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Dynamics of nonlinear dust lattice waves in the presence of ion plasma waves
29 July 2002
Physics Letters A 300 (2002) 282–284 www.elsevier.com/locate/pla
Dynamics of nonlinear dust lattice waves in the presence of ion plasma waves P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 12 April 2002; accepted 30 April 2002 Communicated by F. Porcelli
Abstract The parametric interaction between nonlinear dust lattice waves and ion plasma waves in a plasma sheath is considered. It is found that the interaction is governed by a pair of equations comprising a modified Kortweg–de Vries equation for the driven (by the ion plasma wave ponderomotive force) nonlinear dust lattice waves and a nonlinear Schrödinger equation for the modulated ion plasma wave envelops. The pair admits stationary solutions in the form of a dust density cavity in which localized ion plasma wave electric field envelope is trapped. 2002 Elsevier Science B.V. All rights reserved. PACS: 52.25.Zb; 52.27.Lw; 52.35.Mw; 52.35.Sb
About six years ago Melandsø [1] presented linear and nonlinear theories for dust lattice (DL) waves, taking into account the Debye–Hückel potential energy associated with nearest neighbor dust particle interactions. Farokhi et al. [2] examined the properties of dust lattice waves in a plasma crystal accounting for the dependence of the dust grain charge on the grain potential. The linear theory of DL waves has been verified experimentally [3,4]. Recently, Samsonov et al. [5] provided an evidence for the nonlinear DL waves (DL solitary waves) in the sheath of an rf discharge. Since the sheath is ion rich, it possibly contains large amplitude ion plasma (IP) waves. The latter are associated with the rapid motion of ions around negatively
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University,
SE-90187 Umeå, Sweden.
charged dust particulates, and they can be excited due to the two-stream instability [6]. In this Letter, we consider the parametric coupling between large amplitude IP waves and nonlinear DL waves. The coupling occurs owing to the space charge electric field that is created by the ponderomotive force of the IP waves. It turns out that the self-consistent nonlinear interaction between the IP and DL waves is governed by the nonlinear Schrödinger (NLS) equation for the modulated (by the dust lattice perturbations) IP wave envelops and a modified Kortweg– de Vries (K–dV) equation for the nonlinear DL waves which are reinforced by the ponderomotive force of the IP wave envelops. The coupled NLS and modified K–dV equations admit stationary solutions [7–9] in the form of a dust density cavity which traps IP wave electric field envelops. We thus have a new mechanism for creating a dust density caviton due to the presence of pre-existing IP waves in a plasma sheath.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 8 1 7 - 4
P.K. Shukla / Physics Letters A 300 (2002) 282–284
Let us consider a plasma sheath composed of ions and negatively charged dust grains which interact with each other via the Debye–Hückel potential energy UD =
Q2 i
rij
exp(−kD rij ),
(1)
where the dust particle charge Q and the ion Debye wave number kD (inverse of the ion Debye radius) are constant and rij = |ri − rj |, with rk being the dust particle coordinate. The equation of motion for charged dust particles in a linear chain is ∂UD ∂ 2 rk =− + F, (2) 2 ∂t ∂rk where md is the dust particle mass and F is the external force associated with the space charge electric field Es that is created by the ponderomotive force [10] of the IP waves. We have md
F = QEs = −Zd ∇φp ≡ Fp ,
(3)
