fundamental_ion_collision_freq

plasmapy.formulary.collisions.fundamental_ion_collision_freq(T_i: Unit('K'), n_i: Unit('1 / m3'), ion, coulomb_log=None, V=None, coulomb_log_method='classical')

Average momentum relaxation rate for a slowly flowing Maxwellian distribution of ions.

Braginskii [1965] provides a derivation of this as an average collision frequency between ions and ions for a Maxwellian distribution. It is thus a special case of the collision frequency with an averaging factor.

Parameters

T_i (Quantity) – The electron temperature of the Maxwellian test ions.
n_i (Quantity) – The number density of the Maxwellian test ions.
ion (str) – String signifying a particle type of the test and field ions, including charge state information. This function assumes the test and field ions are the same species.
V (Quantity, optional) – The relative velocity between particles. If not provided, thermal velocity is assumed: \(μ V^2 \sim 2 k_B T\) where \(μ\) is the reduced mass.
coulomb_log (float or dimensionless Quantity, optional) – Option to specify a Coulomb logarithm of the electrons on the ions. If not specified, the Coulomb log will is calculated using the ~plasmapy.formulary.Coulomb_logarithm function.
coulomb_log_method (str, optional) – The method by which to compute the Coulomb logarithm. The default method is the classical straight-line Landau-Spitzer method ("classical" or "ls"). The other 6 supported methods are "ls_min_interp", "ls_full_interp", "ls_clamp_mininterp", "hls_min_interp", "hls_max_interp", and "hls_full_interp". Please refer to the docstring of Coulomb_logarithm for more information about these methods.

Returns

nu_i

Return type

Quantity

Notes

Equations (2.36) and (2.122) in Callen [n.d.] provide the original source used to implement this formula, however, in our implementation we use the very same process that leads to the fundamental electron collision rate (2.17), gaining simply a different coefficient:

\[ν_i = \frac{8}{3 * 4 * \sqrt{π}} ν(v_{Ti})\]

Where \(ν\) is the general collision frequency and \(v_{Ti}\) is the ion thermal velocity (the average, for a Maxwellian distribution).

Note that in the derivation, it is assumed that electrons are present in such numbers as to establish quasineutrality, but the effects of the test ions colliding with them are not considered here. This is a very typical approximation in transport theory.

This result is an ion momentum relaxation rate, and is used in many classical transport expressions. It is equivalent to:

\(1/τ_i\) from equation (2.5i) on page 215 of Braginskii [1965]
\(ν_i\) from page 33 of Richardson [2019]

Examples

>>> import astropy.units as u
>>> from astropy.constants import c
>>> fundamental_ion_collision_freq(0.1 * u.eV, 1e6 / u.m ** 3, 'p')
<Quantity 2.868...e-05 1 / s>
>>> fundamental_ion_collision_freq(1e6 * u.K, 1e6 / u.m ** 3, 'p')
<Quantity 1.741...e-09 1 / s>
>>> fundamental_ion_collision_freq(100 * u.eV, 1e20 / u.m ** 3, 'p')
<Quantity 63087.5... 1 / s>
>>> fundamental_ion_collision_freq(100 * u.eV, 1e20 / u.m ** 3, 'p', coulomb_log_method='GMS-1')
<Quantity 63085.1... 1 / s>
>>> fundamental_ion_collision_freq(100 * u.eV, 1e20 / u.m ** 3, 'p', V = c / 100)
<Quantity 9.111... 1 / s>
>>> fundamental_ion_collision_freq(100 * u.eV, 1e20 / u.m ** 3, 'p', coulomb_log=20)
<Quantity 95918.7... 1 / s>