Plasma Oscillations – “Langmuir Wave” approximation derivation

In a previous article we established an approximate estimate for the frequency of oscillations in a “cold plasma” because of small disturbances in the arrangement of the electrons and ions. In this article we will establish a much better model for oscillations in a plasma, which takes into consideration the fact that Plasmas tend to be hot (the thermal factor). To do this we establish a differential equation using facts we can take as a given. Firstly note that we model our plasma as a fixed lattice of ions, with moving electrons. These moving electrons cause the plasma oscillations we are trying to model. The frequency at which a hot plasma oscillates is called the Langmuir frequency, and the method detailed in this article attempts to find an estimate of this frequency.

We start by looking at the conservation of charge equation:

 e \frac{\partial n }{\partial t}+ \frac{\partial (nev) }{\partial x}=0

Where n is the number density of particles in the plasma, and e is the charge on the electron. We notice that the charge of the electron is clearly a constant, so it can be factored out of the derivative with respect to x, and we can divide through by this electron charge to get the following:

 \frac{\partial n }{\partial t}+ \frac{\partial (nv) }{\partial x}=0

It is clear that we have the derivative of a product with respect to x. We can therefore expand this using the product rule:

 \frac{\partial n }{\partial t}+ v\frac{\partial (n) }{\partial x} + n\frac{\partial (v) }{\partial x}=0

We next assume that the variation in velocities and number density is small. Doing so we can state the following:

 n= n_{0}+{n}' and  v= v_{0}+{v}'

Where n_{0} is the constant number density when the plasma is still, and v_{0} is the velocity of the electrons when the plasma is still (0). The terms {n}'  and {v}'  represent small changes in their respective quantities. We now have a general expression for both the number density and the velocity of the electrons in terms of a constant “still state” and small a small variation in each. We can therefore substitute into the conservation of charge equation for both n and v (remembering that v_{0}=0):

 \frac{\partial (n_{0}+{n}') }{\partial t}+{v}'\frac{\partial (n_{0}+{n}') }{\partial x} + (n_{0}+{n}')\frac{\partial ({v}') }{\partial x}=0

 \frac{\partial ({n}') }{\partial t}+{v}'\frac{\partial ({n}') }{\partial x} + (n_{0})\frac{\partial ({v}') }{\partial x}+{n}'\frac{\partial ({v}') }{\partial x}=0

We can now neglect the derivatives in the form n}'\frac{\partial ({v}') }{\partial x} and {v}'\frac{\partial ({n}') }{\partial x}, as these are second order terms and will be negligible due to the assumptions underlying this approximation of the Langmuir frequency:

 \frac{\partial ({n}') }{\partial t}+ (n_{0})\frac{\partial ({v}') }{\partial x}=0

We now move on to the equation of motion of the fluid. To derive this for our plasma we take the general equation of motion for a small volume of fluid:

 \rho _{m}\frac{dv}{dt}=-\nabla \vec{p}+\frac{dF_{ext}}{dV}

This is simply derived from Newton’s second law, F=ma, where \rho _{m} is the mass density, thus the term \rho _{m}\frac{dv}{dt} corresponds to ma, per unit volume. This means that the right hand side corresponds to the force on the fluid, and it does. The term \frac{dF_{ext}}{dV} corresponds to the force on a small volume dV per unit volume. The term -\nabla \vec{p} (which means the vector gradient of the pressure on the fluid) is an extra force the fluid feels due to changes in pressure at the edges of the small volume dV. We can simplify the equation of motion like so:

 nm\frac{\partial v}{\partial t}=-\frac{\partial n}{\partial x}E_{kin}+f

We now only work in the x direction so we can drop vector notation. To get to this equation (above), I substituted for the mass density, in terms of the number density and the mass, simply called the \frac{dF_{ext}}{dV} term f for simplicity, and replaced pressure with nE_{kin} as this is the pressure in a gas. We can therefore take the vector grad (remembering we are only working in the x direction) to get \frac{\partial n}{\partial x}E_{kin}

