Understanding the 3D collective long-term response of beams exposed to resonances is of theoretical interest and essential for advancing high intensity synchrotrons. This study of a hitherto unexplored beam dynamical regime is based on 2D and 3D self-consistent particle-in-cell simulations and on careful analysis using tune spectra and phase space. It shows that in Gaussian-like beams Landau damping suppresses all coherent parametric resonances, which are of higher than second order (the “envelope instability”). Our 3D results are obtained in an exemplary stopband, which includes the second order coherent parametric resonance and a fourth order structural resonance. They show that slow synchrotron oscillation plays a significant role. Moreover, for the early time evolution of emittance growth the interplay of incoherent and coherent resonance response matters, and differentiation between halo and different core regions is essential. In the long-term behavior we identify a progressive, self-consistent drift of particles toward and across the resonance, which results in effective compression of the initial tune spectrum. However, no visible imprint of the coherent features is left over, which only control the picture during the first one or two synchrotron periods. An intensity limit criterion and an asymptotic formula for long-term rms emittance growth are suggested. Comparison with the commonly used non-self-consistent “frozen space charge” model shows that in 3D this approximation yields a fast and useful orientation, but it is a conservative estimate of the tolerable intensity.

中文翻译：

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空间电荷驱动的2D和3D共振的自洽长期动力学

了解暴露于共振的光束的3D集体长期响应在理论上是有意义的，并且对于推进高强度同步加速器至关重要。迄今为止，尚未探索的射束动力学机制的这项研究基于2D和3D自洽单元内粒子模拟以及基于使用调谐谱和相空间的仔细分析。它表明，在类似高斯的光束中，Landau阻尼抑制了所有相干的参量共振，该共振高于二阶（“包络不稳定性”）。我们的3D结果是在示例性的阻带中获得的，该阻带包括二阶相干参数共振和四阶结构共振。他们表明缓慢的同步加速器振荡起着重要作用。而且，对于发射率增长的早期演化，不相干和相干共振响应之间的相互作用很重要，而且晕圈和不同核心区域之间的区别至关重要。在长期行为中，我们确定了粒子朝向和穿过共振的渐进，自洽漂移，这导致初始调谐谱的有效压缩。然而，没有留下相干特征的可见印记，其仅在前一个或两个同步加速器周期内控制图像。提出了强度限制标准和长期均方根发射率增长的渐近公式。与常用的非自洽“冻结空间电荷”模型的比较表明，在3D中，这种近似会产生快速且有用的方向，但这是对可耐受强度的保守估计。