Survey of Collective Instabilities and Beam-Plasma Interactions in Intense Heavy Ion Beams Page: 8 of 33
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this case, it is evident from Figs. 1 and 2 that the nonlinear dynamics of the Harris instability can
play an important role in increasing Tb = M(vi), which could have a deleterious effect on
longitudinal focusing of the beam pulse if Tb increases to sufficiently large values. This would
of course require a sufficiently long beam transport region for the instability to grow to a
C. Electromagnetic Weibel Instability for One-Component Beams
The eigenmode code bEASt and nonlinear of code BEST have been extended to
incorporate slow-wave transverse electromagnetic perturbations (so called Darwin model),
thereby allowing for the possibility of a Weibel-type instability occurring in a one-component
charged particle beam [39,43-45]. Finite-geometry and self-field effects make a precise stability
analysis based on the linearized Vlasov-Maxwell equations difficult analytically. However, for a
anisotropic distribution of beam particles [Eq. (2)], assuming Tb -- 0, k r2 >> 1, and
-pbrb/c2 1, a simplified analytical model gives the simple approximate estimate 
(Im w)max _ 1 " V~b (7)
Of yL wf c
for the maximum growth rate of the Weibel instability. Here, vh = (2Tlb/mb Y2 is the transverse
thermal speed of the beam particles. Making use of Eqs. (3) and (5), it is readily shown that Eq.
(7) can be expressed in the equivalent form
m 1 f . (8)
wf vo ve2 c
Note from Eq. (8) that (Im o)m assumes a maximum value of (Im ). = 0.5 cfrb /c for
i/vo = 1/, = 0.707 (see Fig. 3).
A typical plot of the normalized maximum growth rate (Im)/(wjrb/c) versus
normalized tune i/vo obtained numerically using the linear eigenmode code bEASt is illustrated
by the solid curve in Fig. 4  for the choice of system parameters Tb/ ITb = 0 , r, = 3rb , and
wpbrb /c << 1. Quite remarkably, taking r, = 3rb and comparing Figs. 3 and 4, the value of v/vo
at maximum growth is in very good agreement with the theoretical estimate in Fig. 3, and the
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Davidson, Ronald C.; Dorf, Mikhail A.; Kaganovich, Igor D.; Qin, Hong; Startsev, Edward A.; Rose, David V. et al. Survey of Collective Instabilities and Beam-Plasma Interactions in Intense Heavy Ion Beams, article, June 19, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc898007/m1/8/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.