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When calculating the decoupled temperature of photons using Saha' equation for the following process: \begin{equation} e^- p\longleftrightarrow H\gamma \end{equation} we find that $T_{dec}=3000$ K$=0.25$ eV.

From my understanding, this phenomenon happens when it becomes thermodynamically favourable for protons and electrons to combine into neutral atoms. I was expecting it to be 13.6 eV (Rydberg energy) for this case, which is the Hydrogen's biding energy. Why is it less than that?

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This is because there are hugely many more photons than charge-carriers per unit volume, roughly 10 billion photons for every electron in the universe.

As an example, consider affairs when the universe cooled to a temperature of 1 eV, or around 10,000 K. At this temperature, electrons are no longer relativistic and their density follows the Boltzmann distribution, $$n_e = 2\left(\frac{m_e T}{2\pi}\right)^{3/2} \exp \left(\frac{\mu_e - m_e}{T}\right).$$ At $T = 10^4$ K, the electron density is $n_e \approx 10^4 \,{\rm cm}^{-3}$.

Meanwhile, the number density of photons with an energy in excess of 13.6 eV can be found by integrating the Planck spectrum, $$n_\gamma = \frac{1}{\pi^2}\int^\infty_{13.6}\frac{E^2}{\exp(E/T)-1}\, {\rm d}E,$$ giving $n_\gamma \approx 3 \times 10^9 \, {\rm cm}^{-3}$ at $T = 10^4$ K. In other words, there are around $3\times 10^5$ more photons than electrons per unit volume with energy greater than 13.6 eV! At these temperatures, there is no shortage of energetic photons available to re-ionize neutral hydrogen once it forms. The following illustration helps visualize this:

enter image description here

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  • $\begingroup$ Ok, so If I understand you well: at $T=$13.6 eV most of the present photons are more energetic and therefor the process cannot happen. And so at $T=0.25$ eV the integral you mentioned should be around 0? I got $n_\gamma \approx 10^{-23}$. But when plugging $T=1$ eV I get $10^{-5}$ instead of $10^5$. $\endgroup$
    – devCharaf
    Apr 26, 2020 at 10:22
  • $\begingroup$ You need to write T in Kelvin; sorry, I was not careful with this in my answer. The temperature corresponding to 1 eV of energy is around 10000 K. $\endgroup$
    – bapowell
    Apr 26, 2020 at 14:45
  • $\begingroup$ Okay thank you :) $\endgroup$
    – devCharaf
    Apr 26, 2020 at 15:39

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