# Refractive index of plasma dependance on temperature

How does the refractive index of plasma changes with temperature? Temperature is not high enough for new ionization. Would it be like in gas p/T dependence ?

http://farside.ph.utexas.edu/teaching/em/lectures/node100.html

## 1 Answer

Would it be like in gas p/T dependence ?

No, it is much more complicated than this.

How does the refractive index of plasma changes with temperature?

This is an extremely complicated question for numerous nuanced reasons, including (in no particular order):

1. Plasmas are often in a collisionless or weakly collisional state, meaning their dynamics are not governed by particle-particle collisions like the Earth's atmosphere.
2. They are rarely (if ever) in thermodynamic equilibrium, so it is difficult to characterize them with simple analogies to neutral gases.
3. The positively and negatively charged ions and electrons, respectively, are rarely (if ever) in equipartition of temperature (i.e., generally $T_{i} \neq T_{e}$).
4. Both ions and electrons can have multiple components (i.e., in the solar wind, electrons are composed of a cold dense core, a hot tenuous halo, a magnetic field-aligned beam called the strahl, and higher energy suprathermal components). So there's not just one temperature defining a plasma.
5. There can be multiple ion species and each can have their own properties (e.g., in the solar wind, generally $n_{\alpha}/n_{p} \sim 5%$, where $\alpha$ and $p$ denote alpha particles and protons, respectively, but $T_{\alpha}/T_{p} \gg 1$).
6. The temperature is a pseudotensor (i.e., the components are derived from the pressure tensor divided by the scalar number density) in a plasma, as I mention at https://physics.stackexchange.com/a/218643/59023. The temperature does not just vary as a scalar, it can be anisotropic, as I mention at What is the correct relativistic distribution function?.
7. The index of refraction in a plasma is dependent upon the frequency and wavenumber of a given wave mode and it is energy-dependent as well. Meaning, a low frequency mode like a magnetosonic wave propagates very differently through a plasma than, say, visible light. Thus, there isn't just one index of refraction in the sense that your question seems to imply.

So it's not just a matter of altering the temperature and observing a different index of refraction. All of the above complications can influence the index of refraction. There is still hope so, as I mention below, but treatment of plasmas with finite temperature can be very tricky.

# Simplest Approach

The simple approach involves looking the plasma beta, $\beta$, given by: $$\beta = \frac{2 \mu_{o} \ n_{o} \ k_{B} \left( T_{e} + T_{i} \right) }{B_{o}^{2}} \tag{1}$$ where $T_{s}$ is the average temperature of species $s$, $n_{o}$ is the charged particle number density, $B_{o}$ is the quasi-static magnetic field, $\mu_{o}$ is the permeability of free space, and $k_{B}$ is the Boltzmann constant.

Many wave modes are affected by $\beta$ and some depend upon $\beta_{j,s}$, which is the plasma beta of the $j^{th}$ component of particle species $s$. For instance, whistler mode waves have been found to depend upon the parallel (to the quasi-static magnetic field, $\mathbf{B}_{o}$) core beta of the electrons or $\beta_{\parallel,ce}$, which is given by: $$\beta_{\parallel,ce} = \frac{2 \mu_{o} \ n_{ce} \ k_{B} T_{\parallel,ce} }{B_{o}^{2}} \tag{2}$$ where $n_{ce}$ is the number density of the cold core and $T_{\parallel,ce}$ is the parallel (with respect to $\mathbf{B}_{o}$) component of cold core temperature.

Some modes have stronger/faster growth rates for large $\beta_{j,s}$ while others would damp for the same $\beta_{j,s}$. Short of listing all the scenarios, unfortunately, this is as detailed as the answer can be.

# Fun Side Note

One of the first things discovered by comparing a cold and hot plasma was what are now called Bernstein modes (indirectly discussed at https://physics.stackexchange.com/a/156813/59023). These are a form of cyclotron harmonic wave (for both ions and electrons) that exist solely because the plasma has a finite temperature. In their simplest approximation, they are electrostatic modes propagating orthogonal to $\mathbf{B}_{o}$ with $\Re{\left[ \omega \right]} = n \ \Omega_{cs}$, where $n$ is a single or multiple integers (i.e., they can have a single or multiple frequency peaks), $\Omega_{cs}$ is cyclotron frequency of species $s$. If they propagate at oblique angles to $\mathbf{B}_{o}$, then they become electromagnetic and their frequency falls between integer harmonics of $\Omega_{cs}$.

Some more details can be found at https://farside.ph.utexas.edu/teaching/plasma/lectures1/node91.html.

# Extra Details/References

An example study looked at the effect of a cold core on the growth threshold of the whistler anisotropy instability (i.e., whistler anisotropy instability = whistler mode waves radiated by an instability driven by the free energy due to a temperature anisotropy).

There has been a lot of work done on hot plasmas, including the following recent work:

To determine the index of refraction for a hot plasma, first you follow the same approach as I discussed at https://physics.stackexchange.com/a/138460/59023 and then find the dispersion relation similar to what I described at https://physics.stackexchange.com/a/264526/59023.

For a bi-Maxwellian velocity distribution, you can find all sorts of detailed derivations and results by googling the phrase harris dispersion relation.

When you look into some of these results, you will see why I did not list/write out the resulting index of refraction because it would fill several pages.

• Thanks. What happens if there were to exist a plasma (electrons and ions) in equilibrium and no external B field to confine the plasma? Just optical EM wave going through plasma with thermodynamic temperature as if i is defined for ideal gas. – Anonymous Jul 4 '16 at 18:05
• @Anonymous - Optical light frequencies are generally well above the local electron plasma frequency for most space plasmas (don't recall the values for lab plasmas off the top of my head). This means, those electromagnetic waves are effectively free modes, i.e., they propagate without interacting with the plasma. So to them, the index of refraction goes to ~1 like in vacuum (unless they are circularly polarized, in which case Faraday rotation becomes an issue). – honeste_vivere Jul 4 '16 at 19:49