Calculator
Example data
| Temperature (K) | Total density (m⁻³) | Ionization fraction x |
|---|---|---|
| 5,000 | 1.00e+20 | 0.00057937 |
| 10,000 | 1.00e+20 | 0.88449706 |
| 20,000 | 1.00e+19 | 0.99999805 |
Formula used
In thermal equilibrium, the ionization balance for a single stage can be approximated with the Saha relation:
For a hydrogen-like, singly ionized gas, define the ionization fraction x = nion / ntotal, with ntotal = nneutral + nion. Charge neutrality gives ne = Z x ntotal. Combining these yields:
The calculator solves this equation numerically in a stable closed form to keep 0 ≤ x ≤ 1 across wide ranges of temperature and density.
How to use this calculator
- Select Saha equilibrium to estimate ionization from temperature and density.
- Enter the total number density and pick units (m⁻³ or cm⁻³).
- Provide temperature in kelvin or electron-volts.
- Set ionization energy, plus optional degeneracy and multiplicity factors.
- Press Calculate. Results appear above the form.
- Use Download CSV or Download PDF to save outputs.
Notes and limits
- The Saha model assumes local thermal equilibrium and a single ionization stage.
- Real plasmas may require multi-stage ionization, partition functions, or non-equilibrium kinetics.
- If you already know electron density, switch to Direct from densities.
- For very low densities or high temperatures, x approaches 1.
Ionization fraction in one sentence
Ionization fraction x describes how much of a gas is ionized: x = ne/nt for a single species, where ne is free-electron density and nt is total particle density. Values range from 0 (neutral) to 1 (fully ionized). Many diagnostics and radiation models use x directly.
Saha equilibrium and LTE assumptions
The Saha equation links x to temperature and density under local thermal equilibrium (LTE). It balances ionization and recombination statistically, using the ionization energy χ and a thermal factor that grows rapidly with T. It works best for weakly coupled plasmas where collisions maintain equilibrium.
Why temperature dominates
Because the key term contains exp(−χ/kT), small temperature changes can swing x by orders of magnitude. For hydrogen with χ ≈ 13.6 eV, kT is about 0.86 eV at 10,000 K, so partial ionization is expected. At 20,000 K, ionization increases sharply even at the same density.
Density and pressure effects
Higher total density pushes the equilibrium toward neutrals because electrons and ions are more likely to recombine. For example, at fixed T, decreasing nt from 1×10^20 m⁻³ to 1×10^16 m⁻³ can move x much closer to 1. This is why astrophysical plasmas can be highly ionized at modest temperatures.
Ionization energy and statistical weights
The calculator lets you adjust χ and optional degeneracy factors (g-ratios). Larger χ suppresses ionization: helium’s first stage (≈24.6 eV) needs far higher T than hydrogen. Degeneracy and partition terms typically shift x by factors of a few, useful when comparing different elements or excited-state populations.
Working with Kelvin and electron-volts
Temperature can be entered in kelvin or eV (1 eV ≈ 11,604.5 K). Density can be m⁻³ or cm⁻³ (1 cm⁻³ = 10^6 m⁻³). Internally, the solver uses logarithms and a stable quadratic form to keep 0 ≤ x ≤ 1, avoiding round-off errors at extreme inputs.
Direct mode: when densities are known
If you already know electron density from a probe, spectroscopy, or simulation, use the direct mode: x = ne/nt. This is also helpful when LTE is doubtful, such as in rapidly changing discharges, photoionized regions, or plasmas with strong external fields where the Saha prediction may be misleading.
Typical applications and sanity checks
Ionization fraction supports estimates of conductivity, Debye length, and emission strength. As a sanity check, very low T (hundreds of kelvin) should yield x ≈ 0 for most gases, while very high T (tens of thousands of kelvin) and low density often produce x near 1. Compare outputs against known benchmarks for your species.
FAQs
What does an ionization fraction of 0.2 mean?
It means about 20% of the particles are ionized in the chosen single-stage model, so ne ≈ 0.2·nt. The remaining ~80% are neutral atoms or molecules.
Which formula is used in Saha mode?
It uses the Saha ionization relation for one stage, combining a temperature-dependent quantum factor with an exponential term exp(−χ/kT), then solves for x between 0 and 1.
Can I use this for multi-stage ionization?
This version is single-stage. For elements with multiple ionization steps, compute each stage sequentially with appropriate χ and partition factors, or use a multi-stage model when precision matters.
When should I avoid Saha equilibrium?
Avoid it when the plasma is not in LTE: very low-collision environments, strongly driven RF discharges, laser-produced plasmas, or photoionized regions. In those cases, use direct densities or kinetic models.
How do I choose ionization energy χ?
Use the first ionization energy for the species and stage you want. For example, hydrogen is about 13.6 eV and helium’s first stage is about 24.6 eV. Ensure units match the selected input mode.
Why does changing density affect x so much?
At fixed temperature, higher total density favors recombination and lowers x, while lower density favors ionization and raises x. The Saha relation includes density explicitly, so changes can shift x dramatically.
What’s a quick check that my inputs are reasonable?
Try a neutral limit (low temperature) and an ionized limit (high temperature, low density). The results should trend toward x≈0 and x≈1 respectively. If not, re-check unit selections and χ.