plasmapy.formulary.radiation.thermal_bremsstrahlung(frequencies: Quantity, n_e: Quantity, T_e: Quantity, n_i: Quantity = None, ion: str | Integral | Particle | CustomParticle | Quantity = 'p+', kmax: Quantity = None) ndarray[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.

  • 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 quasi-neutral 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.


spectrum – Computed bremsstrahlung spectrum over the frequencies provided.

Return type:



For details, see Bekefi [1966].