ФИЗИКА ПЛАЗМЫ, 2014, том 40, № 7, с. 676-682

УДК 533.9

ПЫЛЕВАЯ ПЛАЗМА

THERMODYNAMIC PROPERTIES OF STRONGLY COUPLED PLASMA IN PRESENCE OF EXTERNAL MAGNETIC FIELD

© 2014 г. M. Begum*, S. Baruah**, N. Das*

* Department of Physics, Tezpur University, Tezpur-784 028, Assam, India ** Institute of Plasma Research, Bhat, Gandhinagar-382428, Gujrat, India e-mail: mbegumtu@gmail.com, ndas@tezu.ernet.in Поступила в редакцию 31.07.2013 г. Окончательный вариант получен 01.11.2013 г.

Thermodynamic properties of a Yukawa system consisting of dust particles in plasma is studied in presence of an external magnetic field. It is assumed that dust particles interact with each other by modified potential in presence of magnetic field. A molecular dynamics code is developed to calculate this internal energy for the entire system. Based on the values of internal energy given by the code Helmholtz free energy and pressure are calculated for the system.

DOI: 10.7868/S0367292114070026

1. INTRODUCTION

Complex plasma consisting of electrons, ions and charged dust grains usually of micron size provides a rich field for studying some of the fundamental and interesting phenomena of nature like phase transition, interaction mechanism etc. These phenomena become even more complicated in presence of external magnetic field. The presence of magnetic field modifies interaction potential. Yaroshenko et al. [1] have elaborately discussed about various mutual dust-dust interactions in complex plasmas, including the forces due to induced magnetic and electric moments of the grains. Nambu and Nitta [2] have presented a detailed theory of the "Shukla-Nambu-Salimullah" (SNS) potential in a magnetized electron-ion plasma, which is anisotropic in comparison with the effective potential in unmagnetized plasma. In dusty plasma phase transition manifested by the appearance of solid BCC (body-centered cubic) and FCC (face-centered cubic) like crystal structure have been observed. For a certain value of screening constant k, there appears regularized pattern beyond a critical value of coulomb coupling parameter r. Thermodynamics of dusty plasma was studied extensively by Hamaguchi et al. [3-5]. Totsuji et al. [6] have calculated thermodynamic quantities of a two dimensional Yukawa system. They have found results using giant cluster expansion method in weak coupling regime and by using MD (molecular dynamics) method in strongly coupled regime. Ramazanov et al. [7] have found internal energy and excess pressure of Yukawa system using Langevin dynamics. Presence of magnetic field however alters the internal energy of the particulates in plasma and hence affects the points of phase transition from solid to liquid state. In this work we have calculated free energy and pressure of the system in presence of an exter-

nal magnetic field and on the basis of this, thermody-namic state of the system is investigated. Molecular dynamics has been widely used for understanding the phenomena of Coulomb crystallization and phase transition in strongly coupled plasma.

Thermodynamic behaviour of dusty plasma strongly depends on the nature of electrostatic interaction. Presence of magnetic field affects the dynamics of plasma particles and this leads to modification of interaction mechanism among dust particles as well as coupling of dust particles with the background plasma. In fact magnetic field may affect the charge accumulated on the grains and forces acting on them. For the operation of dusty plasma experiments in presence of magnetic field, it is important to identify suitable plasma parameters. A detailed study of thermodynamic properties and phase transition of dusty plasma in presence of magnetic field is important for the point of understanding of fundamental physics as well as for production and control of the properties of dust crystal. It should be possible to see phase transition of dust crystal just by varying the external magnetic field. Here in this manuscript, we have calculated internal energy of the system by considering the fact that interaction potential among dust grains becomes anisotropic and get modified due to the presence of magnetic field. Then thermodynamic properties like free energy, entropy, pressure etc. are evaluated for a wide range of Coulomb coupling parameter r, screening constant k and for different values of magnetic field. From our study, fitting curve for h(k,T) is established. This helps in finding values of internal energy for any values of k and r within the limitation of simulation. In our calculation, it is assumed that magnetic field does not affect charge on dust. This approximation is justified for

relatively weak magnetic field, when electron gyro radius is much smaller than the dust size [8, 9].

The results of our study may give important insight into the physics of crystallization and phase transition in presence of magnetic field. This may find application in various space and laboratory plasma situation.

