Search Results (6 total)

The authors disagree with Dr. D. V. Pestrikov’s assertion that “the results obtained in the commented paper are not true,” based on how Volterra’s integral equations are treated in the paper. The authors would agree, for clarity, that the equal sign (=) in Eqs. (34), (36), and (38) in the paper be replaced by some special symbol or the approximate sign (≈) to indicate the omission of initial conditions. These and similar changes together with the revision to an error found by the authors have been published in a recent erratum.

This paper presents an analytical investigation of the transverse electron-proton (e-p) two-stream instability in a proton bunch propagating through a stationary electron background. The equations of motion, including damping effects, are derived for the centroids of the proton beam and the electron cloud by considering Lorentzian and Gaussian frequency spreads for the particles. For a Lorentzian frequency distribution, we derive the asymptotic solution of the coupled linear centroid equations in the time domain and study the e-p instability in proton bunches with nonuniform line densities. Examples are given for both uniform and parabolic proton line densities.

We consider the e-p instability for long proton bunches in storage rings. We give a simplified treatment of the centroid model for the linear instability for both unbunched and bunched beams. We point out the very large resulting electron oscillation amplitudes which mean that the electrons very quickly become nonlinear. We then propose a phenomenological theory of the instability in the nonlinear electron regime and show that the resulting model gives a slower growth and different form of the proton oscillations than in the linear theory. We then propose a technique that might eliminate the long term secular growth of the proton oscillations.

This paper considers an intense non-neutral ion beam propagating in the z direction through a periodic-focusing quadrupole or solenoidal field with transverse focusing force, -[κ_x(s)xe-^ _x+κ_y(s)ye-^ _y], on the beam ions. Here, s=β_bct is the axial coordinate, (γ_b-1)m_bc² is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy κ_x(s+S)=κ_x( s) and κ_y(s+S)=κ_y( s), where S=const is the periodicity length of the focusing field. The theoretical model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function f_b(x,y,x^′,y^′,s) and the (normalized) self-field potential ψ(x,y,s) in the transverse laboratory-frame phase space (x,y,x^′,y^′). Here, H-^ (x,y,x^′,  y^′,s)=(1/2)  (x^′2+y^′2)+( 1/2)[κ_x( s)x²+κ_y(s)y²]+ψ(x,y,s) is the (dimensionless) Hamiltonian for particle motion in the applied field plus self-field configurations, where (x,y) and (x^′,y^′) are the transverse displacement and velocity components, respectively, and ψ(x,y,s) is the self-field potential. The Hamiltonian is formally assumed to be of order ε, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients κ_x(s) and κ_y(s). Using a third-order Hamiltonian averaging technique developed by P. J. Channell [Phys. Plasmas 6, 982 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms. This leads to a Hamiltonian, H(X̃,Ỹ,X̃^′,Ỹ^′,s)=(1/2)  (X̃^′2+Ỹ^′2)+(1/ 2)κ_f(X̃²+Ỹ²)+ψ  (X̃,Ỹ,s), correct to order ε³ in the “slow” transformed variables (X̃,Ỹ,X̃^′,Ỹ^′). Here, the transverse focusing coefficient in the transformed variables satisfies κ_f=const, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance (σ<π/3=60°, say). Properties of axisymmetric beam equilibrium distribution functions, F_b⁰(H⁰), with ∂/∂s=0=∂/∂Θ, are calculated in the transformed variables, and the results are transformed back to the laboratory frame. Corresponding properties of the periodically focused distribution function f_b(x,y,x^′,y^′,s) are calculated correct to order ε³ in the laboratory frame, including statistical averages such as the mean-square beam dimensions, 〈x²〉(s) and 〈y²〉( s), the unnormalized transverse beam emittances, ε_x(s) and ε_y(s), the self-field potential, ψ(x,y,s), the number density of beam particles, n_b(x,y,s), and the transverse flow velocity, V_b(x,y,s). As expected, the beam cross section in the laboratory frame is a pulsating ellipse for the case of a periodic-focusing quadrupole field or a pulsating circular cross section for the case of a periodic-focusing solenoidal field.