---
title: "What does half ionized mean?"
description: "A half-ionized parcel and a volume split into neutral and fully ionized regions can share an average but recombine at different rates."
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last_modified: "2026-11-12"
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[Starlight and intergalactic hydrogen](https://sandboxphysics.com/stories/reionization/)

# What does half ionized mean?

Sandbox Physics · 2026-11-12

Half the hydrogen can be ionized in two different ways. One half of a region could be almost fully ionized, with the other half neutral. Alternatively, every small parcel could have half its hydrogen ionized. Both arrangements have the same average ionized fraction, but electrons and protons do not encounter one another at the same rate in the two cases.

Start with [How starlight changes intergalactic hydrogen](https://sandboxphysics.com/stories/reionization/starlight-and-hydrogen/). Fractions, volumes and basic algebra are enough for most of this chapter; rate notation is introduced when needed. The aim is to calculate an ionized volume, distinguish it from a local ionized fraction, and choose the appropriate recombination law.

## Counting ionized volume

Divide uniform-density hydrogen into one thousand equal-volume cells. Make a two-phase approximation: hydrogen is nearly fully ionized in one phase and nearly neutral in the other. Let $f_j$ denote the fraction of cell $j$ occupied by the ionized phase. It has no units.

A cell with $f_j=0.25$ represents one quarter of its volume assigned to the ionized phase. That average has not specified the exact boundary within the cell. It has also not assigned a local ionized fraction of one quarter to every parcel in it.

Use this deliberately constructed arrangement to practise the definition:

**Table 2.1** An equal-volume cell example.

| Number of cells | Ionized volume fraction in each |
| --- | --- |
| $500$ | $1$ |
| $200$ | $0.25$ |
| $300$ | $0$ |

The ionized volume equals 550 cell volumes. Its fraction of the total is the volume filling factor, $Q$:

$$
Q=\frac{V_{\mathrm{ion}}}{V}=\frac{1}{1000}\sum_{j=1}^{1000}f_j
=\frac{500+200\times0.25}{1000}=0.55.
$$

The simple arithmetic average works because the cells have equal volumes. Unequal cells require volume weighting. This ratio of ionized volume to total volume is the starting point of a reionization filling-factor model. [Madau, Haardt and Rees, 1999, §3.1](https://arxiv.org/abs/astro-ph/9809058v1)

Each square in the cell diagram represents ten equal-volume cells. The mean volume filling is $Q=0.55$. The uniform parcel has the same numerical microscopic fraction $x=0.55$. Under the assumptions in the text, their recombination rates are proportional to $0.55$ and $0.55^2$.

## A local fraction counts particles

Zoom in on a parcel small enough to treat as uniform. Write its hydrogen-nucleus number density as $n_{\mathrm H}$, in particles per cubic metre, and its ionized-hydrogen number density as $n_{\mathrm p}$. The local ionized fraction is

$$
x=\frac{n_{\mathrm p}}{n_{\mathrm H}}.
$$

In the two-phase approximation, local $x$ is close to one or zero. Its volume average is $Q$. If hydrogen density is also uniform, the fraction of hydrogen nuclei that are ionized equals $Q$ too. With varying density, volume weighting and hydrogen-number weighting can give different answers.

The same average of 0.55 could instead come from $x=0.55$ everywhere. Neutral atoms, electrons and protons would then coexist throughout the volume. The two-phase arrangement concentrates the free particles in the ionized regions. That matters for recombination.

Count hydrogen only, with no extra electrons from other elements. In a parcel, electron and proton number densities both equal $n_{\mathrm H}x$. The recombination rate per volume is therefore

$$
\mathcal R=\alpha_{\mathrm B}n_{\mathrm e}n_{\mathrm p}
=\alpha_{\mathrm B}n_{\mathrm H}^2x^2.
$$

The coefficient $\alpha_{\mathrm B}$ depends on the fixed temperature and has units $\mathrm{m^3\,s^{-1}}$. Under the Case B approximation, an ionizing photon emitted in direct recombination to the ground state is absorbed nearby and ionizes another atom. That local cycle is excluded from the net recombination rate requiring replacement by the source. The rate $\mathcal R$ has units $\mathrm{m^{-3}\,s^{-1}}$. Halving local $x$ halves both encounter partners, reducing the rate to one quarter at the same density and temperature. [Mellema et al., local ionization equation in §2.2](https://arxiv.org/abs/astro-ph/0508416v2)

For total volume $V$ at uniform density, the two-phase arrangement has total recombination rate $\alpha_{\mathrm B}n_{\mathrm H}^2QV$. A uniformly partially ionized gas with $x=Q$ has rate $\alpha_{\mathrm B}n_{\mathrm H}^2Q^2V$. At $Q=0.55$, the two factors are 0.55 and 0.3025. The two arrangements have different encounter rates at the same average fraction.

