At low temperatures it is not a bad model to take a plasma as a collection of ions and electrons, with the ions always stationary, and the electrons oscillating, due to the force upon them due to the opposing charges of the electrons and ions. For this derivation we will assume that the ions ans the electrons have the same charge, and the same number densities. This will make the derivation of the oscillation frequency much simpler. Furthermore we are neglecting all thermal factors, as this is a “cold plasma” oscillation.
Lets consider the ions fixed at the centre of of 3 dimensional space (x,y,z). The ions lie in the y-z plane at x=0. Let’s also assume that the ions are uniformly distributed in this plane. The electrons are distributed throughout our space, but let’s say all electrons with x>0 are initially displaced “d” in the positive direction, and all electrons with x<0 are initially displaced “d” in the negative direction. There will be an excess of charge due to the ions at x=0, creating an electric field with width “2d” and pulling the electrons back towards the origin.
An electron at displacement of “+ or -d” will experience a force towards “x=0” due to the excess positive charge there because of the ions located at “x=0”. This will cause the electron to accelerate towards “x=0”, with this acceleration getting larger the closer it gets to the origin. Assuming the electrons don’t interact with the ions and form some sort of electrically stable particle, and assuming they don’t interact with the other electrons coming from the opposite direction, they will continue to move past the origin. At this point the force on them is again towards the origin, thus accelerating them towards the origin, and slowing them down. At some point (displacement “d”) velocity will be equal to 0 and the particles will then, again, begin accelerating towards the origin. This is oscillatory motion, as the force and hence, acceleration always act towards a centre point. If the planes are said to have a border, towards the border regions, the field lines begin to curve, and so there will be some movement along the y and z axis from the electrons, but this will be small, and the movement will remain symmetrical and oscillate about the plane. However, the model shown in our derivation works for assumptions that the planes continue to infinity, and work as good approximations away from the border of any finite plane.
We can model the ions as a plane. This is because all of the ions are confined to “x=0” and are uniformly distributed in the plane. Using Gauss’s law, we can model the electric field strength of this plane with area A:
We know the number density of the ions, as it is the same as the number density given for the electrons (denoted by the letter n with subscript e). As the number density describes the number of particles per unit volume, given a volume we can work out the number of ions:
Re-writing the volume in terms of an area and “d” the extent of the excess of positive charge in one direction, which we also know is the same as the initial displacement of the electrons (as the width of the excess charge is given as 2d, and this extend for both positive and negative displacements):
The force on a charge in an electric field is given by:
Applying this to an electron:
We can then substitute for E, the field strength, defined before:
In an oscillating system the force is given by the following:
Given that this force is going to cause the oscillatory movement for an electron with initial displacement “d”:
As the only force acting on the electrons is the force due to the electric field, the force causing the oscillatory motion is entirely supplied by the electric field, and it’s possible to equate the two:
Solving for omega will give us the plasma oscillation frequency:
This gives us the following for the plasma frequency (omega subscript pl):
This is only a rough estimate of the oscillation frequency in a plasma, taking into consideration the thermal factor gives you a much better approximation, which we will discuss in an article coming very soon.
This derivation was part of the Princeton University Physics Challenge 2016, undertaken by a few classmates and myself. We were given a week to solve various problems in plasma physics, including this derivation.
**Disclaimer: these solutions represent the work of me and my team from PUPC 2016 and are not endorsed as the official solutions by the PUPC organizers.
For more information email the organizers at firstname.lastname@example.org
Learn more and maybe compete in the PUPC next year: http://pupc.princeton.edu/Online.php
UPDATE: The team and I ranked 17th in the world in this international challenge having found out about it the night before, so we are really happy!!