Accelerator Physics Statistical and Collective Effects II

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Presentation transcript:

Accelerator Physics Statistical and Collective Effects II A. S. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage Jefferson Lab Old Dominion University Lecture 17 G. A. Krafft Jefferson Lab

Beam Temperature K-V has single value for the transverse Hamiltonian No Temperature! Temperature in beam introduces thermal spreads Transverse Temperature Debye length Longitudinal Temperature Landau Damping

Waterbag Distribution Lemons and Thode were first to point out SC field is solved as Bessel Functions for a certain equation of state. Later, others, including my advisor and I showed the equation of state was exact for the waterbag distribution.

Equation for Beam Radius

Debye Length Picture* *Davidson and Qin

Collisionless (Landau) Damping Other important effect of thermal spreads in accelerator physics Longitudinal Plasma Oscillations (1 D)

Linearized In fluid limit plasma oscillations are undamped

Vlasov Analysis of Problem

Initial Value Problem Laplace in t and Fourier in z

Dielectric function Landau (self-consistent) dielectric function Solution for normal modes are

Collisionless Damping For Lorentzian distribution Landau damping rate

Negative Mass Instability Simplified argument: assume longitudinal clump on otherwise uniform beam Particles pushed away from clump centroid If above transition, come back LATER if ahead of clump center and EARLIER if behind it The clump is therefore enhanced! INSTABILITY; particles act as if they have negative mass (they accelerate backward compared to force!) θ

Longitudinal Impedance

Frequency Domain

NMI Simple Analysis

Linearized Continuity Equation

Oscillation Frequency

NMI Growth time rc rb Δz

Stabilization by Beam Temperature?

Dispersion Relation

Landau Damping

LD from another view

Resonance Effect

Inhomogeneous Solution

Oscillators Similtaneously Excited

Oscillation Frequency

NMI Growth time rc rb Δz

Multipass BBU Instability

BBU Theory

Perturbation Theory Works

CEBAF (Design) Simulations Stable Bunch Number

Close to threshold Bunch Number

Unstable Bunch Number

Chromatic (Landau) Damping