Plasma Parameters Calculator

The handful of numbers that decide how a plasma behaves: how far a charge’s field reaches before the plasma screens it, how fast the electrons and ions oscillate, how often they collide, and how tightly a magnetic field holds them. Enter a density and temperature, the field and gas pressure if you have them, and the calculator also gives the electron cyclotron resonance (ECR) field for a microwave source and flags which regime the plasma is in.

Key terms
Electron density
Free electrons per cubic centimeter. Glow and RF processing discharges hold 10⁹–10¹² cm⁻³; the solar wind about 5 cm⁻³; a fusion plasma 10¹⁴ cm⁻³.
Electron temperature
The spread of electron energies, quoted as kT in electronvolts: 1 eV corresponds to 11 605 K. In a low-pressure discharge the electrons are at 1–10 eV while the ions and gas stay near room temperature (0.026 eV).
Debye length
The distance over which the plasma screens out the electric field of a charge or a biased surface. Probes, sheaths at walls, and anything smaller than a few Debye lengths behave differently from the plasma bulk.
Plasma frequency
How fast the electrons oscillate if they are pushed away from the ions and let go. Electromagnetic waves below this frequency cannot travel through the plasma; they are reflected.
Plasma parameter
The number of electrons inside a sphere one Debye length in radius. When it is large, collective behavior dominates over close two-body encounters, which is what makes an ionized gas a plasma.
Collision frequency and mean free path
How often an electron is deflected by an ion or a gas atom, and how far it travels between such events on average.
Gyrofrequency
How often an electron or ion circles a magnetic field line: 28.0 GHz per tesla for electrons. If particles complete many turns between collisions, the field confines them.
Electron cyclotron resonance (ECR)
Where the electron gyrofrequency equals the frequency of applied microwaves, the electrons absorb energy efficiently. ECR ion sources and plasma reactors are built around this field: 0.0875 T for 2.45 GHz, 1.0 T for 28 GHz.

Inputs

Plasma

From a Langmuir probe, interferometer, or a typical value: 10⁹–10¹² cm⁻³ in processing discharges. The ion density is ne/Z (quasi-neutrality).

Temperatures as kT in eV (1 eV = 11 605 K). Ions in a low-pressure discharge are usually close to the gas temperature, 0.03–0.1 eV.

u
e

A single ion species. For a mixture, use the dominant ion, or an average Z and mass.

Field and microwaves (optional)

B gives the gyrofrequencies, gyroradii, and plasma beta. The frequency of a microwave or RF source gives the ECR field and the cutoff density: 2.45 GHz (magnetron), 14.5, 18, or 28 GHz (ECR ion sources), or 13.56 MHz (RF).

Neutral gas (optional)

Default 10 Å² (1 × 10⁻¹⁹ m², 10⁻¹⁵ cm²): the order of magnitude for electrons of a few eV to a few tens of eV in N₂, O₂, and Ar. It depends strongly on energy and gas: argon has a deep Ramsauer–Townsend minimum near 0.24 eV, and helium is closer to 6 Å². For accurate rates, average a measured cross section over the electron energy distribution.

Size (optional)

The chamber or discharge dimension, to compare with the Debye length and the mean free paths in the regime notes.

Results

Electron Debye lengthλDe—
Debye length with ion screeningλD—Both species screen a slow test charge. Ions flowing into a sheath do not, so sheath widths scale with λDe.
Electron plasma frequencyfpe—
Ion plasma frequencyfpi—
Plasma parameter (electrons per Debye sphere)ND—
Coulomb logarithm, electron–ionln Λ—
Electron–ion collision frequencyνei—
Ion–ion collision frequencyνii—
Electron–neutral collision frequencyνen—
Electron thermal speedvte = (kTe/me)1/2—
Electron gyrofrequencyfce—
Ion gyrofrequencyfci—
Electron and ion gyroradii (thermal)re, ri—
Plasma betaβ—
ECR field for the microwave frequencyBECR—
Cutoff (critical) density for that frequencync—

