1856-5 007 Summer College on Plasma Physics 30 July - 4 August, 007 The Structure Formation of Coulomb Clusters O. Ishihara Yokohama National University Yokoama, Japan
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Summer College on Plasma Physics Abdus Salam International Centre for Theoretical Physics (ASICTP) Trieste, Italy August 0 August 4, 007 The Structure Formation of Coulomb Clusters Osamu Ishihara Yokohama National University Yokohama, JAPAN T. Kamimura Meijo University Nagoya, Japan Collaborators: Yoshiharu Nakamura Takashi Yamanouchi Chikara Kojima Yuta Suga YNU YNU YNU Meijo U
Configurations of Coulomb Clusters in Complex Plasma 1. Introduction -historical survey: crystal structure. Dusts confined in a plasma 3. Hamiltonian of our system 4. Spherically symmetric potential CME (configuration of minimum energy) Shell structure 5. Non-spherical potential D flat structure to spindle-like structure 6. Lattice oscillation 7. Conclusions 3
1. Introduction Historical background on crystal structures D. I. Mendeleev 1869 Periodic table of elements J. J. Thomson 1883,1904, 191 Atomic system - as a stable arrangement of a mixture of a positive charge and a number of electrons. E. Madelung 1918 Ionic crystals based on the electrostatic energy E. Wigner 1934 Lattice structure of electrons in a metal as a result of the Coulomb forces acting among electrons. 4
1. Introduction -historical survey: crystal structure Periodic Table of elements Mendeleev 1869-63 elements Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ H Representative element Li Be B C N O F 1-1 Na Mg Al Si P S Cl K Ca 1 Ti V Cr Mn Fe, Co, Ni, Cu -3 (Cu) Zn 3 As Se Br Ru, Rh, Pb, Ag 4 Rb Sr (Yt) Zr Nb Mo 4 3-5 (Ag) Cd In Sn Sb Te I Os, Ir, Pt, Au 6 Cs Ba - Ce - - - 4-7 - - 5 - - - - 8 - - - - Ta W - 5-9 (Au) Hg Tl Pb Bi - 10 - - Tn Ur - 5
1. Introduction -historical survey: crystal structure Periodic Table of Elements Z element No. of electrons K L M N 1 H 1 He Electron configuration 1s 1 1s 3 4 5 6 7 8 9 10 Li Be B C N O F Ne [He] 1 1 3 4 5 6 [He]s 1 s s p 1 s p s p 3 s p 4 s p 5 s p 6 11 1 13 14 15 16 17 18 Na Mg Al Si P S Cl Ar [Ne] 1 1 3 4 5 6 [Ne]3s 1 3s 3s 3p 1 3s 3p 3s 3p 3 3s 3p 4 3s 3p 5 3s 3p 6 19 0 1 3 4 K Ca Sc Ti V Cr [Ar] 1 1 3 5 1 [Ar]4s 1 4s 3d 1 4s 3d 4s 3d 3 4s 3d 5 4s 1 3d 1 4s 1 3d 5 4s1 5 6 7 8 9 30 Mn Fe Co Ni Cu Zn 5 6 7 8 1 3 3d 5 4s 3d 6 4s 3d 7 4s 3d 8 4s 3d 10 4s 1 3d 10 4s 3d 10 4s1 31 3 33 34 35 36 Ga Ge As Se Br Kr 10 1 10 10 3 10 4 10 5 10 6 3d 10 4s 4p 1 3d 10 4s 4p 3d 10 4s 4p 3 3d 10 4s 4p 4 3d 10 4s 4p 5 3d 10 4s 4p 6 6
1. Ishihara,. Phys. 3. Ishihara, 4. Physica 1. Introduction -historical survey: crystal structure J.J. Thomson 1898 First atomic model Plum-pudding model (raison muffin model) -atom is considered of a positive matrix with negatively charged electrons floating inside ~ q 1 r 0 F r r O., 1998, Polygon structures of plasma crystals, Plasmas 5, 357-364. O., 1998, Plasma crystals structure and stability, Scripta T75, 79-83. 7
