Rigorous formulation of space-charge wake function and impedance by solving the three-dimensional Poisson equation

Area of Science:

• Physics
• Accelerator Physics
• Computational Physics

Background:

• Traditional numerical simulations approximate space-charge forces using 2D Poisson equations on discrete beam segments.
• This approach neglects longitudinal force leakage between segments, a consequence of the beam's relativistic factor (γ).
• Frequency-domain calculations of space-charge impedance obscure the time-domain wake function's longitudinal leakage effects.

Purpose of the Study:

• To directly calculate the space-charge wake function in the time domain.
• To investigate the longitudinal leakage of space-charge forces in particle beams.
• To establish criteria for accurate 2D slicing approximations in numerical simulations.

Main Methods:

• Solved the three-dimensional Poisson equation for a longitudinally Gaussian beam in the time domain.
• Calculated the space-charge wake function directly, accounting for longitudinal effects.
• Analyzed the impact of beam length, chamber radius, and segment length on leakage.

Main Results:

• The longitudinal leakage effect was found to be insignificant for beams significantly longer than the chamber radius, provided segment length meets specific criteria.
• A condition for segment length was identified to ensure accurate approximations.
• The study provides a criterion for optimal beam slicing in numerical simulations.

Conclusions:

• The 2D slicing method can approximate the 3D solution accurately under specific conditions.
• The findings offer guidance for optimizing numerical simulations of space-charge effects in particle beams.
• Understanding and controlling longitudinal leakage is crucial for precise beam dynamics modeling.