---
title: "Mixing in a cluster magnetic field and plasma"
description: "Convert microgauss fields, electron densities and keV energies into mixing terms, and see why a plasma resonance removes the energy dependence."
canonical_url: "https://sandboxphysics.com/stories/axion-photon-mixing/mixing-in-cluster-fields-and-plasma/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/axion-photon-mixing/mixing-in-cluster-fields-and-plasma/"
last_modified: "2026-11-12"
---

[Searching for axions with cosmic magnetic fields](https://sandboxphysics.com/stories/axion-photon-mixing/)

# Mixing in a cluster magnetic field and plasma

Sandbox Physics · 2026-11-12 · L2 · Course chapter

Whether a cluster converts X-ray photons efficiently comes down to one comparison: is the photon energy above or below a critical energy set by the plasma, the axion mass, the coupling and the field? Above it the mixing wins and conversion is large and nearly independent of energy. Below it the mismatch between photon and axion wins, and the largest possible conversion falls as the square of the energy. When the axion mass equals the plasma frequency, the mismatch vanishes and the energy dependence disappears altogether.

The previous chapter, [Photon and axion as two mixing states](https://sandboxphysics.com/stories/axion-photon-mixing/photon-and-axion-as-two-states/), derived the two-state probability. This chapter puts cluster numbers into it.

## The plasma frequency of intracluster gas

Free electrons respond to a passing wave and shift its dispersion. For X-rays far above the plasma frequency the photon behaves as if it had mass $\omega_{\rm pl}$, with

$$
\omega_{\rm pl}^2=\frac{4\pi\alpha\,n_e}{m_e}.
$$

Here $n_e$ is the electron number density, $m_e$ the electron mass and $\alpha$ the fine-structure constant. Evaluated with CODATA constants,

$$
\omega_{\rm pl}=3.713\times10^{-11}\,\mathrm{eV}\,\sqrt{\frac{n_e}{1\,\mathrm{cm^{-3}}}}.
$$

[Meyer, Horns and Raue, 2013](https://arxiv.org/abs/1302.1208v2) print $3.69\times10^{-11}$ for the coefficient. Ours is 0.6 percent larger. We kept the value computed from the constants and recorded the difference instead of matching the rounded one.

Cluster gas has densities of roughly $10^{-3}$ to $10^{-1}\,\mathrm{cm^{-3}}$, so $\omega_{\rm pl}$ is a few times $10^{-12}$ eV. At $n_e=10^{-2.5}\,\mathrm{cm^{-3}}$ it is $2.09\times10^{-12}$ eV, and the photon term at 1 keV is $\Delta_{\rm pl}=-0.341\,\mathrm{kpc^{-1}}$. It scales as density over energy.

## One ratio decides the regime

The amplitude factor from the two-state formula can be rewritten with a single ratio. The mismatch is $\Delta_{\rm pl}-\Delta_a=(m_a^2-\omega_{\rm pl}^2)/(2E)$ and twice the mixing is $g_{a\gamma}B_\perp$. Their ratio is $E_{\rm crit}/E$, with

$$
E_{\rm crit}=\frac{\lvert m_a^2-\omega_{\rm pl}^2\rvert}{2g_{a\gamma}B_\perp}.
$$

Substituting gives an exact form for the largest possible conversion:

$$
\left(\frac{2\Delta_{a\gamma}}{\Delta_{\rm osc}}\right)^2=\frac{1}{1+(E_{\rm crit}/E)^2}.
$$

Far above $E_{\rm crit}$ the factor approaches one. Far below it, it falls as $(E/E_{\rm crit})^2$. The same critical energy appears as eq. (6) of Meyer, Horns and Raue, written as $2.5$ GeV times the mass difference in square nanoelectronvolts, divided by the coupling in units of $10^{-11}\,\mathrm{GeV^{-1}}$ and the field in microgauss. Recomputing their prefactor from the constants gives 2.56 GeV, so their 2.5 is a rounded value.

