Survey of Collective Instabilities and Beam-Plasma Interactions in Intense Heavy Ion Beams Page: 6 of 33
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In accelerators, strongly anisotropic beam distributions Tbb <<1 develop naturally
during the acceleration of the charge bunch , and can provide the free energy to drive both
the electrostatic Harris instability [32-42] and the electromagnetic Weibel instability [39, 43-48].
In this section, we summarize theoretical advances in recent analytical and numerical studies of
the Harris instability in intense one-component beams [33-39, 78], assuming electrostatic
perturbations (V x 6E =-0 and IB 0) about an initial anisotropic thermal equilibrium
distribution (Tb < Tb ) described by the self-consistent Vlasov equilibrium
n n 1 (
fb (rp)= b expi - 1 ,exp- Pz (2)
(2xrmbTb) Tb (mbT b) 2mbTb
in the beam frame. The 'primed' notation for beam-frame variables has been dropped in Eq. (2).
In Eq. (2), HL = p2/2mb + (1/2)mbw (x2 + y2) + ebq0(r) is the single-particle Hamiltonian for
the transverse particle motion, pl = (pa + p) )Vis the transverse particle momentum,
r = (x2 + y2)V is the radial distance from the beam axis, and 00(r) is determined self-
consistently from the equilibrium Poisson equation r-(a/Or)[r ag/or] = -4ebf d3pfb (r,p). In
the remainder of Sec. II, it is convenient to introduce the effective depressed betatron frequency
(. defined by 
O = 2 = b - 2, (3)
where Trb is the transverse beam temperature, rb is the root-mean-square beam radius, mb is the
mass of a beam particle, and
-)p = b fdrrnb(r) (4)
is the average beam plasma frequency-squared. The normalized tune depression T/v0 is defined
T - 3, (5)
w0 0) f
where wf = const is the transverse frequency associated with the applied focusing field in 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/m1/6/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.