Excitation of dipole oscillons in a dusty plasma containing elongated dust rods

The dispersion properties and excitation mechanisms of `dipole oscillons' in a dusty plasma containing charged elongated rod-like dust grains are investigated in the presence of streaming plasma particles for cases without and with an external static magnetic field. In a magnetized dusty plasma, a new `oscillon-ion lower-hybrid' mode is found, which can be excited by the equilibrium energy of cross-field drifting ions.


Introduction
Waves and instabilities occupy a significant part of modern plasma physics research because most properties of plasmas are related to the fundamental wave modes in laboratory and space plasma systems. In recent years, numerous studies have been confined to dusty plasmas having electrons, ions, and charged dust grains of spherical shapes [1]- [5]. However, in cosmic environments, dust particles of different sizes and shapes are quite common [2,6]. The observed infrared and submillimetre radiation are attributed to the thermal emission from dust clouds heated by shock waves, the universal ultraviolet radiation, or stellar radiation, etc. Elongated dust grains are assumed to be formed by coagulation of smaller particulates in partially or fully ionized plasmas by some attractive forces [7,8], the details of which are not yet fully understood. However, it is thought that inelastic, adhesive and collective interactions between micron-sized charged dust particles give rise to kilometre-sized bodies, which are known as planetesimals. Results from a microgravity aggregation experiment [9] flown onboard the space shuttle revealed the structure and growth of dust agglomeration. Specifically, Blum et al [9] reported that a thermally aggregating swarm of dust particles evolves very rapidly and forms unexpected open-structure agglomerates.
Obviously, the dust grains formed in laboratory and astrophysical environments by nucleosynthesis and coagulation may have any shape and size. Recently, many authors [10]- [12] have investigated the electrodynamics and dispersion properties of dusty plasmas whose constituents are electrons, ions and elongated, rotating charged dust grains in the absence and/or in the presence of external magnetic fields. There is a distribution of charges on rod-like dust particles, and an inhomogeneous charge distribution on the dust grain surface produces a finite dust dipole moment, which introduces new physics in dusty plasmas. In particular, it has been shown that oscillating dipoles of elongated charged dust rods give rise to a new wave mode, which is referred to as the 'dipole oscillon' [13]. The frequency of the latter is k ⊥ λ D el , where k ⊥ is the wave-vector component of the dipole oscillon perpendicular to the direction of the dust rod alignment [14,15], λ D is the effective plasma Debye radius, el = 4πd 2 n d0 /I 1/2 , d is the magnitude of the dipole moment, n d0 is the equilibrium number density of elongated dust rods, and I is the moment of inertia of the dipole. We note that the dipole oscillon is a compressional mode, since it propagates in a direction nearly perpendicular to the dust rod alignment. Physically, 'dipole oscillons' arise due to the combined action of the restoring force, which comes from the pressure of inertialess electrons and ions, and the moment of inertia of dust dipoles that oscillate around their equilibrium position.
In this paper, we present a rigorous study on possible electrostatic waves and their associated instabilities in a streaming dusty plasma containing electrons, ions, elongated charged dust rods, and neutrals in the presence/absence of an external magnetic field. The paper is organized as follows. In section 2, we consider the unmagnetized case in collisionless and collisional limits. In section 3, the effect of an external uniform magnetic field on the waves and instabilities of dusty plasmas having nonspherical elongated charged dust grains is presented. Finally, a brief discussion of our results is contained in section 4.

Unmagnetized dusty plasmas
Here, we study the dust dipole oscillations in an unmagnetized dusty plasma and their associated instabilities in the presence of streaming plasma particles, in both collisionless and collisional 40.3 limits. The dispersion properties of electrostatic waves (ω, k) in a dusty plasma are governed by where ω and k are the frequency and the wavevector, respectively. The plasma susceptibilities for electrons and ions (j = e, i) are [1] where λ Dj = (T j /4πn j 0 Q 2 j ) 1/2 is the Debye radius, Q j is the charge, T j is the temperature, Z is the plasma dispersion function of Fried and Conte [16], is the thermal speed of the species j , m j is the mass, ν jn is the collision frequency between the species j and neutrals, and u j 0 is the uniform streaming velocity of the j th species. In equation (1), χ d is the dielectric susceptibility of the elongated dust rods [11]- [15] given by −k 2 ⊥ 2 el /k 2 ω 2 . The latter holds if the wave frequency is much larger than the dust plasma frequency, since the motion of elongated dust rods has been ignored. Accordingly, dust oscillons are decoupled from the dust acoustic waves.

