granularity effects
TRANSCRIPT
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Globular cluster Messier 2 by Hubble Space Telescope.. Located in the constellation of Aquarius, also known as NGC 7089.M2 contains about a million stars and is located in the halo of our Milky Way galaxy.
Granularity effects
Marieke de LoosSteinar WoutersEdgar VredenbregtJom Luiten
Eindhoven UniversityThe Netherlands
Bas van der Geer
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Ultracold Electron/Ion Source
2
Trap & CoolMagneto-optical trapDensity ≈ 1016 / m3
RMS size ≈ 1 mmT = 100 µK
IonizeUltracold plasmaIonization radius ≈ 50 µm
AccelerateUltracold sourceBunch energy E = 15 keV
Killian et al., PRL 83, 4776 (1999)
Luiten et al., PRL 95, 164801 (2005)McCulloch et al.,Nat. Phys. 7, 785 (2011)
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Application: Ultrafast electron diffraction
• Structural dynamics• Resolve atomic length and time scales: ~1 Å, ~100 fs
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Application: Focused ion beams (FIB)
SIM
SEM
Cross Section ImagingTEM sample preparation
Machining, sputtering/milling
Beam-induced deposition
Channeling contrast for crystalline grain analysis
(Logo) engraving
http://www.s3.infm.it/fib.html
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www.pulsar.nlLaser-cooled ion source
V
Ionization
Ultra-coldion beam
ExcitationR
a
d
σL
MOT parametersTemperature: 200 µKvth 0.22 m/sn 1018 m–3
Typical current: 10 pAR 13 µmσL 1.4 µm
GeometryV 2 kVd 20 mma 1 mm
Simple theory predicts total collapse of brightnessBut that is without acceleration…
Coulombinteractions
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www.pulsar.nlLaser-cooled ion source
MOT parametersTemperature: 200 µKvth 0.22 m/sn 1018 m–3
Typical current: 10 pAR 13 µmσL 1.4 µm
GeometryV 2 kVd 20 mma 1 mm
E1=1 MV/m
500 kV/m
200
100
50
2010 kV/m
E1=1 MV/m
500 kV/m
200 kV/m
100 kV/m
50 kV/m
20 kV/m
10 kV/m
Atomic beam limited
Disorder induced heating
Brig
htne
ssE
nerg
y sp
read
[eV
]
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www.pulsar.nlCoulomb interactions
x
pxSpace charge
Disorder induced heating
GPT simulations: n=1018 m–3
t=0
t=10 ps
t=20 psAll interactions
Ideal particle-in-Cell
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Analogy of a cold beam
Rhône Glacier 2012: Bas van der Geer
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We need to go inside
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www.pulsar.nlCoulomb 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
Nature PhotonicsVol 2, May 2008
M. Centurion et. al.
And many others…
PRL 93, 094802O.J. Luiten et. al.
GPT simulations
JAP 102, 093501T. van Oudheusden et. al.
PRST-AB 9, 044203S.B. van der Geer et. al.
PRL 102, 034802 M. P. Reijnders et. al.
JAP 102, 094312 S.B. van der Geer et. al.
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www.pulsar.nlDisorder induced heating
Excess potentialenergy U
Momentumspread
Temperature ↑Brightness ↓
Randomprocesses
Coulombinteractions
High U Low U
x
px σpx
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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 15Chandrasekhar, 1943.
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Where are my neighbors?
Average distance:
Most likely position:
Wigner-Seitz radius:
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Potential energy
Average potential energy:
Potential energy at average position:
It’s the difference that matters:
GPT simulations
Simple theory
Tem
pera
ture
[K]
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www.pulsar.nlParadigm shift
Photo/thermionic emission
Space-charge• ‘Shaping’ the beam• Ellipsoidal bunches
Particle-in-Cell• Macro-particles• One species• Fluid assumption• Liouville holds• Convergent rms values
Laser cooled sources
Disorder induced heating• Fast acceleration• Breaking randomness
Tree-codes (B&H, FMM, P3M)• Every particle matters• Ions and electrons• Ab initio• No Liouville to the rescue• Divergent rms values
k Tphotogun >> 0.02 n1/3 q2 / ε0 >> k Tlaser-cooled
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Application: Focused ion beam
• Aim: Lots of current at nm spotsizes• Design disaster: Beams heats up during acceleration
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www.pulsar.nl‘Typical’ simulation code: GPT
Tracks sample particles in time-domain• Relativistic equations of motion• Fully 3D, including all non-linear effects
• GPT solves with 5th order embedded Runge Kutta, adaptive stepsize• GPT can track ~106 particles on a PC with 1 GB memory• Challenge: E(r,t), B(r,t), flexibility without compromising accuracy
Analytical expressions Field-maps
External fields Coulomb interactions
Particle in Cell All interactions
{E,B}=f(x,y,z,t)
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www.pulsar.nlAlgorithms…
All interactions O(N2):• PP Particle-Particle → slow• P3M 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
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www.pulsar.nlBarnes-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.
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www.pulsar.nlGPT hardware requirements
GPT kernel:• Programming language: C and C++• Multi-core functionality: openMP• Distributed scans: MPI
If you can run Microsoft Office, you can track ~1M particles
Mac mini
HP blade server: 16 servers, 128 cores, 1.2 kW of GPT power!
1.2 kW
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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
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Spo
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m]
Vobj
ectiv
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Spo
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Spo
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ectiv
e [k
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Spo
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Further slides are intentionally removed.
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www.pulsar.nlLaser-cooled e– source
Fields:Cavity field 20 MV/m rf-cavityDC offset 3 MV/m
Particles:Charge 0.1 pC (625k e-)Initial density 1018 / m3Ionization time 10 psInitial Temp 1 K
GPT tracking:- All particles- Realistic fields- All interactions
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www.pulsar.nlLaser cooled e– diffraction
GPT results:εx 20 nm (rms)10% slice ~1 nm
Energy 120 keV Spread 1%εz 60 keV fs
Charge 0.1 pC (625,000 e–)
Ultracold Electron Source for Single-Shot, Ultrafast Electron DiffractionMicroscopy and Microanalysis 15, p. 282-289 (2009).S.B. van der Geer, M.J. de Loos, E.J.D. Vredenbregt, and O.J. Luiten
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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
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