We can now make the same substitutions as before (with number density and velocity in terms of small changes in such quantities) to get the following (remembering that derivative of constants is 0):

 (n_{0}+{n}')m\frac{\partial {v}'}{\partial t}=-\frac{\partial (n_{0}+{n}')}{\partial x}E_{kin}+f

 n_{0}m\frac{\partial {v}'}{\partial t}+{n}'m\frac{\partial {v}'}{\partial t}=-\frac{\partial ({n}')}{\partial x}E_{kin}+f

We can again neglect second order terms, like before leaving us with:

 n_{0}m\frac{\partial {v}'}{\partial t}=-\frac{\partial ({n}')}{\partial x}E_{kin}+f

We now have the following two equations, with some similar terms, we can therefore look to make a substitution: 

n_{0}m\frac{\partial {v}'}{\partial t}=-\frac{\partial ({n}')}{\partial x}E_{kin}+f and \frac{\partial ({n}') }{\partial t}+ (n_{0})\frac{\partial ({v}') }{\partial x}=0

If we take the temporal derivative of the modified conservation of charge equation \frac{\partial ({n}') }{\partial t}+ (n_{0})\frac{\partial ({v}') }{\partial x}=0, an interesting substitution becomes possible:

\frac{\partial }{\partial t}(\frac{\partial {n}'}{\partial t}+n_{0}\frac{\partial {v}'}{\partial x}=0)=\frac{\partial^2 {n}'}{\partial t^2}+n_{0}\frac{\partial }{\partial x}(\frac{\partial {v}'}{\partial t})=0

The term \frac{\partial {v}'}{\partial t} now appears in both the equations we previously derived, allowing us to make a substitution. We first get make \frac{\partial {v}'}{\partial t} the subject of the motion equation:

\frac{\partial {v}'}{\partial t}=-\frac{\partial {n}'}{\partial x}\frac{E_{kin}}{n_{0}m}+\frac{f}{n_{0}m}

We can now substitute for \frac{\partial {v}'}{\partial t} in the other equation:

\frac{\partial^2 {n}'}{\partial t^2}+n_{0}\frac{\partial }{\partial x}(-\frac{\partial {n}'}{\partial x}\frac{E_{kin}}{n_{0}m}+\frac{f}{n_{0}m})=0

\frac{\partial^2 {n}'}{\partial t^2}-\frac{\partial^2 {n}'}{\partial x^2}\frac{E_{kin}}{m}+\frac{\partial f}{\partial x}\frac{1}{m}=0

We now have a differential equation in {n}' that we can solve. To do so we have to consider what the f term represents once again. We defined f to be equal to \frac{dF_{ext}}{dV} where F_{ext} is the external force on the charged particles in our plasma. When considering this external force we can use simple electric fields to determine its magnitude as Ee with e being the charge on the electron. Since f=\frac{dF_{ext}}{dV} thus \frac{f}{n_{0}}=\frac{dF_{ext}}{dN}. Since F_{ext}=Ee, and f=n_{0}\frac{dF_{ext}}{dN} we can simplify the differential equation. Also \frac{dF_{ext}}{dN}={n}'_{e}F_{ext} because change in force per particle is equal to the external force multiplied by the small change in electron number density. This is because the change in electron number density is what creates the electric field, thus force. As the plasma is modelled as a static lattice of ions with moving electrons. So:

\frac{\partial^2 {n}'}{\partial t^2}-\frac{\partial^2 {n}'}{\partial x^2}\frac{E_{kin}}{m}+n_{0}{n}'_{e}\frac{\partial Ee}{\partial x}\frac{1}{m}=0

The derivative of the electric field strength with respect to x is given by the following- \frac{\partial E}{\partial x}= \frac{\rho_{e}}{\varepsilon_{0}} (where\rho_{e} is the charge density of the particles in the plasma), thus we can replace this in our equation: 

\frac{\partial^2 {n}'}{\partial t^2}-\frac{\partial^2 {n}'}{\partial x^2}\frac{E_{kin}}{m}+\frac{n_{0}{n}'_{e}e\rho_{e}}{m\varepsilon_{0}}=0

