Nonlinear electrostatic structures in unmagnetized pair-ion (fullerene) plasmas

Transcription

1 Nonlinear electrostatic structures in unmagnetized pair-ion (fullerene) plasmas S. Mahmood Theoretical Plasma Physics Division, PINSTECH Islamabad Collaborators: H. Saleem National Center for Physics, QAU Campus Islamabad and W. Masood, H. Ur-Rehman and N. Imtiaz Theoretical Plasma Physics Division, PINSTECH Islamabad nd International Scientific Spring Conference, March -6, NCP Islamabad

2 Outline of presentation. Introduction of soliton. Brief introduction of pair-ion (fullerene) plasmas 3. Model for electrostatic waves in pair-ion plasmas 4. KdV solitons 5. Double layer structures in pair-ion plasmas 6. Kadomstev-Petviashvili-Burgers (KPB) equation is obtained for solitons and shocks in PI plasmas 7. Conclusion

3 . Soliton Soliton is a nonlinear structure (bell shaped) which conserves its shape and interacts with surroundings or other solitons as an independent particle. Solitons are robust against perturbations. Soliton was first noticed in the month of August 834 by the Scottish engineer John Scott Russell (88-88), while conducting experiments to determine the most efficient design for canal boats along the Union canal linking Edinburg with Glasgow. In 885, Korteweg and de Vries derived a model equation (known as KdV equation) for the dynamics of nonlinear wave as follows, 3 u u u au b 3 t x x Here a and b are non zero constants, which depends on fluid properties. The KdV equation posses a soliton (or solitray wave) solution. 3

4 Soliton (contd.) Fig..: Soliton solutions of stationary KdV equation. The speed of soliton depends on its amplitude. 4

5 Soliton in Plasmas (contd.) Linear waves in v rplasmas are studied by assuming a harmonic wave i( k. ωt ) solution ~ e in the small amplitude limit. When the wave amplitude is sufficiently large the nonlinearities cannot be ignored. Nonlinearities in plasmas enter through Harmonic generation (involving fluid convection), Nonlinear Lorentz force, Ponderomotive force, trapping of the particles in the wave potential etc. Importance in Plasmas The study of soliton is important to understand the particle or energy transport mechanisms in plasmas. 5

6 Nonlinear Plasma Dynamics When wave amplitude grows to such high values that linear perturbation theory cannot be applied to describe the interactions in plasmas. Then instead of applying Fourier analysis, we transform the original set of nonlinear equations, that describe the wave dynamics, into lowest order nonlinear equations (such as KdV equation, Burgers equation and NLS equation) whose properties are well known. For arbitrary amplitude waves, one has to solve the nonlinear differential equations with full nonlinear glory and difficulties. This method has limitations and cannot be used in general and applicable to only some specific problems. 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. For example, the KdV equation is obtained for low amplitude nonlinear wave with 6 appropriate scaling of space and time variables.

7 . Brief Introduction of pair-ion Plasmas In usual electron-ion plasmas, asymmetry in collective phenomenon occur due to large difference of mass between electrons and ions. However, pair plasmas (e-p) consists of positively and negatively charged particles of same mass to keep the space, time symmetry because the mobility of equal mass particles is same in the electromagnetic field. Pair plasma consists of positrons and electrons have been produced in laboratory experiments. But the identification of collective modes is very difficult because the annihilation time is short compared with the plasma period. Therefore, attention is concentrated on the stable generation of a pair-ion plasma in laboratory consisting of positive and negative ions with an equal mass for collective modes identification. It has been found that fullerene (C 6 ) can be adopted as an ion source for pair-ion plasma production, based on the fact that interaction of electrons with fullerenes lead to the production of both negative and positive ions. 7

8 A fullerene is any molecule composed entirely of carbon, in the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are also called buckyballs, and cylindrical ones are called carbon nano-tubes or buckytubes. The discovery of fullerenes greatly expanded the number of known carbon allotropes, which until recently were limited to graphite, diamond, and amorphous 8 carbon such as soot and charcoal.

