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A rigorous theoretical investigation of linear dust ion acoustic (DIA) solitary waves in an unmagnetized complex plasma consisting of ion and ion beam fluids, nonthermal electrons that are Cairns distributed and immobile dust particles were undertaken. It was found out that, for large beam speeds, three stable modes propagated as solitary waves in the beam plasma. These were the “Fast”, “Slow” and “Ion-acoustic” modes. For two stream instability to occur between ion and ion beam, it is shown that or when .

In a complex plasma, ion beam can significantly affect the propagation charateristics of solitary waves [

In the initial study by [

For a population with excess fast particles, the Cairns distribution was introduced by Cairns et al. (1995) to analyse the effect of particles on solitary waves [

The Cairns distribution is often given as [

where

The Cairn’s distribution function is shown in

It can be clearly seen that, for

When the values are above 0.571, the Cairns distribution ceases to be monotonically decreasing. The critical values of

In this model we considered a collisionless, un-magnetized plasma model that consists of non-thermally distributed electrons that follow the Cairns distribution and negatively charged dust particles that are stationary. In addition, it consists of inertial warm ions and ion-beams of equal mass. The massive dust grains are consi- dered stationary with no charge fluctuations and therefore, will only affect the equilibrium charge neutrality.

Charge neutrality at equilibrium requires that

where

beam) and

and

here,

assumption, in Equation (2), and the system of Equations (3), (4), and (5) will be closed by Poisson’s equation expressed as

where

The basic equations i.e., Equations (3), (4), and (5) were first linearlized. During the linearization process, it was assumed that the pertubations vary as

The electron density is obtained by integrating the Cairns distribution function as

where

The ion density,

But

From Poisson’s Equation, we have

Finally,

where

Normalizing k with

where

In the presence of an ion beam, three longitudinal electrostatic waves involving ion motion could propagate; these were, an ion acoustic mode (IA), fast (F) and slow (S) modes. This is in accordance with the experimental

observations of [

The right hand side (RHS) of Equation (11) is a quartic equation in

the RHS

which the wave is unstable. The maximum values exist between

Since the coefficients of Equation (11) are real, there are two complex roots which are complex conjugates to

each other, i.e.,

with a growth rate of

Further theoretical analysis of the dispersion relation (Equation (11), revealed that, in the limiting case of

This implies that the phase speed of an ion acoustic wave increased in the prescence of nonthermal electrons. However, in the abscence of nonthermal electrons, i.e.,

which in the long wavelength limit, i.e.,

For cold ions,

This is similar to the phase speed obtained by [

Numerical examination of the dispersion relation in Equation (11) for the three wave modes propagating along the beam is shown in

The study findings show that on close examination of the derived dispersion relation there are three longitu- dinal electrostatic modes involving ion motion that propagates. These were ion acoustic, fast and slow modes.

Thus for the two stream instability to occur between ion and ion beam, then it’s

while the IA-acoustic mode remained unaffected. Ion beam temperature changes have the same effect but slightly less as compared to ion beam speed. Ion beam density ratio has no effect on the phase speed of all modes while ion beam temperature ratio affected the IA- mode only.

These theoretical findings could be useful in determining onset of instability in laboratory ion beam driven plasmas as well as space plasmas.

Author 1 acknowledges the funding from East African Astronomical Research Network (EAARN) supported by International Science Program (ISP).

I.Habumugisha,S. K.Anguma,E.Jurua,N.Noreen, (2016) Onset of Linear Instability in a Complex Plasma with Cairns Distributed Electrons. International Journal of Astronomy and Astrophysics,06,1-7. doi: 10.4236/ijaa.2016.61001