Design of the gradient dipole magnet for LLICTF

The Lanzhou Light Ion Cancer Therapy Facility (LLICTF) is a compact medical accelerator currently under construction. It is designed to treat cancer using a 230 MeV, 30 mA H+ beam and a 85 MeV/u, 1 mA 3He2+ beam. The facility comprises two ion sources, a low-energy beam-transport (LEBT), a Radio Frequency Quadrupole (RFQ), a medium-energy beam-transport (MEBT), and the main ring accelerating structure. Due to the presence of two ion sources, it is necessary to introduce a dipole magnet which is symmetrically focused as much as possible to meet the focusing requirements of the LEBT beam. Therefore, a gradient dipole magnet has been designed to achieve this symmetrical focusing. This paper discusses the theoretical and simulated symmetric focusing of the gradient dipole magnet. It also analyzes the effect of fringe fields and space charge. Additionally, this paper presents the results of the model design with CST and the multi-particle simulation results with TraceWin.


Introduction
Particle radiotherapy is a relatively mature treatment for tumours [1], which is also currently the most advanced radiation therapy method.There are currently 120 cancer therapy facilities based on hadron accelerators in operation worldwide [2].Many of them are compact and reliable [3] in order to reduce investment and patient costs.
The LLICTF is also a compact medical accelerator under construction dedicated to treating cancer using a proton beam and a helium beam.Figure 1 shows the layout of this LEBT.There are two independent beam lines, merged by a bending dipole magnet, that transport the beam to the RFQ.
In the design of the helium beam line, the dipole magnet is necessary not only because at least one beam direction must be deflected due to the presence of two ion sources but also for separating impurity particles.To meet the requirements for compact design and engineering space, the dipole magnet with a curvature radius of 500mm and a rotation angle of 60 • has been selected.However, the ordinary dipole magnet is an asymmetric focusing element, which causes the symmetrical beam generated by the helium ion source to become asymmetric after passing through it.It does not meet the beam symmetry requirements of the RFQ.It should be noted that when referring to the symmetry of the dipole magnet in this paper, it means that it can provide the equal focusing in horizontal and vertical directions, while the symmetry of the beam refers to the same phase space properties in horizontal and vertical directions.To achieve symmetrical beams, most accelerators, such as HIT [4] and HIMAC [5], use a magnetic quadrupole triplet after the dipole magnet.In order to maintain beam symmetry while keeping the beam line as compact as possible, this paper proposes the use of a gradient dipole magnet instead of an ordinary dipole magnet.
This paper starts by presenting a theoretical explanation for the symmetric focusing achieved by a gradient dipole magnet.Next, a field model is established using CST.Finally, the paper investigates the influence of fringe fields and space charge effects on the symmetry of gradient dipole magnet through simulations.

Design of the gradient dipole magnet 2.1. Equations of motion in a gradient dipole
For any realistic ordinary bending magnet, ignoring the edge angles, it has a focusing effect in either the horizontal or vertical direction, and it can be regarded as a drift in the other direction.After taking the edge angles into account, the vertical (or horizontal) focusing can be compensated, thus allowing for an overall correction.However, it is still not possible to achieve equal focusing in both the horizontal and vertical directions theoretically.
In theory, gradient dipole can achieve perfectly local symmetrical focusing.In a horizontal deflection of dipole, the equation of transverse motion of a particle is: where k x represents the horizontal focus intensity, k y represents the vertical focus intensity, ρ is the curvature radius in the dipole magnet and n x = − r By ∂By ∂r is the magnetic index.Obviously, when energy spread is not taken into account, the dipole magnet has equal focusing in the horizontal and vertical directions as long as k x = k y .In this case, n x = 0.5.Therefore, a gradient 14th International Particle Accelerator Conference Journal of Physics: Conference Series 2687 (2024) 062005 dipole magnet with magnetic index of 0.5 can theoretically achieve perfectly symmetrical focusing.

Model using CST
In this paper, the desired magnetic field gradient is achieved by introducing pole tips with nonzero slopes.For engineering convenience, the slope value of the entire pole is fixed, that is, there is no radial or azimuthal variation in the slope.

Figure 2. The polar slope of the gradient dipole magnet
From the magnetic index, the slope of the pole with the beam pipe height of 100 mm and the curvature radius of 500 mm can be calculated.As shown in Fig. 2, a slope of 0.05008 is the result of a linear approximation to the slightly curved pole shape obtained from calculations based on a 2D field model.This information can serve as a valuable reference for modeling using CST.
Building field models using CST must account for the effect of fringe fields, which is more realistic and cannot be accurately calculated theoretically.Since fringe fields can impact the symmetry of the gradient dipole magnet, it is necessary to consider their effect and adjust the slope of the polar face to 0.043 in order to optimize symmetry and build an accurate model.Figure 3

Study of factors affecting symmetry
With the field model, relatively realistic simulations can be performed.There are many factors that can affect the symmetry of gradient dipole magnet to be analyzed, such as fringe fields, space charge effects, edge angles, and energy spread.In this paper, we analyzed the influences of fringe fields and space charge effects which are expected to be more significant.

