Non-neutral Plasma Research Articles

Nonlinear dynamics of non-neutral Maxwellian plasma in external trapping field

Nonlinear dynamics of collisionless non-neutral plasma in an external electrical trapping field is considered. For such system, the time-dependent solution of nonlinear Vlasov-Poisson equations are obtained. Proceeding from this solution, the influence of initial conditions on the dynamics of charged layer is discussed. Conditions leading to the capture of two-dimensional charged collisionless layer in the external parabolic potential are presented.

Nonlinear effects induced in the normalized ion density in a linear trap system for a large ion cloud.

In this work, we design, by means of a non-neutral plasma method, a linear trapping model for large ion clouds, which will become the core of an atomic clock. We first obtain the geometry and electromagnetic characteristics of the ion trap. We then perform a systematic analysis describing the main parameters of the ion cloud such as size, secular frequency, and ion number per unit length of temperature. The most appropriate operation point of the ion trap in a set of these specific parameters is evaluated, and a thorough discussion is performed about how minor perturbations introduced in these parameters affect, in a nonlinear response, the performance of the trapping system in sensibility, heating, and radiofrequency potential.

Effects of collision on the time-independent states of a non-neutral plasma diode

A theoretical investigation is presented on the steady states of a planar plasma diode where static ions uniformly occupy the inter-electrode region. A cold and purely monochromatic electron beam is injected from the emitter plate, and the emitted electrons are carried to the collector plate through the uniform background of stationary ions. It is considered that the electrons suffer collisions with the other particles (with ions or neutral atoms) during its transportation through the inter-electrode gap. The effects of collisions are incorporated through a simplified one-dimensional model. With the help of Lagrangian description of fluid-Maxwell's equations, time-independent states are explored for arbitrary values of the neutralization parameter. Using the emitter electric field as a characteristic parameter, the steady-state solutions are categorized into “Bursian” and “Non-Bursian” branches in “emitter electric field vs diode gap” parametric space. The Non-Bursian solutions are found to be very sensitive to the dissipative term as they only exist for small, non-zero values of the normalized collision-frequency.

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

We consider a general two-component plasma of classical pointlike charges $+e$ ($e$ is say the elementary charge) and $-Z e$ (valency $Z=1,2,\ldots$), living on the surface of a sphere of radius $R$. The system is in thermal equilibrium at the inverse temperature $\beta$, in the stability region against collapse of oppositely charged particle pairs $\beta e^2 < 2/Z$. We study the effect of the system excess charge $Q e$ on the finite-size expansion of the (dimensionless) grand potential $\beta\Omega$. By combining the stereographic projection of the sphere onto an infinite plane, the linear response theory and the planar results for the second moments of the species density correlation functions we show that for any $\beta e^2 < 2/Z$ the large-$R$ expansion of the grand potential is of the form $\beta\Omega \sim A_V R^2 + [\chi/6 - \beta (Qe)^2/2] \ln R$, where $A_V$ is the non-universal coefficient of the volume (bulk) part and the Euler number of the sphere $\chi=2$. The same formula, containing also a non-universal surface term proportional to $R$, was obtained previously for the disc domain ($\chi=1$), in the case of the symmetric $(Z=1)$ two-component plasma at the collapse point $\beta e^2=2$ and the jellium model $(Z\to 0)$ of identical $e$-charges in a fixed neutralizing background charge density at any coupling $\beta e^2$ being an even integer. Our result thus indicates that the prefactor to the logarithmic finite-size expansion does not depend on the composition of the Coulomb fluid and its non-universal part $-\beta (Qe)^2/2$ is independent of the geometry of the confining domain.

Bernstein modes in a non-neutral plasma column

This paper presents theory and numerical calculations of electrostatic Bernstein modes in an inhomogeneous cylindrical plasma column. These modes rely on finite Larmor radius effects to propagate radially across the column until they are reflected when their frequency matches the upper hybrid frequency. This reflection sets up an internal normal mode on the column and also mode-couples to the electrostatic surface cyclotron wave (which allows the normal mode to be excited and observed using external electrodes). Numerical results predicting the mode spectra, using a novel linear Vlasov code on a cylindrical grid, are presented and compared to an analytical Wentzel Kramers Brillouin (WKB) theory. A previous version of the theory [D. H. E. Dubin, Phys. Plasmas 20(4), 042120 (2013)] expanded the plasma response in powers of 1/B, approximating the local upper hybrid frequency, and consequently, its frequency predictions are spuriously shifted with respect to the numerical results presented here. A new version of the WKB theory avoids this approximation using the exact cold fluid plasma response and does a better job of reproducing the numerical frequency spectrum. The effect of multiple ion species on the mode spectrum is also considered, to make contact with experiments that observe cyclotron modes in a multi-species pure ion plasma [M. Affolter et al., Phys. Plasmas 22(5), 055701 (2015)].

