Alper A. Garren

Electrostatic deflector calculations for the Berkeley 88-inch cyclotron†

Early computor calculations showed that an electrostatic channel would have to be very narrow to cause sufficient deflection at high energy. One alternative considered was a system of three electrostatic channels in series, acting as a one-turn regenerative extractor. Such a system could easily achieve the required deflection, but at the expense of simplicity. A magnet channel to reinforce the electrostatic channel likewise appeared difficult for a variable-energy machine. Detailed studies of a simple electrostatic channel began with development of an analytic-graphical method of calculating channel efficiency. By this method one can estimate acceptance as a function of radial turn separation, amplitude and frequency of radial oscillations, and channel geometry. A computer code was then developed to compute channel shapes and calculate acceptance more accurately by following the fate of a representative sample of orbits in the presence of the channel. Use of this code indicated that highest acceptance would be obtained if the channel entrance is set out at 39 in. provided the resonances at v r = 1 and v r = 2 v z could be safely traversed, as was believed possible. Some calculations on transition through the v r = 2 v z resonance are presented.

INDIVIDUAL PARTICLE MOTION AND THE EFFECT OF SCATTERING IN AN AXIALLY SYMMETRIC MAGNETIC FIELD

Two aspects of the behavior of a collection of charged particles in an axially symmetric magnetic-field configuration were studied. The first involves the trajectories of particles in this field configuration in the absence of collisions with other particles. By analytic and numerical methods these trajectories were studied and several different types observed. The other aspect involves interparticle Coulomb collisions. The equations derived by other authors were numerically solved for several cases of interest. Details of the calculations and a discussion of the results are presented. (auth)

CENTER-REGION GEOMETRY OF THE BERKELEY 88-INCH CYCLOTRON

In order to determine the proper position of the dee edge, ion source, puller, and beam-defining clipper with respect to the magnet center, measurements of the structure of the electric field in the surroundings of the ion source were made with a half-scale model in an electrolytic tank. The acceleration process during the first three revolutions was studied for different geometries and starting phases, by a graphical approximation method for orbit plotting. A more detailed study was then made to determine initial conditions—such as orbit-center distribution, energy spread, axial focusing, and energy bunching—as functions of the center geometry and starting phase. By increasing the particle path length during the first half revolution approximately 20 deg—achieved by azimuthal displacement of the source—good results were obtained. The ion-start location was finally determined so as to bring the beam close to an equilibrium orbit at 10 in. radius. This was done with an orbit code, modified to take account of the increased transit time through the acceleration gap of smaller radii. Some experimental results of changes in the cyclotron beam quality, caused by changes in the ion-starting conditions, are presented.

CALCULATION OF CYCLOTRON TRIM-COIL CURRENTS FOR FIELD OPTIMIZATION BY LINEAR PROGRAMMING METHODS

Determination of the current settings in the circular pole-face windings of fixed-frequency cyclotrons for optimization of the magnetic field may be solved with the mathematical technique of linear programming. The resulting fields are optimal with regard to particle phase lag as well as radial and vertical stability, and the currents do not exceed preassigned limits.

EXTENDIBLE-ENERGY SYNCHROTRONS.

Abstract Synchrotron guide-field arrangements have been developed that provide for a convinient and economical later increase in energy by the addition of components omitted in the initial construction. Criteria for the design of such an accelerator include new considerations such as emphasis on small circumference to reduce initial costs and on magnet configurations for both stages that have similar orbit behavior and that require little realignment of replacement of the original magnets in the conversion. Space added for future components might also be useful to improve the accelerator in other directions. Thus, at modest cost, the initial layout would afford, in addition to specific provision for conversion to higher energy, a more general flexibility for response to future needs and exploitation of future technology.

ORBIT DYNAMICS IN THE SPIRAL-RIDGED CYCLOTRON

Formulas are derived for the equilibrium orbit, isochronous condition, vertical and horizontal betatron frequencies, and for the effects of the 3/3 radial resonance in a three-fold geometry. The magnetic field is represented by a Fourier series in azimuth with amplitudes expanded in a Taylor series about the reference radius. The form is such that the various parameters may be deduced from an arbitrary set of field measurements in the median plane and the results obtained by direct substitution in algebraic formulas. (auth)

COMPUTATIONAL PROCEDURES IN THE ADJUSTMENT OF A CYCLOTRON MAGNETIC FIELD BY TRIM COILS

This report outlines the system of computational codes used to obtain values for the trim-coil currents to effect an optimal adjustment of the magnetic field in the 88-inch cyclotron at Berkeley according to the method described by one of the authors. The machine computation is traced through steps of analysis of measured field data, determination of restraints for a linear model of the problem, construction of the linear-program matrix, solution for values of trim-coil currents, iteration to improve on this solution, and prediction of orbit properties in corrected field.