9 Production of pair-ion (fullerene) Plasmas Chamber Length6 cm, diameter5.7 cm, B.3 T (solenoid coils), gas pressure x -4 Pa E e - 5 ev (electron beam energy) The production process of pair-ion plasma is divided into three regions: Region I: Consists of electron beam region and fullerene ion production region. Positive ions C 6 are produced by electron impact ionization and low energy electrons are produced in connection with this process. Negative ions C 6 are produced by attachment of these low energy electrons produced during the charging process of positive ions. C 6 e C6 e, 4 ( C6) 5 cm C6 e C, 4 6 σ ( C6) cm σ at ev at 5eV 9

10 Production of pair-ion (fullerene) Plasmas (contd.) Region II: Electrons and ions are rapidly separated by a magnetic filtering effect and only and are expected to exist in midmost of the cylinder. C 6 C 6 Charge particle gyroradii ~(mass) / /charge C 6 C 6 (for same thermal velocity in the perpendicular direction of B) Region III: Electron free pair-ion (, ) plasma generated is attained here and plasma parameters are measured by Langmuir probes. The typical values of pair-ion plasma density ~ x 8 cm -3 at E e ev The temperature of the positively and negatively charged fullerene ions lies in the range.3-.5 ev. [W. Oohara, Y. Kuwabara and R. Hatakeyama, PRE 75, 5643 (7)] The temperature of positive ions is found to be different from negative fullerene ions because the charging processes are quite different for both the plasma species.

11 Electrostatic Waves in pair-ion Plasmas W. Ohara, D. Date and H. Hatakeyama [PRL 95,753 (5)], firstly reported the electrostatic waves in paired fullerene-ion plasmas. Three types of electrostatic waves can propagate parallel to the magnetic field in pair-ion plasmas i.e., γt. Ion acoustic wave (IAW) ω C s k, C s (acoustic speed) m. Intermediate frequency wave (IFW) 3. Ion plasma wave (IPW) ω C ω The IFW has the features that group velocity is negative but the phase velocity is positive i.e., the mode resembles as a backward wave. s k p

12 3. Nonlinear Electrostatic Waves in pair-ion Plasmas Model and nonlinear set of equations Consider homogeneous, unmagnetized fullerene (C 6 ) plasma consists of positively and negatively charged ions. The positive and negative ions are considered to be isothermally heated but have different temperatures. Using two fluid theory, the Continuity equations of positive and negative ions are given as n t ( nv ) x () n t ( nv ) x ()

13 Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.) Momentum equations for positively and negatively charged ions are described as, (where φ is electrostatic potential) i.e., φ Poisson equation gives, 4π e n x Equilibrium is defined as, v v e φ T n v t x m x m n x v t v φ x e v x m m n ( n ) ( ) n n n say T n x φ E x (3) (4) (5) (6) Dispersion relation ( ω VT k ) ( ω VT k ) ωp (7) where V T α Tα m and ω p 4π ne m m m - (m say) 3

14 Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.) Linear analysis of the modes in pair-ion plasmas Case (i) If T T - (T say), then Eq.(7) gives ω ω V k p T which is the usual Langmuir waves in pair-ion unmagnetized plasmas. (8) Case (ii) If T T -, then Eq.(7) gives quartic equation as follows where ( ) ( ) ω V k V k ω ω ω V V k k V V 4 4 T T p p T T T T ω ( V k T VT k ωp) ± Δ { } ( ω ) 4 ω ( ) { } Δ V k V k V V k k V V 4 T T p p T T T T (9) () here (ve) and (-ve) sign gives higher and lower frequency Langmuir waves in pair-ion plasmas. 4

15 Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.) Under the approximation of small k (long wave length), the ve root i.e., the fast Langmuir wave gives the relation, Under the approximation of small k (long wave length), the -ve root i.e., the slow Langmuir wave gives the relation, which is low frequency ion acoustic mode in pair-ion plasmas. 5

16 Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.) Nonlinear set of equations in normalized form are written as follows: n t ( nv ) x n t ( nv ) x v Φ n v v t x x n x v Φ μ n v v t x x n x Φ ( n n ) x eφ T where Φ T and μ have been defined. T Normalization of time, space and velocity has been done by V T T m D T 4π ne ω p 4π ne m 6