Study of structure of charged particles In a neutral plasma, available free energy always drives a plasma unstable and recombination makes a plasma hard to be confined by static electric and magnetic fields. 1970s: Nonneutral plasmas: a group of particles with a single sign of charge can be confined by static electric and magnetic fields - a pure electron plasma. The electron-plasma oscillations observed in a plasma involve only the motion of electrons and no ions are involved in the oscillations. The electron oscillations also appear in a group of electrons without ions if they are trapped electromagnetically. Positive ions in a plasma play a role to push displaced electrons back to equilibrium positions, while electric or magnetic potential keeps electrons in a confined space. Rahman, A. and Schiffer, J.P., 1986, Structure of a one-component plasma in an external field: A molecular-dynamics study of particle arrangement in a heavy-ion storage ring, Phys. Rev. Lett. 57, 1133-1136. Dubin, H.E. and O Neil, T.M., 1999, Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states), Rev. Mod. Physics 71, 87-17. 8
Study of structure of charged particles (continued) 1978 Laser cooling technique was successfully used to observe the cooling of Mg+ ions at 40 K in a Penning trap in which a confining magnetic field serves as a neutralizing background to charged particles, and the cooling of Ba+ ions in a Paul trap in which an RF electric field confines ions. 1979: Electrons on the surface of liquid helium were observed to form Wigner crystals Rafac, R., Schiffer, J.P., Hangst, J. S., Dubin, D. H.E. and Wales, D.J., 1991, Stable configurations of confined cold ionic systems, Proc. Natl. Acad. Sci. USA 88, 483-486. 9
Complex Plasma (Dusty Plasma) Dust particles are charged in a plasma. The plasma effectively confine charged dust particles. Dust particles can be seen by naked eyes. Complex plasma provides a novel tool to study formation of structures of charged particles. 10
. Dusts confined in a plasma Complex plasma (Dusty plasma) Ions, electrons, neutrals _ + + _ + + + + + _ + + + + + _ + + _ + _ + + _ + _ + _ + + + _ + _ + + + _ + _ Normalized charge Havnes ordering parameter Coupling parameter Plasma z P = Γ = Z n = d d n Z e d Ze 4πε dk T 0 d 0 B e d 4πε ak T B e exp( κ ) Dusts, Fine Particles Z=~4 P=10-4 ~10 - Γ<<1, Γ>>1 11
. Dusts confined in a plasma ρ n Dusts in a Plasma ( x) Zen( x) zen( x) d = + d d s s s= i, e N 1 x x x N dust particle 4 ( ) = δ ( ) j a π a j= 1 H int N j= 1 d ( ) j = Z eφ x, φ ( x) = ρ ( x) ε 0. Dust charge Q = Z e= 4πε aφ Dust radius = a Dust surface potential = φ d 0 0 0 1
. Dusts confined in a plasma Dust-Plasma Interaction N r x 3 ns () r e d b = d s s i= 1 4πε0 r xi H Z z e d r n s = N Z z e n 4πε d s s 3 dr k D r x i e r x s i= 1 0 i N 3 Zzen d s s 1 3 = d x ns V x V ( ) N H d b= Φeff xi i= 1 ( ) + 4πε kt k D i 3 dr ( φ φ( r) ) s i= 1 0 B s i e r x n () r e Z z e n e Φ eff ( xi ) = Zze d s dr dr 4πε r x 4πε r x s k r x k r x 3 s d s s 3 D i D i 0 i s 0 i k D r x i 13
. Dusts confined in a plasma Effective Confining Potential Φ ( x ) = eff i s Zzen 4πε d s s 0 3 dr e r x k D r x i i 1 4π = r x + i r Y < = 0 m= 1r> * lm ( θ ', φ') Y ( θ, φ), lm V = 4π R 3 3 For x < R, i Zen 1 d d kd xi kdr eff ( xi) 1 1 D i D i D ε 0k x x D 3 ( ) ( ) Φ = e + k + k e + k R For xi > R, Φ ( x ) = eff i 3 Zen d d R e ε 0 3 x i k D x i 14
. Dusts confined in a plasma Parabolic potential near the plasma center. R = k R D ( ) ε / /, Φ=Φeff nd Zde 0k D Hamaguchi et al, 1994, J. Chem. Phys;Totsuji et al, 005, Phys. Rev. E. Ishihara, 007, J. Phys. D: Appl. Phys. 15
. Dusts confined in a plasma Dust particles can be trapped by the background plasma Dust particles, illuminated by green laser, floating in a helium plasma (YD1, photo by C. Kojima) Dust particles, illuminated by red laser, floating in a helium plasma (YD1, photo by M. Kugue) 16
Moving dust particles in argon plasma in YCOPEX (photo by Y. Nakamura) 17
