Betatron Oscillation Frequency Research Articles

Orbits in Fixed Field Alternating Gradient Synchrotron

An analytical investigation has been made of the motion of a charged particle in an axially symmetric steady magnetic field. For a given momentum, expressions for the shape of the equilibrium orbit and betatron oscillation frequencies have been derived in terms of the field index k, the spiral parameter K, and the set of field coefficients gnm. These algebraic results have been programmed for the IBM704 computer. For a 500 MeV fixed field alternating gradient (FFAG) synchrotron designed as an injector,1 computations yield radial and vertical tunes averaged with respect to momentum of Δr = 3.23, ≤z = 2.37 with variations of |νr| Ζ 0.0001, |Δνz| Ζ 0.0004 depending on the exact momentum. For comparison, Runge-Kutta evaluations which require a factor of six longer computing times yield average tunes of νr = 3.22 and νz = 2.49.

Betatron oscillation resonances in an electron synchrotron

Abstract The conditions under which certain betatron resonances of the 2 nd to the 5 th order are excited, and their influence on accelerator operation, have been studied in an electron racetrack. The precise measuring methods (≈ 0.1%) of parameters n and λ which determine the betatron oscillation frequencies and resonance relations, have been elaborated. High speed filming of optical synchrotron radiation has been broadly used in experimental investigations.

Interaction of coherent betatron oscillations with external systems

The accelerated attenuation of betatron oscillations observed in the VI'EPP-2 storage ring cannot be explained by the finite conductivity of the chamber walls. The interaction of a group of particles with various kinds of parasitic dissipative systems was therefore studied. It was found that, for a give frequency of betatron oscillations, there could be either accelerated attenuation or the self-excitation of coherent oscillations, depending on the impedances of the interacting element for the combination frequencies. The increments and decrements thus found may greatly exceed the values due to the finite conductivity of the chamber walls.

Beam Measurement Probes for the MURA 50-MeV Electron Accelerator. XV

The design and construction of probes to measure beam size, orbit shape, betatron oscillation frequencies, and beam intensity are described, together with the mechanical and electrical aspects of the probe drives which move the probes.

Choice of Betatron Oscillation Frequencies in a Spiral Sector FFAG Accelerator

In a fixed-field alternating gradient (FFAG) accelerator, the stability limits and the stability of the machine against field perturbations are very much affected by the choice of the betatron oscillation frequencies, νr and νz. A general criterion for the stability of the accelerator is proposed and, using this criterion, a computer study was carried out on the dependence of the stability on νr and νz. The results of this stability survey seem to indicate a a relatively unique choice of νr and νz to obtain the desired beam intensity.

Electron model studies of particle motions in spiral ridge magnetic fields

An electron model has been used to investigate both radial and vertical trajectories for one turn in a spiral ridge magnetic field. Extensive magnetic field measurements were made and a method of setting up reproducible magnetic fields taking into account remanent field and hysteresis effects was devised. Various plots of beam profile around the machine are discussed; in particular, the effects of harmonics of the magnetic field, higher than the fourth, on the radial betatron oscillation frequency, have been measured.

Measurement of Betatron Oscillation Frequencies on the UCLA 50-MeV Cyclotron

The vertical and radial betatron oscillation frequencies have been measured on the UCLA 50-MeV spiral-ridge cyclotron. The technique used was that of rf knockout, in which the internal beam was passed through a pair of exciter plates which provided an rf electric field. When the frequency was equal to the betatron frequency a detectable increase in betatron oscillation amplitude or ``knockout'' occurred. The knockout as a function of exciter frequency shows a number of resonances rather than a single one. This provides a measurement of radial as well as vertical betatron frequency. Comparisons of the measured frequencies with those calculated from magnetic field measurements are given and show satisfactory agreement.

Feasibility of Accelerating Polarized Protons with the Argonne ZGS

A computing program was developed for simultaneously tracing trajectories and classical, relativistic spin precessions around the zero gradient synchrotron (ZGS). Depolarizing resonances, previously predicted, were found and their properties were computed; there were 36 significant resonances, 12 of which were serious. A method is proposed for avoiding these 12 resonances by using two or four pulsed quadrupole magnets around the ZGS ring. These would rapidly shift νz, the vertical betatron oscillation frequency, at appropriate energies during the acceleration cycle. With this method, a proton beam, injected into the ZGS with P=1, would have P>0.7 at the end of the acceleration cycle.

On the design of an isochronous cyclotron meson factory

A fixed energy high beam intensity isochronous cyclotron for meson production is under consideration at CERN, Geneva. A convenient and rapid method is described for determining suitable spiral ridge fields giving a chosen variation of the linearized vertical betatron oscillation frequency versus energy. The method implies an approximation of the field ridges into a prechosen number of small hard-edged steps. Some computational results are discussed and a preliminary choice of machine parameters given, as well as a conceptional design of the field magnet and the general layout of other major machine components. The problem of shielding is discussed. A series of shielding studies involving experiments carried out at 400 MeV on the CERN Synchro-Cyclotron are described and some results are quoted. Finally, a description is given of a series of irradiation and activation experiments involving samples of various materials placed inside the CERN SC.

