Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameterless integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the rf bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny coupled-bunch interaction makes the beam unstable; such growth rates for all the modes are analytically obtained for arbitrary multipolarity. In practice, however, the finite threshold of this loss of Landau damping is set either by the high-frequency impedance roll-off or intrabeam scattering. Above the threshold, growth of the leading collective mode should result in persistent nonlinear oscillations.

中文翻译：

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弱空间电荷束束的纵模

通过将线性化的 Vlasov 方程简化为无参数的积分方程，分析地描述了具有排斥感性阻抗（跃迁下方的空间电荷或其上方的腔室电感）的束束的纵向集体模式。对于任何多极性，发现频谱的离散部分由无数具有真实调谐的模式组成，其极限点是非相干零幅度频率。换句话说，尽管存在 rf 桶非线性和潜在的井变形，Landau 阻尼会丢失。因此，即使是微小的耦合束相互作用也会使光束不稳定；对于任意多极性，所有模式的这种增长率都是通过分析获得的。然而在实践中，Landau 阻尼损失的有限阈值由高频阻抗滚降或束内散射设置。高于阈值，领先集体模式的增长应导致持续的非线性振荡。