1 Introduction

The High Intensity Heavy Ion Accelerator Facility (HIAF) is under design at the Institute of Modern Physics[1]. HIAF consists of a superconducting electron-cyclotron-resonance ion source (SECR), an ion Linac accelerator (iLinac), a booster ring (BRing), and a spectrometer ring (SRing)[2-4]. It provides an intense ion beam for high-energy-density physics, nuclear physics, atomic mass measurement research, and other applications.

As the main ring of HIAF, BRing accumulates the beams from iLinac to high intensity and accelerates them to high energy. The layout of BRing is shown in Fig. 1, and the basic parameters are listed in Table 1. The design intensity of BRing is 1e11 (²³⁸U³⁴⁺). By calculation, to accumulate beams up to the design intensity of BRing, the injection gain must reach 88, where the injection gain is defined as the ratio of the retained beam current in the ring to the incoming beam current. For a heavy-ion synchrotron accelerator, a typical injection scheme is multiple multiturn injection, which is a combination of one-plane painting injection and an electron cooling system. However, the space charge effect is strong in the high intensity heavy ion synchrotron. In the multiple multiturn injection scheme, the electron cooling system makes the particles shrink to a small size, further increasing the space charge effect. Moreover, during multiple multiturn injection, the electron cooling process takes a long time (typically several seconds [5]). The combination of the strong space charge effect and the long duration causes substantial beam loss, which creates many problems with respect to maintenance, vacuum, and heat. Hence, the multiple multiturn injection scheme is not suitable for BRing.

Figure 1:Full-screen View

(Color online) Layout of BRing.

To inject enough particles into BRing with a low beam loss, a two-plane painting injection scheme is proposed [6]. The two-plane painting injection scheme was first proposed by C.R. Prior for the Heavy Ion Driven Ignition Facility (HIDIF) in 1998, but it has not been applied because HIDIF is not actually constructed [7]. In the two-plane painting injection scheme, the beam is painted simultaneously in the horizontal and vertical phase space in the ring. This scheme makes full use of the acceptance of the ring to accumulate many more particles than a typical one-plane painting injection. It does not need an electron cooling process and hence has a short injection duration of about 1 ms. In addition, it can obtain a more uniform beam distribution, which can weaken the space charge effect [8]. The short injection duration and the weak space charge effect reduce the risk of beam loss.

In this paper, the two-plane painting injection scheme is introduced in Sect. 2. The physical analysis is described in Sect. 3. Optimization and simulation are covered in Sect. 4 and 5 respectively. A summary is given in Sect. 6.

2 The two-plane painting injection scheme

In the two-plane painting injection system, two aspects are set differently from typical one-plane painting injection. First, four horizontal bump magnets and four vertical bump magnets are located in the injection section, as shown in Fig. 2. During the injection process, they generate a locally bumped orbit. Second, the injection septum is set tilted on its corner, as shown in Fig. 3 [9]. Because the injection septum is set tilted on its corner, the circulating beam will survive if it avoids the injection septum in either of the horizontal and vertical directions.

Figure 2:Full-screen View

(Color online) Injection section of BRing. IS is an injection septum; BPh is a horizontal bump magnet; BPv is a vertical bump magnet.

Figure 3:Full-screen View

Injection septum. Θ is the tilt angle of the injection septum.

The injection process is shown in Fig. 4. At the beginning of the injection, a locally bumped orbit is generated near the injection septum, and the first slice of beam is injected into the ring, as shown in Fig. 4(1). The injected beam then undergoes betatron oscillations in both the horizontal and vertical planes. One revolution later, the first slice of beam will once again enter the injection section. Due to betatron oscillations, the first slice of beam will avoid hitting the injection septum this time, as shown in Fig. 4(2). Meanwhile, a new slice of beam will be injected into the ring, and the bumped orbit will descend at the same time. After a few revolutions, when the first slice of beam comes back to the injection septum, the bumped orbit has moved down, and hence the first slice of beam avoids hitting the injection septum and survives in the ring, as shown in Fig. 4(3). The process is repeated until the injection is complete.

