High Energy and Nuclear Physics

Modeling of Wake Fields and Impedances in Accelerators
R. Samulyak

The electromagnetic interaction of an intensive charged particle beam with its vacuum chamber surroundings in an accelerator plays an important role for the beam dynamics and collective beam instabilities. Wake fields, generated by a moving particle in the accelerator pipe and objects such as RF cavities, bellows, stripline monitors, etc., affect the motion of particles in the tail part of the beam causing parasitic loss, beam energy spread, and instabilities. The effect of wake fields is usually of the same order of magnitude as the space charge effect. While the space charge forces approach zero in the ultrarelativistic limit, wake fields remain finite for an ultrarelativistic beam due to resistivity of the accelerator walls and non-smoothness of the chamber (existence of RF cavities, bellows etc.). The effect of wake fields is an increasingly important issue since operating regimes are continually moving towards higher currents and smaller bunches. To avoid collective beam instabilities that limit accelerator performance, an accurate numerical modeling of wake fields and their interaction with the beam is necessary.

In the traditional approach for including wake field forces in an accelerator code, the total impedance budget for the accelerator ring is calculated or experimentally measured and the corresponding forces are applied to tracked particles once per beam turn. Such a cumulative force approach is not sufficient for the simulation of beam instabilities caused by wake fields. It is less accurate than the 3D computation of the space charge that has already been developed in advanced accelerator modeling codes, including the MaryLie/Impact and Synergia codes. We have developed a model that accounts for the fine structure of particle beams and distributes wake fields in the accelerator chamber. The corresponding theoretical model is based on the expansion of the particle beam in terms of the multipole moments and the notion of the wake function, which allows elimination of the complex temporal behavior of the electromagnetic field between the incident charge creating the wake field and the test charge. The wake function describes the response of the accelerator chamber element to a delta-functional, pulse carrying, m-th moment. Wake functions are independent of beam properties and are defined totally by properties of the accelerator chamber.

The wake field algorithm is coded as a parallel Fortran 90 module which performs charge deposition of macroparticle beams on a 3D grid, expansion of the corresponding charge distribution into miltipole moments, computation of wake functions and wake field forces, and interpolation of the wake field forces from the grid to macroparticles. The module has been implemented in the MaryLie/Impact and Synergia codes. In the current numerical implementation, most of the accelerator chamber elements (resistive pipe, RF cavity, etc.) have associated analytical wake field models valid under certain approximations. Analytical wake field models are beneficial for the study of long-range wake fields and their multiturn effect on the beam dynamics in circular accelerators. To study wake field effects in accelerator elements that cannot be accurately approximated by analytical models, wake functions in a tabular format can also be used. The corresponding data can be obtained through accurate numerical solutions of the full Maxwell system of equations using commercial (MAFIA) or public domain electromagnetic codes.

We have recently developed and implemented a subgrid model for the calculation of resistive wake fields on the sub-millimeter length scale. The resolution of such short -range wake fields is very important for the dynamics and energy balance of accelerator beams. The subgrid model was validated using analytical theory of wake fields for Gaussian bunches. We have shown that for a short Gaussian bunch, particles located at 0.5σ ahead of the bunch center lose energy due to wake forces, and particles located at 1.8σ  behind the bunch center gain energy (Figure 1), opposite to the space charge effect. Such calculation was not possible without the subgrid model, since sub-millimeter range wakes in the vicinity of a particle inducing wake fields are responsible for the energy loss of trailing particles.

Figure 1. Normalized longitudinal resistive wake field force of a short Gaussian bunch calculated using the wake field code with a subgrid model for sub-millimeter scale wake fields.

Reference

• [1] Ryne, R. et al. SciDAC advances and applications in computational beam dynamics. J. Physics: Conf. Series 16: 210-214 (2005).

• More detailed description of the project

Last Modified: January 31, 2008
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