Waves in Plasma

In plasma physics, a wave in plasma is a disturbance that propagates through a plasma medium, consisting of charged particles (ions and electrons) and neutral particles. These waves can take many forms, depending on the properties of the plasma, such as its density, temperature, magnetic field, and electric field.

There are many types of plasma waves, but some of the most common ones are:

1. Electromagnetic waves: These waves are also known as light waves or radio waves, and they can be either transverse or longitudinal. They are created by the oscillation of electric and magnetic fields and can travel through a vacuum or a plasma.

2. Langmuir waves: These waves are also known as electron plasma waves and are created by the oscillation of electrons in the plasma. They are usually transverse and have a frequency close to the plasma frequency, which depends on the density of the plasma.

3. Alfvén waves: These waves are created by the oscillation of magnetic fields in the plasma and are named after Hannes Alfvén, who first predicted their existence. They are usually transverse and can travel through a plasma even in the absence of an electric field.

4. Ion acoustic waves: These waves are created by the oscillation of ions in the plasma and are usually longitudinal. They have a frequency close to the ion plasma frequency, which depends on the density and temperature of the plasma.

The study of plasma waves is important for understanding many phenomena in plasma physics, such as plasma heating, particle acceleration, and plasma confinement in fusion devices.

If the electron in a plasma are displaced from a uniform background of ions, electric field will be built up in such a direction as to restore the neutrality of the plasma by pulling the electrons in their original position. Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions with a characteristics frequency is known as plasma frequency. This frequency (oscillation) is so fast that the massive ions do not have time to respond to the oscillating field and may be considered as fixed. The frequency of the plasma oscillation depends on the density of the plasma and the charge of the particles. The plasma frequency is a fundamental property of the plasma, and it describes how fast the electrons will oscillate back and forth when an electric field is applied.

In other words, if you imagine a bunch of ping-pong balls floating in water, and you push them in one direction, they will bounce back and forth, creating a wave. The same thing happens in a plasma, except instead of ping-pong balls, you have electrons.

The plasma oscillation plays a significant role in many phenomena in plasma physics, including plasma heating and particle acceleration. For example, in a plasma wave accelerator, an intense electric field is used to create a plasma oscillation, which can accelerate charged particles to very high speeds. This process has the potential to revolutionize particle accelerator technology and could lead to new discoveries in fields like energy, medicine, and space exploration.

In figure the open rectangles represents typical elements of ions fluid and the shaded rectangles represents displaced elements of electrons fluid. The resulting charge causes a spatially periodic field which tends to restore the electrons to their neutral position.

To derive the plasma frequency, the following assumption are made

1. The electron are at zero temperature i.e there are no thermal motion (KT = 0)

2. There is no collision

3. The ions are stationary

4. The plasma is infinite in extent

5. The electron motions occurs only in x direction

We can write the electron equation of motion is

The Maxwell equation that doesn't involve B is the Poisson equation

In one dimension,

To solve above equations, linearization method is used which means the amplitude of oscillation is small and higher powers of amplitude factors can be neglected. Now separating the variable into two parts

1. The equilibrium part indicated by a subscript 0 and

2. The perturbed part indicated by a subscript 1

Therefore,

In neutral uniform plasma at equilibrium and at rest

Vo = Eo = ∇no = 0

Now, equation 1 becomes,

The equation can be solved for w as

And current is given as

j = nev

So conductivity is

And dielectric constant is given as

Therefore, for dispersion relation is

This equation shows that the plasma frequency is proportional to the square root of the electron density, which means that a higher density plasma will have a higher plasma frequency. It also shows that the plasma frequency depends on the charge and mass of the electrons, as well as the properties of the vacuum.

The plasma frequency is an important quantity in plasma physics because it determines the behavior of electromagnetic waves and plasma oscillations in a plasma. For example, if an electromagnetic wave has a frequency that is lower than the plasma frequency, it will be reflected by the plasma and cannot propagate through it. On the other hand, if the frequency of the wave is higher than the plasma frequency, it can propagate through the plasma, but it will be modified by the presence of the plasma.

This note is a part of the Physics Repository.