NonDestructive Positron Plasma Diagnostics for Antihydrogen Production

Production of antihydrogen atoms by mixing antiprotons with a cold, confined, positron plasma depends on parameters such as the plasma density and temperature. We discuss a nondestructive diagnostic, based on an analysis of excited, low-order plasma modes, that provides comprehensive characterization of the positron plasma in the ATHENA antihydrogen apparatus. The dipole and quadrupole modes of a spheroidal positron plasma are interpreted in the framework of a cold fluid theory. In particular, the excitation and detection of the dipole mode are analytically modeled considering the response of the center-of-mass to a resonant driving perturbation. The model is compared to, and validated by, numerical simulations with a particle-in-cell code. Measurements of the positron plasma properties are discussed. 1 Corresponding author ([email protected]) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Thu, 17 Apr 2014 22:41:33 INTRODUCTION Recently cold antihydrogen atoms where produced in the ATHENA experiment (ApparaTus for High precision Experiments on Neutral Antimatter, or shortly AnTiHydrogEN Apparatus) at CERN (European Organization for Nuclear Research) by mixing low energy antiprotons with a cold dense positron plasma inside an electromagnetic trap [1]. Very low positron and antiproton temperatures (a few K) and high positron density (' 108 particles/cm3) are the two key ingredients necessary to enhance the recombination rate in ATHENA. Under these conditions the positron cloud is in the plasma regime. In order to understand and control the recombination process, several parameters describing the positron plasma in thermal equilibrium should be measured in a non-destructive way to avoid perturbing the system. Harmonically confined one component plasmas at temperatures close to absolute zero take the shape of an ellipsoid characterized by an aspect ratio α = zp/rp, where zp and rp are the semi-major axis and semi-minor axis respectively. A simple analytic model for low-order axisymmetric plasma modes (Trivelpiece-Gould modes) in a spheroidal plasma has been used as a diagnostic in the ATHENA experiment [2, 3], where these modes were excited and detected to gain information about the positron plasma. The two lowest-order (dipole and quadrupole) modes were interpreted in the framework of a cold fluid theory [4]. The mode frequencies depend only on the plasma density n and aspect ratio α. Corrections due to finite temperature have also been calculated [5, 6]. Previous studies [6, 7, 8, 9, 10, 11, 12, 13] have extracted the information contained in the mode frequencies themselves. The plasma density and aspect ratio can be derived by comparing the measured frequencies of the dipole and quadrupole modes with those predicted by theory. But the actual plasma length and radius (or, equivalently, the number of particles) cannot be ascertained by using only frequency data. However, in Ref. [3], we demonstrated that the plasma length can be extracted from a detailed analysis of the power transmitted through the plasma near the resonance of the center-of-mass mode. The model described in Ref. [3] can be numerically validated using a non-neutral plasma equilibrium code EQUILSOR [14] coupled with a two-dimensional (r − z) particle-in-cell (PIC) simulation RATTLE [6]. The first code (EQUILSOR) is dedicated to the evaluation of the plasma thermal equilibrium. The code solves the Poisson-Boltzmann equilibrium equation assuming axisymmetry [15]. The second code (RATTLE) uses the computed equilibria to create initial distributions of particles in (r,z,vz)-space and then simulates the motion of the particles via a standard PIC technique. The same numerical codes were used by Surko and co-workers [6] to investigate the applicability of Dubin’s mode theory [4] to their electron plasma. This paper is organized in the following manner. The following section briefly describes the experimental setup. The third section is a concise review of the model used to describe the center-of-mass (dipole) mode and confirms its validity by means of a comparison with numerical simulations. In the fourth section a possible extension of the model in order to study the plasma response for the quadrupole mode is discussed. The simulation results show that the extension developed in Ref. [3] also works for the quadrupole mode. The concluding section presents an example of the application of the diagnostic. