Saha Ionization Calculator

How the extended Saha ionization tool works

This upgrade keeps the familiar Saha equation while exposing the ingredients you adjust in real analyses: temperature T, electron density nₑ, ionization energy χ, and the statistical weight ratio g_i+1/g_i. The script converts χ from electronvolts to joules, evaluates the full prefactor 2·(g_i+1/g_i)·(2πmₑkT/h²)^3/2, and balances it against your density entry. A logistic transform keeps the ionized fraction stable even when the raw ratio spans hundreds of orders of magnitude.

The classical form of the Saha equation links the population ratio directly to temperature, density, and statistical weights:

1. Select a preset to populate representative values for a stellar or laboratory species, or stay on the custom row to supply your own partition-function ratio.
2. Enter temperature and electron density (scientific notation such as 1e20 is accepted), then adjust χ or the g-ratio if your source uses different data.
3. Optionally provide a target ionized fraction to solve for the electron density that would reach it, or fill in the sweep range to map how quickly the plasma transitions between neutral and ionized states.
4. Review the summary, interpretation bullets, sweep table, and saved-scenario comparison when planning observations, designing a discharge experiment, or writing up coursework.

Preset quick reference

The presets anchor the drop-down menu. Feel free to overwrite them after they load if your environment requires different numbers.

Stage	χ (eV)	gi+1/gi	Typical temperature	Use case
Hydrogen I → H II	13.60	2.0	7,000–15,000 K	Balmer-line photospheres, H II regions
Helium I → He II	24.59	1.0	12,000–25,000 K	Hot B-type stellar atmospheres
Helium II → He III	54.42	2.0	25,000–50,000 K	Extreme-UV sources and white dwarf envelopes
Sodium I → Na II	5.14	2.0	3,000–6,000 K	Cool stellar photospheres, discharge lamps
Calcium II → Ca III	11.87	1.5	6,000–10,000 K	Chromospheric diagnostics
Iron I → Fe II	7.90	1.6	5,000–9,000 K	Line-blanketing studies, laboratory plasmas

Reading the outputs

• Ionization ratio lists n_i+1/n_i and the accompanying log₁₀ value so you can gauge how many decades separate the ionized and neutral populations.
• Ionized fraction converts the ratio into a percentage of atoms that are stripped of one electron. The interpretation panel labels the plasma as highly ionized, ion dominated, mixed, mostly neutral, or predominantly neutral.
• Scenario insight quantifies how sensitive the fraction is to doubling or halving nₑ and to ±20% temperature changes—useful for uncertainty budgets and lab tolerances.
• Target density solver works backward from a desired fraction to the electron density required at the same temperature and species.
• Temperature sweep table (when enabled) holds nₑ constant and steps through temperatures so you can spot the inflection point where the plasma crosses 50% ionization.

Temperature sweep and comparison strategies

Use the sweep inputs to map a stellar layer or an experiment. A handful of rows spanning a few thousand kelvin reveal when log₁₀(n_i+1/n_i) crosses zero, signalling equal populations. After running a sweep, click “Save scenario” on the most relevant temperatures to build a casebook. The comparison log stores up to eight entries—ideal for contrasting main-sequence versus giant-branch photospheres or before-and-after states of a discharge tube.

Worked example: hydrogen in an A-type photosphere

Consider a stellar layer at 10,000 K with an electron density of 1×10²⁰ m^-3. Choose the hydrogen preset (χ = 13.60 eV, g-ratio = 2) and run the calculation. The tool reports an ionized fraction near 50%, classifying the plasma as mixed. Doubling the temperature pushes the fraction toward 90%, while halving the density drops it below 30%. Saving the scenario lets you compare it directly against cooler layers (for example 7,000 K) and identify where Balmer absorption is strongest.

Switch to the helium preset without changing T or nₑ and the ionized fraction plunges, demonstrating why helium lines require hotter stars. Add a sweep from 15,000 to 25,000 K to tabulate how the helium fraction climbs and copy the table into lab notes or an assignment write-up.

Planning checklist

• Confirm that electron densities are in m^-3. If your data use cm^-3, multiply by 10⁶ before entering them.
• Document the source of statistical weights. Term degeneracies vary across ionization stages; note whether you used ground-state approximations or full partition functions.
• Use the comparison log as a reproducible record of the states you investigated. Each row can be copied into lab reports or observing checklists.
• Re-run the calculator with perturbed inputs when you introduce new opacities, radiation fields, or non-LTE corrections so you know how far the LTE baseline sits from your refined model.

Frequently asked questions

What does the g-ratio represent?

The ratio g_i+1/g_i reflects the relative statistical weights (degeneracies) of the ionized and neutral states. Hydrogen doubles its degeneracy after ionization, hence the preset value of 2. Multi-electron atoms often use partition functions evaluated at the working temperature. Setting the ratio to 1 reproduces the textbook simplified form, but adjusting it improves fidelity for species with dense term structures.

How accurate is the Saha equilibrium in real plasmas?

The equation assumes local thermodynamic equilibrium with collisions dominating over radiation. In tenuous nebulae or rapidly changing laser plasmas the result becomes a first-order guide rather than a final answer. If the scenario insight shows large swings when you tweak density or temperature, consider following up with a non-LTE or kinetic model.

What if the calculator reports “≈∞” or “≈0” for the ratio?

That message indicates the logarithmic term is so extreme that the raw ratio overflows standard floating-point numbers. The logistic formulation still produces a meaningful ionized fraction: you are effectively in a fully ionized or fully neutral limit. Narrow the sweep range or adjust density to explore the transition region if you need more granularity.

Can I compare multiple plasmas at once?

Yes. After any calculation, press “Save scenario” to add it to the comparison log. Up to eight rows are retained. Use the “Copy summary” button to grab the current synopsis for lab books or observation planning documents, and clear the log when you start a new project.

How can I convert densities and keep units consistent?

Common conversions include 1 cm^-3 = 10⁶ m^-3 and 1 mm^-3 = 10⁹ m^-3. If you are working from mass densities, divide by the mean molecular weight and multiply by Avogadro’s number before entering the electron number density. Keeping a short conversion table next to the calculator reduces transcription errors.

Next steps for study

Use this calculator as a launchpad for deeper modelling. Try saving states for successive ionization levels of the same element, compare them with observed spectral lines, or feed the ionized fractions into opacity or emissivity codes. The combination of presets, sweep exploration, and casebook logging provides a lightweight workflow for both lecture notes and professional planning.