Electrostatic Double Layer in Dusty Plasma. Ghafran Khan, Zahid Kumail and Younas Khan Department of Physics, Kohat University of Science and Technology (KUST), Kohat, Pakistan Supervisor : Dr.Muhammad Adnan Khalil

Contents Motivation Model Equation Dispersion Relations Nonlinear Structures Conclusion

"Cairns et al. (1995) describe the distribution for non-thermal electrons to study the ion acoustic solitary structures observed by the FREJA satellite. FREJA and VIKING satellite both observe the electric field in space plasma which suggest that they are mostly likely electrostatic in nature [Dovner et al., 1994]. There are basically two types of density structure observed by these satellite. the FREJA satellite observe density known as lower-hybrid cavitons, the first observation of lower hybrid cavities in the auroral zone were reported by Vego at al [1992]. Similar structure have also been observed by VIKING satellite [Bostrom et al.,1988, Bostrom 1992] without associated lower hybrid waves. Maxwell distribution cannot gives the solution to the upper and lower cavities occur in the electrostatic waves. In 1995 R.A. Cairns propose theoretical explanation of these structure which he take to be a large amplitude of ion sound waves. He show that in the presence of a distribution of electron which is non-thermal, with an excess of energetic particles the nature of ion sound solitary structures changes and that is possible to obtain the solution with density depletions and dimensions roughly agreement with those observed freja and Viking satellite. Motivation

Cairns et al. (1995.) describe the distribution for non- thermal electrons. FREJA and VIKING satellite both observe the electric field in space plasma which suggest that they are mostly likely electrostatic in nature. FREJA satellite observe density known as lower- hybrid cavitons. VIKING satellite also observed similar structure without associated lower hybrid waves. Maxwell distribution cannot gives the solution to the upper and lower cavities occur in the electrostatic waves. Motivation GEOPHYSICAL RESEARCH LETTERS, VOL 22,NO 20, PAGES , OCTOBER 15, 1995

Maxwellian distribution In any system, a particle will have a wide range of energy, Graph shows the maxwell-boltzmann distributions having number of particles each having a particular energies. Most particles having a moderate energies Some particles having high energies A few particles have very low energies f(v) Graph rep M.D of a system

Non-thermal distribution Non-thermal distribution of electron which include the population of energetic (fast) as employed by cairns et al where is the parameter determining the number of energetic (non-thermal) electron present in model. The value of is always less then one. If is equall to zero then it recover Maxwell distribution of electron. Ref: Geophysical research letters. Vol.22.No.20.Pages October (A)(A)

Maxwell and non-thermal Graph

Comparison between Our and Samiran Ghosh work Our Work Electron, Ions, Dust plasma Ions are inertial and mobile Electron are massless and mobile Dust particle are stationary Electrostatic structure Non-thermal distribution More realistic Ref : Eur. Phys. J. Appl. Phys. 33, 199–203 (2006) Electron, Ions, Dust plasma Ions are inertial and mobile Electron are massless and mobile Dust particle are stationary Electrostatic structure Maxwellian distribution Less realistic S.Ghosh

History of Double layer Double layers were described in 1929 by the plasma pioneer and Nobel laureate Irving Langmuir. Another Nobel laureate, Hannes Alfvén, described a double layer as, “… a plasma formation by which a plasma — in the physical meaning of this word — protects itself from the environment. It is analogous to a cell wall by which a plasma. In the biological meaning of this word “protects” itself from the environment.”

Double Layer A double layer is structure in plasma consist of two parallel layer with opposite electrical charge. The sheet of charge cause strong electric field and correspondingly sharp change in voltage across the double layer. Ions and electrons which enter the double layer is accelerated, decelerated or reflected by the electric field. Double layer which may be curved rather than flat. Double layer are found in discharge tube to space plasma. Double layer are very thin, with widths ranging from a few millimetres to the thousand of kilometres.

Model Equations Three component dusty plasma

Normalized Equations Normalized Parameter t→ X →

We use the Fourier analysis for the small amplitude of wave Fourier Analysis: Reductive perturbation: method is used when one cannot deal the equations with full nonlinearity and weak nonlinearity are assumed in the system. The stretched variables in space and time are defined and slow time variations are induced by the nonlinearity of the system. Non-Linear Dynamics in Plasma Linear and Non-linear dynamic

Dispersion Relations Charge Neutrality condition Ion Temperature Ratio Dispersion for Maxwell distribution of electron by (samiran Ghosh) Maxwell distribution

Derivation of mKdV equation In order to find the nonlinear mKdV equation for one dimensional electrostatic waves in a unmagnatized dusty plasma having a non-thermal electrons, the stretching of independent variables is given by We use the standard reductive perturbation method (RPM) to solve the set of nonlinear equations.The perturbed quantities in terms of ε can be expanded. A B

Derivation of mKdV equation We introduce (A) and (B) in Normalized equation and equate the term of O(ε²), O(ε³) and O(ε ⁴ ) Similarly for Poisson equation The term is given by (C) (D) (E) (F)

The phase velocity is given by The (D) and (F) combine to give mKdV equation Dispersion in system

Solution of Double Layer (2) where Sagdeev potential V(ψ) is given by (3) (4) (5)

Solution of Double Layer Applying the first two boundary conditions of (5), we obtain : (6) Using (6), the expression of V(ψ) can be rewritten as, The double layer solution of equation (7)

where Δ is the thickness of the double layer. It should be noted from above equations (6,7) that for the existence of a double layer, we must have Solution of Double Layer as for ion acoustic wave in dusty plasma always β>0 (Eq.2). Also the nature of double layer i.e. whether the system will support a compressive or rarefactive double layer depends on the sign of the coefficient of quadratic nonlinear term α ₁. It follows from (6) that a compressive double or rarefactive double layer exists according as

Non-Thermal and maxwellian Graph

Conclusion We show that the potential difference between double layer is broader and is more realistic to the experimental observation.

Acknowledgement ….