Compression of a mixed antiproton and electron non-neutral plasma to high densities

We describe a multi-step “rotating wall” compression of a mixed cold antiproton–electron non-neutral plasma in a 4.46 T Penning–Malmberg trap developed in the context of the AEḡIS experiment at CERN. Such traps are routinely used for the preparation of cold antiprotons suitable for antihydrogen production. A tenfold antiproton radius compression has been achieved, with a minimum antiproton radius of only 0.17 mm. We describe the experimental conditions necessary to perform such a compression: minimizing the tails of the electron density distribution is paramount to ensure that the antiproton density distribution follows that of the electrons. Such electron density tails are remnants of rotating wall compression and in many cases can remain unnoticed. We observe that the compression dynamics for a pure electron plasma behaves the same way as that of a mixed antiproton and electron plasma. Thanks to this optimized compression method and the high single shot antiproton catching efficiency, we observe for the first time cold and dense non-neutral antiproton plasmas with particle densities n ≥ 1013 m−3, which pave the way for an efficient pulsed antihydrogen production in AEḡIS.Graphical abstract

Study on plasma sheath and plasma transport properties in the azimuthator

A physical model of transport in an azimuthator channel with the sheath effect resulting from the interaction between the plasma and insulation wall is established in this paper. Particle in cell simulation is carried out by the model and results show that, besides the transport due to classical and Bohm diffusions, the sheath effect can significantly influences the transport in the channel. As a result, the ion density is larger than the electron density at the exit of azimuthator, and the non-neutral plasma jet is divergent, which is unfavorable for mass separation. Then, in order to improve performance of the azimuthator, a cathode is designed to emit electrons. Experiment results have demonstrated that the auxiliary cathode can obviously compensate the space charge in the plasma.

Application of chaos theory to the particle dynamics of asymmetry-induced transport

The techniques of chaos theory are employed in an effort to better understand the complex single-particle dynamics of asymmetry-induced transport in non-neutral plasmas. The dynamical equations are re-conceptualized as describing time-independent trajectories in a four-dimensional space consisting of the radius r, rotating frame angle ψ, axial position z, and axial velocity v. Results include the identification of an integral of the motion, fixed-point analysis of the dynamical equations, the construction and interpretation of Poincaré sections to visualize the dynamics, and, for the case of chaotic motion, numerical calculation of the largest Lyapunov exponent. Chaotic cases are shown to be associated with the overlap of resonance islands formed by the applied asymmetry.

Two Modes of the Self-Similar Evolution of Charged Plasma

Two self-similar modes of evolution of charged particles’ high-current beam are described analytically. The situation being considered falls within the field of nonneutral plasma electrodynamics. The process is considered in terms of the nonlinear 1D evolution of the charge density w(x, t) in the channel of a longdistance transmission line with nonlinearly distributed resistance R, capacitance C, and inductance L: R = R(w), C = C(w), and L = 0. It is shown that initially the front of w(x, t) accelerates and then slows down. The description of the process in the channel is based on the charge conservation law. An idealized “kinematic” approach is used according to which an equation in two unknowns (charge density and current density in the channel) can be reduced to an equation in one unknown w(x, t). A strongly nonlinear wave process is studied. A discontinuous solution w(x, t) is constructed with a zero boundary condition at infinity. Such a model description can apply only for revealing the main qualitative features of a complex process. Analytical expressions for the variation in the evolution of the front velocity and perturbed area length are derived. An interrelation between the nonlinearity parameter of the process and the amount of charge in the interelectrode gap is suggested based on the experimental data for the evolution of streamers.

