Groß, J.

Herfurth, F.

Steinmann, J.

Zwicknagel, G.

GSI Repository

New approach for the induced charge calculation for cylindrical electrodes∗J. Steinmann1,2, J. Groß2, F. Herfurth1, and G. Zwicknagel31GSI Helmholtzzentrum, Darmstadt, Germany; 2University of Applied Sciences, Darmstadt, Germany; 3University,Erlangen-Nuernberg, GermanyThe HITRAP facility is part of the GSI accelerator com-plex and it is designed to store and cool up to 105 bareuranium ions. One of the cooling mechanisms is the resis-tive cooling where energy is dissipated in an external RLCcircuit, which is connected to the so called pick-up elec-trodes. These electrodes pick up the induced charge andthe related current caused by the ion dynamics and give afeedback to the ion dynamics due to an imbalance of thepotential on these electrodes. But since there exists no the-oretical treatment of resistive cooling which is able to accu-rately predict cooling time and cooling behavior numericalsimulations are necessary. In [1] we presented a model forthe simulation of resistive cooling of ion clouds in cylin-drical penning traps, which includes the complete ion-ioninteraction. To handle the large computational effort this isdone on graphic boards using parallel programming tech-niques [2]. The induced current is the time derivative ofthe induced charge and can be calculated by a finite ele-ment solver. This allows to model precisely the real geom-etry. But due to the increasing number of necessary ele-ments, the simulation time increases strongly with the re-quired numerical accuracy, and this method gets inefficientfor a time discrete integration scheme in which these cal-culations must be repeated some billion of times. Based onthe solution of the Poisson equation, we therefore workedout an analytical formula for the charge Q which is inducedon the electrodes by an ion:Q =qπ∞∫0I0(v · rR)v · I0(v) ·1∑j=0(−1)j sin(v · ΔzjR)dv (1)Here, I0 is the modified Bessel function of order zero, R isthe trap radius, Δzj = z − zj with z0 and z1 are the leftand right axial coordinate of the pick-up electrode and q isthe charge of the ion at position (r, z). The total chargeinduced on the pick-up electrode caused by an ion cloud iscalculated by superimposing the contributions of all singleions. Since we did not find an analytical solution of (1)some further simplification is needed to ensure a fast eval-uation. Approximating now the term I0(v · rR)/I0(v) inexpression (1) leads to an analytical formula which allowsfast calculation and provides high accuracy:Q ≈ q2kmax∑k=0βk1∑j=0(−1)j[erf(Δzjγk)+r2Δzj√π γ3ke−“Δzjγk”2]Here erf is the error function, kmax the order of approxi-mation, γk = 2Rαk, αk and βk are numerical parameters.∗Work supported by FE, Project-Number DAGROS1012.The calculated charge induced on single electrodes isshown in figure 1. The approximation error depends onFigure 1: Induced charged at five cylindrical electrodes.The bins from left to right: Analytical solution (gray), ap-prox. kmax = 2 (red), approx. kmax = 1 (green), Comsol(blue), calculation scheme of Ref.[3] (yellow). Electrode 2and 4 are the pick-up electrodesthe choice of kmax, see figure 2. Already at kmax = 2the results of (1) and the analytical approximation are ingood agreement. The resulting absolute error for the pick-up electrodes of less than 5 ·10−3 will be sufficiently accu-rate for our simulations.Figure 2: Approximation error per unit charge versus theorder of approximation kmaxReferences[1] J. Steinmann et. al., AIP Conference Proceedings 1521, 2013,pp. 240-249.[2] J. Steinmann et. al., “Simulation of Ion Clouds in Penningtraps using graphic boards”, GSI Scientific Report (2012).[3] G. Maero et. al., Applied Physics B: Lasers and Optics 107(2012) 1087.‘GSI SCIENTIFIC REPORT 2013 PNI-AP-16doi:10.15120/GR-2014-1-PNI-AP-16 173

New approach for the induced charge calculation for cylindrical electrodes

http://repository.gsi.de/record/67879/files/PNI-AP-16.pdf

New approach for the induced charge calculation for cylindrical electrodes

Abstract

Similar works

Full text

Available Versions

GSI Repository