## Three cases from the Lab presets

**Default tangled field.** With $g_{a\gamma}=10^{-11.5}\,\mathrm{GeV^{-1}}$, $B_\perp=5\,\mu\mathrm G$, $n_e=10^{-2.5}\,\mathrm{cm^{-3}}$ and $m_a=10^{-13}$ eV, the axion mass is negligible next to the plasma frequency. The formula gives $E_{\rm crit}=7.04$ keV. Meyer's rounded prefactor gives 6.88 keV, a 2 percent difference that comes entirely from the rounding. Across the 1 to 9 keV band the amplitude factor climbs from about 0.02 to above 0.6, so conversion grows with energy.

**Mass matches the plasma.** Set $m_a=\omega_{\rm pl}=2.09\times10^{-12}$ eV. The diagonal entries are equal, the amplitude factor is one at every energy and $\Delta_{\rm osc}=2\Delta_{a\gamma}$. With $g_{a\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$, $B_\perp=5\,\mu\mathrm G$ and a uniform 96 kpc path,

$$
\Delta_{a\gamma}L=7.64\times10^{-3}\times96=0.733,\qquad P_{\gamma\to a}=\sin^2(0.733)=0.448.
$$

Unpolarized survival is 0.776, the same at 0.5 keV as at 10 keV. In a real cluster the density changes along the path, so a resonance like this would hold only in a thin shell where $\omega_{\rm pl}$ happens to equal $m_a$.

**Heavier axion.** Raise the mass to $10^{-11}$ eV with $g_{a\gamma}=10^{-11}\,\mathrm{GeV^{-1}}$. Now the axion term dominates and $E_{\rm crit}=49.0$ keV. The amplitude factor is $4.2\times10^{-4}$ at 1 keV and $1.7\times10^{-2}$ at 6.5 keV, a forty-fold rise. The spectrum is untouched at low energy and dips at high energy. A low-energy cutoff of this kind is how X-ray searches lose sensitivity as the mass grows.

## The Perseus profile at the published limit

Reynolds and colleagues describe the Perseus electron density with a two-component profile from earlier X-ray work, their eq. (2):

$$
n_e(r)=\frac{3.9\times10^{-2}}{[1+(r/80\,\mathrm{kpc})^2]^{1.8}}+\frac{4.05\times10^{-3}}{[1+(r/280\,\mathrm{kpc})^2]^{0.87}}\,\mathrm{cm^{-3}}.
$$

In their field model B the magnetic pressure is a fixed fraction of the gas pressure, the field is about 7.5 microgauss at 25 kpc, and treating the cluster as isothermal makes the field fall as the square root of the density. [Reynolds et al., 2020](https://arxiv.org/abs/1907.05475v4) Leave out their random cell structure and evaluate only that radial scaling at the coupling they exclude, $g_{a\gamma}=7.9\times10^{-13}\,\mathrm{GeV^{-1}}$:

**Table 3.1** Critical energy along the model-B radial profile, for a very light axion.

| Radius | Electron density | Field | Critical energy |
| --- | --- | --- | --- |
| 25 kpc | $3.70\times10^{-2}\,\mathrm{cm^{-3}}$ | $7.5\,\mu\mathrm G$ | 220 keV |
| 100 kpc | $1.08\times10^{-2}\,\mathrm{cm^{-3}}$ | $4.1\,\mu\mathrm G$ | 119 keV |
| 200 kpc | $3.93\times10^{-3}\,\mathrm{cm^{-3}}$ | $2.4\,\mu\mathrm G$ | 72 keV |

Every critical energy sits far above the Chandra band. At this coupling Perseus mixes X-rays only weakly: a single coherent cell converts well under one percent. A percent-level signal builds up only because the photons cross many cells. At 5 keV the oscillation length runs from about 8 kpc near the centre to about 74 kpc at 200 kpc, close to the cell sizes, so each cell's contribution changes phase with energy. That is where the ripples come from.