Figure 4:Full-screen View

(Color online) Injection process.

3 Physical analysis

During the injection process, the positions of the injected particles are determined by the following equation [10]:

where Xn, Yn are the horizontal and vertical positions of the injected particles when they pass by the injection septum at the nth turn. X_{co, n} and Y_{co, n} represent the horizontal and vertical amplitudes of the bumped orbit at the nth turn. Ax and Ay are the horizontal and vertical betatron oscillation amplitudes. ϕ_x0 and ϕ_y0 are the horizontal and vertical initial phases, Qx and Qy are the horizontal and vertical tunes, and n is the number of turns. Ax, Ay, ϕ_x0, and ϕ_y0 are determined by the position and direction of the incoming beam with respect to the bumped orbit as the beam is injected into the ring and by the Twiss parameters of the ring at the injection point.

The injection septum can be described as:

where a and b meet the equation

It can be seen that if the inequality:

holds, the particle will hit the injection septum or else enter the injection channel and be impacted by the electric field of the injection septum. These particles will be lost in the injection septum or downstream after the injection septum. Substituting Eqs. (1) and (2) into Eq. (5):

For convenience of discussion, L_{co, n} and L_{osc, n} are defined as:

and

where m is the number of circulating particles and iL_{osc, n} is the L_{osc, n} of the ith particle. The physical meaning of L_{co, n} and L_{osc, n, max} is shown in Fig. 5.

Figure 5:Full-screen View

(Color online) Physical meaning of L_{co, n} and L_{osc, n, max}. The origin coordinates (0, 0) represent the ideal orbit.

When L_{co, n} and L_{osc, n, max} satisfy the inequality:

beam loss occurs. To avoid beam loss, the amplitude of the bumped orbit must satisfy:

As the value of L_co decreases, the linear displacement between the incoming beam and the bumped orbit increases. When the increased displacement makes it impossible to capture the incoming beam fully into the acceptance of the ring, the two-plane painting injection process ends. Obviously, to achieve high injection gain, the injection parameters and the bump curves should be optimized to slow down the decrease in amplitude of the bumped orbit.

4 Optimization

The injection parameters and bump curves were optimized using a linear particle-tracking code developed in MATLAB. In this code, the space charge effect was considered as a fixed tune shift per turn for each particle. The value of the tune shift was calculated using the ORBIT code [11, 12].

In the two-plane painting injection scheme, the angle of incoming beam relative to the bumped orbit affects the injection. Therefore, the direction of the incoming beam need to be optimized, while the direction of the bumped orbit is kept parallel to the ideal orbit. For the same reason, the position of the bumped orbit need to be optimized, while the position of the incoming beam is kept immediately beside the injection septum. In summary, the injection parameters to be optimized include the direction of the incoming beam, x′ and y′, the Twiss parameters of the incoming beam, βxi, βyi, αxi and αyi, the Twiss parameters of the ring at the injection point, βxm, βym, αxm, αym, the machine working point, Wpx and Wpy, the tilt angle of the injection septum, Θ, and the bump curves.

The injection gain can be considered as a function of all the injection parameters mentioned above. In this research, it is found that how the value of the function changes with respect to each injection parameter is affected by other injection parameters. So it is impossible to achieve the highest injection gain by tuning each of the injection parameters separately. To achieve the highest injection gain, it is necessary to tune all the injection parameters simultaneously to find the optimal injection parameter combination. For an optimization problem with dozens of variables, the computing load to perform an overall comparison of all possible combinations is obviously unacceptable. Instead, a genetic algorithm was used in this optimization[13-15]. The genetic algorithm is an optimization algorithm that is effective in optimizing multi-variable objective functions and offers fast search capability.

The optimized injection parameters are listed in Table 2, and the optimized bump curve is shown in Fig. 6. As the optimized injection parameters are given, a new lattice is generated to match the twiss parameters and working point by tuning a few quadrupoles in WINAGILE program. All the results described below are based on this information.

