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The growth of the two-stream instability in one-dimensonal (1D) spherical geometry is analyzed using Laplace and Fourier transforms. An analytic expression for the asymptotic growth is obtained and compared to the well-known planar formula. Stronger linear growth is predicted for a converging beam than for its corresponding 1D planar counterpart at the same initial density. Calculations based on the analytic model are compared to 1D particle-in-cell simulations and excellent agreement is obtained.
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Genoni et al. (2004) studied this question.
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