17 Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.) Using reductive perturbation method, the perturbed quantities can be expanded in the power series of ε as follows: ( ) ( ) n εn ε n... j j j ( ) ( ) v εv ε v... j j j ( ) ( ) Φ εφ ε Φ... The stretching of independent variables are introduced in the following fashion: ξ ε ( x t), τ ε 3 t where <ε is an expansion parameter characterizing the strength of nonlinearity Now using above relations in normalized set of equations and collecting the lowest order terms of continuity and momentum equations of ions (~ε 3/ ) and Poisson equation (~ ε) gives after some simplification, i.e., ( ) ( μ) or μ () 7

18 Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.) Now collecting the next higher order terms of continuity and momentum equations of ions (~ε 5/ ) and Poisson equation (~ ε ) and using relation of parameter given in Eq.() after some simplification, we obtain the KdV equation as follows: ( ) ( ) ( ) 3 ( ) Φ AΦ Φ BΦ τ ξ ξ () A B ( 3 )( μ) ( 3 μ) ( ) ( )( μ) ( μ) ( ) 3 3 ( ) ( μ) ( ) ( ) μ (coefficient of nonlinear term) (coefficient of dispersive term) Note: The soliton solution will be formed only when T T - holds. 8

19 Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.) If T T -, then soliton solution of Eq.() is given by where ( ) η Φ φ m sec h W φ m 3u / A (3) (amplitude of the soliton) W 4B/ u (width of the soliton) Transformed variable η ξ u τ has been defined. The velocity of the transformed frame u has been normalized with thermal velocity of positive ion. 9

20 Numerical solutions in pair-ion plasmas Fig.: The electrostatic potential dips are shown for μ< i.e., the T >T - for μ.9 (dotted curve), μ.8(broken curve) and μ.7 (solid curve) at u.8. The potential dip is increased with the increase in positive ion temperature. F h Fig.: The electrostatic potential humps are shown for μ> i.e., the T <T - for μ. (dotted curve), μ. (broken curve) and μ.3 (solid curve) at u.8. The potential hump is increased with the decrease in positive ion temperature. F h

21 Double Layer in Plasmas 5. Double Layers in Pair-ion Plasmas A double layer is a structure in a plasma consists of two parallel layers with opposite electrical charge. The sheets of charge cause a strong electric field and a correspondingly sharp change in voltage (electrical potential) across the double layer. Ions and electrons, which enter the double layer are accelerated, decelerated or reflected by the electric field. In general, double layers (which may be curved rather than flat) separate regions of plasma with quite different characteristics. Mathematical Description of Double Layer in Pair-ion Plasmas For double layers structures, using the stretching for independent variables as follows, 3 ξ ε ( x t),... τ ε t The perturbed quantities are expanded in the powers of ε as defined earlier. Now collecting the higher order terms of continuity and momentum equations of ions (~ε 3 ) and Poisson equation (~ ε 4 ) and using relation of parameter given in Eq.() after some simplification, we obtain the mkdv equation as follows:

22 Double layer in Pair-ion Plasmas (contd.) ) ( ) ( μ A ) ( 3 ) ( ) (3 ) ( 3 ) ( ) (3 μ μ μ B 3 3 ) ( ) (3 ) ( ) (3 μ μ Q τ ξ η U The mkdv equation is described as 3 3 ξ ξ ξ τ B Q A where the coefficients are defined as (4) Now using the transformation the above mkdv equation

23 Double layer in Pair-ion Plasmas (contd.) can be described in the form of energy integral equation as follows, η V ( ) (5) where the Sagdeev potential is defined as UA Q 3 B V ( ) 3 4 For the formation of DLs, the Sagdeev potential must satisfy the following conditions, 4 (6) V d dv ), at and m. d ( V d Q < at and m 3 B m U B A m (7) The above conditions are satisfied only if, coefficients Q and U described as, 3

24 Double layer in Pair-ion Plasmas (contd.) The Sagdeev potential satisfying the DLs boundary conditions as follows, V ( B 4 ) ( m ) (8) Using Eq.(8), the solution of energy integral equation (5) can be written as, m - B tan h m ( η Uτ ) 8 (9) where amplitude and width of the DLs are defined as, m Q 3 B - B 8 Width where B < m must hold For Q>, compressive DLs and for Q< rarefactive DLs are formed. where μ T and μ T - have been defined. 4

25 Double layer in Pair-ion Plasmas (contd.) Numerical solutions of rarefactive DLs (μ>) in pair-ion plasmas.. V F -. F x (solid curve) and (dashed curve) 5

