Molecular Dynamics Simulations Of Dust Crystals

Transcription

1 Molecular Dynamics Simulations Of Dust Crystals 2nd WORKSHOP Diagnostics and Simulation of Dusty Plasmas, September Patrick Ludwig, Hanno Kählert, Torben Ott, Henning Baumgartner and Michael Bonitz ITAP, University of Kiel, Germany

2 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

3 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

4 Strongly correlated (cooperative) behaviour Coulomb-Korrelationen in Halbleitern Interaction makes life interesting Coulomb interaction classical statistics ideal classical plasma quantum statistics U ab r =e a e b / r ng o r st c b lom u Co structure formation Degeneracy parameter ideal quantum plasma ela r r o ns tio strong correlations r n dim / r dim =h / 2 m k B T Coupling parameter classical systems ( 1 ): = U / k B T q 2 / r k B T quantum systems ( 1 ):, r s 1 : strongly correlated behavior Coulomb (Wigner) crystallization r s r / a B U / E F ( a B - Bohr radius) ( E F - Fermi energy)

5 Strongly Coupled Mesoscopic Systems in Traps ('Artifical Atoms') N=8; (1,7) config. N=20; (1,7,12) config. 2D Electron Dimples on the Surface of Liquid 4He (Leiderer et al., Surface Science 113, 405 (1982)) rs=100 2D Ca+ Ion Crystals in a Paul Trap (Werth et al., University Mainz, Germany (2000)) rs=20 2D Electron 'Wigner' Crystals in Quantum Dots (A. Filinov, M. Bonitz, Yu.E. Lozovik, PRL (2001)) 2D Finite Dust Crystals in a rf Plasma Trap (Lin I et al., Taiwan (1999))

6 Mendeleyev table for 2D Coulomb Clusters N=10 Ca+ ions in Paul trap G. Werth, Uni Mainz Dusty Plasma, Melzer et al.

7 Mendeleyev table for 2D Coulomb Clusters N=20 N=10 Electrons on a liquid helium surface, P. Leiderer (1982) Dusty Plasma, Melzer et al.

8 Mendeleyev table for 2D Coulomb Clusters N=19 Wigner Crystal of Electrons in a Quantum Dot A. Filinov et al., 2001 Dusty Plasma, Melzer et al.

9 Mendeleyev table for 2D Coulomb Clusters Experimental ground state configurations of 2D finite dust crystals (Lin I et al., Taiwan) 2D Mesoscopic Systems (Artificial Atoms): - shell structure (strong N-dependence) - Mendeleev-type tables (packing rules) - characteristic occupation numbers: shell closures and unusually stable 'magic' configurations - pronounced symmetry and boundary effects

10 Open Question 2D 3D?

11 3D Spherical Ca+ ion crystals strongly coupled Coulomb clusters in traps fluorescence pattern of 300, 700, and 1400 laser cooled ions in a Paul trap (Mortensen et al., University of Aarhus, Denmark, PRL 2006) at temperature T 5mK, coupling parameter Г=Q2/ r kbt 400 observation of higly ordered crystalline (shell) structures one-component plasma: Coulomb interaction

12 3D Spherical Dust Crystals strongly coupled Coulomb clusters in traps dusty plasma crystal consisting of several tens to hundreds dust particles (white dots) (Arp, Block, Piel, Melzer, Phys. Rev. Lett. (2004)) room temperature, coupling parameter Г 1000 particles arrange themselves on nested distinct shells multi-component plasma: electrons, ions, neutral (e.g. argon) atoms, and dust grains (3.5µm sized)

13 3D Spherical Dust Crystals strongly coupled Coulomb clusters in traps N=190 dusty plasma crystal consisting of several tens to hundreds dust particles (white dots) (Arp, Block, Piel, Melzer, Phys. Rev. Lett. (2004)) room temperature, coupling parameter Г 1000 particles arrange themselves on nested distinct shells multi-component plasma: electrons, ions, neutral (e.g. argon) atoms, and dust grains (3.5µm sized)

14 Dust Crystals: Artifical Classical Atoms Strongly coupled plasma: strong influence of inter-action between particles Open Questions: - internal crystal structure, finite size effects, magnetic fields, temperature, charge fluctuations... - excitations: normal modes, non-linear effects - short time dynamcis: time-dependent crystallization MD Simulation (100 particles) - Non-Newtonian dynamic of dust in the streaming plasma - phase transitions, melting behaviour - transport in macroscopic systems: diffusion, heat conduction - and many more... Experiment:: 3D Yukawa Ball Computer Experiments : MD Simulations!

