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The code benchmarking for hadron linac using the 3D Particle-In-Cell (PIC) code is an important task in the European framework “High Intensity Pulsed Proton Injector” (HIPPI). PARMILA and HALODYN are two of the codes involved in this work. Both of these codes have been developed and used for linac design and beam dynamics studies. In this paper, the simulation results of the beam dynamics were compared and analyzed. As predicted by two codes, the simulation results show some agreements. The physical design strategy which was adopted in two codes was also discussed.

One main task of the HIPPI beam dynamics work group is to compare the validation of the 3D linac codes in the high current regime. It demands to find how much agreement the tracking simulations have with each other and to analyze the code validation based on these results. Some codes are available and currently run for such simulations. Some comparison works have been performed [

As one knows, various approaches were used to describe the space charge effects and different lattice modeling may pose severe problems to understand the source of discrepancies when the tracking simulations are run at high current status. For this reason the code benchmarking has been divided in three steps. The first two steps are the static comparison of space charge calculation and tracking simulation with a zero-current beam current. In the second step, a common Gaussian distribution as the initial conditions was input for comparison among codes.

The third step is the tracking simulation with high current. It also uses a common Gaussian distribution. However in this step, two cases are considered [

In this study, the 3D PIC tracking simulations have been performed with HALODYN and PARMILA. As the predictions of two codes, these simulation results show some agreement compared with each other. The reasons of emittance growth are also explained in details. And the different physical models of two linac codes are discussed based on these simulation results.

The code PARMILA (Version 2.35) [

PARMILA is a scalar code developed in Los Alamos (LANL) [_{SC} and longitudinal directions DZ_{SC} are both 0.05 cm, and the number of mesh intervals in these directions N_{R}, N_{Z} are 20 and 40 respectively. Thus the maximum extent of the space charge mesh is (initially) N_{R}DR_{SC} in the radial direction and N_{Z}DZ_{SC} in the longitudinal direction. The numbers of beam bunch both ahead and behind of the interest bunch is one.

HALODYN is also a Particle-In-Cell code. In the DTL sections, the transport elements are described by a sequence of drift, quadrupole and thin-lens RF cavities, whose accelerating voltage is calculated from the mean value of the accelerating electric fields. The RF cavities are modelled by using the thin lens approximation and the off-axis fields are derived with the modified Bessel functions. The space charge field is computed based on a 3D PIC spectral Poisson solver with closed boundary conditions defined on a rectangular pipe [

Additional HALODYN code has been developed under the UNIX environments system, but PARMILA runs in the Windows platform. More details of the individual codes can be found in the references cited above. Both of these codes treat the space-charge distribution as a round ellipsoid. Both codes use z as the independent variable to transport the beam particles through the linac described as a sequence of elements. All the codes use hard-edged quads. None of the codes included fringe-field effects.

As part of our effort to compare the beam simulation results from two different codes, we choose to start with the UNILAC Alvarez DTL tanks (

Based on the previous simulations, the initial particle distribution was spherical and the energy spread was one order of magnitude lower than the measured one. The following simulations track a particle distribution with longitudinal emittance one order of magnitude larger than the transverse ones, resulting in a much lower longitudinal tune depression (case2). The latter choice is motivated by the fact that the space-charge routine SCHEFF implemented in PARMILA is a 2D r-z solver, and its interest is to test how robust is this approximation for an asymmetric bunched beam. The input distribution is as usual a 6D-Gaussian (truncated in each phase space at 3σ), representing a ^{238}U^{+28} beam of 37 mA with kinetic energy W = 1.4 MeV/u. Other Twiss parameters and lattice parameters are listed in

The phase advances are computed over the first focusing period. The depressed phase advance is computed assuming a 6D uniform space-charge distribution. No coupling exists between the planes.

x-x′ | y-y′ | z-z′ | |
---|---|---|---|

α | 0 | 0 | 0 |

β (mm/mrad) | 0.6 | 1 | 1 |

γ (mrad/mm) | 1.67 | 1.01 | 1.0 |

ε_{n} (mm-mrad) | 0.167 | 0.167 | 1.55 |

Bare phase advance σ_{0} | 45˚ | 45˚ | 42˚ |

Depressed phase advance σ | 30˚ | 30˚ | 37˚ |

Tune depression | 0.67 | 0.67 | 0.88 |

By further studying we can find that the main contribution of the emittance growth comes from the first DTL tank. For example, the emittance growths ratios at the first tank exit are 1.24 and 1.33 for HALODYN and PARMILA code simulation in the horizontal plane, respectively. They are the 39.2% and 41.7% of the total growth ratio. It may be explained by the focusing structure of the DTL. The focusing lattice is designed as the DFFD in the first DTL. The emittance growth occurs at the period structure in which the focusing strength is not smooth. For instance in the seventh period, the focusing strengths of the first two quadrupole were 1709.4 G/cm, but the rest two in the same period were 1681.3 G/cm. Thus the evolutions of the phase advance along the DTL tanks are not enough smooth.

