Hard-edge Model Research Articles

Study of magnetic fringe fields and interference effects on beam dynamics for proton facility gantry

ABSTRACT Proton therapy facilities require a beamline that effectively delivers particles to a nozzle that forms symmetric and achromatic beam spots. In the gantry lattice of the SC200 facility, the magnetic components are designed with a compact layout due to space constraints. This makes it important to properly deal with fringe fields and their interference when more accurate simulations of beam dynamics are required. In this paper, we establish field distributions based on real extracted magnetic maps and analyse the influences of linear fields on beam qualities by combining a perturbation model with Monte Carlo code in BDSIM. To verify the results, COSY INFINITY with the Enge function was used to quantify the beam variations at the isocenter; the results were in good agreement with those of BDSIM. Compared with the conventional hard edge model, beam mismatch at the isocenter is evidence of the impacts of fringe fields, which pose challenges to clinical dose planning. Hence, linear optical optimization on the gantry is also necessary to meet safety and precision requirements. These efforts will guide the commissioning of the beamline.

Effects of linear quadrupole fringe field overlap on transverse matching to an radio frequency quadrupole

Along many low energy beam transport (LEBT) systems transverse hadron beam matching to a subsequent radio frequency quadrupole (RFQ) is performed by magnetic quadrupole multiplets. Large apertures combined with short distances between the quadrupoles lead to overlap of individual singlet fields and significantly impact on the effective beam optics. Systematic measurements of beam transmission through an RFQ have been done and compared to beam dynamics simulations. For the latter, simplified hard-edge quadrupole fields were replaced by measured field data. It turned out that the hard-edge model is not sufficiently accurate to achieve acceptable agreement between calculated and measured beam properties.

Consideration of fringe field overlap of quadrupoles in a compact THz-FEL prototype

Huazhong University of Science and Technology has built up a compact terahertz free electron laser prototype. In the transport line, three quadrupoles with aperture of 45 mm are densely packed and the iron to iron distance are merely 148 mm (Q1-Q2) and 123 mm (Q2-Q3) respectively. This compact arrangement inevitably causes fringe field overlap between the quadrupoles. As the iron to iron induced fringe field reduction has been studied extensively, here we focus on the change in focusing properties of the quadrupole triplet when the fringe field overlap is presented. The three quadrupoles have been measured independently with a Hall probe system and the measurement data are used to do the slice analysis. Comparison among the conventional hard-edge model, the equivalent hard-edge model and the slice model is addressed to evaluate the influence of the fringe field overlap. It is found that the equivalent model is a much better approximation than the conventional model though it has slight difference compared to the slice model. The equivalent model is preferred for beam matching calculation during a physical design phase and the slice model is beneficial for current adjustment in beam commissioning.

Variable-energy isochronous accelerator with cotangential orbits for proton beam therapy

We have devised a new isochronous accelerator for proton beam therapy, which can control beam energy from 70 MeV to 235 MeV without using degraders. A feature of this accelerator is that the equilibrium orbits are cotangential, the same as for the classical microtron. This configuration of orbits leads to a small turn separation region on one side and a large turn separation region on the opposite side. Each beam between 70 MeV and 235 MeV is kicked by a massless septum installed in the large turn separation region, and it is extracted from the vicinity of the small turn separation region. Here we show an optical design of this accelerator, which is based on and extended from the hard edge model of azimuthally varying field cyclotrons. We found that the maximum and minimum magnetic fields are 2.45 T and 0.83 T, respectively, turn separation between 70 MeV and 235 MeV in the small turn separation region is 12.4 mm, and horizontal and vertical tunes over 10 MeV are between 1.0 and 1.5 and between 0.5 and 1.0, respectively. We calculated orbit deviations by a one-milliradian kick while varying the kick position and found that the large turn separation region is appropriate for installing the massless septum; this finding supports the consistency between the optical design and the extraction scheme.