Accelerators with a general magnetic field

The linear motion of a particle moving in an accelerator having a general magnetic field is investigated. A solution is found which is valid for a wide class of accelerators, which includes the AG synchrotron, FFAG accelerators and the fixed-frequency cyclotron. Results are given for the equilibrium orbit, the betatron oscillation frequencies, the betatron oscillations and other linear orbit properties.

Theory of the alternating-gradient synchrotron

The equations of motion of the particles in a synchrotron in which the field gradient index n=−( r B ) ∂B ∂r varies along the equilibrium orbit are examined on the basis of the linear approximation. It is shown that if n alternates rapidly between large positive and large negative values, the stability of both radial and vertical oscillations can be greatly increased compared to conventional accelerators in which n is azimuthally constant and must lie between 0 and 1. Thus aperture requirements are reduced. For practical designs, the improvement is limited by the effects of constructional errors; these lead to resonance excitation of oscillations and consequent instability if 2 ν x or 2 ν z or ν x + ν z is integral, where ν x and ν z are the frequencies of horizontal and vertical betatron oscillations, measured in units of the frequency of revolution. The mechanism of phase stability is essentially the same as in a conventional synchrotron, but the radial amplitude of synchrotron oscillations is reduced substantially. Furthermore, at a “transition energy” E 1 ≈ ν x Mc 2 the stable and unstable equilibrium phases exchange roles, necessitating a jump in the phase of the radiofrequency accelerating voltage. Calculations indicate that the manner in which this jump is performed is not very critical.

Accelerators with similar orbits

We obtain the conditions that must be satisfied by a magnetic system in order that the frequencies of radial and vertical betatron oscillations be independent of the particle momenta (in this case the orbits are called dynamically similar). In such systems mere should in principle be no excitation of betatron oscillations associated with synchrotron oscillations and other phenomena. A magnetic field\(H_0 \left( {\theta } \right)\left( {{{r_0 } \mathord{\left/ {\vphantom {{r_0 } r}} \right. \kern-\nulldelimiterspace} r}} \right)^{\Pi _0 } \) with n0 = const produces both geometric and dynamic similarity of the orbit. In weak-focusing accelerators with segments (race tracks) and in strong-focusing proton synchrotrons, the orbits are not dynamically similar. In order to obtain this kind of similarity in the first case, in addition to n0 = const it is necessary that the magnet sectors have a common center. Different types of annular synchrocyclotrons are considered. In the first type the centers of neighboring magnet sectors are located on different sides of the doughnut and in the second type at the same point (at the center of the accelerator). In the second type the orbits are dynamically similar, unlike those of the first. It is shown mat it is possible to design an annular synchrocyclotron in which the particles can move with stability simultaneously in both directions within the doughnut.

Linear Theory of Betatron Oscillations in Sectorial Cyclotrons

In a sectorial cyclotron the ``force functions'' and the frequencies of the linearized normal and axial betatron oscillations are given as power series in the inverse-square of the number of sectors. The coefficients in these series are expressed either in terms of parameters characterizing the magnetic field or in terms of those specifying the family of equilibrium orbits. The first-order terms agree with those in existing approximate expressions. Among other features of the higher order terms, they show that it is possible to design magnetic fields with momentum compactions varying over wide ranges and frequencies of betatron oscillations remaining constant. Together with a scheme proposed earlier of transferring ions continually between circular accelerators, this opens the possibility of accelerating ions continually in several stages of constant-frequency sectorial cyclotrons to multi-Bev energies. This method of analysis can also be applied to the study of nonlinear betatron oscillations.

Betatron Oscillations in the Synchrotron

Betatron oscillations in the University of Michigan synchrotron have been investigated by exciting these oscillations with a transverse electric field. The results indicate that the betatron oscillation is composed of a group of component frequencies which are separated by the frequency of synchrotron phase oscillation. The suggested explanation of the observed splitting is the modulation of the betatron oscillation frequency by the synchroton oscillations.

The Dynamics of a Synchrotron with Straight Sections

The equations describing the ``betatron'' and ``synchrotron'' oscillations of a particle being accelerated in a synchrotron having straight sections are obtained. In particular, the relative increase in the ratio of either betatron-oscillation frequency to the frequency of revolution is found to be approximately half the ratio of the total length of straight sections to the orbit circumference. The effect of magnetic fringing fields is considered. With regard to the synchrotron phase oscillation, it is found that the criterion of non-relativistic adiabatic damping is no longer that n exceed ⅔ but that it exceed ⅔ by at least ⅓ of the ratio of the total length of the straight sections to the orbit circumference.