15 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

16 Classical Particle-Based 'Bottom-Up' Approaches to the Many-Body-Problem Molecular Dynamics vs. Monte Marlo Molecular dynamics: - deterministic simulation model: exploration of the (classical) position-momentum phase-space ( real many-particle dynamcis!) Metropolis Monte Carlo method: - random number based simulation model: applied to efficiently sample the high-dimensional configuration space ( equilibrium properties only, but direct implementation of T, direct extension to quantum systems)

17 Historical Review

18 Historical Review hard spheres: collisions/h prod. run coll. simulation time IBM KFLOPS, 4-32KB RAM days years

19 Molecular dynamics motion Simulations by means of first mainframe computers by B.J. Alder & T.E. Wainwright in the late 1950's quasi-long-range ordered solid phase short-range ordered fluid phase liquid-vapor region shown are the trajectories of hard sphere particles with periodic boundary conditions valueable insights concerning the collective behaviour of interacting many-body systems: phase transitions!

20 Why Computer Simulations? Molecular Dynamics simulations are a central part in the different directions in sciences due to the fact that: computer simulations are cheap and do not require extensive laboratory facilities, computer experiments can be repeated arbitrarily often and easily modified, comprehensive scans of large parameter sets and ranges can help to discover the conditions of exceptional physical phenomena or optimal settings for physical experiments, computer-simulated experiments can provide maximum information at the microscopic level and therefore help to give a deeper understanding of measurements.

21 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - in molecular dynamics the time propagation is achieved by high-precision numerical integration of the N coupled Newtonian differential equations

22 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - in molecular dynamics the time propagation is achieved by high-precision numerical integration of the N coupled Newtonian differential equations - these equations are splitted into a coupled system of first-order ODEs

23 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - in molecular dynamics the time propagation is achieved by high-precision numerical integration of the N coupled Newtonian differential equations - these equations are splitted into a coupled system of first-order ODEs - total force this formulation is equivalent to the Hamilton equations, where

24 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - once the positions and velocities of all particles are known, their dynamical propagation through the position-momentum space can be directly computed

25 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - once the positions and velocities of all particles are known, their dynamical propagation through the position-momentum space can be directly computed simplest approach: keep integrand const. in the interval Euler Method (Simplest integration scheme)

26 The Basics: Time propagation of the N-particle Hamiltonian Integration of the Equation of Motion - once the positions and velocities of all particles are known, their dynamical propagation through the position-momentum space can be directly computed simplest approach: keep integrand const. in the interval Euler Method (Simplest integration scheme) Result: trajectories in 6N dimensional phase space calculate all (dynamical) classical properties

27 The Basics: Time propagation of the N-particle Hamiltonian MD program structure initialize positions and velocities set time compute the forces acting on the particles integrate equation of motion (Euler method) increment time save data loop the timeloop

28 MD Simulations: Computer Experiments 3D Coulomb cluster of 12 classical identical particles in external parabolic potential (Γ=5000)

29 The Basics: Time propagation of the N-particle Hamiltonian Choosing the time step too small: program runs too slow (covering only small phase space) too large: instability! the choice of the integrator scheme and step size is of crucial importance!

30 Advanced Numerical Time Integration Schemes Adaptive stepsize controlled algorithm based on the fifth-order Runge-Kutta formula Requires Six (force) function evaluations to advance the solution through the interval h from xn to xn+1 xn+h Another combination of the six functions ki yields an embedded fourth-order Runge-Kutta formula ai, bij, ci and ci* are the Cash-Karp coefficients The difference between the fourth- and fifth-order accurate estimates y(x+h) gives an estimate of the local truncation error The desired degree of predetermined accuracy is achieved with minimum computational effort

31 Numerical Equipment Classical and Quantum Simulations 37nodes; in total 128 cores (most are dual/quad pcs, 3 nodes with 16 cpus) 4GB...128GB RAM per node 4.5 TB RAID-Array

