Groß, J.

Herfurth, F.

Steinmann, J.

Zwicknagel, G.

GSI Repository

Simulation of Resistive Cooling in cylindrical Penning Traps∗J. Steinmann1,2, J. Groß2, F. Herfurth1, and G. Zwicknagel31GSI Helmholtzzentrum, Darmstadt, Germany; 2University of Applied Sciences, Darmstadt, Germany; 3University,Erlangen-Nuernberg, GermanyThe simulation of ion clouds in traps taking ion-ion inter-action into account is associated with large computationaleffort. This applies especially for the simulation of cool-ing methods like e.g. resistive cooling, where the coolingforce has a complex structure. In [1] a simulation methodfor the resistive cooling was presented, using a particle incell code. There the ion-ion interaction was treated by amean-field approximation which, however, neglects the in-fluence of ion-ion correlations on the dynamics. We thusdeveloped an algorithm for the simulation of resistive cool-ing including the complete ion-ion interactions.Charged particles moving between two conductive elec-trodes induce a surface charge Q on both of them and causea potential imbalance u. Connecting the electrodes to anexternal RLC-circuit leads to an electrical current i. En-ergy dissipation in the resistor of the circuit then leads tochanges in the induced charges on the electrodes and therelated electric field which acts back on the ions as a cool-ing force ~FD. This non-trivial coupling between the circuitand the ion motion is described by the equations:d2udt2+RLdudt+1LCu =1C(RLdidt+ i)(1)i =N∑k=1ddt·ΔQ(~rk)~¨rj =1mj·(~FE(~rj) + ~FB(~˙rj) + · · · (2)+ ~FC(~r1, · · · , ~rN ) + ~FD(~rj , u))R represent the resistance, C the capacitance and L the in-ductance of the circuit which is driven by the current i andΔQ is the difference of the induced charges on the Pick-Up electrodes. ~FE , ~FB , ~FC and ~FD := −qj ~∇VD repre-sents the forces from the electrical field, magnetic field, theCoulomb interaction and the induced Potential. N is thenumber of ions and ~rj the position vector of the j-th parti-cle. The induced charge on one of the Pick-Up electrodesis given by the equation:Q = a ·N∑k=1∫ z>z<∫ 2π00∂V (r, φ, z; rk, φk, zk)∂r∣∣∣∣adφdz,(3)where z<, z> representing the z coordinates of the elec-trode, Vk the potential of the k-th ion using cylindricalcoordinates and a the radius of the trap.∗Work supported by FE, Project-Number DAGROS1012.In figure 1, the workflow of the resistive cooling algo-rithm is shown. The algorithm starts with an inital dis-tribution of the positions and velocities of the ions, whilethe electrical Potential VD caused by the induced current isneclegted at first. Next equation (2) is solved numericallyto calculate a new set of positions, velocities and acceler-ations from which the right hand side of equation (1) isevaluated. After solving equation (1), finally the electricalpotential VD is calculated and serves together with the newpositions and velocities as a starting point for the subse-quent cycle.Figure 1: Workflow of the resistive cooling algorithmCurrently we work on the implementation of this algo-rithm into our existing code [2]. First simulations donewith single ions show the expected cooling behavior. Fig-ure 2 shows the simulation result of a single 12C5+ ionin a cylindrical geometry with radius of 3.5 mm. Theparameters of the circuit were chosen as (R, L, C) ≈(10 Ω, 1.22 mH, 14.22 pF ). The simulation was doneon a GPU device using an parallelized Adam-Bashforthmethod of fourth order.Next steps are the comparsion of the simulation resultswith experimental data were the circuit parameters are wellknown and investigating the cooling behavior for an in-creasing ion number.Figure 2: Left: Amplitude of the axial motion, Right: Totalaxial energy of the ion during resistive cooling.References[1] G. Maero , Springer , Applied Physics B , vol. 107 , p. 1087-1096.[2] J. Steinmann , GSI Report 2009-1 , p. 307.GSI SCIENTIFIC REPORT 2012 PNI-ACC-09365

Simulation of Resistive Cooling in cylindrical Penning Traps

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Simulation of Resistive Cooling in cylindrical Penning Traps

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