Dark-matter-like solutions to Einstein’s unified field equations

Einstein originally proposed a nonsymmetric tensor field, with its symmetric part associated with the spacetime metric and its antisymmetric part associated with the electromagnetic field, as an approach to a unified field theory. Here we interpret it more modestly as an alternative to Einstein-Maxwell theory, approximating the coupling between the electromagnetic field and spacetime curvature in the macroscopic classical regime. Previously it was shown that the Lorentz force can be derived from this theory, albeit with deviation on the scale of a universal length constant $\ell$. Here we assume that $\ell$ is of galactic scale and show that the modified coupling of the electromagnetic field with charged particles allows a non-Maxwellian equilibrium of non-neutral plasma. The resulting electromagnetic field is "dark" in the sense that its modified Lorentz force on the plasma vanishes, yet through its modified coupling to the gravitational field it engenders a nonvanishing, effective mass density. We obtain a solution for which this mass density asymptotes approximately to that of the pseudo-isothermal model of dark matter. The resulting gravitational field produces radial acceleration, in the context of a post-Minkowskian approximation, which is negligible at small radius but yields a flat rotation curve at large radius. We further exhibit a family of such solutions which, like the pseudo-isothermal model, has a free parameter to set the mass scale (in this case related to the charge density) and a free parameter to set the length scale (in this case an integer multiple of $\ell$). Moreover, these solutions are members of a larger family with more general angular and radial dependence. They thus show promise as approximations of generalized pseudo-isothermal models, which in turn are known to fit a wide range of mass density profiles for galaxies and clusters.

Enhanced Control and Reproducibility of Non-Neutral Plasmas

The simultaneous control of the density and particle number of non-neutral plasmas confined in Penning-Malmberg traps is demonstrated. Control is achieved by setting the plasma's density by applying a rotating electric field while simultaneously fixing its axial potential via evaporative cooling. This novel method is particularly useful for stabilizing positron plasmas, as the procedures used to collect positrons from radioactive sources typically yield plasmas with variable densities and particle numbers; it also simplifies optimization studies that require plasma parameter scans. The reproducibility achieved by applying this technique to the positron and electron plasmas used by the ALPHA antihydrogen experiment at CERN, combined with other developments, contributed to a 10-fold increase in the antiatom trapping rate.

Preface to Special Topic: Collective Effects in Particle Beams and Nonneutral Plasmas

First Page

Large amplitude oscillations in a trapped dissipative electron gas

A collisional trapped non-neutral plasma is described by a hydrodynamical model in one-dimensional geometry. For suitable initial conditions and velocity fields, the Lagrangian variables method reduces the pressure dominated problem to a damped autonomous Pinney equation, representing a dissipative nonlinear oscillator with an inverse cubic force. An accurate approximate analytic solution derived from Kuzmak-Luke perturbation theory is applied, allowing the assessment of the fully nonlinear dynamics. On the other hand, in the cold plasma case, the Lagrangian variables approach allows the derivation of exact damped nonlinear oscillations. The conditions for the applicability of the hot, pressure dominated or cold gas assumptions are derived.

Energy relaxation rate in antiproton–positron nonneutral plasma

Energy relaxation of antiprotons injected in strongly magnetized positron gas is studied by molecular dynamics method. Relaxation rates are calculated for different initial energies and densities of the particles. Influence of strong magnetic field is discussed.

Explicit, analytical radio-frequency heating formulas for spherically symmetric nonneutral plasmas in a Paul trap

We present explicit, analytical heating formulas that predict the heating rates of spherical, one-component nonneutral plasmas stored in a Paul trap as a function of cloud size S, particle number N, and Paul-trap control parameter q in the low-temperature regime close to the cloud → crystal phase transition. We find excellent agreement between our analytical heating formulas and detailed, time-dependent molecular-dynamics simulations of the trapped plasmas. We also present the results of our numerical solutions of a temperature-dependent mean-field equation, which are consistent with our numerical simulations and our analytical results. This is the first time that analytical heating formulas are presented that predict heating rates with reasonable accuracy, uniformly for all S, N, and q. An analytical formula for the Coulomb logarithm in the weak-coupling regime is also presented.

The self-electric field effect on the MRI instability of magnetized rotational flows: Cylindrical model

The self-electric field effect on the magnetorotational instability (MRI) of non-neutral rotating electron-ion plasmas, related to axisymmetric disks, is investigated by using a cylindrical model. Describing the equilibrium state of the rotational system in the presence of a self-electric field, the fluid-Maxwell equations as well as the linear perturbation theory are employed to derive the dispersion relation (DR) of the excited modes in the system. The obtained DR analysis leads to analytical stability conditions, the growth rate of the instability, and the saturation amount of the pure instability excited in the system, depending on the equilibrium velocity difference between the involved components. Moreover, the numerical analysis of the obtained DR shows that the self-electric field has an important role for the MRI in the region where the angular frequency due to drift, , dominates over all the frequencies, in the outer part of non-neutral disks. In this region, the growth rate of the MRI increases dramatically with the ratio of the ion density to electron density until it saturates. On the other hand, in the inner region of the disk in which the gravitational frequency overcomes the others, the growth rate of the instability is found to be much smaller than that of the outer region. These results can be applicable to understand the MRI growth rate of the differentially rotating non-neutral plasma systems existing in the astrophysical environments, particularly in the magnetosphere of a neutron star.

