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: str | Integral | Particle | CustomParticle | Quantity = 'p+', kmax: Unit('m') = 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.
- 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 (particle-like, default:
"p+") – An instance ofParticle, or a string convertible toParticle.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:
Notes
For details, see Bekefi [1966].