High-intensity accelerators are designed with the zero-current phase advance per period (<InlineEquation ID="IEq121"> <EquationSource Format="TEX">$\varvec{\sigma}_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>) kept below 90° to avoid beam degradation. The 4th order single-particle resonance and the 2nd order envelope instability are triggered in a high current beam passing through a periodic channel for <InlineEquation ID="IEq122"> <EquationSource Format="TEX">$\varvec{\sigma}_{0}>90^\circ$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> <mn>0</mn> </msub> <mo>></mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. The 4th order resonance is a single-particle effect, whereas the latter is a collective effect of the entire beam. Both phenomena result in beam degradation, leading to increased beam size and emittance. Although they exhibit overlapping stopbands of emittance increase, there are distinct features that allow differentiation between the two effects. This paper presents a comparative analysis of these effects through detailed numerical simulations using TRACEWIN, employing both Gaussian and KV input beam distributions. Key distinguishing features such as differences in the root mean square (rms) and maximum beam size evolution, beam centroid oscillations, halo formation dynamics, 99.99% emittance growth, and the spatial characteristics of particles are explored. The paper also emphasizes how the choice of beam distribution, whether Gaussian or Kapchinsky-Vladimirsky (KV), affects the onset and severity of these instabilities. This analysis helps in developing a clearer understanding of these effects and their impact on beam dynamics in high-intensity accelerator systems.

Distinct features of the fourth-order single-particle resonance and second-order parametric instability in space-charge-dominated beams

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发表日期：

2026年3月9日

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Rajni Pande

摘要

High-intensity accelerators are designed with the zero-current phase advance per period ( $\varvec{\sigma}_{0}$ $σ_{0}$ ) kept below 90° to avoid beam degradation. The 4^th order single-particle resonance and the 2^nd order envelope instability are triggered in a high current beam passing through a periodic channel for $\varvec{\sigma}_{0}>90^\circ$ $σ_{0} > 90^{\circ}$ . The 4^th order resonance is a single-particle effect, whereas the latter is a collective effect of the entire beam. Both phenomena result in beam degradation, leading to increased beam size and emittance. Although they exhibit overlapping stopbands of emittance increase, there are distinct features that allow differentiation between the two effects. This paper presents a comparative analysis of these effects through detailed numerical simulations using TRACEWIN, employing both Gaussian and KV input beam distributions. Key distinguishing features such as differences in the root mean square (rms) and maximum beam size evolution, beam centroid oscillations, halo formation dynamics, 99.99% emittance growth, and the spatial characteristics of particles are explored. The paper also emphasizes how the choice of beam distribution, whether Gaussian or Kapchinsky-Vladimirsky (KV), affects the onset and severity of these instabilities. This analysis helps in developing a clearer understanding of these effects and their impact on beam dynamics in high-intensity accelerator systems.