Survey of Collective Instabilities and Beam-Plasma Interactions in Intense Heavy Ion Beams Page: 5 of 33
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II. ANISOTROPY-DRIVEN COLLECTIVE INSTABILITIES IN ONE-
COMPONENT NONNEUTRAL BEAMS
A. Nonlinear Stability Theorem
A very important consequence of the nonlinear Vlasov-Maxwell equations is the
existence of a stability theorem (a sufficient conditions for stability) for a one-component
charged particle beams. In particular, for a long, coasting beam in the smooth-focusing
approximation, the stability theorem states that any equilibrium distribution function f1 (H') that
Of (H') 0 (1)
is nonlinearly stable to perturbation with arbitrary polarization [8, 23, 24]. In Eq. (1),
H' = (p'2 + pf + p2)/2mb + mbwOf2r'2/2+ eb0'(r) is the single-particle Hamiltonian in the beam
frame (primed variables), and 0 (r') is determined self-consistently in terms of the beam space-
charge from the Poisson's equation. It follows from Eq. (1) that any isotropic distribution that is
a monotonic decreasing function of energy in the beam frame is nonlinearly stable. The validity
of this theorem has been demonstrated in nonlinear of simulations [67, 83] for beam propagation
over thousands of equivalent lattice periods.
Equation (1) is a sufficient condition for stability. Therefore, a necessary condition for
instability is that the beam distribution function has some distinct nonthermal feature such as an
inverted population in phase space [11-13], or a strongly anisotropic distribution function in the
beam frame. In electrically neutral plasmas, energy anisotropies are well known to provide the
free energy to drive the classical electrostatic Harris instability  and the electromagnetic
Weibel instability . The drive mechanism for instability can be either a temperature
anisotropy or an anisotropy in the relative directed kinetic energy of the plasma components.
B. Electrostatic Harris Instability for One-Component Beams
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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/m1/5/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.