The Lab does not apply a telescope's energy resolution. Ripples narrower than the instrument resolution would be partly smoothed in a real spectrum, and a real analysis has to pass the model through the instrument response.

## Lab tasks

1. Predict. On the default preset, open the advanced panel and read the critical energy. Say whether a 2 keV photon or an 8 keV photon converts more on average.
2. Change one condition. Raise the density slider by one decade. Predict how the critical energy changes, then check.
3. Compare. Choose the plasma-resonance preset and drag the energy cursor across the whole band. The solid curve should stay flat.
4. Explain. Choose the heavier-axion preset. Find roughly where the dashed mean first drops below 0.95. It happens near 3.6 keV. Explain why this is more than ten times below the critical energy: each of the 24 cells adds a small conversion, and the small amounts accumulate.

## Exercises

1. A cluster has $n_e=10^{-3}\,\mathrm{cm^{-3}}$ and $B_\perp=2\,\mu\mathrm G$. For a very light axion and $g_{a\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$, estimate $E_{\rm crit}$.
2. At fixed coupling and field, how does $E_{\rm crit}$ change if the density doubles? If instead the axion mass rises well above the plasma frequency and then doubles?
3. Why does the resonant case lose its energy dependence even though $\Delta_{\rm pl}$ itself still depends on energy?

Answers

1. $\omega_{\rm pl}=3.713\times10^{-11}\times\sqrt{10^{-3}}=1.17\times10^{-12}$ eV, so $\omega_{\rm pl}^2=1.38\times10^{-24}\,\mathrm{eV^2}$. With $2g_{a\gamma}B_\perp=2\times10^{-21}\times3.91\times10^{-8}=7.81\times10^{-29}\,\mathrm{eV}$, the critical energy is $1.77\times10^{4}$ eV, about 18 keV.
2. The critical energy is proportional to $\omega_{\rm pl}^2$, which is proportional to density, so doubling the density doubles it. When the mass dominates, it scales as $m_a^2$, so doubling the mass quadruples it.
3. Only the difference $\Delta_{\rm pl}-\Delta_a$ enters the probability. Both terms scale as one over energy, so when $m_a=\omega_{\rm pl}$ they cancel at every energy, and the mixing term $g_{a\gamma}B_\perp/2$ has no energy in it.

[Try it in the Lab](https://sandboxphysics.com/labs/axion-photon-conversion/): Compare Mass matches the plasma with Heavier axion, low-energy cutoff, and read the critical energy in the advanced panel.

## Sources

1. [Meyer, Horns and Raue, 2013](https://arxiv.org/abs/1302.1208v2)
2. [Reynolds et al., 2020](https://arxiv.org/abs/1907.05475v4)

## Keep reading

Searching for axions with cosmic magnetic fields · Reading path

### From two mixing states to a published limit

1. [Article **How could an invisible particle change a distant spectrum?**](https://sandboxphysics.com/stories/axion-photon-mixing/invisible-particles-and-distant-spectra/)
2. [Article **Photon and axion as two mixing states**](https://sandboxphysics.com/stories/axion-photon-mixing/photon-and-axion-as-two-states/)
3. [You are here **Mixing in a cluster magnetic field and plasma**](https://sandboxphysics.com/stories/axion-photon-mixing/mixing-in-cluster-fields-and-plasma/)
4. [Article **Random domains, polarization and stellar cooling**](https://sandboxphysics.com/stories/axion-photon-mixing/random-domains-polarization-and-stars/)
5. [Article **Recompute a cluster sightline from saved inputs**](https://sandboxphysics.com/stories/axion-photon-mixing/recompute-a-cluster-sightline/)
6. [Article **Reading the Chandra NGC 1275 axion limit**](https://sandboxphysics.com/stories/axion-photon-mixing/reading-the-chandra-ngc-1275-limit/)