different particle densities. As the theory predicts, the process of resonance crossing depends significantly on its speed. For a fast crossing, neither changes in the beam size nor particle capture into separatrices are observed (Fig. 15a). As the time of frequency change decreases (Fig. 15b), an increase in the beam size directly at the time of reso nance crossing and an initial dragging of particles into resonance are observed. However, the dragging of particles into resonance separatrices is most pronounced at an even larger slowdown in frequency tuning (Fig. 16). The capture
of part of the beam into a resonance island and its recession from the beam axis are clearly seen. The transverse beam distribution (Gaussian fit) at the various resonance crossing times corresponding to the section lines in Fig. 16 is shown in Fig. 17. A qualitatively different case of resonance crossing takes place at α0 = 0. In this case, no additional stabil ity regions emerge and the trajectories of all particles in the beam at exact resonance are unstable. Obvi ously, there can be no particle capture into resonance in this crossing regime, but there can be intensity losses, depending on the crossing speed. This property is used for the resonant ejection of particles from the accelerator. An estimate of the crossing speed ensuring that the entire beam will be lost (ejected from the vac uum chamber of the accelerator), an analogue of the adiabaticity condition, gives [20] dν 1 2 (9) y A n ε y , dθ 8π where An is the perturbation harmonic and εy is the vertical beam emittance. The regime with α0 = 0 was tuned with the octupole lenses. The phase trajectory near the resonance mea sured using a vertical coherent kick is shown in Fig. 18. When the resonance was crossed, the particle losses were measured by a synchrotron radiation sensor and the scintillation counter inserted into the vacuum chamber. The characteristic pattern of losses when an unstable thirdorder resonance is crossed is shown in Fig. 19. The betatron frequency changed within the range 0.6687 0.6653 in 1 s. The irregular structure rel. units
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Fig. 14. Particle phase trajectory within the resonance island in variables y, y' (a) and I, ϕ (b).
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Fig. 15. Change in frequency νy = 0.6608 time of change is 40 ms (a) and 0.3 s (b).
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Fig. 16. Resonance crossing; the time of frequency change is 3 s.
The theory developed in [20] allows the plot of par ticle losses to be constructed when a thirdorder reso nance is crossed for α0 = 0. In particular, the theory predicts that, depending on the crossing direction, one of the edges of the plot of losses will have a sharper drop. Indeed, such a behavior was observed experi mentally. Figure 20 shows plots of the measured and calculated particle losses when an unstable thirdorder resonance was crossed in 1 s. However, the coinci dence of the measured and calculated shapes of the dependences of particle losses when the resonance was crossed was accompanied by a significant difference in the absolute value of the losses. Radiative damping (disregarded by the theory) and residual nonlinearity (α0 ≠ 0) for particles with large oscillation amplitudes are currently considered as an explanation of this fact. Measurements of the phase trajectories and the dynamic aperture allow the strength (harmonic ampli tude) of the resonance 3νy = 23 typical of our experi ments to be estimated as A3 ≈ 0.02–0.07 mm–1/2 (see Hamiltonian (7)). The skew sextupole field compo nent corresponding to the resonance under consider ation can be produced by rotating the normal sextu pole magnet (compensating for the natural chromatic ity) around the beam axis. However, estimates of the resonance strength give an unrealistically large rota tion angle—an order of magnitude larger than that provided by a geodetic alignment of the magnets. Sim ulations show that the harmonic amplitude of the skew sextupole can be explained by the existence of a linear betatron coupling that determines the vertical beam emittance εy = κεx, where κ is the coupling coefficient
(κ ≈ 2–4% for VEPP4M). In this case, not the median plane of the magnet but the axes of the normal oscillation modes are rotated, with the rotation angle of the axes increasing through a coefficient of the order of (νx – νy – n)–1, which takes into account the frequency detuning from the nearest coupling reso nance. For VEPP4M with a noninteger part of the 250 200
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of the plot of particle losses is probably related to insta bility (though small, ~10–4) of the power supplies of the quadrupole magnets.
0 150 100 50 0 150 100 50 0
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Fig. 17. Transverse particle density distribution in the beam corresponding to the resonance crossing times marked in Fig. 16.
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Fig. 18. Phase trajectory near a thirdorder resonance at α0 = 0.
