Ionization

Formation of electrons and ions by energy, temperature, or radiation. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Ionization occurs when an atom or molecule stops being electrically neutral because one electron becomes free. In a gas, that turns a set of neutral particles into a mixture of $n_0$, $n_i$, and $n_e$. The central issue is not only whether one particle can ionize another one, but what fraction of the whole gas becomes ionized: the $x_i$. That fraction depends on available energy, electronic binding, collisions, radiation, and recombination. This is why $T$ and $E_ion$ are the main controls: one sets the thermal energy landscape, while the other sets the minimum gate that must be crossed.

🟢 Essential level

The $x_i$ tells what part of the gas has lost electrons and no longer behaves as a purely neutral gas.

When $T$ increases, particles have more available thermal energy, so a collision is more likely to transfer enough $E$ to overcome $E_ion$. If $E_ion$ is large, the electron is more tightly bound and ionization is harder: the gas needs higher $T$, higher $\nu$, or more energetic incident particles. Photoionization adds a quantum idea: intense light is not enough unless each photon carries enough energy. Recombination works in the opposite direction, because free electrons and ions can join again. Thus an ionized gas is a dynamic balance: $n_i$ and $n_e$ are created, but they can also be destroyed while the gas remains partly neutral.

The solar photosphere, at a $T$ of about 5800 K, illustrates this balance well: hydrogen's $E_ion$ is 13.6 eV, far above the typical thermal energy at that temperature, so only a tiny fraction of photospheric hydrogen is ionized. In deeper, hotter stellar layers, $x_i$ rises sharply over a narrow temperature range, exactly as the Saha equation predicts.

🔵 Formal level

Basic formulas

The first formal criterion is energetic: direct impact ionization can start only when the transferable $E$ reaches the $E_min$, identified with the species $E_ion$. This relation decides possibility, not the total amount ionized.

Energetic condition for ionization

For radiation, the threshold is written with $\nu$, $\nu _0$, $\lambda _0$, $h$, and $c$. Light ionizes when the quantum energy of each photon exceeds the threshold, so increasing $\nu$ favors ionization, while a larger $\lambda _0$ describes a less demanding threshold.

Photoionization threshold

Real student questions

Why does not everything ionize exactly when $T$ “matches” $E_ion$? Because $T$ sets an average scale, not an identical energy for every particle.

Why does $n_total$ reduce ionization in Saha? Because it favors combined states and recombination over free species.

Can red light ionize if it is very intense? If $\nu$ is below $\nu _0$, each photon lacks enough energy in this photoionization model.

Does positive $dn_i$ mean a stable plasma? No: it means net ionization is increasing at that instant; it may still change later.

Final synthesis

Ionization is a negotiation between a microscopic threshold and a statistical population. $T$ pushes particles toward ionizing energies; $E_ion$ resists; $n_total$ and recombination regulate the result. $x_i$ is the collective outcome of that competition.