Figure 6:Full-screen View

(Color online) Optimal route of the bumped orbit and bump curves.

5 Simulation

The injection process based on the optimized parameters given above was simulated using the ORBIT code. 1000 macro particles per turn are injected into the ring to model the injected beam. Transverse Gaussian distribution is taken. As the RF system of BRing doesn’t work during injection, the longitudinal uniform distribution is taken for the injected beam in this simulation. The momentum deviation is 5 and a Gaussian momentum distribution is used. The space charge effect was included and calculated using the PIC method [16, 17]. An injection gain of up to 110.3 with a beam loss rate of 2.3% was achieved. The transverse distribution of the retained beam is shown in Fig. 7. It is apparent that the retained beam is relatively uniformly distributed in the acceptance of the ring. The relatively uniform particle distribution weakens the space charge effect and then reduces the tune shift of retained particles.

Figure 7:Full-screen View

(Color online) Transverse distribution of retained particles. The top left figure is the beam distribution in transverse real space. The top right figure is the beam distribution in the horizontal phase space. The bottom left figure is the beam distribution in the vertical phase space. In the three figures, X is the horizontal position, Y is the vertical position, X′ is the horizontal angle, and Y′ is the vertical angle.

Figure 8 shows that the horizontal and vertical tunes of the retained particles are shifted by about -0.015, -0.020 from the machine operating point and are spread over an area with horizontal size 0.008 and vertical size 0.016. The low tune shift reduces the probability of crossing the resonance line and then lowering the beam loss.

Figure 8:Full-screen View

(Color online) Tune of the retained particles. The red dots represent the particles. The horizontal coordinate of a dot is the horizontal tune of the particle, and the vertical coordinate of the dot is the vertical tune of the particle.

After injection, the retained particles continued to be tracked for 70 turns. The beam loss as a function of the number of turns is shown in Fig. 9. Clearly, some beam loss happened after injection, although all particles were initially injected into the acceptance of the ring. This phenomenon was due to the space charge effect. The beam loss distribution along the ring is shown in Fig. 10.

Figure 9:Full-screen View

(Color online) Beam loss during injection.

Figure 10:Full-screen View

(Color online) Beam loss distribution along the ring. ’Distance’ is the distance from the injection point in the downstream direction. The zero point of ’Distance’ represents the injection point into the ring.

Simulations were performed with various values of the main injection parameters. Figure 11 shows the relationship between injection gain and the tilt angle of the injection septum. A smaller injection septum tilt angle was found to decrease beam loss due to the large horizontal profile presented. However, it also increased beam loss due to the large vertical profile. This experiment showed that the two-plane painting injection scheme has a large tolerance for variations in the tilt angle of the injection septum. A tilt angle between 36 ° and 75 ° can meet the requirement of BRing, with 66 ° being the best choice.

Figure 11:Full-screen View

(Color online) Relationship between injection gain and injection septum tilt angle

Figure 12 shows the relationship between injection gain and the direction of the incoming beam. The angle between the direction of the incoming beam and the direction of the bumped orbit determines the initial phase, ϕ₀, of the injected beam and thereby affects the injection gain. Figure 12 shows that the tolerance of the incoming beam direction is about 0.5 mrad.

Figure 12:Full-screen View

(Color online) Relationship between injection gain and incoming beam direction

Figure 13 shows the relationship between injection gain and machine working point. Clearly, there are many possible choices of working point to achieve a high injection gain. This provides convenience for the lattice design of BRing.

Figure 13:Full-screen View

(Color online) Relationship between injection gain and machine working point

6 Summary

In this paper, the two-plane painting injection scheme has been studied. The injection parameters of the two-plane painting injection have been optimized. The resulting injection process has been simulated using the ORBIT code. An injection gain of up to 110.3 with a loss rate of 2.3% has been achieved. The results show that a two-plane painting injection scheme can meet the requirement of beam accumulation in the BRing of the HIAF.