# cold_plasma_permittivity_SDP¶

plasmapy.formulary.cold_plasma_permittivity_SDP(B: Unit("T"), species, n, omega: Unit("rad / s"))

Magnetized Cold Plasma Dielectric Permittivity Tensor Elements.

Elements (S, D, P) are given in the “Stix” frame, ie. with B // z.

The $$\exp(-i \omega t)$$ time-harmonic convention is assumed.

Parameters: B (Quantity) – Magnetic field magnitude in units convertible to tesla. species (list of str) – List of the plasma particle species e.g.: [‘e’, ‘D+’] or [‘e’, ‘D+’, ‘He+’]. n (list of ~astropy.units.Quantity) – list of species density in units convertible to per cubic meter The order of the species densities should follow species. omega (Quantity) – Electromagnetic wave frequency in rad/s. sum (~astropy.units.Quantity) – S (“Sum”) dielectric tensor element. difference (~astropy.units.Quantity) – D (“Difference”) dielectric tensor element. plasma (~astropy.units.Quantity) – P (“Plasma”) dielectric tensor element.

Notes

The dielectric permittivity tensor is expressed in the Stix frame with the $$\exp(-i \omega t)$$ time-harmonic convention as $$\varepsilon = \varepsilon_0 A$$, with $$A$$ being

$\begin{split}\varepsilon = \varepsilon_0 \left(\begin{matrix} S & -i D & 0 \\ +i D & S & 0 \\ 0 & 0 & P \end{matrix}\right)\end{split}$

where:

\begin{align}\begin{aligned}S = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega^2 - \Omega_{c,s}^2}\\D = \sum_s \frac{\Omega_{c,s}}{\omega} \frac{\omega_{p,s}^2}{\omega^2 - \Omega_{c,s}^2}\\P = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega^2}\end{aligned}\end{align}

where $$\omega_{p,s}$$ is the plasma frequency and $$\Omega_{c,s}$$ is the signed version of the cyclotron frequency for the species $$s$$.

References

• T.H. Stix, Waves in Plasma, 1992.

Examples

>>> from astropy import units as u
>>> from numpy import pi
>>> B = 2*u.T
>>> species = ['e', 'D+']
>>> n = [1e18*u.m**-3, 1e18*u.m**-3]