The Altar-Appleton-Hartree dispersion relation is given by: $$ begin{align} n^{2} & = 1 - frac{ omega_{pe}^{2} }{ omega^{2} } frac{ 2 left( omega^{2} - omega_{pe}^{2} right) }{ 2 left( omega^{2} - omega_{pe}^{2} right) - Omega_{ce}^{2} sin^{2}{theta} pm Omega_{ce} Delta } tag{0a} Delta & = sqrt{ Omega_{ce}^{2} sin^{4}{theta} + 4 left( frac{ omega^{2} - omega_{pe}^{2} }{ omega } right)^{2} cos^{2}{theta} } tag{0b} end{align} $$ where $n^{2}$ is the index of refraction squared, $omega$ is the wave/mode angular frequency, $omega_{pe}$ is the electron plasma frequency, $Omega_{ce}$ is the electron cyclotron frequency, and $theta$ is the wave normal angle (i.e., the angle of the wave vector, $mathbf{k}$, with respect to the quasi-static magnetic field, $mathbf{B}_{o}$).

Has anyone come across such a plot before...

Yes, it is a standard plot in most plasma physics text books [e.g., see Stix, 1992].

Obviously it's not an accurate plot you can take measurements from...

Why not? It's a valid theoretical plot within the limits of the approximation.

...could you show me where to find it or even give me a bit more information on the topic.

Yes, look up plasma physics text books like Stix [1992] or Gurnett and Bhattacharjee [2005].

In the plot it showed at different plasma frequencies the various types of plasma phenomena such as Alfven waves, Helicon waves, the cyclotron frequency etc.

I am a bit confused by this though. The Altar-Appleton-Hartree dispersion relation is specific to high frequency, cold plasma modes. You can see this from Equations 0a and 0b above which only have electron terms, no ion terms. Alfven waves are very low frequency modes. Are you sure they lecturer was not presenting a more generic thing called a Clemmow-Mullay-Allis diagram (or CMA diagram)? These cover all waves/modes in cold plasmas.

It wasn't a continuous function, it had a vertical asympitote at the cyclotron frequency.

Of course not, there are resonances and cutoffs in these types of dispersion relations. A cutoff occurs when $n^{2} rightarrow 0$ and a resonance occurs when $n^{2} rightarrow infty$.

The x-axis didn't increase linearly either due to the massive differences in plasma frequency for different phenomena.

Okay, this is definitely sounding more like a CMA diagram than one derived from the Altar-Appleton-Hartree dispersion relation unless it was in the limit where $omega_{pe} gg Omega_{ce}$ (i.e., sometimes called the high density limit), in which case there can be a rather large separation between cyclotron and plasma modes.

References

• D.A. Gurnett and A. Bhattacharjee, Introduction to Plasma Physics: With Space and Laboratory Applications, Cambridge University Press, The Edinburgh Building, Cambridge CB2 8RU, UK, 2005.
• T.H. Stix, Waves in Plasmas, Springer-Verlag New York, Inc., New York, NY, 1992.