Regime notes

    How it is calculated

    All quantities are in SI units with temperatures as energies (\(T\) in joules is \(e\) times the value in eV). With the ion density \(n_i = n_e/Z\) (quasi-neutrality):

    \[ \begin{gathered} \lambda_{De} = \sqrt{\frac{\varepsilon_0 T_e}{n_e e^2}}, \qquad \frac{1}{\lambda_D^2} = \frac{1}{\lambda_{De}^2} + \frac{1}{\lambda_{Di}^2}, \quad \lambda_{Di} = \sqrt{\frac{\varepsilon_0 T_i}{n_i Z^2 e^2}}, \\[8pt] \omega_{pe} = 2\pi f_{pe} = \sqrt{\frac{n_e e^2}{\varepsilon_0 m_e}}, \qquad \omega_{pi} = \sqrt{\frac{n_i Z^2 e^2}{\varepsilon_0 m_i}}, \qquad N_D = \frac{4\pi}{3} n_e \lambda_{De}^3. \end{gathered} \]

    Some texts call \(\Lambda = 4\pi n_e \lambda_{De}^3 = 3N_D\) the plasma parameter; either way it must be much larger than 1 for the plasma to be weakly coupled. In practical units \(f_{pe} \approx 8.98\ \text{kHz} \times \sqrt{n_e/\text{cm}^{-3}}\) and \(\lambda_{De} \approx 7.43\ \text{m} \times \sqrt{(T_e/\text{eV})/(n_e/\text{cm}^{-3})}\).

    Collisions

    Coulomb collisions are counted with the Coulomb logarithm. The calculator uses the NRL Plasma Formulary’s expressions, with \(n_e\) in cm⁻³ and \(T\) in eV:

    \[ \begin{gathered} \ln\Lambda_{ei} = 23 - \ln\!\left(n_e^{1/2} Z\, T_e^{-3/2}\right) \ \ (T_e < 10Z^2\ \text{eV}), \\[6pt] \ln\Lambda_{ei} = 24 - \ln\!\left(n_e^{1/2}\, T_e^{-1}\right) \ \ (T_e > 10 Z^2\ \text{eV}), \\[8pt] \ln\Lambda_{ii} = 23 - \ln\!\left[\frac{Z^2}{T_i}\left(\frac{2 n_i Z^2}{T_i}\right)^{1/2}\right]. \end{gathered} \]

    The electron–ion and ion–ion collision frequencies are the inverse Braginskii collision times,

    \[ \nu_{ei} = \frac{4\sqrt{2\pi}\, n_e Z e^4 \ln\Lambda_{ei}}{3\,(4\pi\varepsilon_0)^2\, m_e^{1/2}\, T_e^{3/2}}, \qquad \nu_{ii} = \frac{4\sqrt{\pi}\, n_i Z^4 e^4 \ln\Lambda_{ii}}{3\,(4\pi\varepsilon_0)^2\, m_i^{1/2}\, T_i^{3/2}}, \]

    which for Z = 1 are the Formulary’s \(\nu_e = 2.91\times10^{-6}\, n_e \ln\Lambda\, T_e^{-3/2}\) s⁻¹ and \(\nu_i = 4.80\times10^{-8}\, \mu^{-1/2} n_i \ln\Lambda\, T_i^{-3/2}\) s⁻¹ (\(\mu = m_i/m_p\)). The electron–ion mean free path is \(v_{te}/\nu_{ei}\) with \(v_{te} = \sqrt{T_e/m_e}\). Electron–neutral collisions use a constant momentum-transfer cross section \(\sigma_m\) (your input) and the mean electron speed:

    \[ n_n = \frac{p}{k_B T_g}, \qquad \nu_{en} = n_n \sigma_m \bar v_e, \quad \bar v_e = \sqrt{\frac{8 T_e}{\pi m_e}}, \qquad \lambda_{en} = \frac{1}{n_n \sigma_m}. \]