3. Hamiltonian of our system Hamiltonian for our numerical study H F = k r Φ ~ ax + by + cz k k = mω = αmω ( α 0) 1 Φ ( r ) = mω r ( 1 α) z i i i N = Φ ( x ) + in x-y plane Q i i 4πε0 i, j xi xj in z direction. N ( j> i) 1 1 Q = Zde, Φ = mω0 ( xi + yi + zi ) κ 1 z x y 18
3. Hamiltonian of our system (Q 4 πε mω ) 1 3 0 0 0 E N N 1 1 Q 1 ω0 ( i i i ) i κ 4πε0 i, j i j x x H= m x + y + z + mω / 0 0 0 ( j> i) H = x + y z + + 1 1 N N N i i i i= 1 κ i= 1 i= 1 i j x x H = H( κ, N) Κ=elongation parameter N=number of dust particles 19
3. Hamiltonian of our system Numerical solution d ri dri =, ( 1~ ) ih ν i= N dt dt 0
4. Spherically symmetric potential Stable structure and metastable structure (κ=1) Stable structure H :total energy Metastable structure 9(9,0) 9(8,1) N :H for a shell, :H for a shell with a dust particle at the center :H for the outer cluster, :H for the inner cluster 1
4. Spherically symmetric potential κ = 1 Shell structure (onion-like structure) T. Yamanouchi, M. Shindo, O. Ishihara and T. Kamimura, Thin Solid Films 506-507, 64-646 (006). T. Yamanouchi(006)
4. Spherically symmetric potential Nn Configuration, total energy and energy state 1 (1,0) 0.0 S H=total energy S : Stable M : Metastable 3 4 5 6 7 8 9 (n 1,n ) (,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0) (9,0) (8,1) 1.5 H 3.93111 7.1433 11.594 15.943 1.4493 7.5478 34.8804 34.5874 1 (10,0) 41.6499 0 (9,1) 41.86979 1 (11,0) 49.64603 1 (10,1) 49.75467 1 (1,0) 58.0676 (11,1) 58.5173 S/M S S S S S S S S M S M S M S M 1 (1,1) 67.16838 S 3 (13,0) 67.3417 M 1 4 (13,1) 76.808 (14,0) 76.85183 1 (14,1) 86.88141 5 (15,0) 87.01687 1 (15,1) 97.49474 6 (16,0) 97.68994 1 (16,1) 108.6064 7 (17,0) 108.8744 1 (17,1) 10.189 8 (18,0) 10.5539 1 (18,1) 13.3188 9 (19,0) 13.7687 S M S M S M S M S M S M 0 1 3 (19,1) (18,) (0,0) (0,1) (19,) (1,0) (1,1) (0,) (,0) (1,) (,1) (0,3) (3,0) 144.9436 145.089 145.3899 157.9699 158.068 158.59 171.5003 171.535 17.109 185.46 185.4771 185.5668 186.033 Compiled by T. Yamanouchi S M M S M M S M M S M M M 3
4. Spherically symmetric potential: CME (configuration of minimum energy) Coulomb Crystal CME (Configuration of Minimum Energy) State zi Φ = xi + yi + ( κ = 1) κ n=3 n=4 n=5 n=8 Triangle --------------------- Polyhedron ------------------------ 4
4. Spherically symmetric potential: CME (configuration of minimum energy) Fundamental CME (unit) configurations in a spherical harmonic potential (κ=1) N=4 N=6 N=8 5
4. Spherically symmetric potential: CME (configuration of minimum energy) Tetrahedron (N=4),Octahedron(N=6),--- N=4: A tetrahedron. two sets of particles at right angles to each other - projection -a square. N=6: An octahedron. two sets of 3 particles, forming an equilateral (regular) triangle each, twisted by the angle 60 degrees - a regular hexagon N=8: two sets of 4 particles, twisted by an angle 45 degrees - projection - a regular octagon. The essential features of the configurations for N=4, 6 and 8 are the fact that there are two sets of separated configurations in parallel plane and they make angles each other by the amount θ = π N θ = π = π, π = π, π = π 4 6 3 8 4 6
Non-spherical potential z y i i i Φ = x + + κ κ << 1, κ >> 1 7
5. Non-spherical potential : D flat structure to spindle-like structure N=3 N=4 κ = 0.5 κ = 0.5 κ = 0.5 κ = 1.0 κ = 1.4 κ = 1.0 κ =.4 κ = 4. κ =.4 8
5. Non-spherical potential : D flat structure to spindle-like structure Non-spherical potential (κ<<1, κ>>1,) N=4 κ < 0.68 i i yi Φ = x + + z κ 1.67 < κ <.86 0.68 < κ < 1.5.86 < κ < 4.17 1.5 < κ < 1.67 κ >4. 17 9
