Under the influence of a background plasma microparticles introduced into the plasma can selforganize in various ways. Differences in selforganization are induced by interparticle interaction and mobility. When dust particles have high kinetic energies, they are in a gaseous state. By ”cooling” it down, the microparticles system starts to self-arange and liquify until the particles crystalize.
In the gaseous state particles have higher kinetic energies and lower level of order. When liquifying the particle cloud, electrostatic interactions overcome their kinetic energy and particles arrange towards a hcp lattice structures while more crystalized structures arrange in f cclattice types.
The structural analysis plays a very important role in the characterization of complex plasmas. There are many examples where changing the self-organization of microstructures influence mechanical and optical properties of materials. By detailed reconstruction of
2.4 Structures 21 particles positions and their correlation with neighboring particles the local order can be evaluated observing the coupling parameter, pair correlation function or by comparison with different ideal lattice types.
2.4.1
Coupling parameter
Γ
One of the fundamental parameters of large systems of dust particles is the coupling pa- rameter Γ. It is the ratio of the potential energy of the interaction between two particles to their kinetic energy. For a simple Coulomb interaction between the particles,
Γ = Z
2e2
dTd (2.32)
whered∼=n−d1/3 is the characteristic average interparticle spacing with number density nd
of the particles and Td characterizes their kinetic energy.
For the Yukawa interaction the screening of the particles by the surrounding plasma modifies the coupling ratio by a factor exp(−k) wherek=d/λdis the structure (or lattice) parameter - the interparticle spacing normalized by the effective screening length, leading to
Γ = Z
2e2
dTd exp(−d/λd) (2.33)
Depending on the value of Γ, the system can be strongly or weakly coupled. Usually there are strongly coupled systems if Γ≥1.
There are plenty numerical works devoted to the resolving of the phase of the systems characterizing it with two independent dimensionless parameters Γ and k =d/λd. Differ- ent phases and possible lattice types are presented in (Γ, k) plane summarizing available numerical results. The phase diagram of the Debye-H¨uckel system is presented in figure 2.2 obtained by numerical modeling [101]. Three different phases were found depending on the values of coupling and structure parameters. For Γ above the melting curve two solid phases are found, bccfor small k and f cc for biggerk and fluid phase for Γ below melting curve.
Another interesting correlation can be made from the number of particles with 6 neigh- bors and the coupling parameter. By calculating the percentage of particles with 6 nearest neighborsN6 fromN particles in the system preliminary estimations of coupling parameter can be made [102]. Figure 2.3 shows the dependence of N6/N on Γ for different types of confinement from [102].
0 2 4 6 8 10 102 103 104 solid (fcc) solid (bcc) fluid
!
"
Figure 2.2: Phase diagram of Debye-H¨uckel systems, obtained from numerical modelling.
Figure 2.3: Relative number of microparticles N6/N that have six neighbors (6-fold cells) vs. the coupling parameter Γ for different types of confinements presented in insets (a) for potential-well and (b) for parabolic confinement.
2.4.2
Pair correlation function - g(r)
The structural analysis of the systems of particles in the plasma starts with a studies of single layers of particles. In this case fast and simple information of the interparticle separation could be extracted by calculating the pair correlation functiong(r).
2.4 Structures 23 by distance r, and it measures the translational order in the system.
g(r) = 1 N N & i=0 Nj & j=0 δ|ri−rj|/Nj (2.34)
where ri and rj are the coordinates of two particles separated by distance r+ ∆r, ∆r is a bin thickness andNj is the number of particles in the bin. An example ofg(r) for different
Γ is presented in figure 2.4. The position of the first peak corresponds to the particle separation.
There are many publications where pair correlation function is used for structural char- acterization of the system [103], [104], [105], [102].
r/a g(r) 0 0.5 1 1.5 2 0 0.5 1 1.5 !!" r/a g(r) 0 0.5 1 1.5 2 0 0.5 1 1.5 !!" r/a g(r) 0 0.5 1 1.5 2 0 0.5 1 1.5 !!"#
Figure 2.4: Pair correlation factor, g(r), for different coupling parameters.
2.4.3
Local Order Analysis
For a detailed characterization of the system in an ordered state, a much more precise method for the structural analysis should be used. In a liquid or crystal phase, particles have well defined positions creating different lattice types. In these states the method should evaluate the position of each particle with respect to its nearest neighbors and compare this results with ideal lattice types.
To determine the local order of the particles a bond order parameter method is used [106]. In the framework of this method, the local rotational invariants for each particle are calculated and compared with those for perfect lattice types like f cc/hcp/bcc/ico (see Fig. 2.5).
Figure 2.5: Different types of lattice structures commonly found in complex plasma sys- tems.
each particle i by usingM nearest neighborsNb(i):
ql(i) = 4π (2l+ 1) m=l& m=−l | qlm(i)|2 1/2 (2.35) wl(i) = & m1,m2,m3 m1+m2+m3=0 l l l m1 m2 m3 qlm1(i)qlm2(i)qlm3(i), (2.36) where qlm(i) = 1 Nb(i) N&b(i) j=1 Ylm(rij) (2.37)
and Ylm - are the spherical harmonics, rij = ri − rj, where ri are the coordinates of
i-th particle. In equation (2.36)
l l l
m1 m2 m3
are the Wigner 3j-symbols, and the summation in the latter expression is performed over all indices mi =−l, ..., l, that satisfy the condition: m1 +m2+m3 = 0.
It is important to stress that each lattice type has its own unique set ofql wlrotational invariants. It give us a possibility to identify observed lattice types, by comparing the observed ql, wl values with those qid
l , widl for a perfect lattice.
To define the local order around a particle we usedq4,q6,w4 rotational invariants. The rotational invariants can easily be calculated for perfect fcc/hcp/ico/bcc. For fcc/hcp/ico the number of nearest neighbors Nb = 12 and we have for fcc: qfcc
4 = 0.1909, qfcc6 = 0.5745, wfcc 4 = −0.1593 w6fcc = −0.01316; for hcp: q hcp 4 = 0.0972, q hcp 6 = 0.4847, w hcp 4 = 0.1341, whcp6 = −0.01244 and for icosahedral lattice type (ico): q4ico = 0, qico6 = 0.6633, w4ico =
−0.1593,wico
2.5 Optical emission spectroscopy 25