where Zd is the number of electrons residing on the 2 is the pondust grain surface and φp = e2 |E|2 /2mi ωpi deromotive potential of the IP waves whose electric field E oscillates at the ion plasma frequency ωpi = (4πni0 e2 /mi )1/2 . Here, ni0 (≈ Zd nd0 ) is the unperturbed ion number density, nd0 is the unperturbed dust number density, e is the magnitude of the electron charge, and mi is the ion mass. Following the perturbation technique of Melandsø [1], we expand Eq. (2) by assuming that the particle displacement u(r, t) from the equilibrium position is small, and thereby keep the first nonlinear terms. The resulting nonlinear equation for one-dimensional DL waves in the presence of an IP wave driver is of the form γ ∂t2 u − Cl2 ∂x2 u + Cl2 ∂x (ux )2 − Cl2 L2d ∂x3 ux 2 Cd2 + ∂x |E|2 = 0, (4) 8πni0 Ti where ∂t = ∂/∂t, ∂x = ∂/∂x, ux = ∂u/∂x, Cd = (Zd Ti /md )1/2 , and Ti is the ion temperature. For κ = kD a 1, where a is the inter-particle spacing, we have [1,5] Cl2 = Q2 /md a 3 + 2 ln(κ −1 ) , γ = 11 + 6 ln(κ −1 ) 3 + 2 ln(κ −1 )
283
and
2 3 + 2 ln(κ −1 ) , L2d = 1 kD while for κ 1, we have Cl2 = Q2 /md a (κ 2 + 2κ + 2) exp(−κ), γ = (κ 3 + 3κ + 6κ + 6)/(κ 2 + 2κ + 2), and L2d = a 2 /12. The quantity ux and the dust density perturbation nd1 ( nd0 ) are related by nd1 ux ≈ − (5) ≡ −N. nd0 The interaction between the nonlinear DL waves and a large amplitude IP wave gives rise to envelope of waves which are governed by [11] 2 2iωpi (∂t + vg ∂x )E + 3VT2i ∂x2 E − ωpi NE = 0,
(6)
where VT i = (Ti /mi )1/2 is the ion thermal speed and vg = 3kVT2i /ωpi is the group velocity of the IP waves. In deriving Eq. (6) we have invoked the WKB approximation, viz. |∂t E| ωpi E, and replaced the driven ion number density perturbation by Zd nd1 in order to maintain the quasineutrality in our dusty plasma. Eqs. (4)–(6) are the desired set for studying the dynamics of the nonlinear DL waves in the presence of modulated IP waves in the sheath of an rf discharge. In the following, we seek localized solutions of our set in the frame ξ = x − V τ , where V is the constant velocity, by assuming that u and E are function of ξ and τ . Hence, Eqs. (4)–(6) for unidirectional propagation are written as ∂τ N +
C2 γ Cl Cl ∂ξ N 2 + L2d ∂ξ3 N + d ∂ξ |E|2 = 0 (7) 4 2 2Cl
and i∂τ E +
3VT2i 2 ωpi NE = 0, ∂ E− 2ωpi ξ 2
(8)
where V ∼ Cl ∼ vg and the electric field E is normalized by (16πni0 Ti )1/2 . Eqs. (7) and (8) are modified K–dV and NLS equations, respectively. In the absence of the IP waves, Eq. (7) gives a stationary solution in the form of a compressional dust density solitary wave. The latter is of the form [12] N = N sech2 ϕ,
(9)
284
P.K. Shukla / Physics Letters A 300 (2002) 282–284
where ϕ = (ξ − Vs τ )/δ, Vs = γ Cl N /6, and δ = √ 12/γ N Ld . It turns out that the soliton speed Vs (width δ) is directly (inversely) proportional to the soliton amplitude N . However, when the modulated IP waves interact with the nonlinear DL waves, there appear envelope solitons consisting of a dust density cavity N = −Nm sech2 θ,
(10)
and camel-shaped ion plasma wave intensity distribution 2 sech2 θ tanh2 θ, |E|2 = Em
(11)
which are the stationary solutions [7,8] of Eqs. (7) and (8). Here, Nm is the minimum (normalized) dust density at θ = |ξ − U τ |/λ = 0, Em is the maximum (normalized) value of the IP wave electric field at the center of the dust density cavity, U is a constant, and λ is the width of the localized dust density cavity. The exact forms of Nm , Em and λ can be deduced following Refs. [7–9]. We stress that the dust density cavitation occurs on account of the ponderomotive force (Fp ) of the IP waves. Finally, we note that when the nonlinear and dispersive terms in Eq. (7) are neglected, we then have Cd2 ∂ξ W 2 , (12) 2Cl where we have introduced E = W (ξ ) exp(−iΩτ ). Eq. (12) is coupled with the NLS equation 3VT2i 2 1 ∂ W + Ω − ωpi N W = 0. (13) 2ωpi ξ 2
∂τ N = −
Following Karpman [13,14] one can obtain stationary and nonstationary solutions of Eqs. (12) and (13) in the form of an inverted bell-shaped dust density cavity which traps localized IP wave electric field envelops having a secant-hyperbolic profile. To summarize, we have considered the parametric interaction between the nonlinear DL waves and large amplitude IP waves in a plasma whose constituents are weakly coupled ions and strongly correlated dust grains. It is found that the self-consistent interaction between DL and IP waves produces a dust density cavity which traps an envelope of IP waves. Hence, the
present investigation provides a novel mechanism for creating a dust density caviton by the ponderomotive force of pre-existing IP waves in the sheath of an rf plasma discharge.
