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Dynamics of Ions in an Electrostatic Ion Beam Trap Daniel Zajfman Dept. of Particle Physics Weizmann Institute of Science Israel and Max-Planck Institute for Nuclear Physics Heidelberg, Germany Oded Heber Henrik Pedersen ( MPI) Michael Rappaport Adi Diner Daniel Strasser Yinon Rudich Irit Sagi Sven Ring Yoni Toker Peter Witte (MPI) Nissan Altstein Daniel Savin (NY) Charles Coulomb (1736-1806)

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The most common traps: The Penning and Paul trap Penning trap DC electric + DC magnetic fields Paul trap DC + RF electric fields Ion trapping and the Earnshaw theorem: No trapping in DC electric fields

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A new class of ion trapping devices: The Electrostatic Linear Ion Beam Trap Physical Principle: Photon Optics and Ion Optics are Equivalent Photons can be Trapped in an Optical Resonator Ions can be Trapped in an Electrical Resonator? R R L V1V1 V2V2 V 1

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Photon Optics Optical resonator Stability condition for a symmetric resonator: Symmetric resonator

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Optical resonatorParticle resonator Trapping of fast ion beams using electrostatic field Photon optics - ion optics L M VV E k, q V>E k /q

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L=407 mm Entrance mirror Exit mirror Field free region Phys. Rev. A, 55, 1577 (1997).

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V1V1 V4V4 V2V2 V3V3 VzVz V1V1 V4V4 V2V2 V3V3 VzVz Field free region Trapping ion beams at keV energies No magnetic fields No RF fields No mass limit Large field free region Simple to operate Directionality External ion source Easy beam detection Why is this trap different from the other traps? Detector (MCP) EkEk Neutrals

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Beam lifetime The lifetime of the beam is given by: n: residual gas density v: beam velocity : destruction cross section Destruction cross section: Mainly multiple scattering and electron capture (neutralization) from residual gas.

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Does it really works like an optical resonator? f V z (varies the focal length) Left mirror of the trap Step 1: Calculate the focal length as a function of V z

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Number of trapped particles as a function of V z. Step 2: Measure the number of stored particles as a function of V z

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Step 3: Transform the V z scale to a focal length scale

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Physics with a Linear Electrostatic Ion Beam Trap Cluster dynamics Ion beam – time dependent laser spectroscopy Laser cooling Stochastic cooling Metastable states Radiative cooling Electron-ion collisions Trapping dynamics

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E k, m, q W0W0 Pickup electrode WnWn E k =4.2 keV Ar + (m=40) 2W n 280 ns T 2930 ns (f=340 kHz) Induced signal on the pickup electrode. Digital oscilloscope

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Time evolution of the bunch length The bunch length increases because: Not all the particles have exactly the same velocities ( v/v 5x10 -4 ). Not all the particles travel exactly the same path length per oscillation. The Coulomb repulsion force pushes the particles apart. After 1 ms (~350 oscillations) the packet of ions is as large as the ion trap

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Time evolution of the bunch width ΔT: Characteristic Dispersion Time

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V1V1 V1V1 Characteristic dispersion time as a function of potential slope in the mirrors. ΔT=0 No more dispersion?? Steeper slope Flatter slope WnWn How fast does the bunch spread?

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T=15 msT=5 msT=1 ms T=30 msT=50 msT=90 ms

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“Coherent motion?” Expected Observation: No time dependence! Shouldn’t the Coulomb repulsion spread the particles? What happened to the initial velocity distribution? Dispersion No-dispersion

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Asymptotic bunch length WnWn n Injection of a wider bunch:Critical (asymptotic) bunch size? Bunch length ( s) Oscillation number 0 1 2 3 X 10 4 0 0.5 1 1.5 Self-bunching?