2. DESCRIPTION OF THE THEORETICAL MODEL

We consider a 3D dusty plasma of identical, spherical particles of mass md and charge Q immersed in a neutralizing background plasma subjected to an external magnetic field B applied along z-direction. The dynamics of the plasma particles is affected by the magnetic field and as a result the interaction potential among dust particles become anisotropic in presence of the field. In the xy -plane, perpendicular to the magnetic field, the interaction potential around a test dust particle with a charge Qd is [10, 11] taken as

= .Qd , exp I ~ 4neorf ^ p

(1)

where f = (1 + f) and f = (aP/aci), ®pt and beings the ion plasma and ion gyro-frequencies respectively, ps ~4fikDe = (Cs/©„■) is the ion-acoustic gyro-radius, XDe is the electron Debye radius, and Cs = ^De®pi is the ion-acoustic speed. The interaction potential has been taken as normal Debye—Huckel along z-direction.

The thermodynamic properties of dusty plasma is characterized by two important parameters: Coulomb coupling parameter r and screening constant k defined as

r =

ryl 2

Z e

4neoaKBTd'

K

(2)

(3)

r =r =

Qd

f 4ne oaKBTdfl

K P s f De

3. THERMODYNAMIC PROPERTIES

In this section we develop a model to investigate thermodynamic properties of strongly coupled dusty plasma column system in presence of an external magnetic field. The Hamiltonian for a system of N-parti-cles may be written as

N I |2

H = V + u,

^ 2m

j = i

(4)

where pj is the momentum of the jth particle and u is the excess energy of the dust particles and background plasma particles. Following the procedure of Hamaguchi et al. [3] and using equation (2) as the interaction potential, excess energy Uex may be calculated as

Ue-

=1 - rj) -

NQln NQlkD

' * j

Idok,

8ne0

Q¡N 8ns 0

y exp(-kDL\n\)

\n\L '

w ■+ n II

(5)

where, a = (3/4nnd ) is the mean inter-particle distance, Td is the temperature of the dust grains, nd is the dust number density. Depending on the values of r or k, dusty plasma may come into strongly coupled regime. The presence of magnetic field affects the shielding of dust particles and this indirectly leads to modification of the two controlling parameters r and k to r m and km respectively, where

where kD = 1/^ D. In equation (5), the first term on the right-hand side represents the interaction potential between dust grains, the second term represents the free energy of the background plasma that, on average, neutralizes the charge on the particulates. The third term represents the free energy of each sheath, and the fourth term represents the energy of interaction of every particulates and its own images under periodic boundary conditions. In presented problem, the space, mass, time, velocity, energy, and external magnetic field

strength are normalized by XD, md, \](mdX2D)/(KBTd),

4(md /KBTd), KBTd, and 7^/4^60).

In presence of magnetic field equation (5) is modified due to the Yukawa type of interaction potential given by equation (1).

The thermal component of potential energy is defined as

u,H(K r) = U(k, r) - U„(k), (6)

where u„(k) is the Madelung energy per particle in units of KBTd. Madelung energy in units of r can be written as

u(k,r)

E(k) = lim

r-

(7)

For a system of N negatively charged dust particles at temperature T in a volume V, Helmholtz free energy per particle in units of KBTd can be expressed as

F

f =

NKBTd

and the pressure is

P =

PV

NKBTd

(8)

(9)

a

Free energy and pressure can be determined [4] as a function of (k, r) by using the following relations

f = u(k,r)

dr r '

Further, entropy can be obtained by using the same relation

(10)

* = -f + .

dr

(11)

Free energy for fluid and solid state [4] may be calculated with the help of equation (11) in following way

ffluid(K r) = foi(k) + fr(k) + fideal(r),

(12)

where

foi = J

u(k, r ')

r '

d r ',

fir = J

u(K,r ') r '

d r '

and fideai = 3lnr + |ln(kT)Ry -1 + lnA

The total free energy in solid state may be written as

fsolid(K,r) = fanharm(K,r) + fharm((K^\ (13)

where the anharmonic component of potential energy in units of KBTd is

r

fanharm(K r) J"

uth«r) - ^

dr ' r ' '

d \

mda-± = Fj(t) dt

(14)

where F,(t) = -Qd Z V^j) + Qvi(t) x B for i = 1, 2, 3,

..., N and j # i. Here md is mass of the dust grain, rt is

the position of the grain i, F{ is the force acting on the

ith particle and O ij represents Debye-Huckel type of interaction potential. For our MD simulation we have chosen the values of dust density, ion density, dust radius,magnetic field etc. that are consistent with the data of typical laboratory plasma produced by radio-frequency discharge [13]. We have taken dust grain

mass md = 4.0 x 10 15 Kg, ion mass m{ = 1.6726 x 10 Kg, dust density nd = 3.74 x 1010 m-3, ion density n = 1.0 x 1014 m-3, dust charge qd = 1.77952 x 10-16 C

and dust grain radius rd = 2.0 x 10-6 m. The normalized size of simulation box is taken equal 9.792616. The simulation is performed with 686 particles for BCC crys

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