thermal_bremsstrahlung

Calculate the bremsstrahlung emission spectrum for a Maxwellian plasma in the Rayleigh-Jeans limit \(ℏ ω ≪ k_B T_e\).

\[\frac{dP}{dω} = \frac{8 \sqrt{2}}{3\sqrt{π}} \bigg ( \frac{e^2}{4 π ε_0} \bigg )^3 \bigg ( m_e c^2 \bigg )^{-\frac{3}{2}} \bigg ( 1 - \frac{ω_{pe}^2}{ω^2} \bigg )^\frac{1}{2} \frac{Z_i^2 n_i n_e}{\sqrt(k_B T_e)} E_1(y)\]

where \(E_1\) is the exponential integral

\[E_1 (y) = - \int_{-y}^∞ \frac{e^{-t}}{t}dt\]

and \(y\) is the dimensionless argument

\[y = \frac{1}{2} \frac{ω^2 m_e}{k_{max}^2 k_B T_e}\]

where \(k_{max}\) is a maximum wavenumber approximated here as \(k_{max} = 1/λ_B\) where \(λ_B\) is the electron de Broglie wavelength.

Parameters:

frequencies (Quantity) – Array of frequencies over which the bremsstrahlung spectrum will be calculated (convertible to Hz).
n_e (Quantity) – Electron number density in the plasma (convertible to m^-3).
T_e (Quantity) – Temperature of the electrons (in K or convertible to eV).
n_i (Quantity, optional) – Ion number density in the plasma (convertible to m^-3). Defaults to the quasineutral condition \(n_i = n_e / Z\).
ion (particle-like, default: "p+") – An instance of Particle, or a string convertible to Particle.
kmax (Quantity) – Cutoff wavenumber (convertible to radians per meter). Defaults to the inverse of the electron de Broglie wavelength.

Returns:

spectrum – Computed bremsstrahlung spectrum over the frequencies provided.

Return type:

Quantity

Notes

For details, see Bekefi [1966].

Examples

>>> import astropy.units as u
>>> import numpy as np
>>> thermal_bremsstrahlung(10**15 * u.Hz, 1e10 * u.cm**-3, 2e7 * u.K)  # solar flare
<Quantity 8.17560238e-23 kg / (m s2)>
>>> thermal_bremsstrahlung(
...     10 ** np.arange(15, 16, 0.1) * u.Hz, 1e22 * u.cm**-3, 1e2 * u.eV
... )
<Quantity [ 79.59052452, 117.73282254, 127.85119908, 127.12505588,
       121.01549498, 112.02367743, 101.45553309,  90.04503155,
        78.23475796,  66.32227273] kg / (m s2)>
>>> thermal_bremsstrahlung(
...     1e17 * u.Hz, 1e16 * u.cm**-3, 1e4 * u.eV, ion="Fe-56 12+"
... )
<Quantity 2.16932808e-10 kg / (m s2)>