    Magnetic field and microwaves

    \[ \begin{gathered} \omega_{ce} = \frac{eB}{m_e}, \qquad \omega_{ci} = \frac{ZeB}{m_i}, \qquad r_e = \frac{v_{te}}{\omega_{ce}}, \qquad r_i = \frac{\sqrt{T_i/m_i}}{\omega_{ci}}, \qquad \beta = \frac{2\mu_0 (n_e T_e + n_i T_i)}{B^2}, \\[8pt] B_\text{ECR} = \frac{2\pi f\, m_e}{e}, \qquad n_c = \frac{\varepsilon_0 m_e (2\pi f)^2}{e^2}. \end{gathered} \]

    \(B_\text{ECR}\) is the field where the electron gyrofrequency equals the microwave frequency: \(f/(27.992\ \text{GHz})\) teslas, so 2.45 GHz resonates at 87.5 mT and 28 GHz at 1.000 T. Above the cutoff density \(n_c\) (where \(f_{pe} = f\)) an ordinary wave cannot propagate into the plasma: 7.4 × 10¹⁰ cm⁻³ at 2.45 GHz and 9.7 × 10¹² cm⁻³ at 28 GHz. ECR sources reach higher densities with waves that travel along the field from the high-field side or by mode conversion.

    Typical values

    From the formulas above (Z = 1, ne in cm⁻³):

    PlasmaneTeλDefpeND
    Solar wind near Earth510 eV10.5 m20 kHz2.4 × 10¹⁰
    Ionosphere (F layer)10⁶0.1 eV2.4 mm9.0 MHz5.4 × 10⁴
    RF or ECR processing plasma10¹¹5 eV53 µm2.8 GHz6.1 × 10⁴
    Fluorescent lamp10¹¹1 eV24 µm2.8 GHz5.4 × 10³
    Tokamak core10¹⁴10 keV74 µm90 GHz1.7 × 10⁸
    Laser-produced plasma10²¹1 keV7.4 nm280 THz1.7 × 10³

    The densities and temperatures are round, representative values (compare the NRL Plasma Formulary’s table of typical plasma parameters), not measurements of a particular device.

    Worked example

    The default inputs: an argon plasma with ne = 10¹¹ cm⁻³, Te = 5 eV, Ti = 0.05 eV, in 1 mTorr of gas at 20 °C, at the 87.5 mT resonance of a 2.45 GHz source, in a 10 cm chamber.

    Assumptions and limits

    References

    Primary sources and standards

    1. A. Beresnyak, 2023 NRL Plasma Formulary, Plasma Physics Division, U.S. Naval Research Laboratory, Washington, DC (2023), nrl.navy.mil (the Coulomb logarithms; the collision rates νe and νi; the practical-unit plasma frequencies, Debye length, gyrofrequencies, and gyroradii used as checks).
    2. S. I. Braginskii, “Transport processes in a plasma,” in Reviews of Plasma Physics, vol. 1, M. A. Leontovich (ed.), Consultants Bureau, New York (1965), pp. 205–311 (the electron and ion collision times τe and τi).
    3. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed., Springer (2016), doi:10.1007/978-3-319-22309-4 (Debye shielding, the plasma parameter, plasma oscillations, cutoffs, and cyclotron motion).
    4. M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed., Wiley (2005) (electron–neutral collisions and momentum-transfer cross sections in processing discharges; ECR discharges).
    5. Y. Itikawa, M. Hayashi, A. Ichimura, K. Onda, K. Sakimoto, K. Takayanagi, M. Nakamura, H. Nishimura, and T. Takayanagi, “Cross sections for collisions of electrons and photons with nitrogen molecules,” J. Phys. Chem. Ref. Data 15, 985–1010 (1986), doi:10.1063/1.555762 (the N₂ momentum-transfer cross section, about 10⁻¹⁵ cm² for electrons of a few eV to tens of eV).
    6. R. Geller, Electron Cyclotron Resonance Ion Sources and ECR Plasmas, Institute of Physics Publishing (1996) (the resonance condition and the high-field ECR sources).
    7. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Rev. Mod. Phys. 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002 (e, mₑ, ε₀, μ₀, kB, and u).

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