thermal_speed

plasmapy.formulary.speeds.thermal_speed(
T: Annotated[Quantity, Unit('K')],
particle: str | int | integer | Particle | CustomParticle | Quantity,
method: Literal['most_probable', 'rms', 'mean_magnitude', 'nrl'] = 'most_probable',
mass: Annotated[Quantity, Unit('kg')] = None,
ndim: int = 3,
) → Annotated[Quantity, Unit('m / s')][source]

Calculate the speed of thermal motion for particles with a Maxwellian distribution. (See the Notes section for details.).

Aliases: vth_

Lite Version: thermal_speed_lite

Parameters:

  • T (Quantity) – The temperature of the particle distribution, in units of kelvin or energy.

  • particle (Particle) – Representation of the particle species (e.g., "p" for protons, "D+" for deuterium, or "He-4 +1" for singly ionized helium-4).

  • method (str, optional) – (Default "most_probable") Method to be used for calculating the thermal speed. Valid values are "most_probable", "rms", "mean_magnitude", and "nrl".

  • mass (Quantity) – Mass override in units convertible to kg. If given, then mass will be used instead of the mass value associated with particle.

  • ndim (int) – (Default 3) Dimensionality (1D, 2D, 3D) of space in which to calculate thermal speed. Valid values are 1, 2, or 3.

Returns:

vth – Thermal speed of the Maxwellian distribution.

Return type:

Quantity

Raises:

  • TypeError – The particle temperature is not a Quantity.

  • UnitConversionError – If the particle temperature is not in units of temperature or energy per particle.

  • ValueError – The particle temperature is invalid or particle cannot be used to identify an isotope or particle.

Warns:

  • RelativityWarning – If the ion sound speed exceeds 5% of the speed of light.

  • UnitsWarning – If units are not provided, SI units are assumed.

Notes

There are multiple methods (or definitions) for calculating the thermal speed, all of which give the expression

\[v_{th} = C_o \sqrt{\frac{k_B T}{m}}\]

where \(T\) is the temperature associated with the distribution, \(m\) is the particle’s mass, and \(C_o\) is a constant of proportionality determined by the method in which \(v_{th}\) is calculated and the dimensionality of the system (1D, 2D, 3D). The \(C_o\) used for the thermal_speed calculation is determined from the input arguments method and ndim, and the values can be seen in the table below:

Values for \(C_o\)

↓ method

ndim →

1

2

3

"most_probable"
\[0\]
\[1\]
\[\sqrt{2}\]

"rms"
\[1\]
\[\sqrt{2}\]
\[\sqrt{3}\]

"mean_magnitude"
\[\sqrt{2/π}\]
\[\sqrt{π/2}\]
\[\sqrt{8/π}\]

"nrl"
\[1\]

The coefficients can be directly retrieved using thermal_speed_coefficients.

The Methods

In the following discussion the Maxwellian distribution \(f(\mathbf{v})\) is assumed to be 3D, but similar expressions can be given for 1D and 2D.

  • Most Probable method = "most_probable"

    This method expresses the thermal speed of the distribution by expressing it as the most probable speed a particle in the distribution may have. To do this we first define another function \(g(v)\) given by

    \[\int_{0}^{∞} g(v) dv = \int_{-∞}^{∞} f(\mathbf{v}) d^3\mathbf{v} \quad \rightarrow \quad g(v) = 4 π v^2 f(v)\]

    then

    \[\begin{split}g^{\prime}(v_{th}) = \left.\frac{dg}{dv}\right|_{v_{th}} = 0\\ \implies v_{th} = \sqrt{\frac{2 k_B T}{m}}\end{split}\]

  • Root Mean Square method = "rms"

    This method uses the root mean square to calculate an expression for the thermal speed of the particle distribution, which is given by

    \[v_{th} = \left[\int v^2 f(\mathbf{v}) d^3 \mathbf{v}\right]^{1/2} = \sqrt{\frac{3 k_B T}{m}}\]

  • Mean Magnitude method = "mean_magnitude"

    This method uses the mean speed of the particle distribution to calculate an expression for the thermal speed, which is given by

    \[v_{th} = \int |\mathbf{v}| f(\mathbf{v}) d^3 \mathbf{v} = \sqrt{\frac{8 k_B T}{π m}}\]

  • NRL Formulary method = "nrl"

    The NRL Plasma Formulary [Richardson, 2019] uses the square root of the Normal distribution’s variance as the expression for thermal speed.

    \[v_{th} = σ = \sqrt{\frac{k_B T}{m}} \quad \text{where} \quad f(v) \sim e^{v^2 / 2 σ^2}\]

Examples

>>> import astropy.units as u
>>> thermal_speed(5*u.eV, 'p+')
<Quantity 30949.6... m / s>
>>> thermal_speed(1e6*u.K, particle='p+')
<Quantity 128486... m / s>
>>> thermal_speed(5*u.eV, particle='e-')
<Quantity 132620... m / s>
>>> thermal_speed(1e6*u.K, particle='e-')
<Quantity 550569... m / s>
>>> thermal_speed(1e6*u.K, "e-", method="rms")
<Quantity 674307... m / s>
>>> thermal_speed(1e6*u.K, "e-", method="mean_magnitude")
<Quantity 621251... m / s>

For user convenience thermal_speed_coefficients and thermal_speed_lite are bound to this function and can be used as follows.

>>> from plasmapy.particles import Particle
>>> mass = Particle("p").mass.value
>>> coeff = thermal_speed.coefficients(method="most_probable", ndim=3)
>>> thermal_speed.lite(T=1e6, mass=mass, coeff=coeff)
np.float64(128486.57...)