plasmapy.formulary.radiation.thermal_bremsstrahlung(frequencies: Unit("Hz"), n_e: Unit("1 / m3"), T_e: Unit("K"), n_i: Unit("1 / m3") = None, ion_species: plasmapy.particles.particle_class.Particle = 'H+', kmax: Unit("m") = None) → numpy.ndarray

Calculate the Bremsstrahlung emission spectrum for a Maxwellian plasma in the Rayleigh-Jeans limit \(\hbar\omega \ll k_B*T_e\)

\[\frac{dP}{d\omega} = \frac{8 \sqrt{2}}{3\sqrt{\pi}} \bigg ( \frac{e^2}{4 \pi \epsilon_0} \bigg )^3 \bigg ( m_e c^2 \bigg )^{-\frac{3}{2}} \bigg ( 1 - \frac{\omega_{pe}^2}{\omega^2} \bigg )^\frac{1}{2} \frac{Z_i^2 n_i n_e}{\sqrt(k_B T_e)} E_1(y)\]

where E_1 is the exponential integral

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

and y is the dimensionless argument

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

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

  • frequencies (Quantity) – Array of frequencies over which the bremsstrahlung spectrum will be calculated (convertable to Hz).
  • n_e (Quantity) – Electron number density in the plasma (convertable 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 (convertable to m^-3). Defaults to the quasi-neutral conditon n_i=n_e/Z.
  • ion (str or Particle, optional) – An instance of Particle, or a string convertible to Particle.
  • kmax (Quantity) – Cutoff wavenumber (convertable to u.rad/u.m). Defaults to the inverse of the electron de Broglie wavelength.

spectrum – Computed bremsstrahlung spectrum over the frequencies provided.

Return type:



For details, see “Radiation Processes in Plasmas” by Bekefi. ISBN 978-0471063506.