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Extra info for Collisionless Relaxation in Beam-Plasma Systems [thesis]
41) "geq and for the initially uniform beam The analysis above predicts that the emittance growth depends on the initial mismatch parameter . Note that "eq > "0 in both cases, and the minimum emittance growth is achieved for a matched beam with = 1. The value of the minimum emittance change is equal to ("geq ="0 )min ' 1:023 for the initially gaussian beam and is given by ("ueq ="0 )min ' 1:047 for a beam with a uniform initial pro le. To test the accuracy of the estimates, the analytical predictions for the nal rms beam properties and emittance growth are compared with the PIC simulation and previous estimates of Ref.
Explicitly, for a beam with an initial gaussian density (Eq. 7) xo Thus, up to a numerical coe cient, is equal to the mismatch parameter for a beam with the size a and the temperature u that are used to normalize the system (Eq. 3)). The goal of this work is to model the evolution of the rms beam properties. The use of the simple envelope equation model has demonstrated that the coupling between di erent distribution moments is essential and needs to be accounted for to accurately describe the dynamics of the rms beam size.
The dynamical CME predictions (3), theoretical estimates ( ) (given by Eqs. 40)) and PIC results (4) for an initially gaussian distribution are plotted in Fig. 14. The dynamical CME results are obtained by the numerical integration until stationary values of x and v are achieved. It can be seen that the dynamical CME results o er an improvement in the accuracy compared to the analytical estimates of Sec. 3. The numerical and analytical predictions for the asymptotic rms values of an initially uniform beam are presented in Fig.