Acknowledgement This work was partially supported by the European Commission (Brussels) through the contract No. HPRN-CT2000-00140 for carrying out the task of the Human Potential Research Training Network entitled “Complex Plasmas: The Science of Laboratory Colloidal Plasmas and Mesospheric Charged Aerosols”.
References [1] F. Melandsø, Phys. Plasmas 3 (1996) 3890. [2] B. Farokhi, P.K. Shukla, N.L. Tsintsadze, D.D. Tskhakaya, Phys. Lett. A 264 (1999) 318; B. Farokhi, P.K. Shukla, N.L. Tsintsadze, D.D. Tskhakaya, Phys. Plasmas 7 (2000) 814. [3] G. Morfill, H.M. Thomas, M. Zuzic, in: P.K. Shukla, D.A. Mendis, T. Desai (Eds.), Advances in Dusty Plasmas, World Scientific, Singapore, 1997, pp. 99–142. [4] A. Homann, A. Melzer, S. Peters et al., Phys. Lett. A 242 (1998) 173. [5] D. Samsonov, A.V. Ivlev, R.A. Quinn, G. Morfill, S. Zhdanov, Phys. Rev. Lett. 88 (2002) 095004. [6] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol, 2002. [7] K. Nishikawa, H. Hojo, K. Mima, H. Ikezi, Phys. Rev. Lett. 33 (1974) 148. [8] V.G. Makhankov, Phys. Lett. 50A (1974) 42. [9] R.K. Varma, N.N. Rao, Phys. Lett. 79A (1980) 311; N.N. Rao, P.K. Shukla, Phys. Plasmas 4 (1997) 636. [10] P.K. Shukla, Phys. Rev. Lett. 84 (2000) 5328. [11] P.K. Shukla, L. Stenflo, G. Morfill, IEEE Trans. Plasma Sci. 29 (2001) 198. [12] R.C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972; L. Stenflo, N.L. Tsintsadze, T.D. Buadze, Phys. Lett. A 135 (1989) 37. [13] V.I. Karpman, Plasma Phys. 13 (1971) 477; V.I. Karpman, Phys. Scripta 11 (1975) 263. [14] V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, Oxford, 1975, pp. 165–169.
Physics Letters A 300 (2002) 282–284 www.elsevier.com/locate/pla
Dynamics of nonlinear dust lattice waves in the presence of ion plasma waves P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 12 April 2002; accepted 30 April 2002 Communicated by F. Porcelli
Abstract The parametric interaction between nonlinear dust lattice waves and ion plasma waves in a plasma sheath is considered. It is found that the interaction is governed by a pair of equations comprising a modified Kortweg–de Vries equation for the driven (by the ion plasma wave ponderomotive force) nonlinear dust lattice waves and a nonlinear Schrödinger equation for the modulated ion plasma wave envelops. The pair admits stationary solutions in the form of a dust density cavity in which localized ion plasma wave electric field envelope is trapped. 2002 Elsevier Science B.V. All rights reserved. PACS: 52.25.Zb; 52.27.Lw; 52.35.Mw; 52.35.Sb
About six years ago Melandsø [1] presented linear and nonlinear theories for dust lattice (DL) waves, taking into account the Debye–Hückel potential energy associated with nearest neighbor dust particle interactions. Farokhi et al. [2] examined the properties of dust lattice waves in a plasma crystal accounting for the dependence of the dust grain charge on the grain potential. The linear theory of DL waves has been verified experimentally [3,4]. Recently, Samsonov et al. [5] provided an evidence for the nonlinear DL waves (DL solitary waves) in the sheath of an rf discharge. Since the sheath is ion rich, it possibly contains large amplitude ion plasma (IP) waves. The latter are associated with the rapid motion of ions around negatively
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University,
SE-90187 Umeå, Sweden.
