# 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
• test_particle (ParticleLike) – The test particle streaming through a background of field particles.

• field_particle (ParticleLike) – The background particle being interacted with.

• v_drift (Quantity) – The relative drift between the test and field particles. Cannot be negative.

• T_b (Quantity) – The temperature of the background field particles in units convertible to degrees Kelvin.

• n_b (Quantity) – The number density of the background field particles in units convertible to $$\frac{1}{m^{3}}$$.

• Coulomb_log (Quantity) – The value of the Coulomb logarithm for the interaction.

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.