## Follow a parcel after the light goes out

Keep the density and temperature of one parcel fixed, remove its source, and allow recombination alone. The notation $\mathrm dx/\mathrm dt$ means the change in local ionized fraction per unit time. Recombination removes protons, so

$$
\frac{\mathrm dx}{\mathrm dt}=-\alpha_{\mathrm B}n_{\mathrm H}x^2.
$$

If $x_0$ is the initial fraction and time starts when the source turns off, the solution is

$$
x(t)=\frac{x_0}{1+\alpha_{\mathrm B}n_{\mathrm H}x_0t}.
$$

The decline slows as free electrons and protons become less abundant. Define $t_{\mathrm{rec},0}=1/(\alpha_{\mathrm B}n_{\mathrm H})$, a timescale based on fully ionized gas. Starting from $x_0=1$, the parcel has $x=1/2$ after one such time.

A volume prescription may instead remove fully ionized volume at a fixed rate, $\mathrm dQ/\mathrm dt=-Q/t_{\mathrm{rec}}$, giving an exponential decrease. It describes a changing amount of the ionized phase. It has not followed the electrons in this same parcel or specified how a spatial boundary moves. Applying that exponential to the parcel would replace the correct one-half with about 0.368 after one recombination time. Keeping the definitions separate prevents this error.

## Problems with explained answers

1. Of one thousand equal-volume cells, 250 are fully ionized and 500 have half their volume assigned to the ionized phase. The remainder are neutral. Find $Q$ and explain whether it requires identical local $x$ everywhere.
2. Two equal-volume regions have hydrogen number densities $n$ and $3n$. Only the denser region is fully ionized. Find the volume filling factor and the ionized fraction weighted by hydrogen nuclei.
3. A fixed-density parcel starts with $x_0=0.5$. Find its local fraction one $t_{\mathrm{rec},0}$ after its source turns off, using the fully ionized timescale defined above.

For the first problem, the ionized volume is $250+500/2=500$ cell volumes, so $Q=0.5$. Local values in the two-phase picture remain mostly one and zero; $Q$ states how much space the phases occupy.

For the second, the ionized region occupies half the volume, giving $Q=0.5$. It contains the fraction $3n/(n+3n)=0.75$ of all hydrogen nuclei. Removing the uniform-density assumption separates the two weightings.

For the third, substitution gives $x=0.5/(1+0.5)=1/3$. Both encounter-partner densities decrease with $x$, so a parcel initially half ionized does not follow a scaled copy of the initially fully ionized parcel's decline.

Next, restore the sources and work out [the photon supply in an expanding volume](https://sandboxphysics.com/stories/reionization/photons-in-an-expanding-box/).

## Sources

1. [Madau, Haardt and Rees, 1999, §3.1](https://arxiv.org/abs/astro-ph/9809058v1)
2. [Mellema et al., local ionization equation in §2.2](https://arxiv.org/abs/astro-ph/0508416v2)

## Keep reading

Starlight and intergalactic hydrogen · Reading path

### Photons, ionized volume and a radio signal

1. [Article **How starlight changes intergalactic hydrogen**](https://sandboxphysics.com/stories/reionization/starlight-and-hydrogen/)
2. [You are here **What does half ionized mean?**](https://sandboxphysics.com/stories/reionization/ionized-volume-and-local-fraction/)
3. [Article **Supplying photons to an expanding volume**](https://sandboxphysics.com/stories/reionization/photons-in-an-expanding-box/)
4. [Article **Where the photons go when ionized bubbles meet**](https://sandboxphysics.com/stories/reionization/when-ionized-bubbles-overlap/)
5. [Article **Reading ionized bubbles through neutral hydrogen**](https://sandboxphysics.com/stories/reionization/reading-a-neutral-hydrogen-map/)
6. [Article **Recompute a fixed-source ionization history**](https://sandboxphysics.com/stories/reionization/recompute-a-fixed-source-history/)
7. [Article **Reading HERA's 21 cm upper limits**](https://sandboxphysics.com/stories/reionization/reading-hera-upper-limits/)