Collisionless streaming plasmas
Here, we consider electrons having a Boltzmann distribution while the ions are streaming with a uniform velocity u i0 ẑ parallel to the direction of the dust rod alignment. The streaming ion motion is caused by the presence of a dc electric field, as in the sheath region of a laboratory rf discharge. We focus on waves with kv td |ω| ν en kv te and |ω − k u i0 + iν in | kv ti , where ν en and ν in are collision frequencies of electrons and ions with neutral atoms/molecules. Accordingly, from equations (1) and (2) we obtain where k is the z-component of the wavevector k, and the z axis is parallel to the direction of the alignment of dust dipole rods.

Collisional dusty plasmas
We now consider waves with |ω|, ν en kv te and |ω − k u i0 | ν in kv ti . Here, hot electrons follow a Boltzmann distribution and we have χ e ≈ 1/k 2 λ 2 De . On the other hand, the ions are collisional. In such a situation, equations (1) and (2) give where C re = el λ De and C s = ω pi λ De . Letting ω = ω r + iω i in equation (6), where ω i ω r , we obtain and Equation (10) exhibits instability if u i0 > ω r /k .

Effect of collisions on the ion-dust two-stream instability
For streaming ions, hot electrons, and cold dust rods, the conditions ν en |ω| kv te , k u i0 and ν in k u i0 , are usually satisfied in a collisional plasma. Here, we consider the case of a strong ion flow compared to the conditions described in the previous subsection. Using, equations (1) and (2), and the conditions mentioned above, we obtain the following dispersion relation Neglecting collisions, namely ν in 0, and assuming k u i0 |ω|, equation (11) gives a Buneman-type instability with ω r ∼ ω i , where where A = 1 + 1/k 2 λ 2 De . On the other hand, for ν in |ω| and ν in k u i0 , we obtain from equation (11) which predicts an oscillatory instability in a collisional dusty plasma.

Magnetized plasmas
Here, we consider a dusty plasma containing nonspherical elongated rod-like grains in an external magnetic field B 0 = B 0ẑ . The dispersion properties of electrostatic waves in our dusty magnetoplasma are governed by (ω, k) ≡ 1 + χ e + χ i + χ d = 0, where the plasma susceptibilities for electrons and ions (j = e, i) are now given by [1] In equation (15), ξ jn = (ω − k · u j 0 + iν jn − nω cj )/ √ 2k v tj , ω cj = |Q j B 0 /m j c| is the gyrofrequency, c is the speed of light in vacuum, n = I n (b j ) exp(−b j ), with I n being the modified Bessel function of order n, b j = k 2 ⊥ ρ 2 j and ρ j = v tj /ω cj is the thermal gyroradius.

Cold plasma limit
In the cold plasma limit for magnetized electrons and ions, the approximations k v te |ω − k y u 0 + iν en,in | ω ci ω ce with b e , b i 1 are usually valid. Then, the dispersion relation becomes Neglecting the streaming of electrons and ions and collision frequencies, and assuming k k ⊥ and ω 2 pi ω 2 ci , we obtain from equation (16) a new mode which we call the 'oscillonion-lower-hybrid (OILH)' wave whose frequency is The damping rate of the OILH mode including electron-neutral collisions is For negligible electron-neutral collision frequency, namely |ω − k y u 0 | ν en and assuming ω ≡ k y u 0 + δ with δ k y u 0 , equation (16) reduces to where and Balancing the first and third terms in equation (19), we have k y u 0 ω osc and ω r = el ω ci /ω pi for k ⊥ ∼ k. This is the real part of the OILH mode frequency. Balancing the small second and fourth terms, we obtain the growth rate of the OILH mode as