26 Double layer in Pair-ion Plasmas (contd.) Numerical solutions of compressive DLs (μ<) in pair-ion plasmas.. V F F x : (solid curve) and (dashed curve) 6

27 7 ) ( ) ( v n y u n x t n x u x n n x y u v x u u t u η y u y n n y y v v x v u t v η ) ( ) ( v n y u n x t n x u x n n x y u v x u u t u η β y v y n n y y v v x v u t v η β 6. Dissipative shocks and solitons in Pair-ion Plasmas Kadomstev-Petviashvili-Burgers (KPB) Equation in PI Plasmas Electrostatic structures are studied in unmagnetized, weakly dissipative PI plasmas in the presence of weak transverse perturbations. The dissipation in the system is incorporated by taken into account the kinematic viscosity of both the species. (Continuity equation for ve ions) (Momentum equations for ve ions) (Continuity equation for -ve ions) (Momentum equations for -ve ions) Normalized set of dynamic equations are written as

28 Dissipative shocks and solitons in Pair-ion Plasmas The Poisson equation is written as follows, n n x y where β T / T μ μ, η - and η ( ) have been defined. v D s In fully ionized gas, the coefficient of viscosity in the absence of magnetic field is given by 5 / / 5 T± Ai μ ±. gm / cm sec Braginskii (96) 4 Z lnλ Using reductive perturbation method, the perturbed quantities can be expanded about their equilibrium values in powers of ε such that v D s n v α () () 3 εnα ε nα ε nα (3)... () () 3 (3) α εuα ε uα ε uα... 3/ () 5 / () 7 / (3) α ε vα ε vα ε vα u For weak dissipation, () () 3 (3) ε ε ε... η ε / η ± ± where η ± is... () 8

29 9 3 3 χ ξ ξ ξ τ ξ D C B A ] ) ( ) )[( )( ( ] ) )( (3 ) )( [3 3 3 β β β β A ] ) ( ) [( ) ( ) ( β β B ] ) ( ) [( ) ( ) ( β η β η C D / Dissipative shocks and solitons in Pair-ion Plasmas t y t x 3 / /, ), ( ε τ ε χ ε ξ The stretching of the independent variables for -D electrostatic waves in PI plasmas is defined as Now collecting the terms of different powers of ε, we finally obtained the KPB equation as follows where the coefficients as defined as, (Nonlinear coefficient) (Dispersive coefficient) (Dissipative coefficient) (weakly dispersive coefficient) ()

30 Dissipative shocks and solitons in Pair-ion Plasmas Using transformation ζ k( ξ χ Uτ ),the solution of KPB equation is given by, here ( ζ, χ, τ ) k C B Soliton solution C tan h AB C sec h AB C B C B 6C ζ χ ( 5B 6C ζ χ ( 5B D) τ D) τ (wave number of nonlinear wave) has been defined. () The KP equation is obtained by putting C in KPB equation, which gives soliton solution such as where B ( ζ, χ, τ ) sech A U ( 4B D), [ ζ χ Uτ ] β and T β T - () have been defined. 3

31 Dissipative shocks and solitons in Pair-ion Plasmas T β - T For β.4 (left) and β.6 (right) T > T For β.8 (left) and β.6 (right) T > T η. 5/ η η (β ) 3

32 Dissipative shocks and solitons in Pair-ion Plasmas For β.4 (left) and β.6 (right) T > T For β.8 (left) and β.6 (right) T > T 3

33 7. Conclusion Linear and nonlinear electrostatic waves are studied in unmagnetized pair-ion plasmas. Korteweg-de Vries (KdV) equation is obtained using reductive perturbation method. Both electrostatic potential hump and dip solitons are obtained in unmagnetized pair-ion plasmas. The electrostatic potential humps are formed, when the temperature of negative ion species in greater than the positive ions, while in reverse temperature conditions potential dips are obtained. Both compressive and rarefactive potential double layers are obtained depending on the ratio between difference in positive and negative ion species. 33

34 7. Conclusion Similarly, both compressive and rarefactive shocks and solitons are obtained from KPB equation depending on the temperature ratio between positive and negative ion species. No electrostatic structures are obtained, when the temperature of both positive and negative ions are same in unmagnetized pair-ion plasmas. 34

35 Thank You 35