Simulation result
Firstly, a method to quantify the symmetry of the gradient dipole magnet is required before conducting the simulation.This can be done by sending a symmetric beam into the gradient dipole magnet and measuring the mismatch parameter M between horizontal and vertical directions after the beam comes out.The expression for M can be found in Reference [6], and a value of M close to 0 indicates that the symmetry of the gradient dipole magnet is better.The selection of an appropriate symmetric beam is a critical decision due to the infinite number of available options.A natural choice is the matched solution of the perfect gradient dipole whose envelope remain constant.The envelope equation is explained in detail in Reference [7], and solving it allows us to obtain the TWISS parameters of a matched beam.Specifically, when the space charge effect is not taken into account, the TWISS parameters can be calculated as α = 0 and β = 0.7071 mm/π.mrad for an emittance of 16.4 π.mm.mrad.The correctness of the above theory is also verified using the matrix model in TraceWin, where the matrix model is continuously perfectly focusing.As shown in Fig. 5, the matched beam stay the same after passing through the gradient dipole magnet.
After taking into account the space charge effect with 1 mA, the TWISS parameters of the matched beam to be α = 0 and β = 0.8685 mm/π.mradcan also be determined.
In this paper, the multi-particle mode and matched beam in TraceWin are used to simulate the effect of fringe fields in the field model on the symmetric focusing of the gradient dipole magnet.Regardless of the space charge effect, a matched beam with a Gaussian distribution of 100,000 macro-particles is used at the entrance of the gradient dipole, where the TWISS parameters of the matched beam are calculated from the above calculation.Figure 6 shows the phase space distribution at the of the gradient dipole when simulated using the matrix model and the field model, respectively.It can be calculated that the M for the beam at the outlet obtained using the matrix model is 0, while the M using the field model is 0.0076, which is a small value for practical purposes.For simplicity of analysis, a cylindrical coordinate system is used here, and Figure 7 shows the conversion relationship between Cartesian and cylindrical coordinate systems in this gradient dipole magnet.In this coordinate axis, the corresponding rotation angle of the magnet center The conversion relationship between Cartesian and cylindrical coordinate systems is 60 • , and the corresponding angles at both ends of the magnet are 30 • and 90 • , respectively.Figure 8 shows the magnetic field distribution at different rotation angles at radius R of 450 mm, 500 mm and 550 mm, respectively.It can be seen that the magnetic field of gradient dipole magnet gradually weakens at the edge.And Figure 9 shows the magnetic field distribution at different curvature radius at rotation angles of 60 • , 80 • and 82 • , respectively.It shows that by scanning the magnetic field at different rotation angles, the magnetic field begins to become significantly nonlinear within 8 degrees of the edge.This is one of the main reasons for asymmetric focusing of field model.By comparing the M value of the beam at the outlet with or without the space charge effect, the effect of it on the symmetry of the gradient dipole was characterized.The matched beams with Gaussian distributions at the entrance of the gradient dipole are 0mA and 1mA, respectively, and the TWISS parameters are calculated from the above.The phase space distribution at the outlet of the gradient dipole is shown in Fig. 10.It can be calculated to Figure 10.Phase space distribution at gradient dipole outlet at beam intensity of 0 mA and 1 mA, respectively obtain their M values of 0.0076 and 0.0123, respectively.It is evident that the presence of the space charge effect will increase the asymmetry in the beam.

Summary and further work
To ensure that the helium beam line in the LEBT of LLICTF meets the requirements for symmetric beam, this paper proposed using a gradient dipole magnet with n=0.5 for achieving symmetric focusing, instead of an ordinary dipole magnet.A successful model of this magnet has been developed and presented in this paper.The effects of fringe fields from the field model and space charge effects on the symmetry of the gradient dipole magnet have been analyzed in this paper.The results indicated that these effects only cause very small asymmetries in beam transport.
Future work will include attempts to offset part of the fringe fields by introducing two specific edge angles, which was inspired by the matrix model in TraceWin.Additionally, the effects of energy spread, beam angular momentum, and three-dimensional space charge on the symmetry of the gradient dipole magnet will be analyzed in subsequent work.The goal is to identify ways to maintain maximum symmetry focus under the most realistic conditions.

Figure 1 .
Figure 1.The layout of this LEBT

Figure 3 .
Figure 3.The model of the gradient dipole magnet by CST

Figure 4 .
Figure 4.The field of the gradient dipole magnet by CST

Figure 5 .
Figure 5.The variation of α (left) and β (right) of the matched beam in gradient dipole magnet

Figure 6 .
Figure 6.Phase space distribution of the gradient dipole outlet obtained using the matrix model (left) and the field model (right), respectively

Figure 7 .
Figure 7.The conversion relationship between Cartesian and cylindrical coordinate systems

Figure 8 .Figure 9 .
Figure 8. Magnetic field distribution at different rotation angles at radius R of 450 mm, 500 mm and 550 mm, respectively