. N, kHz 1.2 1.1 1.0 20 10 0 0.666
0.667
0.668 νy
Fig. 19. Particle losses measured by the synchrotron radia tion sensor (a) and the scintillation counter (b). The verti cal line corresponds to the exact resonance.
. N, rel. units 1.0 0.8
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REFERENCES
. N, rel. units 1.0 2
(a)
(b)
0.2 1 0.667
νy
0 0.666
tron radiation, the physics of charged particle beams in accelerators is regularly investigated at the VEPP4 facility. Owing to the guidingmagneticfield stabilization of the VEPP4M accelerator with a feedback system and the use of reference generators of an accelerating RF voltage, the depolarizer, and the NMR magne tometer from a rubidium frequency standard, a record (1–3) × 10–9 accuracy of measuring the resonant depolarization frequency has been achieved in super fine scanning experiments. The experimental dependence of the Touschek elec tron count rate on the beam energy has been measured for the first time at the same facility in the beam energy range from 1.85 to 4.00 GeV. The result obtained can be demanded for the designed Super Cτ and Bfactories in estimating the background conditions. We measured the beam energy spread for the VEPP4M accelerator, which determines the energy resolution in experiments on precise particle mass measurements. A comparative analysis of the various methods for measuring the energy spread was per formed. We carried out experimental studies of the beam dynamics as betatron resonances were crossed. They are of considerable interest in the physics of accelera tors, since the problem of resonance crossing and the accompanying particle losses or perturbations of the beam distribution may turn out to be very topical for future lowemittance electron–positron accelerators.
2
1 0.667
νy
Fig. 20. Measured (1) and calculated (2) particle losses when the resonance is crossed from bottom to top, y = 0.6653 0.6685 (a), and from top to bottom, y = 0.6687 0.6653 (b).
betatron frequencies νx = 0.56 and νy = 0.62, this coef ficient is approximately equal to 20. 6. CONCLUSIONS In addition to experiments on highenergy physics and nuclear physics and experiments using synchro
1. A. G. Shamov, KEDR Collab., V. V. Anashin, V. M. Aulchenko, E. M. Baldin, A. K. Barladyan, A. Yu. Barnyakov, M. Yu. Barnyakov, S. E. Baru, I. V. Bedny, O. L. Beloborodova, A. E. Blinov, V. E. Bli nov, A. B. Bobrov, V. S. Bobrovnikov, A. V. Bogo myagkov, A. E. Bondar, D. V. Bondarev, A. R. Buzykaev, S. I. Eidelman, Yu. M. Glukhovchenko, V. V. Gulevich, S. E. Karnaev, G. V. Karpov, S. V. Karpov, V. A. Kiselev, S. A. Kononov, T. A. Kozlova, K. Yu. Kotov, E. A. Kravchenko, V. F. Kulikov, G. Ya. Kurkin, E. A. Kuper, E. B. Levichev, D. A. Maksimov, V. M. Malyshev, A. L. Maslennikov, A. S. Medvedko, O. I. Meshkov, S. I. Mishnev, I. I. Morozov, N. Yu. Muchnoi, V. V. Neufeld, S. A. Nikitin, I. B. Nikolaev, I. N. Okunev, A. P. Onuchin, S. B. Oresh kin, I. O. Orlov, A. A. Osipov, S. V. Peleganchuk, S. S. Petrosyan, S. G. Pivovarov, P. A. Piminov, V. V. Petrov, A. O. Poluektov, I. N. Popkov, G. E. Pospe lov, V. G. Prisekin, A. A. Ruban, V. K. Sandyrev, G. A. Savinov, A. G. Shamov, D. N. Shatilov, B. A. Shwartz, E. A. Simonov, S. V. Sinyatkin, Yu. I. Skovpen, A. N. Skrinsky, V. V. Smaluk, A. M. Su kharev, E. V. Starostina, A. A. Talyshev, V. A. Tayursky, V. I. Telnov, Yu. A. Tikhonov, K. Yu. Todyshev, G. M. Tumaikin, Yu. V. Usov, A. I. Vorobiov, A. N. Yush kov, V. N. Zhilich, V. V. Zhulanov, and A. N. Zhuravlev, Nucl. Phys. B, Proc. Suppl. 181–182, 311 (2008).
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