Simulations of Beam Quality in a 13 MeV PET Cyclotron
A. Pramudita*, E. Mulyani, I.A. Kudus
Center for Accelerator Science and Technology, National Nuclear Energy Agency (BATAN), Jl. Babarsari, Yogyakarta 55281, Indonesia
A R T I C L E I N F O A B S T R A C T AIJ use only:
Received date Revised date Accepted date Keywords:
13 MeV cyclotron fourth harmonics
negative hydrogen ion (H−) Runge-Kutta (RK4) Scilab 5.3.3 Opera3d/Tosca
Simulation of negative hydrogen ion (H−) beam trajectory in a 13 MeV PET cyclotron (DECY-13) was carried out by using Runge-Kutta (RK4) approximation method and Scilab 5.4.1. The magnetic and electric field were calculated using Opera3d/Tosca software at 1 mm resolution. The cyclotron is a fourth harmonics type, meaning that the acceleration is four times per cycle, at a radiofrequency (rf) of 77.66 MHz and 40 kV amplitude. The calculations and simulations show that the maximum distance between ion source and puller is about 6 mm, the maximum width of the beam at 13 MeV is about 10 mm, and the initial phase between the rf and the beam is 10° to 10°, yielding about 10% of the beam from the ion source is accelerated to 13 MeV.
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INTRODUCTION
One of the activities at the Center for Accelerator Science and Technology, National Nuclear Energy Agency (BATAN) is development of accelerator technology, and one of which is the development of a 13 MeV proton cyclotron for PET (Positron Emission Tomography), namely DECY- 13 (Development of Experimental Cyclotron in Yogyakarta - 13 MeV), which is planned to be completed in 2019 [1]. One important thing in the cyclotron design is to know the proton beam’s trajectories and properties from the ion source to the target. The proton beam’s trajectories are determined by the distribution of the rf (radiofrequency)’s electric fields that accelerate the beam, the magnetic fields that bend and focus the beam, and the initial positions and phases of the beam. The distributions of the electric and magnetic fields can be calculated in 3 dimensions by using Opera3D software and Tosca module (for static electric and magnetic fields) or Soprano module (for rf electric field) or RELAX3D (for static electric field).
The proton beam’s trajectory simulation had been carried out by using Scilab, an open source software similar to Mathlab, the latter is a commercial one. The simulation can be used to evaluate the design of the cyclotron whether it works alright and reaches the intended beam energy
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and quality, or requires still further design improvements.
The beam’s trajectory simulation was carried out for electron in a linear electrostatic accelerator by using a program written in Pascal [2], and the static electric field distribution was calculated using SOR (Simultaneous Over Relaxation [3] or Succesive Over Relaxation [4]) iteration methods.
The beam’s trajectory simulation in a cyclotron was carried out using Scilab 5.3.3 in 2012 for H− beam trajectory reaching up to 13 MeV. Trajectory calculation was done using equation of motion F = dp/dt = mγdv/dt, F is the Lorentz force, p momentum, m mass, γ relativistic correction,t time, velocity v = ds/dt = vo + dv, position x = xo + vdt, vo
and xo are initial velocity and position, respectively.
The spatial step ds is fixed at 1 mm, the same with the accuracy of the electric and magnetic field simulations. Using this method, however, the calculated beam energy increase were not linear [5].
Beam trajectory simulation was also developed for cyclotron design in Korea by using pwheel program [6] in Fortran, which employed Runge-Kutta (RK4) numerical solution at fixed time step dt. This method have been applied in the program for beam trajectory simulation in Scilab 5.4.1 [7].
THEORY
The trajectory calculation of an ion of electric charge q velocity v is based on the Lorentz force F acting on the charged particle in an electric
field E and magnetic field B. The electric field E is a function of position (x,y,z) and time (t), while the magnetic field B is a function of position only. The values of both fields were obtained from calculation or simulation or from measurement (mapping).
The components of the equations of motion can be written as
dpx/dt = Fx = q(Ex + vyBz − vzBy),
dpy/dt = Fy = q(Ey + vzBx − vxBz), (1) dpz/dt = Fz = q(Ez + vxBy − vyBx).
The solutions of each component of equation (1) solved by using 4th-order Runge-Kutta method (RK4) are [8] (i = x,y,z)
pi(to + dt) = pio + (ki1 + 2ki2 + 2ki3 + ki4)/6, (2) where
ki1 = Fi(to, po)dt,
ki2 = Fi(to + dt/2, po + ki1/2)dt,
ki3 = Fi(to + dt/2, po + ki2/2)dt, (3) ki4 = Fi(to + dt, po + ki3)dt.
