Session GO05: Fundamental Plasmas: Nonneutral Plasmas

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Trapped ion crystals are employed in a variety of applications ranging from N-body quantum simulations to studies of strongly-coupled plasmas. Here the normal modes of an ion crystal are derived using an approach based on the Hermitian properties of the system dynamical matrix. This approach is equivalent to the standard Bogoliubov method, but for classical systems it is arguably simpler and more general in that canonical coordinates are not required. The general theory is developed for stable, unstable, and neutrally-stable systems. The method is then applied to develop reduced eigenvalue problems for the large magnetic field limit, where the spectrum breaks into ExB drift modes, axial modes, and cyclotron modes. Thermal fluctuation levels in these modes are analyzed and shown to be consistent with the Bohr-van-Leeuwen theorem. Rotational as well as vibrational motions are considered, and an expression for the rotational inertia of the crystal is derived. A magnetic contribution to this inertia which dominates in large magnetic fields is described, and an unusual limit is discovered for the special case of spherically-symmetric confinement, in which the rotational inertia does not exist.   

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We will discuss the first direct measurements of enhanced parallel velocity slowing (and diffusion) rates due to 1D long-range collisions in Mg$^{\mathrm{+}}$ ion plasmas. The un-neutralized magnetized ion plasmas are contained in near-thermal-equilibrium states in cylindrical Penning-Malmberg traps, with densities $n\sim 10^{7}\mbox{cm}^{-3}$ and temperatures $10^{-5} [Preview Abstract]

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We have developed non-destructive temperature measurements in cold electron plasmas, by tracking the ratio of the frequencies for two radial TG-eigenmodes. These TG modes are magnetized plasma waves with finite $k_{z} $ and $k_{\bot } $ quantized by the finite cylindrical plasma column length $L_{p} $ and radius $r_{p} $ with a frequency $f_{TG} (k_{\bot } ,k_{z} )$ proportional to (a fraction) the plasma frequency$f_{p} $. The modes frequencies are shifted upward by finite temperature, since plasma pressure increases the wave restoring force. These thermal shifts in frequency are well understood, and the ratio of the frequencies of two radial eigenmodes can be expressed as $(f_{TG1} /f_{TG2} )^{2}=(k_{2} /k_{1} )^{2}(1+\alpha k_{1} T)/(1+\alpha k_{2} T)$ . Here, $\alpha^{-1}=4\pi e^{2}n_{e} /3$ stands for an ``electrostatic'' plasma pressure. If the wave vectors $k$'s are known well enough for a given plasma column, then solving this frequency ratio equation for $T$ gives the absolute values of $T(t)$. However, the Bessel function solutions for finite length plasma columns are a rather crude approximation, and it is beneficial to use a known temperature evolution such as initial cyclotron cooling to calibrate the Bessel function coefficients. Applying this technique and using up to the fourth radial eigenmode we have measured the absolute temperature evolution down to the 20 $m$eV range, with an estimated accuracy of 10{\%}.   

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Studies of cross-magnetic-field heat transport are critical to further our understanding of astrophysical and fusion plasmas. Incisive heat transport experiments can be performed on nonneutral plasmas due to their quiescent nature and excellent confinement properties. Utilizing a cylindrical pure electron plasma confined in a Penning-Malmberg trap we present measurements of cross-field thermal diffusivity over a range of magnetic fields from B$=$1 kG to B$=$13 kG. The plasma has density n$\approx $10$^{\mathrm{7}}$ cm$^{\mathrm{-3}}$ and temperature T\textless 1 eV resulting in r$_{\mathrm{c}}$\textless \textless $\lambda_{\mathrm{D}}$. Heat transport in this regime is expected to be dominated by 1D long-range collisions with impact parameter r$_{\mathrm{c}}$\textless $\rho $\textless $\lambda_{\mathrm{D}}$, and the predicted long-range thermal diffusivity, $\chi_{\mathrm{L}}$, is notably independent of magnetic field, i.e. $\chi _{\mathrm{L}}\propto $n$^{\mathrm{0}}$B$^{\mathrm{0}}$. In contrast, classical 3D short-range collisions with impact parameter $\rho $\textless r$_{\mathrm{c}}$, yield a classical thermal diffusivity, $\chi _{\mathrm{C}}\propto $n$^{\mathrm{1}}$B$^{\mathrm{-2}}$. Experimentally, the thermal diffusivity is calculated from the radial heat flux which is derived from measurements of the temporal evolution of the radial temperature and density profiles. Our measurements of thermal diffusivity are in excellent agreement with the long-range prediction and verify the magnetic-field independence of the heat transport. Furthermore, it is found that the measured diffusivity exceeds the classical prediction by four to six orders of magnitude over the range of tested magnetic fields.   