We know that charge density (\rho_{e}) is the charge per unit volume thus \rho_{e}=en_{e} (where n_{e} is the electron number density) allowing us to find an interesting result:

 \frac{\partial^2 {n}'}{\partial t^2}-\frac{\partial^2 {n}'}{\partial x^2}\frac{E_{kin}}{m}+n_{0}{n}'_{e}(\frac{e^2n_{e}}{m\varepsilon_{0}})=0

The term \frac{e^2n_{e}}{m\varepsilon_{0}} is the square of the cold plasma oscillation frequency we worked out here (\omega_{pl}) we can therefore substitute this in, and once finished we will have an approximation of the Langmuir frequency, in terms of the cold plasma frequency, allowing us to see how the thermal energy of the plasma affect the oscillating frequency:

 \frac{\partial^2 {n}'}{\partial t^2}-\frac{\partial^2 {n}'}{\partial x^2}\frac{E_{kin}}{m}+n_{0}{n}'_{e}\omega_{pl}^2=0

We now need to solve the differential equation, which ends up being very simple. We just have to assume that the changes in electron number density can be modelled as a plane wave, {n}'_{e}=Ae^{i(\omega t-kx)} (where k is the wave vector). We also recognise that the other changes in number densities in the equation also must correspond to changes in electron number densities, as explained above- it’s the changes in electron number density that matter. We can therefore find the required partial derivatives of the plane wave to substitute in for the \frac{\partial^2 {n}'}{\partial t^2} and \frac{\partial^2 {n}'}{\partial x^2} terms, as these are simple to work out:

\frac{\partial^2 {n}'}{\partial t^2}= \frac{\partial^2 }{\partial t^2}(Ae^{i(\omega t-kx))})= i^2\omega^2Ae^{i(\omega t-kx)}=-\omega^2Ae^{i(\omega t-kx)}

\frac{\partial^2 {n}'}{\partial x^2}= \frac{\partial^2 }{\partial x^2}(Ae^{i(\omega t-kx))})= i^2k^2Ae^{i(\omega t-kx)}=-k^2Ae^{i(\omega t-kx)}

Then we can substitute this all back into our differential equation:

-\omega^2Ae^{i(\omega t-kx)}+k^2Ae^{i(\omega t-kx)}(\frac{E_{kin}}{m})+Ae^{i(\omega t-kx)}(n_{0}\omega_{pl}^2)=0

The plane wave term(Ae^{i(\omega t-kx)}) is now common to all the terms so we can cancel it:

k^2(\frac{E_{kin}}{m})-\omega^2+(n_{0}\omega_{pl}^2)=0

\omega^2=k^2(\frac{E_{kin}}{m})+(n_{0}\omega_{pl}^2)

We now have a function for the Langmuir frequency we are trying to derive, but we can make a few further simplifications. The kinetic energy of the electrons in a hot plasma is equal to \frac{1}{2}mv_{therm}^2 where v_{therm} is the thermal velocity of the electrons (velocity due to thermal energy). This means that if we substitute the kinetic energy in terms of the thermal energy, the mass of the electron cancels. Furthermore n_{0}, which was the number density of the electrons at an equilibrium point, is equal to Zn_{i} where Z is the atomic number of the ions in the plasma, and n_{i} is their number density. This can be said because at equilibrium the number of ions is equal to the number of electrons in the plasma. Thus:

\omega^2=k^2(\frac{v_{therm}^2}{2})+(Zn_{i}\omega_{pl}^2)

Therefore an approximation to the Langmuir frequency of oscillation of a plasma can be given by the following:

\omega=\sqrt[]{k^2\frac{v_{therm}^2}{2}+Zn_{i}\omega_{pl}^2}

 

**Disclaimer- This represents my own work as part of a physics challenge. I cannot be sure of the validity of each step of my derivation, so I encourage anybody reading this to give me any tips if something isn’t quite correct. Thank you.

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