Granularity effects Bas van der Geer

Transcription

1 Granularity effects Bas van der Geer Marieke de Loos Steinar Wouters Edgar Vredenbregt Jom Luiten Eindhoven University The Netherlands Globular cluster Messier 2 by Hubble Space Telescope.. Located in the constellation of Aquarius, also known as NGC M2 contains about a million stars and is located in the halo of our Milky Way galaxy.

2 Ultracold Electron/Ion Source Trap & Cool Magneto-optical trap Density / m 3 RMS size 1 mm T = 100 µk 2 Ionize Ultracold plasma Ionization radius 50 µm Killian et al., PRL 83, 4776 (1999) Accelerate Ultracold source Bunch energy E = 15 kev Luiten et al., PRL 95, (2005) McCulloch et al., Nat. Phys. 7, 785 (2011)

3 Application: Ultrafast electron diffraction Structural dynamics Resolve atomic length and time scales: ~1 Å, ~100 fs

4 Application: Focused ion beams (FIB) Cross Section Imaging TEM sample preparation Machining, sputtering/milling SIM Beam-induced deposition SEM Channeling contrast for crystalline grain analysis (Logo) engraving

5 Laser-cooled ion source Ionization MOT parameters Temperature: 200 µk v th 0.22 m/s n m 3 σ L Excitation R Ultra-cold ion beam Typical current: 10 pa R 13 µm σ L 1.4 µm V a d Coulomb interactions Geometry V d a 2 kv 20 mm 1 mm Simple theory predicts total collapse of brightness But that is without acceleration

6 Laser-cooled ion source Brightness Atomic beam limited Disorder induced heating 10 kv/m 20 E 1 =1 MV/m 500 kv/m MOT parameters Temperature: 200 µk v th 0.22 m/s n m 3 Typical current: 10 pa R 13 µm σ L 1.4 µm Energy spread [ev] E 1 =1 MV/m 500 kv/m 200 kv/m 100 kv/m 50 kv/m 20 kv/m 10 kv/m Geometry V d a 2 kv 20 mm 1 mm

7 Coulomb interactions Disorder induced heating t=0 p x Space charge x t=10 ps All interactions t=20 ps Ideal particle-in-cell GPT simulations: n=10 18 m 3

8 Rhône Glacier 2012: Bas van der Geer Analogy of a cold beam

9 We need to go inside

10 Coulomb interactions Macroscopic (mean field): Space-charge Average repulsion force Bunch expands Deformations in phase-space Governed by Poisson s equation Microscopic: Disorder induced heating Neighbouring particles see each other Potential energy momentum spread Stochastic effect Governed by point-to-point interactions PRL 93, O.J. Luiten et. al. JAP 102, T. van Oudheusden et. al. PRST-AB 9, S.B. van der Geer et. al. PRL 102, M. P. Reijnders et. al. JAP 102, S.B. van der Geer et. al. GPT simulations Nature Photonics Vol 2, May 2008 M. Centurion et. al. And many others

11 Disorder induced heating Random processes Excess potential energy U High U Low U Coulomb interactions p x σ px Momentum spread x Temperature Brightness

12 Nearest neighbor w(r) dr: Probability that the nearest neighbor is between r and r+dr. assuming a uniform random distribution with number density n. Hence: Stochastic problems in Physics and astronomy, Reviews of Modern Physics 15 Chandrasekhar, 1943.

13 Where are my neighbors? Average distance: Most likely position: Wigner-Seitz radius:

14 Temperature [K] Potential energy Average potential energy: Potential energy at average position: GPT simulations It s the difference that matters: Simple theory

15 Paradigm shift Photo/thermionic emission Laser cooled sources Space-charge Shaping the beam Ellipsoidal bunches Disorder induced heating Fast acceleration Breaking randomness Particle-in-Cell Macro-particles One species Fluid assumption Liouville holds Convergent rms values Tree-codes (B&H, FMM, P 3 M) Every particle matters Ions and electrons Ab initio No Liouville to the rescue Divergent rms values k T photogun >> 0.02 n 1/3 q 2 / ε 0 >> k T laser-cooled

16 Application: Focused ion beam Aim: Lots of current at nm spotsizes Design disaster: Beams heats up during acceleration

17 Typical simulation code: GPT Tracks sample particles in time-domain Relativistic equations of motion Fully 3D, including all non-linear effects GPT solves with 5 th order embedded Runge Kutta, adaptive stepsize GPT can track ~10 6 particles on a PC with 1 GB memory Challenge: E(r,t), B(r,t), flexibility without compromising accuracy External fields Coulomb interactions Analytical expressions Field-maps Particle in Cell All interactions {E,B}=f(x,y,z,t)

18 Algorithms All interactions O(N 2 ): PP Particle-Particle slow P 3 M Particle-Particle Particle-Mesh Accuracy traded for speed: B&H Barnes&hut tree: O(N log N) FMM Fast-Multipole-Method: O(N) Imaga credit: Southern European observatory

19 Barnes-Hut Hierarchical tree algorithm: Includes all Coulomb interactions O(N log N) in CPU time User-selectable accuracy Division of space Tree data structure J. Barnes and P. Hut, Nature 324, (1986) p. 446.

20 GPT hardware requirements GPT kernel: Programming language: C and C++ Multi-core functionality: openmp Distributed scans: MPI Mac mini If you can run Microsoft Office, you can track ~1M particles 1.2 kw HP blade server: 16 servers, 128 cores, 1.2 kw of GPT power!

21 Application: Focused ion beam Aim: Lots of current at nm spotsizes Design disaster: Beams heats up during acceleration Multi-objective global optimization Kalyanmoy Deb et. al., A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, IEEE Transactions On Evolutionary Computation, Vol. 6, No. 2, A

22 Vobjective [kv] Spotsize [nm]

23 Vobjective [kv] Spotsize [nm]

24 Vobjective [kv] Spotsize [nm]

25 Vobjective [kv] Spotsize [nm]

26 Further slides are intentionally removed.

27 Laser-cooled e source Fields: Cavity field DC offset 20 MV/m rf-cavity 3 MV/m Particles: Charge 0.1 pc (625k e - - -

28 Laser cooled e diffraction GPT results: ε x 20 nm (rms) 10% slice ~1 nm Energy 120 kev Spread 1% ε z 60 kev fs Charge 0.1 pc (625,000 e ) Ultracold Electron Source for Single-Shot, Ultrafast Electron Diffraction Microscopy and Microanalysis 15, p (2009). S.B. van der Geer, M.J. de Loos, E.J.D. Vredenbregt, and O.J. Luiten

29 Conclusion Granularity effects are becoming increasingly relevant Physics: Nearest neighbor interaction Design simulations need: Sub-nm precision Lots of particles for good statistics All pair-wise Coulomb interactions Start-to-end in 3D in complicated fields Multi-objective optimizations TODO! Break randomness in either space or time