stix

plasmapy.dispersion.analytical.stix_.stix(B: Quantity, w: Quantity, ions: Particle, n_i: Quantity, theta: Quantity)[source]

Calculate the cold plasma dispersion function presented by Stix [1992], and discussed by Bellan [2012]. This is an analytical solution of equation 8 in Bellan [2012]. See the Notes section below for additional details.

Parameters:

  • B (Quantity) – The magnetic field magnitude in units convertible to T.

  • w (Quantity, single valued or 1-D array) – Wavefrequency in units convertible to rad/s. Either singled valued or 1-D array of length \(N\).

  • ions (a single or list of particle-like object(s)) – A list or single instance of particle-like objects representing the ion species (e.g., "p" for protons, "D+" for deuterium, ["H+", "He+"] for hydrogen and helium, etc.). All ions must be positively charged.

  • n_i (Quantity, single valued or 1-D array) – Ion number density in units convertible to m-3. Must be single valued or equal length to ions.

  • theta (Quantity, single valued or 1-D array) – The angle of propagation of the wave with respect to the magnetic field, \(\cos^{-1}(k_z / k)\), in units convertible to radians. Either single valued or 1-D array of size \(M\).

Returns:

k – An array of wavenubmers in units rad/m (shape \(N \times M \times 4\)). The first dimension maps to the w array, the second dimension maps to the theta array, and the third dimension maps to the four roots of the Stix polynomial.

  • k[0] is the square root of the positive quadratic solution

  • k[1] = -k[0]

  • k[2] is the square root of the negative quadratic solution

  • k[3] = -k[2]

Return type:

Quantity of shape (N, M, 4)

Raises:

  • TypeError – If applicable arguments are not instances of Quantity or cannot be converted into one.

  • ValueError – Particles in ions are not positively charged ions.

  • ValueError – The size of n_i is not the same as the length of ions.

  • ValueError – If of B or n_i is negative.

  • ValueError – If w is negative or zero.

  • ValueError – If w or theta are not single valued or a 1-D array.

  • UnitTypeError – If applicable arguments do not have units convertible to the expected units.

Notes

The cold plasma dispersion function is defined by Stix [1992] in section 1-3 (and present by Bellan [2012] in equation 8) to be

\[(S\sin^{2}(\theta) + P\cos^{2}(\theta))(ck/\omega)^{4} - [ RL\sin^{2}(\theta) + PS(1 + \cos^{2}(\theta)) ](ck/\omega)^{2} + PRL = 0\]

where,

\[\begin{split}\mathbf{B_o} &= B_{o} \mathbf{\hat{z}} \\ \cos \theta &= \frac{k_z}{k} \\ \mathbf{k} &= k_{\rm x} \hat{x} + k_{\rm z} \hat{z}\end{split}\]
\[\begin{split}S &= 1 - \sum_{s} \frac{\omega^{2}_{p,s}}{\omega^{2} - \omega^{2}_{c,s}}\\ P &= 1 - \sum_{s} \frac{\omega^{2}_{p,s}}{\omega^{2}}\\ D &= \sum_{s} \frac{\omega_{c,s}}{\omega} \frac{\omega^{2}_{p,s}}{\omega^{2} - \omega_{c,s}^{2}}\end{split}\]
\[R = S + D \hspace{1cm} L = S - D\]

\(\omega\) is the wave frequency, \(k\) is the wavenumber, \(\theta\) is the wave propagation angle with respect to the background magntic field \(\mathbf{B_o}\), \(s\) corresponds to plasma species \(s\), \(\omega_{p,s}\) is the plasma frequency of species \(s\), and \(\omega_{c,s}\) is the gyrofrequency of species \(s\). The derivation of this dispersion relation assumed:

  • zero temperature for all plasma species (\(T_{s}=0\))

  • quasi-neutrality

  • a uniform background magntic field \(\mathbf{B_o} = B_{o} \mathbf{\hat{z}}\)

  • no D.C. electric field \(\mathbf{E_o}=0\)

  • zero-order quantities for all plasma parameters (densities, electric-field, magnetic field, particle speeds, etc.) are constant in time and space

  • first-order perturbations in plasma parameters vary like \(\sim e^{\left [ i (\textbf{k}\cdot\textbf{r} - \omega t)\right ]}\)

Due to the cold plasma assumption, this equation is valid for all \(\omega\) and \(k\) given \(\frac{\omega}{k_{z}} \gg v_{Th}\) for all thermal speeds \(v_{Th}\) of all plasma species and \(k_{x} r_{L} \ll 1\) for all gyroradii \(r_{L}\) of all plasma species.

The relation predicts \(k \to 0\) when any one of P, R or L vanish (cutoffs) and \(k \to \infty\) for perpendicular propagation during wave resonance \(S \to 0\).

Examples

>>> import astropy.units as u
>>> from plasmapy.particles import Particle
>>> from plasmapy.dispersion.analytical.stix_ import stix
>>> inputs = {
...     "B": 8.3e-9 * u.T,
...     "w": 0.001 * u.rad / u.s,
...     "ions": [Particle("H+"), Particle("He+")],
...     "n_i": [4.0e5,2.0e5] * u.m**-3,
...     "theta": 30 * u.deg,
... }
>>> stix(**inputs)
<Quantity [ 6.03817661e-09-0.j, -6.03817661e-09+0.j,
    6.97262784e-09-0.j, -6.97262784e-09+0.j] rad / m>