32 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

33 Simulation of 3D Coulomb Clusters Hamiltonian for N identical (classical) dust particles N=190 screening independent parabolic confinement (Arp et al., Confinement of Coulomb balls, Phys. Plasmas 12, (2005)) neglect effects of the ion flow (3D dust cloud is in the plasma bulk) consider isotropic dust-dust interaction potential - Coulomb interacting particles - statically screened Coulomb potential (Yukawa potential) - linear isotropic screening Arp et al., PRL (2004) length and energy units,

34 Ground States of Spherical 3D Coulomb Clusters Goal: cluster ground states from first principle molecular dynamics simulations via simulated annealing Probability of ground and metastable states Difficulty: energetically very close metastable states 'slow' cooling N=31 'fast' cooling Realization: - systematic velocity scaling (reduction) of all particles cooling to T=0 (25,6) - optimal cooling rate (26,5) - large number of independent runs (26,5) (26,5) Finding the global minimum on the 3N dimensional potential energy surface is conceptually simple, but in practice a challenging problem! metastable states (27,4) ground state H. Kählert, P. Ludwig, H. Baumgartner, M. Bonitz, D. Block, S. Käding, A. Melzer, and A. Piel, Phys. Rev. E 78, (2008)

35 Mesoscopic 3D crystals with Coulomb potential Concentric spherical shells Mendeleyev table Closed shells (+), magic numbers: + N=12 (12,0) N=13 (12,1) [Rafac et al. '91]... + N=57 (45,12) N=58 (45,12,1) N=59 (46,12,1) + N=60 (48,12) [Tsuruta/Ichimaru '93]... N=160 (102,45,12,1) - Ludwig, Kosse, Bonitz, Structure of Spherical 3D Coulomb crystals, Phys. Rev E 71, (2005) - Arp et al., J. Phys.: Conf. Ser. 11, 234 (2005) - Bonitz et al., PRL 96, (2006) + N=154 (98,44,12) N=155 (98,44,12,1)... + N=310 (165, 94, 41,10) N=311 (166, 89, 43,12,1)

36 Comparison to Hasse & Avilov [Phys. Rev. A (1991)] N structure radius Correct shell structure and energy energy Γ

37 Experiment vs. Simulation N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD with Coulomb potential): ground state configuration: (115, 56, 18, 1) Small, but significant differences Arp et al., PRL (2004)

38 Simulation of Coulomb/Yukawa Balls Hamiltonian for N identical (classical) dust particles N=190 screening independent parabolic confinement (Arp et al., Confinement of Coulomb balls, Phys. Plasmas 12, (2005)) neglect effects of the ion flow (3D dust cloud is in the plasma bulk) fluctuations negligible (charge, dust grain size) (Baumgartner et al., Contrib. Plasma Phys. 47, 281 (2007)) consider isotropic dust-dust interaction potential - Coulomb interacting particles - statically screened Coulomb potential (Yukawa potential) - linear isotropic screening Arp et al., PRL (2004) length and energy units,

39 Screening Induced Structural Changes (Example N=25) Screening reduced repulsion - crystal compression - particles move inward - evolution towards a bulk-like closed-packed symmetry

40 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1)

41 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1) Yukawa-Potential: κ r

42 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1) Yukawa-Potential: κ r

43 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1) Yukawa-Potential: κ r

44 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1) Yukawa-Potential: κ r

45 Experiment vs. Simulation (2) N=190 Experimental data for N=190 shell configuration: (107, 60, 21, 2) Theory (MD simulation): ground state of Coulomb cluster: (115, 56, 18, 1) Yukawa-Potential: κ r Exact agreement with experimental configuration! Bonitz, Block, Arp, Golubnychiy, Baumgartner, Ludwig, Piel, Filinov, PRL 96, (2006) Other models: Totsuji model (PRE 2005): selfconfinement: cancellation of screening effects Coulomb result!

46 Experiment vs. Simulation Shell radii in units of mean interparticle distance Experiment (symbols): 43 clusters, N= Molecular dyamics with Coulomb and Yukawa potential shell 1 shell 2 shell 3 shell 4 shell 5 Bonitz et al., PRL 96, (2006) Excellent agreement without free parameters! (but means nothing...)