charged dust particulates, and they can be excited due to the two-stream instability [6]. In this Letter, we consider the parametric coupling between large amplitude IP waves and nonlinear DL waves. The coupling occurs owing to the space charge electric field that is created by the ponderomotive force of the IP waves. It turns out that the self-consistent nonlinear interaction between the IP and DL waves is governed by the nonlinear Schrödinger (NLS) equation for the modulated (by the dust lattice perturbations) IP wave envelops and a modified Kortweg– de Vries (K–dV) equation for the nonlinear DL waves which are reinforced by the ponderomotive force of the IP wave envelops. The coupled NLS and modified K–dV equations admit stationary solutions [7–9] in the form of a dust density cavity which traps IP wave electric field envelops. We thus have a new mechanism for creating a dust density caviton due to the presence of pre-existing IP waves in a plasma sheath.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 8 1 7 - 4
P.K. Shukla / Physics Letters A 300 (2002) 282–284
Let us consider a plasma sheath composed of ions and negatively charged dust grains which interact with each other via the Debye–Hückel potential energy UD =
Q2 i
rij
exp(−kD rij ),
(1)
where the dust particle charge Q and the ion Debye wave number kD (inverse of the ion Debye radius) are constant and rij = |ri − rj |, with rk being the dust particle coordinate. The equation of motion for charged dust particles in a linear chain is ∂UD ∂ 2 rk =− + F, (2) 2 ∂t ∂rk where md is the dust particle mass and F is the external force associated with the space charge electric field Es that is created by the ponderomotive force [10] of the IP waves. We have md
F = QEs = −Zd ∇φp ≡ Fp ,
(3)
where Zd is the number of electrons residing on the 2 is the pondust grain surface and φp = e2 |E|2 /2mi ωpi deromotive potential of the IP waves whose electric field E oscillates at the ion plasma frequency ωpi = (4πni0 e2 /mi )1/2 . Here, ni0 (≈ Zd nd0 ) is the unperturbed ion number density, nd0 is the unperturbed dust number density, e is the magnitude of the electron charge, and mi is the ion mass. Following the perturbation technique of Melandsø [1], we expand Eq. (2) by assuming that the particle displacement u(r, t) from the equilibrium position is small, and thereby keep the first nonlinear terms. The resulting nonlinear equation for one-dimensional DL waves in the presence of an IP wave driver is of the form γ ∂t2 u − Cl2 ∂x2 u + Cl2 ∂x (ux )2 − Cl2 L2d ∂x3 ux 2 Cd2 + ∂x |E|2 = 0, (4) 8πni0 Ti where ∂t = ∂/∂t, ∂x = ∂/∂x, ux = ∂u/∂x, Cd = (Zd Ti /md )1/2 , and Ti is the ion temperature. For κ = kD a 1, where a is the inter-particle spacing, we have [1,5] Cl2 = Q2 /md a 3 + 2 ln(κ −1 ) , γ = 11 + 6 ln(κ −1 ) 3 + 2 ln(κ −1 )
283
and
2 3 + 2 ln(κ −1 ) , L2d = 1 kD while for κ 1, we have Cl2 = Q2 /md a (κ 2 + 2κ + 2) exp(−κ), γ = (κ 3 + 3κ + 6κ + 6)/(κ 2 + 2κ + 2), and L2d = a 2 /12. The quantity ux and the dust density perturbation nd1 ( nd0 ) are related by nd1 ux ≈ − (5) ≡ −N. nd0 The interaction between the nonlinear DL waves and a large amplitude IP wave gives rise to envelope of waves which are governed by [11] 2 2iωpi (∂t + vg ∂x )E + 3VT2i ∂x2 E − ωpi NE = 0,
(6)
where VT i = (Ti /mi )1/2 is the ion thermal speed and vg = 3kVT2i /ωpi is the group velocity of the IP waves. In deriving Eq. (6) we have invoked the WKB approximation, viz. |∂t E| ωpi E, and replaced the driven ion number density perturbation by Zd nd1 in order to maintain the quasineutrality in our dusty plasma. Eqs. (4)–(6) are the desired set for studying the dynamics of the nonlinear DL waves in the presence of modulated IP waves in the sheath of an rf discharge. In the following, we seek localized solutions of our set in the frame ξ = x − V τ , where V is the constant velocity, by assuming that u and E are function of ξ and τ . Hence, Eqs. (4)–(6) for unidirectional propagation are written as ∂τ N +
C2 γ Cl Cl ∂ξ N 2 + L2d ∂ξ3 N + d ∂ξ |E|2 = 0 (7) 4 2 2Cl
