A. Yu. Ivanov1, E. S. Dzlieva1, A. I. Eikhval’d1, V. Yu. Karasev1,2
1 Institute of Physics, St. Petersburg State University, Ulianovskaya 1, Peterhof, St. Petersburg,
198504 Russia
2 REC 13 “Plasma” , Lenina 33, Petrozavodsk, Karelia, 185640, Russia
Investigation of phase transitions in dusty-plasmas in rf-discharge was held in [1,2], where phase states
of 2D structures characterized by autocorrelational functions. Behavior of dusty structures in magnetic
fields investigated in [3-7], where results of dependence of structure rotation mode on value of
induction of applied magnetic field value were obtained. Here we present results of investigation of
phase states of dusty structures in glow discharge with magnetic field applied held by means of
correlational functions.
Experimental setup and its modifications is shown in [6,8-9]. Method and conditions of observations
are same as in [6]. We investigate two dimensional horizontal sections of dusty-plasma structures
which are perpendicular to vector of magnetic field B. In absence of magnetic field and at it’s
induction value of B0=160 gauss layers are motionless, magnetic field of different induction values
provides rotation of structure. General view of structure and sections’ positioning are illustrated at
figure 1 in [6]. Parameters of particles are: undefined (random) shape, 1 through 4 μm and density 4.6
g/sm3 (LiNbO3).
Fig 1. Image of horizontal section dusty-plasma structure in glow discharge. Vertical dimension of
structure is 8.6 mm. Conditions: Ne gas at pressure of P=0.7 torr, discharge current i=2.5 mA, particles
of LiNbO3 with dimension of 1 through 4 m and undefined shape, magnetic field equals zero.
Vertical profile of this structure is shown on figure 1 in [6].
Typical image of one of central horizontal structure sections is shown on figure 1. All the particles are
wrapped in the area of illumination, its depth is about 0.7 mm less then 2 interparticle distances in
vertical direction. Thus figure 1 shows single horizontal layer of structure. Positioning of particles is
subpixel. In the case of high rate particle rotation [6] its position was determined by averaging in time.
Experimental videomaterials are saved as videofiles each containing more then 100 frames. A film for
each of 12 values of induction of magnetic field was made at several sections of the structure. Special
software was developed for processing of images, these provided for identification of particles’
positions and building of correlation functions for phase state definition.
For the purposes of this study, it was necessary to determine and monitor the degree of order of the
structure under investigation. In studies of dusty plasma, three functions are used for this purpose: the
pair correlational function g(r) (or the pair distribution function n(r)), which determines the
translational order; the function of the orientational order (in a hexagonal structure, g6(r)); and the
structural factor S [1]. The phase state of a two-dimensional section is determined by the parameters of
the two functions g(r) and g6(r) simultaneously [10,1]. However, the use of only one function, namely,
the pair distribution function g(r), showed that the application of magnetic field causes rearrangement
of the structure (Fig. 2).

Fig. 2. The pair distribution function of particles for different values of the induction of a magnetic
field: (a) 0, (b) 28, (c) 80, (d) 136, (e) 160, (f) 250, (g) 300, and (h) 380 gauss. The function is
determined for the ( ) upper, ( ) lower, and ( , ) intermediate sections. Gas discharge conditions: Ne
under a pressure of 0.9 torr; discharge current, 2.4 mA; LiNbO3 particles.
In studies of strata in which the objects formed are inhomogeneous and have relatively small
dimensions, the function g(r), having the meaning of the density of particles at a distance r from some
selected particle, is used. It was shown in [11] that the informativeness of this function is maximal
upon processing of particles from the central area of a section (the experimental conditions of [11]
with respect to the discharge current and particle size, as well as the pressure and type of the gas, were
similar to ours). Upon construction of g(r), we used averaging over the central particles of each
section. In strong magnetic fields, g(r) was also averaged over time (over several video frames).