Magnetic field distortion and particle optics in quadrupole magnets when placed close to dipole magnets

The field quality of quadrupole magnets (QPMs) in the ring is affected when accompanied by various corrector dipole magnets (CDMs). Variations of integrated quadrupole field strength in presence of CDMs are measured at various field excitations and by varying the distance between the magnets, using a rotating coil. Experimental results are compared with the results obtained from 3D simulations for better understanding of the results. Possibility of studying the interference effect by scanning the field with a Hall probe is explored. Dependence of field interference on the distance between magnets, pole gap and the steel length are studied. Based on these experimental results the basic cause of such field distortion is discussed. Effects of the adjacent magnets on the higher order multipoles of QPMs are also measured. After knowing the field distortion and having the detailed fringe field data, the trajectory of the charged particle while passing through a QPM is found by solving the differential equations in the first order approximation. The fringe field is utilised to solve the particle trajectory equations and to calculate various lens parameters. The equivalent hard edge models are derived and compared with the conventional hard edge approximation. Presence of the CDM may break the magnetic field symmetry of a QPM. Appropriate technique is developed to treat this asymmetric field and convert this into equivalent hard edge models to be used as input of any existing beam dynamics code. The effect of such field modification of QPM in presence of adjacent magnet in the ring and the variation of different optical parameters are reported.

Spherical aberration from trajectories in real and hard-edge solenoid fields

For analytical, real and hard-edge solenoidal axial magnetic fields, the low-energy electron trajectories are obtained using the third-order paraxial ray equation. Using the particle trajectories, it is shown that the spherical aberration in the hard-edge model is high and it increases monotonously with hard edginess, although the focal length converges, in agreement with a recent field and spherical aberration model. The model paved the way for a hard-edge approximation that gives correct focal length and spherical aberration, which is verified here by the trajectory method. In essence, we show that exact hard-edge fields give infinite spherical aberrations.

The effects of magnetic fringe fields on beam dynamics in a beam transport line of a terahertz FEL source

The transport line used in a terahertz FEL device has to transport electron beam through the entire system efficiently and meet the requirements of the beam parameters at the undulator entrance. Due to space limitations, the size of the magnets (five quadrupoles and two bending magnets) employed in the transport line was limited, and some devices were densely packed. In this paper, analyses of the effect of fringe fields and magnetic interference of magnets are presented. 3D models of these magnets are built and their linear optical properties are compared with those obtained by hard edge models. The results indicated that the effects of these factors are significant and they would cause a mismatch of the beam at the exit of the transport line under the preliminary lattice design. To solve this problem, the beam was re-matched using the particle swarm optimization algorithm.

A model of field and spherical aberration in soft/hard edge solenoid magnets

A solenoid magnetic field model is presented that describes the on axis field by a parameter of its hard edginess and axial half-width at half-maximum field, which universally define its spherical aberration without solving the ray equation. The model shows an increase in spherical aberration from real soft edge fields to hard edge models, as used in beam tracking. It compares well with existing field models. It simply and accurately finds the spherical aberration in many types of solenoids.

Analytical transfer matrix of a quadrupole fringe

The analytical linear transfer matrices for different quadrupole fringes including quadratic, high order power and exponential models are deduced in this paper. As an example, the transfer matrices of the quadrupole BEPCII 105Q are computed for the above three models and compared with hard edge and slice-by-slice models in cases of near 60° and 90° FODO cells. These models' results are much better than the hard edge model's, and can meet the requirement of accurate calculation.

Effect of magnetic fringe field and interference on beam matching in a Medium Energy Beam Transport line of the Spallation Neutron Source linac

A Medium Energy Beam Transport (MEBT) line is employed in the Spallation Neutron Source linear accelerator (linac) to match the beam from a radio frequency quadrupole to a drift tube linac and to perform other functions. The MEBT lattice consists of 14 electromagnetic quadrupoles and other devices. The quads have very small aspect ratios (steel length over aperture diameter), and they are densely packed in the lattice. Significant fringe fields and magnetic interference cause difficulties in beam matching. We have performed 3D modeling of the magnets, expanded simulation data to 3D multipoles, computed their linear optical properties, and compared their performance with that predicted by simple hard-edge models. This paper reports our findings and a proposed general solution to the problem.

