Field emission of Electrons from Negatively Charged Cylindrical Particles with Nonlinear Screening in a Dusty Plasma

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1 Reseach & Reviews: Jounal of Pue and Applied Physics Field emission of Electons fom Negatively Chaged Cylindical Paticles with Nonlinea Sceening in a Dusty Plasma Gyan Pakash* Amity School of Engineeing and Technology, Amity Univesity, Lucknow, India Reseach Aticle Received date: 13/07/016 Accepted date: 7/08/016 Published date: 9/08/016 *Fo Coespondence Gyan P, Amity School of Engineeing and Technology, Amity Univesity, Lucknow, India. gpakash@lko.amity.e ABSTRACT In this communication, electic field emission of electons fom negatively chaged cylindical paticles in the egime of nonlinea sceening has been discussed. Using the suitable epesentation of the nonlinea sceening by Guevich and the matix method, the tunnelling pobability of an electon though the potential enegy baie aound the chaged cylindical paticle has been evaluated as a function of its adial enegy; and the coesponding values of the cuent density have also been calculated. The note completes with the discussion of numeical esults. This investigation is useful in sty plasma and laboatoy in case of metallic wies and nanotubes. Keywods: Nonlinea sceening, Tunnelling effect, Dusty plasma, Electon emission INTRODUCTION Tunneling phenomena is most inteesting phenomena which is much widely discussed in a vaiety of aeas of physics. It has been pointed out by Mendis et al. [1], Mendis [], Shukla [3], Shukla and Mamun [4] and Fotov et al. [5] that thee ae many situations in space and laboatoy, the lage negative chage and small adius of the st paticles cause a sufficiently lage electic field at the suface, which is enough fo significant field emission of electons to occu fom the paticle. The electic field emission of the electons is the dominant pocess of electon emission in the absence of photoelectic o themionic emission of electons fom the suface of the paticles. It has been specifically pointed by Mendis et al. [1] and Mendis [] that the electic field emission of electons fom the st paticles should be taken into account in electon/ion kinetics in sty plasma. Sodha et al. [6] have ecently fomulated a quantitative theoy and concluded that in some situations the incopoation of the electic field emission of the electons in the fomulation has significant effect on the electon/ion kinetics and the chaging of the st paticles. Recently, Sodha and Dixit [7] have made a compaison of the esults of the popula Fowle [8] and Fowle and Nodheim [9] theoy fo the tunneling pobability of an electon with those, based on exact solution [10] of the Schodinge s equation. It was pointed out [7] that the esults of the popula theoy [8,9] ae at consideable diffeence with those based on the exact [10] theoy; futhe the fomulation of Ghatak et al. [11] and Roy et al. [1] gives almost exact values of the tunneling coefficient fo symmetic potentials. In most of studies (e.g. Shukla and Mamun [4] ), the electical field emission fom cuved sufaces has been calculated fom the expession given by Fowle and Nodheim [9] fo plane sufaces and putting the values of the electic field, as that at the cuved suface. It is well known that the electic field aound a cuved suface is fa fom being unifom and hence it is not appopiate to use the expession of Fowle and Nodheim [9], based on the assumption of a unifom electic field. Besides as pointed out in the peceding paagaph this expession is a poo appoximation, even fo a plane suface. Sodha and Kaw [13] and Sodha and Dubey [14] obtained expessions fo the tunneling pobability and field emission cuent density fom spheical [13] and cylindical [14] sufaces by solving Schodinge s equation, coesponding to appopiate expession fo the potential enegy of the electon, using the JWKB appoximation. Dubey [15] made a simila analysis fo the spheical suface, taking into account the image foce on the electon. Sodha et al. [16] and Sodha and Dixit [17] have ecently investigated the electic field emission fom spheical [16] and cylindical [18] sufaces, using the fomalism of Ghatak et al. [11] and Roy et al. [1] ; the sceening of the paticle by the electons, ion, and othe paticles was neglected in these investigations. Sodha and Dixit [19] have also studied the effect of Debye shielding (linea sceening) on electic field emission fom spheical and cylindical sufaces and found that sceening inceases the tunneling pobability. RRJPAP Volume 4 Issue 3 Mach, 016 5

