# SingleParticleCollisionFrequencies

class plasmapy.formulary.collisions.frequencies.SingleParticleCollisionFrequencies(test_particle: , field_particle: , *, v_drift: Unit('m / s'), T_b: Unit('K'), n_b: Unit('1 / m3'), Coulomb_log: Unit(dimensionless))

Bases: object

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

Parameters
Raises

ValueError – If the specified v_drift and n_b arrays don’t have equal size.

Notes

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

momentum loss: $$\frac{d\textbf{v}_{α}}{dt}=-ν_{s}^{α\backslashβ}\textbf{v}_{α}$$

transverse diffusion: $$\frac{d}{dt}\left(\textbf{v}_{α}-\overline{\textbf{v}}_{α}\right)_{⊥}^{2}=ν_{⊥}^{α\backslashβ}v_{α}^{2}$$

parallel diffusion: $$\frac{d}{dt}\left(\textbf{v}_{α}-\overline{\textbf{v}}_{α}\right)_{∥}^{2}=ν_{∥}^{α\backslashβ}v_{α}^{2}$$

energy loss: $$\frac{d}{dt}v_{α}^{2}=-ν_{ϵ}^{α\backslashβ}v_{α}^{2}$$

These equations yield the exact formulas:

momentum loss: $$ν_{s}^{α\backslashβ}=\left(1+\frac{m_{α}}{m_{β}}\right)ψ\left(x^{α\backslashβ}\right)ν_{0}^{α\backslashβ}$$

transverse diffusion: $$ν_{⊥}^{α\backslashβ}=2\left[\left(1-\frac{1}{2x^{α\backslashβ}}\right)ψ\left(x^{α\backslashβ}\right)\ +ψ'\left(x^{α\backslashβ}\right)\right]ν_{0}^{α\backslashβ}$$

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

energy loss: $$ν_{ϵ}^{α\backslashβ}=2\left[\left(\frac{m_{α}}{m_{β}}\right)ψ\left(x^{α\backslashβ}\right)-ψ'\left(x^{α\backslashβ}\right)\right]ν_{0}^{α\backslashβ}$$

where,

$$ν_{0}^{α\backslashβ}=\frac{4\pi e_{α}^{2}e_{β}^{2}λ_{αβ}n_{β}}{m_{α}^{2}v_{α}^{3}}$$,

$$x^{α\backslashβ}=\frac{m_{β}v_{α}^{2}}{2k_B T_{β}}$$,

$$ψ\left(x\right)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}t^{\frac{1}{2}}e^{-t}dt$$,

$$ψ'\left(x\right)=\frac{dψ}{dx}$$,

and $$\lambda_{\alpha \beta}$$ is the Coulomb logarithm for the collisions, $$n_\beta$$ is the number density of the field particles, $$v_\alpha$$ is the speed of the test particles relative to the field particles, $$k_B$$ is Boltzmann’s constant, and $$T_\beta$$ 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), $$\psi$$ asymptotes to zero or one respectively. For simplified expressions in these limits we encourage the curious reader to refer to p. 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>


Coulomb_logarithm

Evaluates the Coulomb logarithm for two interacting electron species.

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 $$\nu_0^{\alpha/\beta}$$ 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.

(see documentation for the SingleParticleCollisionFrequencies class for details)