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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) Ion trapping and the Earnshaw theorem: No trapping in DC electric fields The most common traps: The Penning and Paul trap Penning trap DC electric + DC magnetic fields Paul trap DC + RF electric fields A new class of ion trapping devices: The Electrostatic Linear Ion Beam Trap Physical Principle: Photon Optics and Ion Optics are Equivalent V1 V2 V1<V2 R R Photons can be Trapped in an Optical Resonator L Ek, q Ions can be Trapped in an Electrical Resonator? V V>Ek/q V Photon Optics Optical resonator Stability condition for a symmetric resonator: L f 4 Symmetric resonator Photon optics - ion optics Optical resonator Particle resonator Ek, q V V>Ek/q M Trapping of fast ion beams using electrostatic field L V Field free region Entrance mirror Exit mirror Phys. Rev. A, 55, 1577 (1997). L=407 mm Trapping ion beams at keV energies Neutrals Field free region Ek Detector (MCP) V1 V2 V3 V4 V1 V2 V3 V4 Vz Vz • No magnetic fields Why is this trap different from the other traps? • No RF fields • No mass limit • Large field free region • Simple to operate • Directionality • External ion source • Easy beam detection Beam lifetime The lifetime of the beam is given by: 1 σnv n: residual gas density v: beam velocity : destruction cross section N(t) N0e t Destruction cross section: Mainly multiple scattering and electron capture (neutralization) from residual gas. Does it really works like an optical resonator? L f 4 Left mirror of the trap Vz (varies the focal length) f Step 1: Calculate the focal length as a function of Vz Step 2: Measure the number of stored particles as a function of Vz Number of trapped particles as a function of Vz. Step 3: Transform the Vz scale to a focal length scale L f 4 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 Ek=4.2 keV Ar+ (m=40) Wn Pickup electrode Ek, m, q W0 Induced signal on the pickup electrode. T 2930 ns (f=340 kHz) 2Wn 280 ns Digital oscilloscope Time evolution of the bunch length The bunch length increases because: • Not all the particles have exactly the same velocities (v/v5x10-4). • 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 Time evolution of the bunch width Wn W02 n2ΔT2 ΔT: Characteristic Dispersion Time How fast does the bunch spread? Wn V1 Wn W02 n2ΔT2 V1 Flatter slope Characteristic dispersion time as a function of potential slope in the mirrors. Steeper slope ΔT=0 No more dispersion?? T=1 ms T=5 ms T=15 ms T=30 ms T=50 ms T=90 ms Expected Wn W02 n2ΔT2 “Coherent motion?” Dispersion Observation: No time dependence! No-dispersion Shouldn’t the Coulomb repulsion spread the particles? What happened to the initial velocity distribution? Injection of a wider bunch:Critical (asymptotic) bunch size? 1.5 Bunch length (s) Wn 1 Self-bunching? 0.5 Asymptotic bunch length 0 0 1 2 Oscillation number n 3 X 104 Injection of a “wide” bunch Asymptotic bunch length n Q1: What keeps the charged particles together? Q2: Why is “self bunching” occurring for certain slopes of the potential? Q3: 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) Simple classical system: Trajectory simulation for a 1D system. L Ion-ion interaction: Vij <v>, v qiqj rij const. Higher density Stronger interaction Solve Newton equations of motion W0 Stiff mirrors Soft mirrors “Bound”! non-interacting interacting Trajectory simulation for the real (2D) system. Trajectories in the real field of the ion trap Without Coulomb interaction With Repulsive Coulomb interaction E1>E2 What is the real Physics behind this “strange” behavior? 1D Mean field model: a test ion in a homogeneously charged “sphere”: Nq V(X) Δx p12 p22 Η NqV(x1 ) qV(x2 ) qU(x1 x2 ) 2m1 2m2 q ρ Sphere-trap Ion-trap interaction interaction Ion-sphere interaction L ρ E Ion-sphere interaction (inside the sphere) Δx ~r U(x) 12 k x 2 U0 ~ 1/r2 x ρq k 3ε0 for Δx << L, the equations of motion are: where X is the center of mass coordinate interaction strength ( negative k -> repulsive interaction) Δx Δp/m Δp ΔxqV(X) kx Exact analytic solution also exists. Solving the equations of motion using 2D mapping mapping matrix M: Δx Δx n M Δp n Δp 0 1 - kT 2/m* T/m* M kT 1 T: half-oscillation time m m/η * Interaction strength and P0 dT η T dP0 Phys. Rev. Lett., 89, 283204 (2002) p 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 x Stability and Confinement conditions for n half-oscillations in the trap: Δx Δx n M Δp n Δp 0 Trace(M) 2 Stability condition in periodic systems: k ρq 3ε0 x 0 kT / m 4 For the repulsive Coulomb force: k < 0 2 * Self bunching occurs only for negative effective mass, m* m* m/η 0 P dT Since η 0 T dP0 p dT 0 dP0 English: The system is stable (self-bunched) if the fastest particles have the longest oscillation time! dT 0? dP0 Synchronization occurs only if dT/dp>0 Physics 001 Oscillation period in a 1D potential well: Lm p T 4 2p S dT 4 2Lm 2 dp S p m,p S=“slope” L 2p2 dT if S Lm , dp 0 2 2p dT if S , 0 Lm dp “Weak” slope yields to self-bunching! Oscillation time What is the kinematical criterion dT/dP > 0? v1<v2 Ion velocity slow p=Fc t <v> fast p=Fc t The Coulomb Repulsive Force Fc q1q2 Δz2 Time dT/dv>0 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. Exact solution for any periodic system cos(T ) cos(T ) sin( T ) 4 Trace( M ) cos(T ) (1 ) 2 (T ) 2 (1 )T Attractive where k /m |Trace(M)|<2 Stable exact condition |Trace(M)|≥2 Unstable exact condition Repulsive 0 - kT2η/m 4 k ρq 3ε 0 Impulse approx. works for repulsive interaction (k < 0) Q1: What is the difference between a steep and a shallow slope? Q2: What keeps the charged particles together? Q3: Nice effect. What can you do with it? High resolution mass spectrometry Example: Time of flight mass spectrometry Target (sample) Ek,m,q Detector laser L Time of flight: T L m 2Ek The time difference between two neighboring masses increases linearly with the time-of-flight distance. ΔT L 1 Δm 8mEk The Fourier Time of Flight Mass Spectrometer Camera MALDI Ion Source Laser Ion trap MCP detector Lifetime of gold ions in the MS trap Excellent vacuum – long lifetime! Fourier Transform of the Pick-up Signal Dispersive mode: dT/dp < 0 Resolution: 1.3 kHz, f/f1/300 4.2 keV Ar+ f . Self-bunching mode: dT/dp > 0 tmeas=300 ms Δf/f< 8.8 10-6 f (kHz) <3 Hz Application to mass spectrometry: Injection of more than one mass m<m Ek “Real” mass spectrometry: If two neighboring masses are injected, will they “stick” together because of the Coulomb repulsion? 132Xe+, 131Xe+ FFT Even more complicated: Mass spectrum of polyethylene glycol H(C2H4O)nH2ONa+ H(C2H4O)nH2OK+ 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 Combined Ion trap and Electron Target