Longitudinal splitting of bunches with variable energies in synchrotron

The ever-enriching beam application has put for-ward more demands on accelerators. If multiple bunches of different energies can be extracted for FLASH point scanning in a short time (< 500 ms), the spatial dose distribution advantage of proton Bragg peak can be combined with the temporal effect ad-vantage of FLASH. This paper provides three methods for generating short bunches in synchrotron. By adjusting the pulse parameters of the barrier voltages in the induction synchrotron, the short bunches can be easily generated. The short bunches can be kicked out of the beam reference orbit by kickers. When the pro-cess is repeated for many times, and the energy-changing process is added to the time interval during which the short bunches are continuously generated, the short bunches of multiple energies can be extracted in one working cycle. By finely controlling the changes of rectangular pulses, the number of particles in the short bunches can be better controlled, making an important contribution to further enriching the application of synchrotron.


Introduction
In order to complete the FLASH radiotherapy process of point scanning for 1L volume tumor within 500 ms, a total of 9261 points need to be scanned when the point interval is 5mm, and the scanning time interval of each point is required to be <54 µs.There are 21 energy layers required, and the switching time of adjacent energy layers is less than 25 ms.If the beam is turned off after the set value is reached according to the target dose during point scanning, the current technology cannot meet this requirement.
In order to shorten the time interval of point scanning during FLASH radiotherapy, we used the rectangular pulse voltage to perform beam splitting operation in the synchrotron.By adjusting the pulse voltage to control the number of particles in the short bunch, we avoided the waiting time of turning off the beam according to the dose feedback on the target.In the subsequent research, we improved the beam splitting method so that interval requirements of FLASH point scanning can be met.
In order to shorten the time of energy regulation, the energy is adjusted by rectangular pulse voltage at the time interval of generating short bunches, and finally variable energy beam splitting can be realized.Combined with the scanning method for continuous variable energy beam splitting, which will be introduced in the following articles, variable energy scanning irradiation can be realized, so as to meet the requirements of variable energy during FLASH radiotherapy.
T. Yoshimoto et al. have studied rectangular pulse beam splitting operation [1].In terms of multienergy fast extraction of synchrotrons, Leo Kwee Wah and Takumi Monma et al. proposed a continuous energy sweep scanning method [2].By moving the acceleration voltage, the particles there experience a shortage in the accelerating voltage.Eventually, the particles overflow the bucket region.In this method, particles with large momentum dispersion after overflow bucket are extracted by electrostatic deflector.

Longitudinal dynamics
Synchrotron based on induction cell was proposed by K.Takayama and J.Kishiro of KEK in 2000 [3].A typical induction synchrotron pulse voltage distribution is shown in figure 1, where   is the accelerating voltage,   is the barrier voltage.
The discrete equations are: ) where e is the charges carried by the particle, η is the slip phase factor, β is the relativistic velocity quantity, E is the energy, and ∆E is the energy deviation of the particle.The footmark n and n+1 represents the number of turns.When (ϕ, ∆E/ω0) is the phase space coordinate system, the Hamiltonian equation is as follows [3].
where ω0 is the cyclotron frequency of the particle, and Vac is the acceleration voltage, and ( ') V  is the voltage acted on the particles.
Through inference, the bucket height can be obtained as [4]: where Vbb is the barrier voltage amplitude, τpulse is the barrier voltage pulse width, and T0 is the cyclotron period.It can be seen that when the beam current parameters and synchrotron parameters are given, the bucket height is related to the barrier voltage amplitude Vbb and the pulse width τpulse.
To achieve longitudinal separation of the beam bunch in the induction synchrotron, we add two more barrier voltages.In the following beam splitting operation, the pulsed rectangular is mainly composed of the accelerating voltage   and the barrier voltages  1 and  2 .The two barrier voltages  1 and  2 are used to constrain the main bunch and the split short bunch, respectively.The phase range between the  1 voltages is Δ 1 , which corresponds to the phase range of the main bunch region.The phase range between the two  2 voltages is Δ 2 , which corresponds to the phase range of the short bunch region.The pulse widths of the trap voltages  1 ,  2 are  1 and  2 respectively.The value of the accelerating voltage Vac can be positive or negative.For synchrotron, the amplitude of the accelerating voltage Vac needs to match the rate of change of the bending magnet.The waveform changes of the pulse rectangular in one cycle is shown in Figure 2. VdeB is the demagnetizing voltage to avoid saturation of the magnetic core in the induction cavity.

