2. Beam Capture at Injection Energy

When accelerating the heavy-ion beam in a synchrotron, the bunched beam is needed, which can be accomplished through the capture [Reference Lee21–Reference Eliseev, Meshkov, Mikhailov and Sidorin23]. As one of the potential sources for longitudinal beam quality degradation and beam loss, beam capture requires three principles: (i) the capture must be optimized to keep any beam loss to a minimum to ensure beam intensity and to reduce the radioactive contamination of the machine; (ii) the capture duration should be as short as possible to reduce the machine cycle period, and (iii) the growth of longitudinal emittance ε at the end of capture should be minimal since the growth of ε will increase the demands on the RF system. The RF voltage program V(t) [Reference Lee21] for beam capture is determined by the following equation:

(1) $\begin{array}{}V\left(t\right)=\frac{V_i}{{\left(1-t/t_c\cdot \sqrt{V_c}-\sqrt{V_i}/\sqrt{V_c}\right)}^2},\end{array}$

where V_i and V_c are the initial and final voltage amplitude in the RF cavity, respectively, and t_c is the capture time. V_i, which is the activation value of different electronic circuits in the RF system, is chosen in such a way that the corresponding bucket area generated by it should be much smaller than ε of the initial injected costing beam. The voltage V_c must provide a sufficient bucket area to enclose ε. t_c should be large enough compared to the period of the synchronous oscillation to make a linear variation of the phase space parameters, which will help to preserve ε throughout the capture process.

For the ²³⁸U³⁵⁺ beam at the BRing, the revolution frequency is 0.10 MHz at 17 MeV/u, which is far lower than the lower limit of the working frequency of the RF cavity, the RF cavity has to work in the fourth harmonic (h = 4) of the revolution frequency according to the design, then, the coasting beam will be captured in four stationary buckets, and four bunches will be generated. In general, the stationary bucket generated by V_c according to the following equation (2) [Reference Lee21] should be 1.5 times of ε.

(2) $\begin{array}{}{A}_{\text{SB}}=16\sqrt{\frac{e{V}_c}{2\pi }\frac{{\beta }^{2}E}{h{\omega }_0{}^2\eta }},\end{array}$

where e is the charge, β = v/c, in which v is the velocity of particle moving, E is the energy of the synchronous particle, integer h is known as the harmonic number, ω ₀ is the angular revolution frequency, and η is the phase-slip factor. In this case, ε in the phase space (ϕ, ΔE/ω0) is 323 eVs, the stationary bucket A _SB should be 485 eVs, and the capture voltage V_c of 30.29 kV will be required.

First, simulations of the capture, with different t_c from 0.005 s to 0.035 s with a step of 0.005 at V_c = 30.29 kV, were performed without any errors of the RF and injection energy fluctuation. In these cases, V_i is chosen to be 0.1 kV, and the beam ion distributions in the phase space at the end of capture with different t_c are shown in Figure 2. The rms momentum spread Δp/p _rms and the rms longitudinal emittance ε _rms [Reference Lee21] are qualitatively derived, which are shown in Figure 3.

Figure 3: Δp/p _rms (black) and ε _rms (red) at the end of capture with t_c =0.005 s, 0.01 s, 0.015 s, 0.02 s, 0.025 s, 0.03 s, and 0.035 s at V_c = 30.29 kV.

As can be seen from Figure 3, Δp/p _rms is ranged from 0.00149 to 0.00157, which is very similar to each other; however, ε _rms decreases from 70.36 eVs to 53.04 eVs as t_c increases from 0.005 s to 0.035 s; among them, ε _rms of about 55 eVs is almost irrespective of t_c when t_c ≥ 0.002 s. The maximum ε _rms of 70.36 eVs occurs when t_c = 0.005 s; the reason is the very rapid RF voltage ramping leading to beam filamentation. To balance ε _rms and the duration of the machine cycle, t_c of 0.02 s will be applied in the capture process.

Apart from t_c, V_i is also a critical parameter to be controlled. Simulations were also performed for V_i from 0.08 kV to 0.14 kV with a step of 0.01 at V_c = 30.29 kV and t_c = 0.02 s, and ε _rms are derived which are shown in Figure 4. The minimum ε _rms occurs when V_i = 0.11 kV.

Figure 4: ε _rms at the end of capture with V_i = 0.08 kV, 0.09 kV, 0.1 kV, 0.11 kV, 0.12 kV, and 0.13 kV at V_c = 30.29 kV and t_c = 0.02 s.