2 Tsytovich et al. [18] have studied the sceening of the chage on the paticle, which is an impotant paamete in the evaluation of the tunneling pobability. The natue of sceening [18] is detemined by the paamete β = za / λ τ= ZZ e ( πn ) ( kt ), di i i i (1) Whee z = Ze / akte, is the adius of the paticles, π ne Z λ = is the ionic Debye adius, 4 i i di kti τ=ti / Te, and Ze is the chage on the paticle, e is the electonic chage, k is the Boltzmann s constant, n i is the ion density, ez i is the ionic chage, T i is the ion tempeatue, T e is the electon tempeatue, In geneal the potential enegy of an electon aound a long cylindical paticle with chage Ze pe unit length and adius a is given by Vs λsc V() = ln( ). Λ() λsc ln( ) a fo > a (a) and V() Vs V0 = fo < a, (b) Vs λ a sc = Ze ln( ) is the potential enegy of an electon at the suface of the paticle. Λ () is a function of, distance fom the axis of the cylindical paticle, chaacteizing the sceening of the chage on the paticle and is the suface enegy baie. When β <<1, the sceening is linea and the sceening paamete Λ() is given by [18] Λ () = exp[( a) λ d ] fo > a (3) d = λdi + Ti Te ne nz i i λ [1 ( )( )], n e is the electon density and λ d is the usual Debye adius in the plasma. When β <<1, the sceening coesponding to Guevich sceening potential is nonlinea and the sceening paamete Λ() is given by [18] 4 Λ ( ) = [1 ( ) ] fo < < ( + ), a λ sc a a λ sc and Λ()=0 fo > (a+λ sc ), (4a) (4b) λ = 3.4 λ ( πβ 4) 1 5 sc di RRJPAP Volume 4 Issue 3 Mach,

3 The electic field emission of electons fom the suface of spheical and cylindical paticles has been discussed by Sodha and Dixit [18] fo the case of linea sceening (β <<1), when the sceening is well epesented by Eq.(3). The electic field emission fom the suface of spheical paticles in the egion of nonlinea sceening has been investigated by Gyan Pakash [19]. In this pape the autho has evaluated the tunneling pobability (fo the electons with enegy moe than suface enegy baie (ε (=E / )>1) and the electic field emission cuent fom a negatively chaged cylindical paticle fo the case, when the sceening is nonlinea (β <<1) and epesented by Eqs.(4a) and (4b). Using Eqs. (4a), (4b), (a) and (b) the potential enegy of an electon is given by V s µ 1 V ( ρ) = ln( ). Λ( ρ) ln( µ 1) ρ V ( ρ ) = 0 fo ρ > µ fo 1< ρ < µ (5a) (5b) V V V ( ) 0 s ρ = + fo ρ < 1 (5c) ρ = a ρ 1 ( ) [1 ] µ 1 Λ 4 ρ = and µ = (1 + λ ) sc a Substituting fo V() fom Eq.(b), Schödinge s equation fo an electon inside the paticle ( < a ) may be expessed as ψ(, θ, z) + (8 π m h )[ E+ V0 Vs )] ψ(, θ, z) = 0 (5d) Whee m and E ae the mass and total enegy of the electon and h is Planck s constant. Putting ψ = 1/ ± in icz ue () θ ± e (5e) Whee n and c can be intepeted in tem of P and P (the θ and Z components of linea momentum) as suggested by θ Z Sodha and Kaw [13] and Sodha and Dubey [14]. Using Eq.(5e), the equation (Eq.(5d)) eces to + (8 π mh)[ E + V0 Vs] u= 0 d. Following ealie wokes [13-16,19] the tem ħ /8m can be neglected as compaed to the othe enegy tems. E is the adial enegy of the electon, the above equation eces to β 1[ ε 1 β] u 0 d ρ + + = fo ρ< 1, (6a) β 8π a mv h ε = E V 1 = 0, 0, and β = Vs V0. Similaly, fo egions 1 < ρ < µ and ρ > µ Schodinge s equation may be witten using Eqs.(5a) and (5b), one obtains µ 1 ln( ) ρ ln( 1) 1 ρ β 1{ ε ( β )[1 ] } u = 0 ρ µ µ d fo 1< ρ < µ (6b) RRJPAP Volume 4 Issue 3 Mach,