Knowing the function g(r), one can obtain several quantitative results. The first of them is the value of
the interparticle distance. For the upper part of the structure, it is close to 0.3 mm; for the lower part,
this distance is about 0.4 mm (Fig.2a). The interparticle distance virtually does not change in a
magnetic field Figs. 2b–2h). The second result is the occurrence of different degrees of order in layers
at different heights. Thus, in the absence of a magnetic field, the first three maxima are clearly seen in
the upper section; i.e., the long-range order is present in this case. In the remaining part of the
structure, there is only the short-range order (only the first maximum is observed, Fig. 2a). Upon
application of a magnetic field, the order in the upper sections of the structure is destroyed (Figs. 2b–
2d), whereas the degree of order in the lower part of the structure increases. For example, in Fig. 2e,
the lowest section and the section nearest to it (which is denoted by the triangles in the plot) exhibit the
second order: the ratio of the height of the peak to its width is equal to 3.5. This is the third result
obtained by us. Upon a further increase in the induction of the magnetic field (300–400 G), when the
velocities of rotation become significant, the order is destroyed and the structure becomes flat (Fig.
2g). It cannot be separated into sections, and, in its lowest part, a void forms in the center. The longrange
order is completely absent (Fig. 2h).
Further in this work by the means of developed software an analysis of optimal calculation of function
g(r). At first, a region of structure choice is made providing for particles with respect to which
calculation and averaging are to be held. Secondary, optimal calculation step is found, it’s necessary
because an image of particle is a spot of finite size of several pixels. And moreover lattice period is
different in various sections. An example of calculated function with its three-parameter
approximation [12] is shown at fig. 3a.
a) b)
Fig 3. Pair correlational g(r) and orientational correlational g6(r) functions with approximations. a)
triangles is experimental g(r), solid line – result of three-parameter approximation with parameters:
a=0.42 mm (interparticle distance), b=0.05 mm (width of approximating gaussians), ξ = 0.84 mm
(coefficient in argument of decreasing exponent –r/ξ – length of correlation). Correspondence of
approximation and experiment ascertained by root-mean-square method. b) points are the values of
experimentally built function g6(r), solid line – exponential approximation with argument –r/ξ6, ξ6 =
0.48 mm, dashed line – power law approximation with index η = -0.89.
Orientational correlational function g6(r), defined by neighbor particles, reveals degree of orientational
order of the system. By the means of software orientational function g6(r) and it’s exponential and
power law approximations for experimental video data corresponding to different sections of structures
were calculated. An example is shown at fig. 3b.
According to three-parameter approximation [12] held for g(r) (which contains factor ξ

− changes from 0.1 through 0.7 mm. Parameter η of power law
approximation of function g (r) ~ r −η 6 lies in diapason from 0.7 through 2.Hence, length of
translational order does not exceed three interparticle distances Δ (Δ = 0.35 – 0.42 mm), and length of
orientational order is equal or less then two interparticle distances in all the diapason of magnetic field
According to criteria of two-dimension melting theory [10, 1] proximity of ξ и ξ6 values, and also
η > 0.25 indicate liquid-like structure state. Magnetic field induction of 300 gauss provides ξ6 close to
0.1, probably system tends to disordered gas-like state, and at the same time the video images of
sections shows nonhomogeneous particles density. One might say, that according to theory KTHNY
[10] in experiment conditions [6] with magnetic induction value change in diapason from 0 through
400 gauss state of plasma-dusty structure changes from almost hexatic to gasous one. Certain
exceeding of ξ over ξ6 seems to be bound to inhomogeneousness.
In this paper influence of magnetic field as a method of stimulation of phase transition in dusty-plasma
is investigated. In addition to described above obtained results following may be noticed. Application
of longitudinal magnetic field at glow discharge leads to decrease of radial diffusion of plasma
particles, changes distribution of energy-release, and leads to rotating motion of dusty component. All
three listed reasons may cause phase transition of melting type in dusty-plasma structure.
V.Yu.K. acknowledges financial support from the Ministry of Education and Science of the Russian
Federation and a fellowship from the U.S. Civilian Research & Development Foundation (No. RUX0-
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