# cold_plasma_permittivity_LRP¶

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

Magnetized Cold Plasma Dielectric Permittivity Tensor Elements.

Elements (L, R, P) are given in the “rotating” basis, ie. in the basis $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$, where the tensor is diagonal and 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) – 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. left (~astropy.units.Quantity) – L (“Left”) Left-handed circularly polarization tensor element. right (~astropy.units.Quantity) – R (“Right”) Right-handed circularly polarization tensor element. plasma (~astropy.units.Quantity) – P (“Plasma”) dielectric tensor element.

Notes

In the rotating frame defined by $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$ with $$\mathbf{u}_{\pm}=(\mathbf{u}_x \pm \mathbf{u}_y)/\sqrt{2}$$, the dielectric tensor takes a diagonal form with elements L, R, P with:

\begin{align}\begin{aligned}L = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega\left(\omega - \Omega_{c,s}\right)}\\R = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega\left(\omega + \Omega_{c,s}\right)}\\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]
>>> omega = 3.7e9*(2*pi)*(u.rad/u.s)
>>> L, R, P = permittivity = cold_plasma_permittivity_LRP(B, species, n, omega)
>>> L
<Quantity 0.63333...>
>>> permittivity.left    # namedtuple-style access
<Quantity 0.63333...>
>>> R
<Quantity 1.41512...>
>>> P
<Quantity -4.8903...>