Coherent resonance stop bands in alternating gradient beam transport

An extensive experimental study is performed to confirm fundamental resonance bands of an intense hadron beam propagating through an alternating gradient linear transport channel. The present work focuses on the most common lattice geometry called ``FODO'' or ``doublet'' that consists of two quadrupoles of opposite polarities. The tabletop ion-trap system ``S-POD'' (Simulator of Particle Orbit Dynamics) developed at Hiroshima University is employed to clarify the parameter-dependence of coherent beam instability. S-POD can provide a non-neutral plasma physically equivalent to a charged-particle beam in a periodic focusing potential. In contrast with conventional experimental approaches relying on large-scale machines, it is straightforward in S-POD to control the doublet geometry characterized by the quadrupole filling factor and drift-space ratio. We verify that the resonance feature does not essentially change depending on these geometric factors. A few clear stop bands of low-order resonances always appear in the same pattern as previously found with the sinusoidal focusing model. All stop bands become widened and shift to the higher-tune side as the beam density is increased. In the space-charge-dominated regime, the most dangerous stop band is located at the bare betatron phase advance slightly above 90 degrees. Experimental data from S-POD suggest that this severe resonance is driven mainly by the linear self-field potential rather than by nonlinear external imperfections and, therefore, unavoidable at high beam density. The instability of the third-order coherent mode generates relatively weak but noticeable stop bands near the phase advances of 60 and 120 degrees. The latter sextupole stop band is considerably enhanced by lattice imperfections. In a strongly asymmetric focusing channel, extra attention may have to be paid to some coupling resonance lines induced by the Coulomb potential. Our interpretations of experimental data are supported by theoretical predictions and systematic multiparticle simulations.

Spectral analysis of forced turbulence in a non-neutral plasma

The paper investigates the dynamics of magnetized non-neutral (electron) plasmas subjected to external electric field perturbations. A two-dimensional (2-D) particle-in-cell code is effectively exploited to model this system with a special attention to the role that non-axisymmetric, multipolar radio frequency (RF) drives applied to the cylindrical (circular) boundary play on the insurgence of azimuthal instabilities and the subsequent formation of coherent structures preventing the relaxation to a fully developed turbulent state, when the RF fields are chosen in the frequency range of the low-order fluid modes themselves. The isomorphism of such system with a 2-D inviscid incompressible fluid offers an insight into the details of forced 2-D fluid turbulence. The choice of different initial density (i.e. fluid vorticity) distributions allows for a selection of conditions where different levels of turbulence and intermittency are expected and a range of final states is achieved. Integral and spectral quantities of interest are computed along the flow using a multiresolution analysis based on a wavelet decomposition of both enstrophy and energy 2-D maps. The analysis of a variety of cases shows that the qualitative features of turbulent relaxation are similar in conditions of both free and forced evolution; at the same time, fine details of the flow beyond the self-similarity turbulence properties are highlighted in particular in the formation of structures and their timing, where the influence of the initial conditions and the effect of the external forcing can be distinguished.

A second-order differential equation for a point charged particle

A model for the dynamics of a classical point charged particle interacting with higher order jet fields is introduced. In this model, the dynamics of the charged particle is described by an implicit ordinary second-order differential equation. Such equation is free of run-away and pre-accelerated solutions of Dirac’s type. The theory is Lorentz invariant, compatible with the first law of Newton and Larmor’s power radiation formula. Few implications of the new equation in the phenomenology of non-neutral plasmas is considered.

Analytical investigation of anode heat flux in a magnetoplasmadynamic thruster

An analytical approach has been developed to analyze the anode thermal process in a self-field MPD thruster. Effects of both non-neutral and quasi-neutral plasma regions near the anode surface have been included. The heat flux calculation, which requires the contributions of current density and voltage drop, has been determined by deriving the analytical correlations between these parameters. The solution process has been formulated based on an iterative method to satisfy the condition of current density continuity at the mentioned regions interface. The anode of Princeton benchmark thruster has been investigated for ratios of the squared discharge current to mass flow rate of 16, 36, and 64 (kA)2·s/g. Generally, the calculated heat flux, current density, and voltage drop have shown a good agreement with the corresponding experimental data. Furthermore, the modeling has corroborated that the heat flux becomes maximum at the anode mid-plane where the total voltage drop is lowest. Also, it has been discussed that the second law of thermodynamics has obliged the current density to have a peak-shaped profile. Ultimately, a simple linearized model has been proposed to explain the spotty current effects on the non-neutral plasma structure.