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An unusual Trivelpiece-Gould (TG) mode frequency splitting pattern, due to interaction with an elliptical density perturbation caused by an $m_{\mathrm{\theta \thinspace }}=$ 2 diocotron mode, is observed for the first time in a magnetized pure electron plasma column. A single $m_{\mathrm{\theta \thinspace }}=$ 0,$_{\thinspace }m_{r}$ \textgreater 1 TG mode appears to branch out into frequency triplets as ellipticity increases. Here, $m_{\mathrm{\theta \thinspace }}$and $m_{r}$ are the azimuthal and radial wave numbers, respectively. For sufficiently small elliptical perturbations, the mode splitting $\Delta f_{\mathrm{\thinspace }}$/$f$ is linearly proportional to the plasma density quadrupole moment $q_{\mathrm{2}}$. An explanation of this effect involves mixing of the axisymmetric ($m_{\mathrm{\theta \thinspace }}=$ 0) mode with two non-axisymmetric ($m_{\mathrm{\theta }}\ne $0) nearly-degenerate plasma modes. For example, both the ($m_{\mathrm{\theta \thinspace }}=$ 0, $m_{r} \quad = \quad n)$ and the ($m_{\mathrm{\theta \thinspace }}=$ 2, $m_{r} \quad = \quad n$ -1) modes have frequencies determined by $k_{\mathrm{z}}r_{\mathrm{p}} \quad \approx \quad j_{\mathrm{1}}_{,n}$. We found that an elliptical density perturbation not only shifts the frequencies of the modes, but it also removes the orthogonality, with near-degeneracy allowing strong mixing of the eigenfunctions: the ($m_{\mathrm{\theta \thinspace }}=$ 2) modes pick up the ($m_{\mathrm{\theta \thinspace }}=$ 0) components resulting in the splitting pattern.   

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A plasma wave is said to have positive energy if energy must be added to the plasma when the wave is excited. Likewise, a wave is said to have negative energy if energy must be removed from the plasma when the wave is excited. ~Since energy is reference frame dependent, the sign of wave energy is reference frame dependent. This paper considers weakly damped, electrostatic waves that propagate on a stable non-neutral plasma, and establishes criterion that the waves have negative energy as viewed in the laboratory reference frame.$^{\mathrm{1\thinspace }}$ The criterion for the sign of wave energy is developed by using the symmetry properties of the plasma equilibrium and the fact that Vlasov dynamics is an incompressible flow in phase space, rather than the usual and more difficult procedure of calculating the value of the wave energy directly. [1] Thomas M. O'Neil, Phys. Plasmas 26, 102106 (2019)   

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Two-dimensional nonneutral plasma crystals stored in Penning traps are a leading platform for quantum simulation and sensing experiments. For small amplitudes, the out-of-plane motion of such crystals, which is exploited for quantum information protocols, can be described by a discrete set of normal modes called the drumhead modes. However, experimental observations of crystals with Doppler cooled and even near ground-state cooled drumhead modes reveal an unresolved drumhead mode spectrum. In this work, we establish in-plane thermal fluctuations in ion positions as a major contributor to the broadening of the drumhead mode spectrum. In the process, we demonstrate how the confining magnetic field leads to unconventional properties for the in-plane normal modes. These properties, in turn, have implications for the sampling procedure required to choose the in-plane initial conditions for molecular dynamics simulations. For current operating conditions of the NIST Penning trap, our study suggests that the two dimensional crystals produced in this trap undergo in-plane potential energy fluctuations in the range of 10 mK. Our study therefore motivates the need for designing improved techniques to cool the in-plane degrees of freedom.   

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Dynamic friction is used to cool nonrelativistic hadrons via copropagation with a strongly-magnetized electron beam. Some electron-ion collider designs require cooling at relativistic energies. In the beam frame, particle motion is nonrelativistic, and the essential physics is a single ion drifting briefly through a magnetized electron plasma. Previous analytic work and parametric models are not accurate in this parameter regime. We present simulations of the longitudinal friction force, showing agreement with theoretical calculations in the high- and low-velocity limits. Agreement is found with previous work in the high-velocity limit, as expected. A two-parameter function captures the simulated results and shows a scaling with ion charge number that differs from previously published results.   

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Trapped nonneutral plasmas are an excellent environment to investigate the nonlinear dynamics of collective systems like two-dimensional inviscid fluids. Starting from a single, axisymmetric column of a single-component plasma in a Penning-Malmberg trap (fluid vortex), a Kelvin-Helmholtz perturbation of any wavenumber can be induced in the vortex by a proper multipolar rotating field applied to the azimuthal patches of a sectored electrode. A linear theory demonstrates that modes with arbitrarily high wavenumber can be excited even when a limited number of sectors (typically four or eight) is available. In experiments and particle-in-cell simulations we observed the resonant wave growth to the nonlinear regime, where saturation and collapse to lower-order modes can occur. Mode frequency dependence on the wave amplitude, excitation strength and initial density profile affect width and shifting of the resonance peak. Opportunities for accurate control of the wave amplitude are also discussed.   

