Realistic simulations of a cyclotron spiral inflector within a particle-in-cell framework

We present an upgrade to the particle-in-cell ion beam simulation code opal that enables us to run highly realistic simulations of the spiral inflector system of a compact cyclotron. This upgrade includes a new geometry class and field solver that can handle the complicated boundary conditions posed by the electrode system in the central region of the cyclotron both in terms of particle termination, and calculation of self-fields. Results are benchmarked against the analytical solution of a coasting beam. As a practical example, the spiral inflector and the first revolution in a $1\mathrm{MeV}/\mathrm{amu}$ test cyclotron, located at Best Cyclotron Systems, Inc., are modeled and compared to the simulation results. We find that opal can now handle arbitrary boundary geometries with relative ease. Simulated injection efficiencies and beam shape compare well with measured efficiencies and a preliminary measurement of the beam distribution after injection.

Figure 1

Schematic of a spiral inflector with particle trajectories from an opal simulation. The beam enters axially (from the top) through an aperture (grey) and is bent into the midplane by a combination of the electrostatic field generated by the spiral electrodes (green and blue, voltage typically symmetric at $+{\mathrm{V}}_{\text{spiral}}$ and $-{\mathrm{V}}_{\text{spiral}}$) and the cyclotron’s main magnetic field. Then it is accelerated by the two Dees (copper, Dummy-Dees not shown). Color online.

Figure 2

Representative example of a boundary left of $\mathbf{x}$ in $x$-direction. ${\mathbf{x}}_{L,x}=\mathbf{x}-h_x·{\widehat{\mathbf{e}}}_x$, ${\mathbf{x}}_{B,x}=\mathbf{x}-s_{L,x}·{\widehat{\mathbf{e}}}_x$, $\mathbf{x}$ and ${\mathbf{x}}_{R,x}=\mathbf{x}+h_x·{\widehat{\mathbf{e}}}_x$ are the point outside of $\mathrm{\Omega }$, boundary point, near-boundary point, and interior point, respectively. Similar for the $y$ and $z$-directions.

Figure 3

The four stages of opal geometry preparation: (a) Initial CAD model of the electrodes in the system. (b) An inverted solid is created by subtraction of all elements in (a) from a “master volume” (in this case a cylindrical chamber). (c) The inverted geometry is saved in stp format and a mesh is generated using GMSH [14]. (d) During initialization, opal creates a voxel mesh to speed up the tests that have to be performed every time-step (cf. text).

Figure 4

The boundary $\partial \mathrm{\Omega }$ is discretized by a set of triangles $T$. Shown is the line segment/triangle intersection test with the line segment $({\mathbf{X}}_0,{\mathbf{X}}_1)$, the triangle $t\in T$ and the intersection at $\mathbf{I}\in \partial \mathrm{\Omega }$.

Figure 5

Cross-sectional view of beam off-centered by $\xi$ along the x-axis. A mirror line charge of $-{\rho }_L$ is also depicted at $\frac{r_p^2}{\xi }$. Adapted from [18 p 229].

Figure 6

${\chi }_{\mathrm{red}}^2$ of the calculated $\varphi$ compared to the simulated values. It can be seen that the results are slightly worse for a beam close to the beam pipe. Also, for number of mesh cells in x-direction $>256$, only marginal improvement can be seen. These simulations were performed with a total number of particles $\mathrm{NP}=8000·\mathrm{MX}$ where MX denotes the number of mesh cells in x-direction (the independent variable in the plot).

Figure 7

$\varphi$ and $E_x$ along the x-axis for different offsets $\xi$. Dots are values calculated by OPAL, while solid lines are the analytical solution. Excellent agreement can be seen with $0.01<{\chi }_{\mathrm{red}}^2<0.02$ for the potential and $0.03<{\chi }_{\mathrm{red}}^2<1.3$ for the field.

Figure 8

Schematic of the $\mathrm{DAE}\delta \mathrm{ALUS}$ facility. ${\mathrm{H}}_2^{+}$ is produced in the ion source, transported to the $\mathrm{DAE}\delta \mathrm{ALUS}$ Injector Cyclotron (DIC) and accelerated to $60\mathrm{MeV}/\mathrm{amu}$. Ions are subsequently extracted from the cyclotron and injected into the $\mathrm{DAE}\delta \mathrm{ALUS}$ Superconducting Ring Cyclotron (DSRC), where they are accelerated to $800\mathrm{MeV}/\mathrm{amu}$. During the highly efficient stripping extraction, 5 emA of ${\mathrm{H}}_2^{+}$ becomes 10 emA of protons which impinge on the neutrino production target.

Figure 9

Initial distribution for injection through the spiral inflector, obtained by carefully simulating the LEBT. The beam is slightly hollow and has a clearly visible halo. Both are caused by overfocused protons, as discussed in [10]. The length of the bunch corresponds to one full rf period at 49.2 MHz and injection energy of 62.7 keV, centered at the synchronous phase, thus, to first order, representing a DC beam.

Figure 10

Measurement and simulation of a 5-finger probe (for description cf. text) in the first turn of the BCS test cyclotron central region, 45° after the exit of the spiral inflector. Each finger has a main peak and a tail toward lower radius. These are the particles that are not sufficiently accelerated. To guide the eye and to compare with the simulations, the data for each finger is fitted with two Gaussians.

Figure 11

Radial positions of the peak of each finger of the five-finger probe (cf. text). Blue (circles): Finger 2, orange (triangles): Finger 3, green (squares): Finger 4, solid: FFT, dashed: SAAMG.