After the new momentum pi was obtained, the new velocity vi = pi/γm and the new position xi = xio + vidt can be determined.
As the magnetic and electric potential field data are given at mm accuracy, the field value for each calculated position is rounded to the nearest mm. In trajectory calculation using pwheel, 5-point bspline interpolation was employed, and electric potential data were provided at accuracy of 0.25 mm, 0.5 mm, 1 mm, 2 mm, and 4 mm at 256×256×5 data size [6]. In this paper bspline interpolation, which is also available in Scilab, was not employed, as it would require much longer calculation time.
CALCULATION
Program for H− beam trajectory was written in Scilab 5.4.1. The program needs data of the magnetic field which are loaded using program loadB [7]. Because of symmetry, it is sufficient to have only 1/8 of the overall field volume of 480 mm
× 480 mm × 30 mm in size, which is obtained from computattion using Opera3d software and Tosca module [9]. The size of the binary data is about 38 MB for each Bx, By, dan Bz components.
Data on the electric potential at dee voltage of 40 kV toward ground was also computed by using the Opera3d software and the Tosca module.
The data was loaded by using loadV12 and loadV34 programs in reference 7. As there is no symmetry, the data cover all field region of 480 mm × 480 mm
× 30 mm in size. To reduce the size of the data, it is devided into 4 quadrants, namely V1, V2 (loadV12), V3, and V4 (load V34). The size of binary data is about 56 MB for each V1, V2, V3, and V4. The electric fields Ex, Ey, and Ez for each quadrant were calculated from the potential difference at spatial distance ds = 1 mm in the direction of each electric field component.
The beam from the ion source (IS) will enter the puller only if it arrives before the puller change the polarity. Hence it should reaches the puller at a time less or equal to T/2, where T is the periode of the rf. If the initial velocity of the beam coming out from the IS ≈ 0, the maximum distance between IS to the puller is approximately s = ½at2 = d, a ≈ eV/md, then d = (qV/2m)½T/2, V = 40,000 volt, T = 1/(77.66 MHz), m mass of H → d ≤ 8.9065 mm. As V varies, the effective value V/(2)½ should be taken, then d ≤ 6.2978 mm ≈ 6 mm. This d value is taken in the simulations done in this paper.
The beam starts at the center of the ion source slit: x = 17.5 mm; y = 9 mm; z = 0. The beam initial velocity is approximated by the increase of the beam energy upon travelling 0.1 mm from the initial posisition. The motion, energy, and position are calculated by using equations (1), (2), and (3). The trajectory is plotted in XY plane, energy and vertical position are plotted as the function of turn-number. As the cyclotron works in 4th harmonics, each turn-number equals to four cycles of the accelerating radiofrequency. The beam position at each beginning of turn-number is plotted (in black) against the beam trajectory (in green), showing the beam phase at each turn. Figure 1 shows the beam trajectory and phase with initial phase 25° [7].
Fig. 1. The beam trajectory (green) for 100 turns with initial beam phase 25°. The black curve at the lower first quadrant shows the beam phase at the beginning of each turn-number [7].
Figure 2 shows the energy at each turn- number. The energy reaches 13.95 MeV at 100th
turn, while the 13 MeV energy is reached at about 92nd turn. Figure 3 shows the beam vertical position (z), which is about 12 mm wide at 100th turn.
Fig. 2. The beam energy (blue) for 100 turns with initial beam phase 25°. The red curve shows the maximum energy (4×40 keV) reachable at each turn [7]. Enlarged graph (inset) shows 13 MeV is reached at about turn-number 92.
Fig. 3. The beam vertical position (blue) for 100 turns with initial beam phase 25°. The curve shows the maximum vertical width of the beam is about 12 mm at 100th turn [7].
Figure 4 is the plot of energy and y position which shows the position of the beam at Y+ axis when the closest beam energy of 13 MeV is reached. The stripper to change the H beam into H+ beam, hence changing the direction of the beam outward for extraction, is located to the left just after the Y+ axis. Since the outmost trajectory is that of the 100th turn, Figure 4 shows that the beam energy of 13 MeV is reached at turn-number 92, in consistent with the one shown in Figure 2. It also shows that ymax ≈ 425 mm at turn-number 92 or at beam energy of 13 MeV. Different values of ymax
for different initial points of the beam will provide estimates on the horizontal width of the beam near the stripper. The beam vertical width can be estimated from Figure 3.