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When cooled to near the Doppler limit, ion plasmas confined in a Penning trap form two and three-dimensional crystals, which provide a useful platform for quantum simulation and sensing experiments. This talk will focus on recent experiments to measure small displacements, and hence weak forces and electric fields, using single-plane crystals consisting of several hundred Be$^{+}$ ions. By coupling the spin and motional degrees of freedom of the ions through the application of a spin-dependent optical dipole force, displacements of $50\,$pm ($40\times$ smaller than the ground-state wavefunction) are measured with a single measurement signal-to-noise ratio of 1. This displacement sensitivity is calibrated by driving the axial motion of the crystal far from the center-of-mass mode frequency, and implies $12\,\mathrm{yN}/\sqrt{\mathrm{Hz}}$ and $77\,(\mathrm{uV/m})/\sqrt{\mathrm{Hz}}$ sensitivities to forces and electric fields, respectively. When driving on-resonance with the center-of-mass mode, the sensitivity to weak forces and electric fields is greatly improved, but new limitations arise from frequency fluctuations of this mode.   

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Creation of a magnetically confined short-Debye-length electron-positron plasma would open a new frontier in experimental plasma physics. We report on progress toward achieving this goal. $10^{10}$ positrons are required for a 10 liter plasma with 10 Debye lengths in the system ($n \approx 10^{12}$ m$^{-3}$ and $T \approx$ 1 eV). Positrons from the NEPOMUC positron source will be trapped, cooled, and accumulated in a buffer-gas-trap, transferred to a high capacity multi-cell trap (using a 5-T magnet), and then delivered to a levitated dipole trap or a stellarator in a series of pulses. The dipole field ($B_{max} \approx$ 1-T) is produced by a light ($< 2$ kg), magnetically levitated superconducting coil ($I >$ 30 kA-t). Hour long levitation using feedback is anticipated. Annihilation gammas will be detected with 48 scintillator detectors. Recent experiments successfully injected positrons into an electron space charge (-58 V) in a prototype trap. First experiments with a levitated dipole will be in 2021. Stellarator design is underway.   

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Electron-positron plasmas are the quintessential "pair plasmas", comprising positively and negatively charged particles of equal mass. Theoretical and computational treatments of such systems go back more than 40 years and include a number of intriguing predictions that have yet to be tested. The EPOS (Electrons and Positrons in an Optimized Stellartor) project --- the latest branch of the APEX (A Positron Electron eXperiment) Collaboration --- aims to create and study electron-positron plasmas in the laboratory, magnetically confined on toroidal flux surfaces specifically designed for that purpose. In this talk, we will review the relevant physics at this unique combination of regime (low temperatures and densities, which are common to non-neutral plasmas but unusual for quasineutral plasmas), geometry (toroidal flux surfaces, which are common for quasineutral plasmas but unusual for nonneutral plasmas), and mass ratio (i.e., unity), as well as discuss the implications for the upcoming design of the device.   

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The APEX collaboration aims to create the first magnetically confined, low-temperature pair plasma with a spatial dimension of several Debye lengths so that collective behavior will be observable. A major limiting factor for this project is the scarcity of positrons, even at the NEPOMUC, the most intense positron source today. A crucial challenge is the accumulation of large numbers of moderated positrons. A device is needed which is capable of storing up to 10$^{\mathrm{11}}$ positrons at low-temperatures ($\sim 0.05\,$eV) for long times ($\ge 1\,$hour). A possible solution is the multi-cell Penning-Malmberg trap (MCT) concept, which separates the space charge of the positrons into multiple radially arranged Penning traps. We present first measurements with electrons stored in a single Penning trap and plans for the development of a dedicated MCT. With the single Penning trap the plasma diagnostics and manipulation techniques were tested, which are crucial for the operation of an MCT. These techniques include the measurement of the plasma parameters and the controlled off-axis displacement of the plasma. The MCT will be used to confine plasmas simultaneously in different off-axis cells, and to investigate the confinement as well as different injection and ejections schemes.   

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We study the expansion of a cold, non-neutral ion plasma into the vacuum. The plasma is made from cold rubidium atoms in a magneto-optical trap (MOT) and is formed via ultraviolet photoionization. We observe the development of both shock shells and ion pair correlations in the plasma as it expands. We also present two different computer simulations of the plasma expansion (a particle trajectory model and a fluid model) that recreate experimental conditions. The simulations not only verify the formation of shock shells and correlations, but they also determine the time- and position-dependent density, temperature, and Coulomb coupling parameter, $\Gamma({\bf{r}},t)$ of the expanding plasma. This analysis concludes that the experimental plasma is strongly coupled ($\Gamma({\bf{r}},t) \geq 1$) over the course of its expansion, even after experiencing disorder-induced heating.