What is the refraction index of ions in plasma ?
And why people do not discuss it in books ? Is it that it is not important for plasma phenomena ?
[Physics] Why in plasma physics usually is given the refractive index of electrons but not for ions
plasma-physics
Best Answer
Cold Plasma Index of Refraction
We start by deriving the expression for the dielectric tensor, as shown at https://physics.stackexchange.com/a/138460/59023.
If we consider the case of a cold uniform plasma with only linear waves, then one can show the dielectric tensor has the form: $$ \begin{align} \overleftrightarrow{\mathbf{K}} & = \left[ \begin{array}{ c c c } S & -i \ D & 0 \\ i \ D & S & 0 \\ 0 & 0 & P \end{array} \right] \tag{0} \end{align} $$ where the $S$, $D$, and $P$ terms are defined as: $$ \begin{align} S & = 1 - \sum_{s} \frac{\omega_{ps}^{2}}{\omega^{2} - \Omega_{cs}^{2}} \tag{1a} \\ & = \frac{1}{2} \left( R + L \right) \tag{1b} \\ D & = \sum_{s} \frac{ \Omega_{cs} \ \omega_{ps}^{2} }{\omega \left( \omega^{2} - \Omega_{cs}^{2} \right)} \tag{1c} \\ & = \frac{1}{2} \left( R - L \right) \tag{1d} \\ P & = 1 - \sum_{s} \frac{ \omega_{ps}^{2} }{ \omega^{2} } \tag{1e} \\ R & = 1 - \sum_{s} \frac{ \omega_{ps}^{2} }{ \omega \left( \omega + \Omega_{cs} \right) } \tag{1f} \\ L & = 1 - \sum_{s} \frac{ \omega_{ps}^{2} }{ \omega \left( \omega - \Omega_{cs} \right) } \tag{1g} \end{align} $$ where $\Omega_{cs}$ is the gyrofrequency (or cyclotron frequency) of species $s$, $\omega_{ps}$ is the plasma frequency of species $s$, and $\omega$ is the wave frequency. These parameters are defined as: $$ \begin{align} \Omega_{cs} & = \frac{ Z_{s} \ e \ B_{o} }{ \gamma \ m_{s} } \\ \omega_{ps} & = \sqrt{\frac{ n_{s} \ Z_{s}^{2} \ e^{2} }{ \varepsilon_{o} \ m_{s} }} \end{align} $$ where $Z_{s}$ is the charge state of species $s$ (e.g., +1 for protons), $e$ is the elementary charge, $B_{o}$ is the quasi-static magnetic field magnitude, $\gamma$ is the relativistic Lorentz factor, $m_{s}$ is the mass of species $s$, $n_{s}$ is the number density of species $s$, and $\varepsilon_{o}$ is the permittivity of free space.
Note that the $S$, $D$, $P$, $R$, and $L$ terms do have physical significances (e.g., $R$ and $L$ terms correspond to right- and left-hand polarized modes, respectively) but here we rely upon them mostly for brevity.
The dispersion relation, $D(\mathbf{k}, \omega)$, is derived from the equation: $$ \mathbf{n} \times \left( \mathbf{n} \times \mathbf{E} \right) + \overleftrightarrow{\mathbf{K}} \cdot \mathbf{E} = 0 $$ where we rewrite this in the tensor form $\overleftrightarrow{\mathbf{D}} \cdot \mathbf{E} = 0$. If $\overleftrightarrow{\mathbf{D}}$ has a determinant that goes to zero, then there is a non-trivial solution for $\mathbf{E}$. This solution is the dispersion relation, $D(\mathbf{k}, \omega)$, which can be simplified down if we assume the index of refraction, $\mathbf{n}$, is parallel to the wave vector, $\mathbf{k}$, then: $$ D\left( \mathbf{k}, \omega \right) = A \ n^{4} - B \ n^{2} + R \ L \ P = 0 \tag{2} $$ where the terms $A$ and $B$ are defined by: $$ \begin{align} A & = S \ \sin^{2}{\theta} + P \cos^{2}{\theta} \tag{3a} \\ B & = R \ L \ \sin^{2}{\theta} + P \ S \ \left( 1 + \cos^{2}{\theta} \right) \tag{3b} \end{align} $$ where $\theta$ is the angle between $\mathbf{k}$ and the quasi-static magnetic field. The dispersion relation in Equation 2 has the unique solutions of: $$ n^{2} = \frac{B \pm F}{2 \ A} \tag{4} $$ where $F$ is defined by: $$ F = \left( R \ L - P \ S \right)^{2} \sin^{2}{\theta} + 4 \ P^{2} \ D^{2} \ \cos^{2}{\theta} \tag{5} $$ We can see that $\Im{ [F] } = 0$, always. Since the terms $A$ and $B$ are real, then we can say that $n^{2}$ must either be purely real ($n^{2}$ $>$ 0) or purely imaginary ($n^{2}$ $<$ 0). If $n^{2}$ $<$ 0, then the wave becomes evanescent.
Warm Plasma Index of Refraction
Note that including a finite temperature for each species complicates things significantly, but does not result in a neglect of ions (often the opposite). There are numerous articles on the warm plasma dispersion relation so I won't go into detail here [e.g., Seough and Yoon, 2009]
Answers
I am not sure why you think only electrons are included in the index of refraction of plasmas. As you can see from my Equation 4 above, the index of refraction for a plasma includes electrons and all ion species.
Gurnett and Bhattacharjee, [2005] have several chapters on the index of refraction of various types of plasmas, and all of them discuss both ions and electrons. In some instances one can neglect ion contributions because, for instance, $\omega \gg \Omega_{ci}$ and $\omega \gg \omega_{pi}$ which results in all the ion contributions in the $S$ (Equation 1a), $D$ (Equation 1c), $P$ (Equation 1e), $R$ (Equation 1f), and $L$ (Equation 1g) terms being much much smaller than the electron contributions.
In summary, good plasma physics books will not ignore the ion contributions unless they are genuinely negligible for very specific circumstances.
No, quite the opposite. For instance, the dispersion relation for electromagnetic ion cyclotron (EMIC) waves is almost entirely dominated by the ion contributions. There are numerous plasma modes that rely heavily upon the ion contributions.
So as you can see, ion contributions to the index of refraction in plasmas can be very important or negligible, it just depends upon the specific circumstance.
References