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Thu, 17 Apr 2014 22:41:33 FIGURE 1. (a) Schematic of the mixing trap electrodes. The positrons are confined in the center group designed to create a harmonic electrostatic field. The central electrode is called the ring electrode and the other 3 pairs are labeled 1, 2 and 3. The driving signal applied to one electrode is shown together with the resistances on the transmitting and receiving electrodes. The shape of the prolate positron ellipsoid is shown schematically. (b) The axial potential of the ATHENA nested trap is shown and the ranges of axial motion of the positrons and of the antiprotons is indicated schematically. EXPERIMENTAL SETUP Electromagnetic traps of the Penning-Malmberg type are used in the ATHENA experiment to confine charged particles. The traps are realized by placing a series of cylindrical electrodes of various lengths and with an inner radius of 1.25 cm inside a uniform 3-Tesla magnetic field parallel to the trap axis and applying static voltages to them. A potential well along the trap (or z−) axis is thereby produced which provides axial confinement for particles having energies lower than the top of the well. The magnetic field ensures radial confinement. The trap structure is installed inside a cryogenic bore and can be cooled to about 15 K. The voltages on these electrodes are manipulated to perform various procedures. The mixing trap [Fig. 1(a)] is composed of 3 groups of electrodes which produce the nested trap configuration [Fig. 1(b)]. Thus, the simultaneous confinement of particles having opposite signs of charge is achieved. The positron confining region is comprised of seven electrodes. The lengths of these electrodes have been chosen, according to Ref. [16], in order to create a harmonic potential when the ratios between the applied voltages are suitably chosen. The axial modes are excited by applying a sinusoidal perturbation to one electrode This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Thu, 17 Apr 2014 22:41:33 with an electromotive force Vt = vte jωt . The resulting oscillation of the plasma induces a current in the pick-up electrode [17, 18, 19] and a voltage Vr = vr(ω)e jωt is detected across the resistance Rr [Fig. 1(a)]. DIPOLE MODE In cold fluid theory [4], the lowest-order mode is a coherent oscillation of the whole plasma along the z axis with a frequency ω1 equal to that of single particle motion inside the trap, ω1 = ωz = √ qV0 md2 . (1) In Eq. (1) q is the particle electric charge and m is its mass, V0 is the potential difference between the ring and the type 3 electrode, and the length d is related to the trap radius rw (d = 1.74 rw for the mixing trap design). A simple model, based on the observation that the dipole mode can be described as a damped oscillation of the center-of-mass of the positron cloud, enables study of the excitation and detection processes. The center-of-mass equation of motion along the trap axis can be written as [3]: z̈cm + γżcm +ω1zcm = q m 〈Ezi〉 , (2) where zcm is the axial position of the cloud center-of-mass and γ describes the damping of the oscillations. In this equation the driving term 〈Ezi〉 is an effective axial electric field acting on the center-of-mass when a potential Vi is applied on the electrode labeled i (i indicates the electrode type, see also Fig. 1(a)). It can be approximated by 〈Ezi〉 = gi(α,zp) Vi 2rw , (3) where the characteristic function gi(α,zp), defined by gi(α,zp) = 3αrw zp ∫ zp −zp dz ∫ zpα−1 √ 1−z/zp 0 rdrFzi(r,z), (4) has been introduced to describe the coupling between the perturbation signal and the center-of-mass response. In Eq. (4), Fzi(r,z) represents the axial component of the electric field at the position (r,z) when a unit potential is applied on the electrode i while the rest of the electrodes are grounded. The factor gi(α,zp) can be numerically evaluated using a truncated Fourier-Bessel series (as in Ref. [3]) or by using directly the EQUILSOR-RATTLE numerical Poisson solver. This coupling function depends not only on the trap geometry and on the type of the electrode used to drive the mode, but also on the size and shape of the plasma. It has a strong dependence on the plasma length, but only weakly depends on the aspect ratio (or, equivalently, on the plasma radius; see Fig. 2). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Thu, 17 Apr 2014 22:41:33 [cm] p z 0 0.5 1 1.5 2 2.5 1