and i∂τ E +
3VT2i 2 ωpi NE = 0, ∂ E− 2ωpi ξ 2
(8)
where V ∼ Cl ∼ vg and the electric field E is normalized by (16πni0 Ti )1/2 . Eqs. (7) and (8) are modified K–dV and NLS equations, respectively. In the absence of the IP waves, Eq. (7) gives a stationary solution in the form of a compressional dust density solitary wave. The latter is of the form [12] N = N sech2 ϕ,
(9)
284
P.K. Shukla / Physics Letters A 300 (2002) 282–284
where ϕ = (ξ − Vs τ )/δ, Vs = γ Cl N /6, and δ = √ 12/γ N Ld . It turns out that the soliton speed Vs (width δ) is directly (inversely) proportional to the soliton amplitude N . However, when the modulated IP waves interact with the nonlinear DL waves, there appear envelope solitons consisting of a dust density cavity N = −Nm sech2 θ,
(10)
and camel-shaped ion plasma wave intensity distribution 2 sech2 θ tanh2 θ, |E|2 = Em
(11)
which are the stationary solutions [7,8] of Eqs. (7) and (8). Here, Nm is the minimum (normalized) dust density at θ = |ξ − U τ |/λ = 0, Em is the maximum (normalized) value of the IP wave electric field at the center of the dust density cavity, U is a constant, and λ is the width of the localized dust density cavity. The exact forms of Nm , Em and λ can be deduced following Refs. [7–9]. We stress that the dust density cavitation occurs on account of the ponderomotive force (Fp ) of the IP waves. Finally, we note that when the nonlinear and dispersive terms in Eq. (7) are neglected, we then have Cd2 ∂ξ W 2 , (12) 2Cl where we have introduced E = W (ξ ) exp(−iΩτ ). Eq. (12) is coupled with the NLS equation 3VT2i 2 1 ∂ W + Ω − ωpi N W = 0. (13) 2ωpi ξ 2
∂τ N = −
Following Karpman [13,14] one can obtain stationary and nonstationary solutions of Eqs. (12) and (13) in the form of an inverted bell-shaped dust density cavity which traps localized IP wave electric field envelops having a secant-hyperbolic profile. To summarize, we have considered the parametric interaction between the nonlinear DL waves and large amplitude IP waves in a plasma whose constituents are weakly coupled ions and strongly correlated dust grains. It is found that the self-consistent interaction between DL and IP waves produces a dust density cavity which traps an envelope of IP waves. Hence, the
present investigation provides a novel mechanism for creating a dust density caviton by the ponderomotive force of pre-existing IP waves in the sheath of an rf plasma discharge.
Acknowledgement This work was partially supported by the European Commission (Brussels) through the contract No. HPRN-CT2000-00140 for carrying out the task of the Human Potential Research Training Network entitled “Complex Plasmas: The Science of Laboratory Colloidal Plasmas and Mesospheric Charged Aerosols”.
References [1] F. Melandsø, Phys. Plasmas 3 (1996) 3890. [2] B. Farokhi, P.K. Shukla, N.L. Tsintsadze, D.D. Tskhakaya, Phys. Lett. A 264 (1999) 318; B. Farokhi, P.K. Shukla, N.L. Tsintsadze, D.D. Tskhakaya, Phys. Plasmas 7 (2000) 814. [3] G. Morfill, H.M. Thomas, M. Zuzic, in: P.K. Shukla, D.A. Mendis, T. Desai (Eds.), Advances in Dusty Plasmas, World Scientific, Singapore, 1997, pp. 99–142. [4] A. Homann, A. Melzer, S. Peters et al., Phys. Lett. A 242 (1998) 173. [5] D. Samsonov, A.V. Ivlev, R.A. Quinn, G. Morfill, S. Zhdanov, Phys. Rev. Lett. 88 (2002) 095004. [6] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol, 2002. [7] K. Nishikawa, H. Hojo, K. Mima, H. Ikezi, Phys. Rev. Lett. 33 (1974) 148. [8] V.G. Makhankov, Phys. Lett. 50A (1974) 42. [9] R.K. Varma, N.N. Rao, Phys. Lett. 79A (1980) 311; N.N. Rao, P.K. Shukla, Phys. Plasmas 4 (1997) 636. [10] P.K. Shukla, Phys. Rev. Lett. 84 (2000) 5328. [11] P.K. Shukla, L. Stenflo, G. Morfill, IEEE Trans. Plasma Sci. 29 (2001) 198. [12] R.C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972; L. Stenflo, N.L. Tsintsadze, T.D. Buadze, Phys. Lett. A 135 (1989) 37. [13] V.I. Karpman, Plasma Phys. 13 (1971) 477; V.I. Karpman, Phys. Scripta 11 (1975) 263. [14] V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, Oxford, 1975, pp. 165–169.