Fig. 4. The beam energy is plotted against y position for 100 turns (shown the final 12 turns) with initial beam phase 25°, showing that 13 MeV is reached at about turn-number 92 and ymax ≈ 425 mm.
RESULTS AND DISCUSSION
After simulations with various beam initial phase, it turned out that the beam reached the intended 13 MeV energy if the initial phase were between 10° and 10°. The position of the beam trajectory in the puller was also better (centered) if the slit of the ion source was shifted 1 mm to the right, as shown in Figure 5. The slit center position became x = 18.67 mm, y = 9.23 mm, z = 0.
Fig. 5. The beam trajectory with initial beam phase 30°, after the initial beam position was shifted 1 mm to the right. Inset figure shows the beam grazes the lower puller, before the shift.
The slit of the ion source for DECY-13 is having a size of 4 mm height and 0.55 mm width.
All beam trajectory coming out from the slit of the ion source were represented as starting at 9 points:
at z = 2, 0, 2 mm, 3 points each along vertical axis of the slit (x,y = 18.67, 9.23), and 3 points each 0.275 mm to the left (x,y = 18.40, 9.16) and right (x,y = 18.94, 9.30) of the slit. The turn-number and ymax of the beam reaching 13 MeV from those different initial positions in the slit of the ion source, and different initial phases (10°, 0°, 10°), are tabulated in Table 1. Differences of ymax will provide estimates on the horizontal width of the beam near the stripper.
Table 1. Table of turn-number/ymax (mm) at 13 MeV, or maximum energy reached < 13 MeV, at various initial phases and initial positions in the ion source slit. Points (a) and (c) is 0.275 mm to the left and right, respectively, from the vertical axis of ion source slit. Point (b) is on the vertical axis.
Initial phase Turn-number/ymax (mm) at 13 MeV
z = 0 z = 2 z = 2
10°
a 99/430 12.9 MeV 12.9 MeV
b 96/426 100/432 99/432
c 95/421 100/424 100/432
0°
a 97/428 100/433 100/442*
b 96/426 100/426 12.8 MeV
c 93/421 96/426 12.9 MeV
10°
a 96/427 12.9 MeV 12.9 MeV
b 96/423 12.9 MeV 12.8 MeV
c 96/423 100/427 12.9 MeV
Table 1 shows that at initial beam phase between 10° and 10°, the difference between the lowest and highest ymax is 12 mm (excluding one point with * mark), hence the approximate vertical width of the beam at 13 MeV. The lowest turn- number at reaching 13 MeV is 92, the highest is 100. From Figure 3 the vertical width at these turn- numbers is about 10 mm, suggesting that the approximate beam width at 13 MeV is about 10 mm, in accord with the size of stripper designed for DECY-13.
Table 1 also shows that the beam with initial phase 0° coming out from points (b) and (c) did not reach the intended energy of 13 MeV. The beam pulled out from the slit of the ion source is having initial phase of 180° span. Allowing initial beam phase between 10° and 10° (20° phase width) with some parts of the beam do not reach 13 MeV or out of stripper size, suggesting that only about 10% of the beam coming out from the ion source will arrive at the stripper at 13 MeV. The 20° phase width is better than previous simulations [7], and similar to those found in KIRAMS [10]. The 6 mm maximum distance between the ion source and puller also agrees with those found at several PET cyclotrons: KIRAMS (3 mm, 45 kV dee peak voltage)[11], Siemens Eclipse (3 mm, 39 kV dee voltage)[12], and GE (General Electric) Minitrace (1.5 mm, 30 keV dee voltage)[13].
CONCLUSION
At the accelerating radiofrequency (rf) of 77.66 MHz and 40 kV of dee peak voltage, the maximum distance between the ion source and puller is 6 mm. This agrees with those found at
several PET cyclotrons. The advantage of closer distance can be further simulated, however, it will also increase the chance of spark incidence, which may reduce the lifetime of an ion source.
The beam initial phase acceptance of 20° is better than the result of previous simulations (16°), yielding estimate that 10% of the ion source output will reach the sripper at the intended energy of 13 MeV. The beam width at this energy was estimated to be about 10 mm which agrees with the size of the beam stripper.
ACKNOWLEDGMENT
The authors thank Mr. Hari Suryanto from PTRR-BATAN and Mr. Hyun Wook Kim from Sungkyuwan University, Korea, for providing information on the puller to ion source distances.
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