If V_c is below 30.29 kV, the generated stationary bucket area cannot constrain the injected beam completely, and the beam loss will be inevitable. On the contrary, V_c higher than 30.29 kV will induce large momentum spread which may exceed the limited momentum acceptance of the BRing, and the additional beam loss may occur. After a series of calculation, simulation, and optimization, V_i = 0.11 kV, V_c = 30.29 kV, and t_c = 0.02 are determined for ²³⁸U³⁵⁺ beam capture. Δp/p _rms of the captured beam is 0.0015, and almost all particles can be captured after 1984 machine turns.

For the ²³⁸U³⁵⁺ beam, the designed intensity will exceed 3.0 × 1010. Such high beam intensity in the BRing is expected to cause an additional collective effect. As one of the important high beam intensity effects, the space charge [Reference Lee21, Reference Boine-Frankenheim24, Reference Spiller, Blasche, Hülsmann, Krämer, Ramakers and Reich-Sprenger25] plays an important role for the beam dynamics especially. The equation of the longitudinal phase space motion indicates the dependence of particle distribution on the time variations of the RF field, and this external RF field, or voltage, is modified by the field due to the space charge, which consequently modifies the particle distribution and eventually leads to the beam loss or beam quality degradation.

The space charge is an important issue only in low- and medium-energy accelerators because the space charge voltage per turn scales to 1/βγ ². In this paper, the space charge effect with various beam intensities N ranging from 3.0 × 1010 to 1.0 × 1011 at the end of capture was estimated, and Δp/p _rms and ε _rms are derived, which are shown in Figure 5.

Figure 5: Δp/p _rms (black) and ε _rms (red) with different N = 3.0 × 1010, 4.0 × 1010, 5.0 × 1010, 6.0 × 1010, 7.0 × 1010, 8.0 × 1010, 9.0 × 1010, and 1.0 × 1011.

It is concluded that the maximum ε _rms of 56.10 eVs occurs when N = 7.0 × 1010, which is increased only by 1.71% than the case of N = 3.0 × 1010. So, the space charge effect on the emittance growth is insignificant if the beam intensity is less than 1.0 × 1011.

3. Beam Acceleration

After capture, the stationary buckets will be transformed into moving ones to start acceleration through synchronous phase shifting [Reference Lee21, Reference Rybarcyk26, Reference Hotchi, Kinsho and Hasegawa27]. Since the ionization cross sections and space charge effect decrease largely with the increase of beam energy, the injected low-energy heavy-ion beam should be accelerated to high energy as fast as possible to minimize the significant ionization beam loss, stabilize the dynamic residual gas pressure, and suppress the strong space charge effect; then, a rapid acceleration with a maximum bend magnetic field ramping rate over time $\dot{B}$ of 12 T/s is proposed.

During the beam acceleration process, the moving bucket area which is determined by equation (3) should remain approximately invariant of around 485 eVs in (ϕ, $\Delta E/\omega 0$ ). The required voltage amplitude V_s and synchronous phases φ _s for the RF acceleration system can be calculated with the following equations (3) and (4) [Reference Lee21]:

(3) $\begin{array}{}{A}_{\text{AB}}=16\sqrt{\frac{e{V}_s}{2\pi }\frac{{\beta }^{2}E}{h{\omega }_0^2\eta }}\left(\frac{1-\sin {\phi }_s}{1+\sin {\phi }_s}\right),\end{array}$