To further compare the calculate results of space-charge effects between two codes, we also plot the distribution in phase space as shown in

For a quantitative comparison, we also show the output radial distribution of the particles in

x-x' | y-y' | z-z' | ||
---|---|---|---|---|

HALODYN | α | −0.42 | −0.18 | −0.027 |

β (mm/mrad) | 1.87 | 4.92 | 8.33 | |

−α/β | 0.22 | 0.037 | 0.003 | |

γ (mrad/mm) | 0.63 | 0.21 | 0.12 | |

ε_{n,rms} (mm・mrad) | 0.269 | 0.244 | 4.01 | |

PARMILA | α | −0.58 | −0.31 | 0.098 |

β (mm/mrad) | 1.98 | 4.96 | 8.35 | |

−α/β | 0.29 | 0.06 | −0.01 | |

γ (mrad/mm) | 0.67 | 0.22 | 0.12 | |

ε_{n,rms} (mm・mrad) | 0.298 | 0.247 | 3.93 |

i.e. halo region of the bunch. The difference around the extreme outer edges of the beam is at the level of <0.04 mA.

More peculiar behaviour is the evolution of the longitudinal emittance as shown in

By comparing the evolution of the longitudinal phase space predicted by both codes, it was possible to conclude that they describe the same longitudinal beam dynamics. It was thought that the reason of the large discrepancies in the emittance curves, as well as of the different predictions in term of beam loss, might be in the different definition of particle loss in the longitudinal plane. After the code adjustments carried out as in the previous case, it has been observed that results are highly sensitive to the definition of particle loss in the longitudinal plane. Few particles far from the RF bucket area, if included in the computation of the RMS quantities, might drive to an unrealistic overestimation. While this definition in the transverse coordinates is rather straight- forward.

There are some reasons to explain it. One reason is the longitudinal mismatch from the entrance of UNILAC. This initial mismatch drives a beam dilution through the space-charge. Another is the fact that the synchronous phase jumps from −30˚ to −25˚ when particles transport from tank3 to tank4. The bucket area shrinks at the same time. The longitudinal emittance experiences a weak growth due to the large bucket area in the first three thanks. When entering in tank 4, the phase jump accompanying the bucket area shrinks makes the bunch tails cross the separatrix, leading eventually to a even larger phase space dilution.

In

The calculated RMS emittance and growth ratios were summarized in

These study results presented here show remarkable agreement between the predictions from 3D PIC code HALODYN and PARMILA. Generally the simulation results performed by two codes get agreement both in the transverse and longitudinal plane. The reason of the emittance growth was also analyzed in details.

These results demonstrate that the codes with the 3D FFT in HALODYN and 2D Poisson solver SCHEFF in PARMILA have the robustness of the r-z approximations for the beam parameters under consideration. It leads to conclusion that no gross errors have been made in the physics or methods of the codes. Of course, these studies should be investigated further with other codes.

Emittance (mm・mrad) | INPUT (x/y/z) | OUTPUT (x/y/z) | Emittance growth ratios |
---|---|---|---|

PARMILA | 0.167/0.167/1.55 | 0.298/0.247/3.93 | 1.78/1.48/2.54 |

HALODYN | 0.167/0.167/1.55 | 0.269/0.244/4.01 | 1.61/1.46/2.59 |

The author thanks Dr. Prof. Ingo Hofmann for helpful discussion. We are also grateful to Dr. Andreas. Sauer (IAP. Frankfurt University) for his continue support on PARMILA.

We also acknowledge the support of the European Community-Research Infrastructure Activity under the “FP6” Structuring the European Research Area program.

XuejunYin,WolfgangBayer,AndreaFranchi, (2015) Linac Beam Dynamics Code Benchmarking. Journal of Modern Physics,06,1044-1050. doi: 10.4236/jmp.2015.68108