Particle optics of quadrupole doublet magnets in Spallation Neutron Source accumulator ring

The Spallation Neutron Source ring employs doublet quadrupoles and dipole correctors in its straight sections. The electromagnetic quadrupoles have a large aperture, small aspect ratio, and relatively short iron-to-iron distance. The corrector is even closer to one of the quads. There have been concerns on the magnetic fringe field and interference in the doublet magnets and their assemblies. We have performed 3D computing simulations to study magnetic field distributions in the doublet magnets. Further, we have analyzed the particle optics based on the $z$-dependent focusing functions of the quads. The effect of the magnetic fringe field and interference, including the third-order aberrations, on the particle motion are investigated. The lens parameters and the first-order hard edge models are derived and compared with the parameters used in the ring lattice calculations.

RMS envelope matching of electron beams from “zero” current to extreme space charge in a fixed lattice of short magnets

We present detailed calculations of RMS-envelope matching over a broad range of beam intensities for the University of Maryland Electron Ring (UMER). Containment of beams from zero current to extreme space charge, all without changing the strength of external focusing in the periodic lattice, is possible thanks to the high density of quadrupoles in UMER. In turn, the small-aspect ratio of the UMER magnets results in gradient or field profiles that are ``all edges,'' thus requiring special treatment when constructing accurate hard-edge models. Further, the results of matching calculations, for both symmetric and asymmetric FODO (alternating gradient) schemes, are compared with calculations from simple general expressions valid in the uniform-focusing approximation of the periodic lattice. Finally, some aspects of the source-to-FODO matching calculation/optimization problem are discussed, together with sensitivity studies of the matching solutions under realistic conditions. The examples from the UMER project, which include experimental results, emphasize the practical aspects of beam envelope matching.

Possibilities of confinement of betatron frequencies of a relativistic cyclotron

The possibility of partial compensation of the relativistic variation in the betatron frequency of an isochronous cyclotron, using a special radial dependence of the magnetic field structure, has been investigated with the framework of a hard-edge model for a particular class of radial sector cyclotrons with N = 6 and a FODO type magnetic system. The radial change in the depth of the azimuthal field variation has been found to be an effective means of confining the frequency Q x within the limits from 1 to 2 and from 2 to 2.5, for the energy range (protons) from 1 to 1000 and from 380 to 1500 MeV, respectively.

Calculations and measurements for a magnetic quadrupole lens with a large aperture and a bell-shaped field distribution

A magnetic quadrupole lens of an aperture radius of 12.5 cm and a physical over-all length of 12 cm is described. The fringe field is linearized by means of rounded-off pole pieces and parallel end plates. The distribution of the transverse field components along the axis is nearly bell-shaped. The field strength necessary to focus parallel incoming electrons of an energy of 20 keV has been measured. The effect of the field distribution on the motion of electrons through the quadrupole lens has been calculated in first order theory using a hard-edge and a bell-shaped model. The bell-shaped model gives 4 and 3% larger fields than the hard-edge model, respectively, for two different focal lengths. There is a good agreement between measurement and bell-shaped model calculations for paraxial rays. For beams crossing the axis at large angles the fringe field linearization reduces the third order aberration by a factor of 4 compared to a pure bell-shaped quadrupole field. Moreover, the focusing fields were obtained by a stepwise numerical integration of the trajectory equations using the measured field distribution. The agreement with the measurements is satisfactory. Finally the stigmatic focusing of parallel beams by a quadrupole doublet consisting of two of these lenses has been considered. The bell-shaped model yields better results than the hard-edge model as is shown by numerically calculated trajectories using the measured field distributions and the field gradients obtained by the models.