5. Non-spherical potential : D flat structure to spindle-like structure Total potential energy does not change substantially, while the structure changes drastically. To identify quantitatively the structural change, we separate potentials into radial component and vertical component: Φ =Φ +Φ N R Z ( ) ( r z x y ) Φ = = + R i i i i i i z N i Φ = Z 1 i α N 30
5. Non-spherical potential : D flat structure to spindle-like structure N=4 3 Φ Φ +Φ R Z Φ Z 1 Φ R 0 0 1 3 4 5 κ 31
5. Non-spherical potential : D flat structure to spindle-like structure N=4 TETRAHEDRON RHOMBUS CHAIN κ 3
5. Non-spherical potential : D flat structure to spindle-like structure Force balance determines the length of sides. N=4 f + f = f cos ( π /4) atr 1 f1 = 1/b f 1 = 1/ b fatr = ( b/ ) f1 f ( ) 1/3 b = 4 + / 4 = 1.106 33
5. Non-spherical potential : D flat structure to spindle-like structure 1 1 1 H= x + y + z + H = N N N i i i i κ i j i j r r 0 (1) For κ < ( 4 + )/ 8 = 0.6768, N=4 Φ = + = R Φ Z = () For 0.68 < κ < 1.5 (3) For 1.5 < κ < 1.68 /3 (1 ).447 0 /3 κ Φ R = κ 1 1/3 κ 1 /3 Φ Z = 1 ( κ 1) 4. 4/3 /3( ) /3 Φ R = 5 κ κ ( ) ( ) 1/3 /3 Φ Z = κ κ 34
5. Non-spherical potential : D flat structure to spindle-like structure N=4 κ 35
(1) N=6, κ=4 () N=8, κ=.0 (A) (B) 4 y x Twisted by 90 z x (A) (B) 6 Twisted by 60 (A) (B) (3) N=10, κ=01.49 (4) N=6, κ=3.50 8 Twisted by 45 4(6 x4) 36
N = 6, κ = 4 N = 8, κ = N = 10, κ = 1.5 N = 6, κ = 3.5 37
N = 10, κ = 1.5 38
N =, κ = 8.3 39
N =, κ = 8.3 40
N = 6, κ = 3.5 41
N =, κ = 46 4
N = 4, κ = 46 43
N = 4, κ = 46 44
CHANGING THE ELONGATION PARAMETER N = 4, κ = 97 For κ=97 a straight chain. 45
N =, κ = 46 8 46
N = 3, κ = 16 5 47
N = 4, κ = 49 40 48
5. Lattice Oscillation Lattice oscillation δx j X δz j j-1 j j+1 Z d r / λd E( r)= - φ( r)=e r 1+ e r λd Q E( r + δr) E( r) + δr E( r) r longitudinal r transverse φ( r)= Q r λ e r / D Fr ()= κδr O. Ishihara and S.V.Vladimirov, Phys. Rev. E 57, 339 (1998). 49
5. Lattice Oscillation Longitudinal Oscillation d j-1 j j+1 δz j-1 δz j δz j+1 Z [ ( )] δz j = δz0exp i ωt jkd E E QE( r j ) = Q ( δzj -δz j-1) (Z δ j+1 -Z) δ j r d r m d δ Z j d = ( ) Q E δzj δzj δzj dt r 1 + 1 d ω Q d d d kd = 1+ + exp sin 3 md d λd λd λd Q kd ω sin λd > 3 md d d 50
5. Lattice Oscillation Transverse Oscillation QE(r j,j+1 ) QE(r j,j-1 ) r j,j-1 r j,j+1 δx j δx j-1 θ j-1 θ j+1 δx j+1 Er ( ) Er ( ) Ed ( ) j, j 1 j, j+ 1 [ ( )] δx j = δx0exp i ωt jkd QE( r j ) = Q E j-1 j j+1 X d δx j δx j 1 δx j δx j+ 1 [ ( r )cosθ E( r )cosθ ] = Q E( r ) E( r QE( r ) = QE( d) j j, j 1 j 1 j, j+ 1 j+ 1 j, j 1 j, j+ 1) ω ω 1 j+ 1 δxj δx j δx d κ 4Q d d kd = 1+ exp sin 3 md mdd λd λd κ 4Q kd sin λ > 3 D md mdd d d 51
5. Lattice Oscillation Lattice Oscillation ω = a L ( d ) sin kd, a L ( d ) = Q m d d 3 1 + d + d λ D λ D exp d λ D κ kd ω = a ( d) sin, T m d a T ( Q d d )= 4 d exp md + 1 3 λ λ d D D Plasma parameters Te =1 ev ne= 10 9 cm -3 (λ D =0.3 mm) Dust particle a (radius)=1 µm ρ (density) =1 g/cm -3 Q (charge) =-4,000 e d (interparticle distance) = 0.3mm ---> ω/π = 11 Hz longitudinal frequency 5
Conclusions Complex plasma provides a good tool to study structures of charged particles. We studied the CME configurations of dust particles in a plasma Elongation parameter κ controls the con- figuration from flat D to spindle-like like structure. There are fundamental structures with small numbers of particles. For large κ,, we identify that the lattice oscillation (longitudinal /transverse) could be simulated by varying the parameter κ. REF: O. Ishihara, J. Phys. D: Appl. Phys. 40, R11-R147(007). 53