Paper  Title  Page 

TUZBC03  SelfConsistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits  899 


Funding: Supported by DOE grant DEFG0299ER41104 and contracts DEAC0205CH11231 and DEAC0276SF00515.
We discuss our progress on integration of the coupled VlasovMaxwell equations in 4D. We emphasize Coherent Synchrotron Radiation from particle bunches moving on arbitrary curved planar orbits, with shielding from the vacuum chamber, but also include space charge forces. Our approach provides simulations with lower numerical noise than the macroparticle method, and will allow the study of emittance degradation and microbunching in bunch compressors. The 4D phase space density (PSD) is calculated in the beam frame with the method of local characteristics (PF). The excited fields are computed in the lab frame from a new double integral formula. Central issues are a fast evaluation of the fields and a deep understanding of the support of the 4D PSD. As intermediate steps, we have (1) developed a parallel selfconsistent code using particles, where an important issue is the support of the charge density*; (2) studied carefully a 2D phase space Vlasov analogue; and (3) derived an improved expression of the field of a 1D charge/current distribution which accounts for the interference of different bends and other effects usually neglected**. Results for bunch compressors are presented.
* Self Consistent Particle Method to Study CSR Effects in Bunch Compressors, Bassi, et.al., this conference.** CSR from a 1D Bunch on an Arbitrary Planar Orbit, Warnock, this conference. 

Slides  
THPAN084  Self Consistent Monte Carlo Method to Study CSR Effects in Bunch Compressors  3414 


Funding: Supported by DOE grant DEFG0299ER41104 and contract DEAC0276SF00515. We report on the implementation of a self consistent particle code to study CSR effects on particle bunches traveling on arbitrary planar orbits. Shielding effects are modeled with parallel perfectly conducting plates. The "vertical" charge distribution is assumed to be stationary. The macroscopic Maxwell equations are solved in the lab frame while the equations of motion are integrated in the beam frame interaction picture where the dynamics is governed by the self fields alone. We study different methods to construct a smooth charge density from particles, e.g. gridless nonparametric curve estimation and charge deposition plus filtering. We present numerical results for bunch compressors. In particular, we study different initial distributions. The transverse initial distribution is Gaussian and we study different initial longitudinal distributions: Gaussian, parabolic and nonlinear chirp. A parallel version of the code has been implemented and this will speed up parameter analysis and allow microbunching studies. 

FRPMN099  Equilibrium Fluctuations in an NParticle Coasting Beam: Schottky Noise Effects  4318 


Funding: Supported by DOE grant DEFG0299ER41104
We discuss the longitudinal dynamics of an unbunched beam with a collective effect due to the vacuum chamber and with the discretness of an Nparticle beam (Schottky noise) included. We start with the 2N equations of motion (in angle and energy) with random initial conditions. The 2D phase space density for the NParticles is a sum of delta functions and satisfies the Klimontovich equation. An arbitrary function of the energy also satisfies the Klimontovich equation and we linearize about a convenient equilibrium density taking the initial conditions to be independent, identically distributed random vaiables with the equilibrium distribution. The linearized equations can be solved using a Laplace transform in time and a Fourier series in angle. The resultant stochastic process for the phase space density is analyzed and compared with a known result*. Work is in progress to study the full nonlinear problem. To gain further insight we are studying three alternative approaches: (1) a BBGKY approach, (2) an approach due to Elskens and Escande** and (3) the 'threelevelapproach' of Donsker and Varadhan (see "Entropy, Large Deviations and Statistical Mechanics'', by R. S. Ellis).
* V. V. Parkhomchuk and D. V. Pestrikov, Sov. Phys. Tech. Phys. 25(7), July 1980 ** "Microscopic Dynamics of Plasmas and Chaos", Y. Elskens and D. Escande, IoP, Series in Plasma Physics, 2003. 