(4) $\begin{array}{}{\phi }_s={\sin }^{-1}\frac{V_s}{V_m},\end{array}$

in which $V_m=L·\rho ·{\dot{B}}_{\max }$ is the voltage that is used to keep the synchronous particle on its ideal orbit, L is the circumference of the BRing, ρ is the bending radius of the dipole magnet, B _˙ is the ramping rate of the dipole magnetic field over time, ${\dot{B}}_{\max }$ is the maximum value of $\dot{B}$ available, V_s and φ _s are the needed voltages and synchronous phases, respectively, and E is the particle kinetic energy which is B-dependent. The acceleration cycle will be divided into three stages based on $\dot{B}$ . The first stage is $\dot{B}$ increasing from 0 T/s to the maximum value of 12 T/s in 0.05 s, which is determined by hardware equipment such as the magnet and power supply. The second stage is the process of B increasing with constant $\dot{B}$ of 12 T/s. The third stage is B increasing to the required value while $\dot{B}$ decreasing from 12 T/s to 0 T/s in 0.05 s. After the three stages, B will increase from 0.18 T corresponding to the injection energy of 17 MeV/u to 1.58 T corresponding to the extraction energy of 830 MeV/u. The programmed $\dot{B}$ (green solid line) and B (green short dot line) curves are shown in Figure 6. Similarly, V_s and φ _s are divided into three stages. At the first stage, V_s increases from 30.29 kV up to its maximum amplitude of 268.00 kV, and φ _s increases from 0 rad to 0.58 rad, and in this stage, E rises from 17 MeV/u to 108.87 MeV/u. At the second stage, V_s drops from 268.00 kV to 212.24 kV, and φ _s continues to increase from 0.58 rad to 0.764 rad, and in this stage, E rises from 108.87 MeV/u to 595.91 MeV/u. At the last stage, V_s continues to drop from 212.24 kV to 4.52 kV, and φ _s drops from 0.764 rad to 0 rad, and in this stage, E rises from 595.91 MeV/u to 830 MeV/u, and the acceleration efficiency is nearly 99%.

Figure 6: RF voltage programs V and φ _s of the RF cavity at different operating frequencies, B, $\dot{B}$ , and E during the whole process from capture to bunch merging. Black solid line and black short dash dot line are V_sand φ_sof the main RF cavity with f = 1.79 MHz (h = 4), respectively. Red solid line and blue solid line are V ₂ and V ₃, respectively. Beam capture: 0–0.02 s; acceleration: 0.02–0.1856 s; the 4 : 2 bunch merging: 0.1856–0.2106 s; the 2 : 1 bunch merging: 0.2106–0.2406 s.

4. Multiple Bunch Merging at the Extraction Energy Platform

After acceleration, a 4 : 2 and 2 : 1 two-step bunch merging [Reference Zipfel, Klingbeil, Laier, Ningel and Thielmann28, Reference Bozsik, Hofmann and Jahnke29] process will be used to transform four bunches into one to meet the extraction requirement. The first step is to merge four bunches into two bunches; during this step, the voltage amplitude of one RF cavity operating at h = 4 and the frequency f ₁ = 1.79 MHz is decreased linearly from V ₁ = 4.52 kV to 0.10 kV over merging time t; concurrently, the voltage amplitude of another RF cavity operating at h = 2 and the frequency f ₂ = 0.895 MHz is increased linearly from 0.10 kV to its final value V ₂. t and V ₂ will affect the longitudinal emittance ε of the merged bunches, and it will inevitably further affect the following 2 : 1 bunch merging. The simulations, with different t from 0.005 s to 0.035 s with a step of 0.005 at different V ₂ = 1 kV, 2 kV, and 3 kV, are performed, and ε _rms and beam loss are derived, which are shown in Figure 7(a). The second step is to merge two bunches into one. In this step, V ₂ is decreased linearly from 2 kV to 0.10 kV over time t ₂; meanwhile, the voltage of the RF cavity working at h = 1 is increased linearly from 0.10 kV to V ₃. The ε _rms dependence on t ₂ ranging from 0.005 s to 0.035 s and with different V ₃ = 1 kV, 2 kV, and 3 kV is derived, which is shown in Figure 7(b).

Figure 7: ε _rms with different merging times at V ₂ = 1 kV (red), 2 kV (blue), and 3 kV (green) at the end of 4 : 2 bunch merging (a) and 2 : 1 bunch merging (b).

According to the 4 : 2 bunch merging simulation result, it was found that ε _rms tends to decrease with the increase of t; however, when t is greater than 0.025 s, ε _rms tends to be fixed at around 60 eVs. So, the RF program of t = 0.025 s and V ₂ = 2 kV for the 4 : 2 bunch merging step is determined, the phase φ _s is fixed at −1.571 rad, the beam distribution in the longitudinal phase space is shown in Figure 8(a), and ε _rms is 60.19 eVs.

Figure 8: Beam ion distributions in the phase space at the end of the 4 : 2 bunch merging (a) and the 2 : 1 bunch merging (b).

For the 2 : 1 bunch merging simulation result, it can be known that ε _rms decreases with the increase of t ₂, and similar results are obtained with V ₃ = 2 kV and 3 kV. However, ε _rms increases with the increase of V ₃, so the RF program of t ₂ = 0.03 s and V ₃ =1 kV is determined, the phase φ _s is fixed at 0.785 rad, the beam distribution in the longitudinal phase space is shown in Figure 8(b), and ε _rms is 60.19 eVs.