thermal_bremsstrahlung

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 \(ℏ ω ≪ k_B T_e\)

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

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 quasi-neutral condition \(n_i = n_e / Z\).

  • ion (str or Particle, optional) – 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 “Radiation Processes in Plasmas” by Bekefi. ISBN 978\-0471063506.