47 Experiment vs. Simulation (cont.) Shell populations Experiment (symbols): 43 clusters, N= shell 1 Molecular dyamics with Coulomb and Yukawa potential shell 2 shell 3 shell 4 shell 5 Bonitz et al., PRL 96, (2006) Systematic screening dependence! Non-invasive diagnostics for plasma parameter in experiment: outer (4th) shell: 3rd shell: κ =0. 62±0. 23 κ =0. 58±0. 43

48 Radial Density Profile in Spherical Trap Ground state density of a confined Yukawa plasma Crosses: 3D crystal from MD simulation, (shell averaged density) Lines: fluid model (mean field theory, LDA with correlations) 1:1 correspondence between the discrete particle model (exact correlations) and continuous density profile (fluid model) C. Henning, P. Ludwig, A. Filinov, A. Piel, and M. Bonitz, Ground state of a confined Yukawa plasma including correlation effects, Phys. Rev. E 76, (2007) Density profile of a confined Yukawa plasma is inhomogeneous! explains change of occupation numbers with κ

49 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

50 Langevin Dynamics Simulation Stochastic Langevin Equation:, accounts for friction with neutral gas (friction coefficient ν, temperature T) initial conditions: weakly correlated state at high temperature (e.g. by laser-heating)

51 Relevance of ground state configuration? Effect of finite temperatures Shell populations for different temperatures at κ=0.67 vs. Coulomb potential (T=0) 2 independent verifications: - Langevin-Equilibrium-MD - Thermodynamic Monte Carlo Excellent agreement between both methods T effect negligible compared to screening H. Baumgartner et al, Contrib. Plasma Phys. 47, (2007)

52 Relative Probability of States: Effect of Friction Probability of ground and metastable states for the N = 31 cluster. ground state configuration energetically lowest metastable state General trend: slow cooling favors the ground state (27, 4) over metastable states. H. Kählert et al., Phys. Rev. E 78, (2008) Recent experiments: D. Block et al., Physics of Plasmas 15, (2008) S. Käding et al., Physics of Plasmas 15, (2008) - cluster destroyed and created by fast switch of confinement potential - creation of metastable states

53 Molecular Dynamics Simulations Of Dust Crystals Outline of the talk: 1) Introduction: Wigner Crystals in Traps (Artificial Atoms) finite size and surface effects of mesoscopic Coulomb systems in traps 2) Basics of Molecular Dynamics Simulation of Strongly Coupled Systems historical review, numerical solution of the classical equation of motion 3) Ground States of Spherical 3D Plasma Crystals MD simulation idea, effect of screening, comparison with experiments 4) Finite Temperature Simulations of Dust Yukawa Balls occurrence of metastable states, effect of friction, melting of finite systems 5) Summary and Outlook T. Ott, P. Ludwig, H. Kählert, and M. Bonitz, Molecular dynamics simulations of dusty plasmas, Tutorial lecture in the book Introduction to Complex Plasmas, M. Bonitz, N. Horing, J. Meichsner, and P. Ludwig (eds.) Springer Series "Atomic, Optical and Plasma Physics", Springer, Berlin (2009)

54 Anisotropic and non-monotonic (wake) potentials Structure of wake potential Mach numbers _ + _ supersonic ion flow Contour plot of φij(r, z) M=1.1 M. Lampe, G. Joyce, and G. Ganguli, IEEE trans. on Plas. Science 33, 57 (2005) M=2 Te/Ti=50 ρ=50 mtorr The dust grain is located at the origin of the coordinate system; and denote the ion streaming direction. F. Jenko, G. Joyce, and H. M. Thomas, Physics of Plasmas 12, (2005)

55 How does crystallization happen in time? Cooling of a weakly coupled state to low temperatures Talk First results: outer shell builds up first Short-time behavior of spherical energy oscillations suppressed Yukawa balls by H. Kählert for high damping Time-evolution of various energy contributions Density profile of a Coulomb system with N=150 and ν/ω0=1 during the cooling process Screening dependence for ν/ω0=1 Influence of the damping coefficient for Coulomb interaction

56 Transport in Macroscopic Dusty Plasmas Anomalous diffusion in 2D Yukawa plasmas time The mean-squared displacement follows the relation friction Poster Superdiffusion in 2D dusty plasmas by T. Ott et al. this afternoon.