Simulation code and results
Simulations based on the parameters of a small heavy ion synchrotron are carried out to further illustrate the method of longitudinally manipulating to produce short bunches.When rectangular pulses are used to generate a short bunch, its bucket is shown in Figure 3, where the Bucket1 and Bucket2 regions are the phase regions where the main bunch and the short bunch are located.The short bunch is stably captured in the Bucket2.Then we can use the kicker to kick the short bunch out of the reference orbit.After the short bunch is kicked out, the energy of the bunch can be boosted by   .Repeating the manipulation many times, the continuous extraction of short bunches of different energies can be realized.
The simulations are based on the parameters of Au 31+ .The method is the same for longitudinal manipulation of protons or other ions, except that if the initial energy of protons is low, the effect of space charge effect needs to be considered.The beam parameters and synchrotron parameters used in the simulation calculation are shown in Table 1.The initial beam is uniformly distributed in the ring in the longitudinal direction, and the momentum dispersion conforms to the Gaussian distribution law.Only the particle momentum dispersion and phase change, i.e. the change of longitudinal motion parameters, are considered in the simulation calculation.It is considered that when the particle momentum dispersion exceeds ±6‰, the particle will be lost due to the coupling of transverse and longitudinal motion.Since the particles are initially uniformly distributed in the ring, it is necessary to limit the particle distribution to the Bucket1 region by the capture process.If the chopper is used in practical engineering, the initial particles distributed in the specific phase region can be obtained without the capture process.During capture, allow the potential well voltage on both sides of Bucket 1 to rise from 0 to 1500 V in 10 ms (5300 turns).After the capture, the particles are mainly distributed in the Bucket1 region, but there are still a few particles that do not get enough phase shift due to the phase shift velocity close to 0. An acceleration voltage of 1000 V was applied to the particles distributed in Bucket 1 until the particle energy reached 7.5 MeV/u and then the particles were manipulated longitudinally to produce short bunches.The pulse voltage parameters adopted in the simulation calculation are shown in table 2. Due to the large amplitude setting of the potential well voltage, the dispersion of the beam current is greatly changed during the capture and beam splitting process, resulting in beam losses and beam halos."Beam outflow method" means that by closing  1 and − 2 in Figure 2, it is equivalent to cancel the right barrier of Bucket1 and the left barrier of Bucket2, and let some particles move from Bucket1 to the longitudinal phase stable region of Bucket2.The original particles were only in Bucket1.After the number of particles entering Bucket2 reaches the design value,  1 and − 2 are enabled, the barrier pulses on both sides of the bucket are restored, and the particles are recaptured into the bucket area.In this method, the short bunches seem to open the valve from Bucket1 and flow out into Bucket2, which we call the "beam outflow method", as Figure 4 shows.
The phase velocity (per cycle) of the particle is: where η is the slip phase factor and δ is the momentum deviation of the particles.According to the above equation, the positions of the particles of different momentum deviation layers after a certain number of turns can be further calculated, thus providing an important reference for accurately controlling the number of particles in the split bunch and the time required for splitting."outflow" of the short bunch."Cutting bunch method" means that in the large bunch originally constrained by − 1 and  2 , directly enable  1 , − 2 , the Bucket1 and Bucket2 generated at this time capture the particles in their respective phase regions, respectively, resulting in Bucket1 and Bucket2.In this method, the short bunches look like they are cut from the original long bunch by the potential barriers generated by  1 and − 2 , as Figure 5 shows.We call it the "cutting bunch method".By controlling the phase values of  1 and − 2 , the number of particles in the split bunches can be adjusted.
"Translation bucket method" means that the voltage of − 1 ~2 in Figure 2 is set at the beginning, and the Bucket1 and Bucket2 regions are established.As a result of translating the bucket, the translated bucket will capture two bunches in the new phase range, as Figure6 shows.In this method, the particles recaptured after shifting the bucket, we call it "translation bucket method".By controlling the amplitude of the phase jump value   , the number of particles in the splitting bunches can be adjusted.In order to fully see the effect of bucket translation, a larger translation of 40° is taken.The simulation results show that the momentum dispersion range of the short bunches is larger than that of the particles in bucket 1 before splitting.The main reason is that the amplitude and phase range of the potential well voltage used in the splitting process have great influence on beam stability.The influence of the beam can be reduced by selecting proper parameters of the potential well voltage.Since the parameters of the beam splitting process can be controlled according to the phase shift velocity of particles with different momentum, "beam outflow method" has advantages in accurately controlling the number of particles in short bunches."Cutting bunch method" and "translation bucket method" are suitable for splitting bunches in a short time.Compared with the traditional RF manipulation, these methods greatly reduce the time of splitting to generate short bunches, and has an incomparable advantage in the flexibility of splitting bunches.

Conclusion
The variable energy beam splitting method in this paper provides an idea for FLASH point scanning.This paper mainly introduces three methods to generate short bunches by longitudinal dynamic manipulation in the synchrotron, and gives the principle of controlling the number of particles in short bunches by each method.In the next step, we will further optimize the parameters of the splitting methods, and combine the beam measurement to provide a more complete particle number control method.Thus, the point scanning method without dose feedback control is realized, and the time interval between different scanning points is shortened during the point scanning process to meet the requirements of FLASH point scanning.By adding the energy transformation process to the time interval of continuous generation of short bunches, short bunches of different energies can be extracted within a working cycle.With special scanning method, variable energy scanning can be achieved, so as to meet the requirements of variable energy during FLASH point scanning radiotherapy.

Figure 1 .
Figure 1.Schematic diagram of beam and pulse voltage distribution of induction synchrotron.

Figure 2 .
Figure 2. Schematic diagram of the voltages of the rectangular splitting the bunch.

Figure 3 .
Figure 3. Bucket under the action of pulsed rectangular.

Figure 4 .
Figure 4. Longitudinal phase space distribution of the beam before (left image) and after (right image)"outflow" of the short bunch."Cutting bunch method" means that in the large bunch originally constrained by − 1 and  2 , directly enable  1 , − 2 , the Bucket1 and Bucket2 generated at this time capture the particles in their respective phase regions, respectively, resulting in Bucket1 and Bucket2.In this method, the short bunches look like they are cut from the original long bunch by the potential barriers generated by  1 and − 2 , as Figure5shows.We call it the "cutting bunch method".By controlling the phase values of  1 and − 2 , the number of particles in the split bunches can be adjusted.

Figure 5 .
Figure 5. Longitudinal phase space distribution of the beam before (left) and after (right)"cutting" long bunch.

Figure 6 .
Figure 6.Longitudinal phase space distribution of the beam before (left) and after (right) translating the bucket.

Table 1 .
Parameters of Small Proton Synchrotrons

Table 2 .
Key Parameters of Pulse Voltage in Simulation Calculation