SingleParticleCollisionFrequencies

Bases: object

Compute collision frequencies between test particles (labeled ‘a’) and field particles (labeled ‘b’).

Parameters:

test_particle (particle-like) – The test particle streaming through a background of field particles.
field_particle (particle-like) – The background particle that test_particle is interacting with.
v_drift (Quantity, keyword-only) – The relative drift velocity between the test and field particles. Cannot be negative.
T_b (Quantity, keyword-only) – The temperature of the background field particles in units convertible to kelvin or eV per particle.
n_b (Quantity, keyword-only) – The number density of field_particle in units convertible to m^-3.
Coulomb_log (Quantity, keyword-only) – The value of the Coulomb logarithm for the interaction.

Raises:

ValueError – If the specified v_drift and n_b arrays do not have equal size.

Notes

The frequency of collisions between a test particle (subscript \(α\)) and a field particle (subscript \(β\)) each with mass \(m\) and charge \(e\) are given by four differential equations:

momentum loss: \(\frac{d\mathbf{v}_α}{dt} = - ν_{s}^{α \backslash β} \mathbf{v}_α\)

transverse diffusion: \(\frac{d}{dt} \left( \mathbf{v}_α - \overline{\mathbf{v}}_α \right)_⊥^2 = ν_⊥^{α \backslash β} v_α^2\)

parallel diffusion: \(\frac{d}{dt} \left(\mathbf{v}_α-\overline{\mathbf{v}}_α\right)_∥^2 = ν_∥^{α \backslash β} v_α^2\)

energy loss: \(\frac{d}{dt} v_α^2 = -ν_ϵ^{α \backslash β} v_α^2\)

These equations yield the exact formulas:

momentum loss: \(ν_{s}^{α \backslash β} = \left( 1 + \frac{m_α}{m_β} \right) ψ \left( x^{α \backslash β} \right) ν_{0}^{α \backslash β}\)

transverse diffusion: \(ν_⊥^{α \backslash β} = 2 \left[ \left( 1-\frac{1}{2x^{α \backslash β}} \right) ψ \left( x^{α \backslash β} \right) \ + ψ' \left( x^{α \backslash β} \right) \right] ν_0^{α \backslash β}\)

parallel diffusion: \(ν_{||}^{α \backslash β} = \left[ \frac{ψ \left( x^{α \backslash β} \right) }{x^{α \backslash β}} \right] ν_{0}^{α \backslash β}\)

energy loss: \(ν_ϵ^{α \backslash β} = 2\left[ \left( \frac{m_α}{m_β} \right) ψ \left( x^{α \backslash β} \right) - ψ' \left(x^{α \backslash β} \right) \right] ν_0^{α \backslash β}\)

where,

\(ν_{0}^{α \backslash β} = \frac{4π e_α^2 e_β^2 λ_{αβ} n_β}{m_α^2 v_α^3}\),

\(x^{α \backslash β} = \frac{m_β v_α^2}{2 k_B T_β}\),

\(ψ \left( x \right) = \frac{2}{\sqrt{π}} \int_0^x t^{ \frac{1}{2} } e^{-t} dt\),

\(ψ' \left( x \right) = \frac{dψ}{dx}\),

and \(λ_{α β}\) is the Coulomb logarithm for the collisions, \(n_β\) is the number density of the field particles, \(v_α\) is the speed of the test particles relative to the field particles, \(k_B\) is Boltzmann’s constant, and \(T_β\) is the temperature of the field particles.

For values of \(x ≪ 1\) (the ‘slow’ or ‘thermal’ limit) or \(x ≫ 1\) (the ‘fast’ or ‘beam’ limit), \(ψ\) asymptotes to zero or one respectively. For simplified expressions in these limits, please refer to page 31 of Richardson [2019].

Examples

>>> import astropy.units as u
>>> v_drift = 1e5 * u.m / u.s
>>> n_b = 1e26 * u.m**-3
>>> T_b = 1e3 * u.eV
>>> Coulomb_log = 10 * u.dimensionless_unscaled
>>> frequencies = SingleParticleCollisionFrequencies(
...     "e-", "e-", v_drift=v_drift, n_b=n_b, T_b=T_b, Coulomb_log=Coulomb_log
... )
>>> frequencies.energy_loss
<Quantity -9.69828719e+15 Hz>

See also

Coulomb_logarithm

Attributes Summary

`Lorentz_collision_frequency`	The Lorentz collision frequency.
`energy_loss`	The energy loss rate due to collisions.
`momentum_loss`	The momentum loss rate due to collisions.
`parallel_diffusion`	The rate of parallel diffusion due to collisions.
`phi`	The parameter phi used in calculating collision frequencies calculated using the default error tolerances of `quad`.
`transverse_diffusion`	The rate of transverse diffusion due to collisions.
`x`	The ratio of kinetic energy in the test particle to the thermal energy of the field particle.

Attributes Documentation

Lorentz_collision_frequency

The Lorentz collision frequency.

The Lorentz collision frequency (see Ch. 5 of Chen [2016]) is given by

\[ν = n σ v \ln{Λ}\]

where \(n\) is the particle density, \(σ\) is the collisional cross-section, \(v\) is the inter-particle velocity, and \(\ln{Λ}\) is the Coulomb logarithm accounting for small angle collisions.

See Equation (2.86) in Callen [n.d.].

The Lorentz collision frequency is equivalent to the variable \(ν_0^{α/β}\) on p. 31 of Richardson [2019].

This form of the Lorentz collision frequency differs from the form found in MaxwellianCollisionFrequencies in that \(v\) is the drift velocity (as opposed to the mean thermal velocity between species).

energy_loss: The energy loss rate due to collisions.

momentum_loss: The momentum loss rate due to collisions.

parallel_diffusion: The rate of parallel diffusion due to collisions.

phi

The parameter phi used in calculating collision frequencies calculated using the default error tolerances of quad.

For more information refer to page 31 of Richardson [2019].

transverse_diffusion: The rate of transverse diffusion due to collisions.

x

The ratio of kinetic energy in the test particle to the thermal energy of the field particle.

This parameter determines the regime in which the collision falls.