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Injection of a “wide” bunch Asymptotic bunch length n

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Q 1 : What keeps the charged particles together? Q 2 : Why is “self bunching” occurring for certain slopes of the potential? Q 3 : Nice effect. What can you do with it? There are only two forces working on the particles: The electrostatic field from the mirrors and the repulsive Coulomb force between the particles. It is the Repulsive Coulomb forces that keeps the ions together. + - (Charles Coulomb is probably rolling over in his grave)

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Simple classical system: Trajectory simulation for a 1D system. W0W0 Solve Newton equations of motion, v L Higher density Stronger interaction Ion-ion interaction: interacting non-interacting Stiff mirrors “Bound”! interacting non-interacting Soft mirrors

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Trajectory simulation for the real (2D) system. Trajectories in the real field of the ion trap Without Coulomb interaction With Repulsive Coulomb interaction E1>E2E1>E2

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1D Mean field model: a test ion in a homogeneously charged “sphere”: interaction strength ( negative k -> repulsive interaction) for Δx << L, the equations of motion are: where X is the center of mass coordinate Exact analytic solution also exists. Ion-sphere interaction What is the real Physics behind this “strange” behavior? L ΔxΔx V(X) q Nq ρ Ion-trap interaction Sphere-trap interaction Ion-sphere interaction (inside the sphere) E x ΔxΔx ρ

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mapping matrix M : Interaction strength The mapping matrix produces a Poincaré section of the relative motion as it passes through the center of the trap: Self-bunching: stable elliptic motion in phase space T: half-oscillation time and Solving the equations of motion using 2D mapping Phys. Rev. Lett., 89, 283204 (2002)

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Stability and Confinement conditions for n half-oscillations in the trap: Stability condition in periodic systems: For the repulsive Coulomb force: k < 0 Since Self bunching occurs only for negative effective mass, m* The system is stable (self-bunched) if the fastest particles have the longest oscillation time! English:

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Oscillation period in a 1D potential well: L m,p S=“slope” Synchronization occurs only if dT/dp>0 “Weak” slope yields to self-bunching! Physics 001

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What is the kinematical criterion dT/dP > 0? Ion velocity Oscillation time v1

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Is dT/dP>0 (or dT/dE>0) a valid condition in the “real” trap? Negative mass instability region dT/dE is calculated on the optical axis of the trap, by solving the equations of motion of a single ion in the realistic potential of the trap.

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Impulse approx. works for repulsive interaction (k < 0) Exact solution for any periodic system |Trace(M)|<2 Stable exact condition |Trace(M)|≥2 Unstable exact condition Repulsive Attractive

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Q 1 : What is the difference between a steep and a shallow slope? Q 2 : What keeps the charged particles together? Q 3 : Nice effect. What can you do with it? High resolution mass spectrometry Example: Time of flight mass spectrometry laser E k,m,q Time of flight: L The time difference between two neighboring masses increases linearly with the time-of-flight distance. Target (sample) Detector

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The Fourier Time of Flight Mass Spectrometer MALDI Ion Source Camera Laser Ion trap MCP detector

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Lifetime of gold ions in the MS trap Excellent vacuum – long lifetime!

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Fourier Transform of the Pick-up Signal. Resolution: 1.3 kHz, f/f 1/300 4.2 keV Ar + ff Dispersive mode: dT/dp < 0

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f (kHz) Self-bunching mode: dT/dp > 0 <3 Hz t meas =300 ms Δf/f< 8.8 10 -6

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Application to mass spectrometry: Injection of more than one mass FFT m

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Mass spectrum of polyethylene glycol H(C 2 H 4 O) n H 2 ONa + H(C 2 H 4 O) n H 2 OK + Even more complicated:

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Combined Ion trap and Electron Target Future outlook: Complete theoretical model, including critical density and bunch size Peak coalescence Can this really be used as a mass spectrometer? Study of “mode” locking Transverse “mode” measurement Stochastic cooling Transverse resistive cooling Trap geometry Atomic and Molecular Physics

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