4 and βε 1 u 0 d ρ + = fo ρ µ >. (6c) Using the analytical solution of Eqs.(6a) and (6c) and numeical solution of Eq.(6b) by the well-known matix method and matching u and at ρ=1 and µ, the tunneling pobability of an electon T(ε ) has been evaluated as a function of ε. Details of d ρ this appoach may in addition to othe places be seen in the book by Ghatak and Loknathan [10]. The computations made by the Sodha and Dixit [7] fo a plane suface indicate that thee is no significant dependence of the cuent density on the tempeatue of the metal at least when kt/e F < Hence in the pesent investigation only the case T=0K is consideed; in this case the cuent density is given by [16,19]. 4π me J =. V 3 0 Φ ( β1, β, εf ) h (7) ε, Φ ( β, β, ε ) = ( ε ε ) T( β, β, ε ) dε 1 F 1 ε1 ε = EF V 0, F E F is the Femi enegy of the metal and ε 1 and ε ae the limits of ε of electons taking pat in the field emission ; these ae ε = ε 1+ β 1 F and ε = 0 Figue 1 displays the dependence of the tunneling pobability T(ε ) on the adial enegy ε of the electon. It can be seen that the tunneling pobability inceases with inceasing ε and deceasing value of µ (coesponding moe intense sceening); this is because the width of the enegy baie deceases with inceasing ε and deceasing µ. This is also clea fom the Figue 1 that enegy of the electon is inceasing and eaching the suface baie o beyond, the tunneling pobability T(ε ) is tending towads the unity. Figue shows the dependence of T(ε fo diffeent values of β =V s/. It is seen that T(ε ) inceases with inceasing ε and deceasing β.this is e to the fact fo any value of ε, the width of the enegy baie is lage fo highe values of β (=V s/ ). Figue 3 illustates that T(ε ) inceases with deceasing β1 (α ). This is again because fo a given value of ε and β (=V V ), the s/ 0 width of the enegy baie deceases with deceasing β1 (α ). Finally Figue 4 shows that the field emission cuent inceases with inceasing β (=V s/ ) and deceasing µ ( inceasing sceening). This is a consequence of the dependence of T(ε and ε 1 on β and µ, as displayed by Figues 1-3. Eq.(7) along with the computed values of ɸ is an essential input fo the investigation of the kinetics of complex plasmas, when the ole of electic field emission of electons by the paticle is significant. Figue 1. Effect of Guevich shielding ( =1 + λsc/a) on the dependence of tunneling pobability T( ε) on ε fo β1=150, β=0.95. The cuves a, b, c and d coespond to =3, 4, 5 and 7 espectively. RRJPAP Volume 4 Issue 3 Mach,

5 Figue. Dependence of Tunneling pobability T(ε fo diffeent values of β, fo β 1 =150, µ=3. The cuves a, b, c and d coespond to β =0.80, 0.85, 0.90 and 0.95 espectively. Figue 3. Dependence of Tunneling pobability T(ε fo diffeent values of β 1, fo β =0.95, µ=3. The cuves a, b, c, d and e coespond to β 1 =100, 15, 150, 175 and 00 espectively. Figue 4: Effect of Guevich shielding (µ=1 + λ sc /a) on the dependence of field emission cuent density φ on the chage on the cylindical paticles β =V s/ fo β =150, ε F =0.75. The cuves a, b, c and d coespond to µ=3, 4, 5 and 6 espectively. REFERENCES 1. Mendis DA, et al. Physics of Dusty Plasma. AIP New Yok Mendis DA. Plasma Souces Science and Technology. 00; Shukla PK. Dust ion-acoustic shocks and holes. Physics of Plasmas. 000;7: Shukla PK and Mamun AA. Intoction to Dusty Plasma. IOP, Bistol Fotov VE, et al. Dusty plasmas. Phys Usp. 004;47: Sodha MS, et al. Effect of electic ield emission on chaging of st paticles in a plasma. J Plasma Physics. 009;76: Sodha MS and Dixit A. A citical look at electic field emission of electons fom metallic sufaces. J Appl Phys. 008;104. RRJPAP Volume 4 Issue 3 Mach,

6 8. Fowle RH. Statistical Mechanics, the theoy of the popeties of matte in equilibium. Cambidge Univesity Pess, Cambidge. 1955; Fowle R and Nodheim L. Electon Emission in Intense Electic Fields. Poc Roy Soc London Se A. 198;119: Ghatak AK and Loknathan S. Quantum Mechanics: Theoy and Applications Ghatak AK, et al. Electomagnetic wave popagation in a adially and axially non-unifom medium: optics of SELFOC fibes and ods; wave optics consideations Jounal of Physics D: Applied Physics. 1971;4. 1. Sodha MS, et al. Effect of solid paticles on electomagnetic popeties of ocket exhausts. B J Appl Phys J Phys D. 1968;1: Sodha MS and Dubey PK. Field emission fom wies. Jounal of Physics D: Applied Physics Dubey PK. Effect of image foce on emission of electons fom negatively chaged solid paticles. J Phys D. 1970;3: Sodha MS, et al. Electic field emission of electons fom spheical metallic st paticles. Can J Phys. 009;87: Sodha MS and Dixit A. Field emission of electons fom cylindical metallic sufaces. J.Appl.Phys. 008; Tsytovich VN, et al. Elementay Pocesses in Complex Plasmas. Spinge, Belin. 008; Sodha MS and Dixit A. Effect of Debye shielding on electic field emission fom spheical and cylindical sufaces. J Appl Phys. 009; Gyan P. Electic field emission of electons fom negatively chaged spheical paticles in a sty plasma in the egime of nonlinea sceening. Can J Phys. 010;88: RRJPAP Volume 4 Issue 3 Mach,