57 Summary: 1) Molecular Dynamics is a powerful tool to study charged many particle systems 2) 3D dusty plasma crystals have an interesting internal structure and are an unique test system for correlations effects - High sensivity of shell occupation numbers to screening (explained by inhomogeneous density profile) - Details of shell structure well reproduced by parabolic confinement and Yukawa pair potential - Effect of temperature and charge fluctuations negligible compared to screening - Friction determines the occurrence probability of ground and metastable states

58 Summary: 1) Molecular Dynamics is a powerful tool to study charged many particle systems 2) 3D dusty plasma crystals have an interesting internal structure and are an unique test system for correlations effects - High sensivity of shell occupation numbers to screening (explained by inhomogeneous density profile) - Details of shell structure well reproduced by parabolic confinement and Yukawa pair potential - Effect of temperature and charge fluctuations negligible compared to screening - Friction determines the occurrence probability of ground and metastable states Many thanks for your attention!

59 Effect of charge fluctuations Dust grain size (and charge) variations among particles experiment: highly charged particles (Z~10,000) small size fluctuations (<1%, Gauss distribution) effect of charge fluctuations on ground state negligible compared to screening Initial charges are Gaussian distributed H. Baumgartner et al, Contrib. Plasma Phys. 47, (2007)

60 _ + _ First principle Langevin Dynamics simulation N-particle Hamiltonian, Dynamic dielectric function supersonic ion flow Contour plot of φij(r, z) Outlook: Dynamical Screening and Wake Effects Non-reciprocal interactions The upstream grain exerts an attractive force on the downstream grain, while itself feeling a repulsive force from the downstream grain! M. Lampe, G. Joyce, and G. Ganguli, IEEE trans. on Plas. Science 33, 57 (2005)

61 Melting of Few-Particle Systems Relative Interparticle Distance Fluctuations (IDF) as function of temperature Strong dependence of melting temperature on precise particle number V. Golubnychiy, H. Baumgartner, M. Bonitz, A. Filinov, and H. Fehske, J. Phys. A 39, 4527 (2006) Melting: process of abrupt loss of spatial correlations, i.e., increase of relative interparticle distance fluctuations ( Lindemann -criterion) Small N: broad melting region. Results depend on computational scheme.

62 Melting of Few-Particle Systems (cont.) Top: Relative Interparticle Distance Fluctuations as function of temperature Bottom: Variance of Interparticle Distance Fluctuations J. Böning, A. Filinov, P. Ludwig, H. Baumgartner, M. Bonitz, and Yu.E. Lozovik, Phys. Rev. Lett. 100, (2008) Block averaged IDF vs. block number for N=8 charged bosons (PIMC) Melting: process of abrupt loss of spatial correlations, i.e., increase of relative interparticle distance fluctuations ( Lindemann -criterion)

63 Melting of Few-Particle Systems (cont.) Top: Relative Interparticle Distance Fluctuations as function of temperature Bottom: Variance of Interparticle Distance Fluctuations J. Böning, A. Filinov, P. Ludwig, H. Baumgartner, M. Bonitz, and Yu.E. Lozovik, Phys. Rev. Lett. 100, (2008) Melting: process of abrupt loss of spatial correlations, i.e., increase of relative interparticle distance fluctuations ( Lindemann -criterion)

64 Phase Diagram of Yukawa Balls Critical temperatures for solid-liquid transition high Tm stable cluster General Trends: - Melting occurs at lower T with increased screening - small cluster are more stable (higher Tm) (Thesis, H. Baumgartner)

65 Periodic Table of Structural Transitions in Yukawa Balls Diagram of structural transitions Ground state configurations as functions of particle number N and screening parameter к. The numbers in brackets give the configuration of the inner shells. - The number of particles in the inner shell growths with increasing N and κ. - For high values of κ the configuration converges to a close packed lattice. 2 types of anomalies in the shells filling: (1) (white circles) The inner shell loses 1 particle to the outer when N is increased by 1. (2) (black circles) Increasing κ makes 2 particles change from the outer to the inner shell at once. number of particles on inner shell H. Baumgartner, D. Asmus, V. Golubnychiy, P. Ludwig, H. Kählert, and M. Bonitz, New Journal of Physics (2008)