thermal_bremsstrahlung

plasmapy.formulary.radiation.thermal_bremsstrahlung(frequencies: Quantity, n_e: Quantity, T_e: Quantity, n_i: Quantity = None, ion: = 'p+', kmax: Quantity = None) [source]

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:
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)>