Talk:Saha ionization equation

Units

can someone tell me what the exact units of the individual variables should be? as we all know there are many different forms, different units, for the variables. thank you.

I will fix the equations so its clear what each variable means, but its not necessary to specify the exact units of each variable as long as they are consistent. For example, the electron mass is in units of mass, and we might choose the MKS system of units, in which case the mass would be kilograms. Or we could choose the CGS system of units, in which case the mass would be in grams. The equation is true in either system, so we don't need to specify which system, all we need to do is to say the electron mass is in units of mass, which is pretty obvious, so thats not even needed either. PAR 22:06, 12 August 2005 (UTC)

Ionization energy

The definition of epsilon_i says it is "the energy required to remove an electron from an (i-1)-level ion, creating an i-level ion", but the usage in the equation is that of the total ionization energy of state i, since the equation should have the energy to go from state i-1 to state i (or, as written in the equation, from state i to state i+1), and that is written explicitly as an energy for state i+1 minus the energy for state i.

Please correct me if I'm wrong, but I think that either the "- epsilon_i" should be removed from the exponential argument in the equation, or else epsilon_i should be defined as the total ionization energy above the neutral state.

Sorry for any clumsiness in submitting this comment; I have not done it before, and I don't want to attempt to edit the page myself.

Incidentally, the Saha equation can be put into a form completely analogous to the Boltzmann equation for the distribution over excitation states, i.e., the ratio of the fraction in a given state to the total, rather than the fraction relative to the next lower state. This form is both faster to compute and more accurate than the form relating consecutive states and eliminates the need to compute a "dominant potential" estimate for the most populated state (i.e., to avoid computing the ratio for states in which the denominator is essentially zero). I published the derivation in 1970 as a graduate student in a little short-lived backwater journal named the "Maryland Astronomical Journal", to which all grad students were strongly requested to contribute. Although I received several reprint requests, the improved form did not achieve traction, as I see it nowhere in the literature today. I would be happy for Wikipedia to be the first place for it to be widely available. If you would like to pursue this, please contact me at jwf@ipac.caltech.edu or whatever way you prefer.

Regards,

John W. Fowler

Spitzer Science Center 134.4.23.49 18:33, 9 September 2006 (UTC)

Infrared Processing and Analysis Center, J.W. Fowler

Saha equation normalized to Total atomic number density September 2012

• Source arXiv

87.211.116.227 (talk) 10:52, 23 April 2025 (UTC)

California Institute of Technology

Can you send me a reprint, preferably in PDF format? PAR 15:19, 11 September 2006 (UTC)

mimeograph stencilmachine 87.211.116.227 (talk) 08:39, 23 April 2025 (UTC)

The Maryland Astronomical Journal was a home-brewed journal in the Astronomy Program at the University of Maryland circa 1970. It was hand-typed by the chairman's wife, typos scrawled over by hand, and copies were produced on the new-fangled Xerox machine (we had just gotten over mimeograph). My paper, "Ionization Equilibrium in Stellar Plasmas", was 15 pages long, of which the first six are relevant here. I can scan those and convert them to PDF, but I'm not sure where to email the result. Please let me know how to do that and whether you prefer the entire paper. I gather there is some reluctance to use regular email, perhaps a Wikipedia policy? If not, a message to my email address above would allow me to send the PDF by return email.

Regards,

John W. Fowler

Reduced mass?

In Oxenius' Kinetic Theory of Particles and Photons (Template:ISBN), Appendix E.2, it is stated that the reduced mass of the electron (with respect to the electron-ion pair that may combine) should be used in evaluating the thermal de Broglie wavelength of the electron in the Saha equation. Is that wrong or applicable but just for some special case, or should this be changed in the article? --Tardis 23:28, 10 January 2007 (UTC)

"Detailed Derivation from University of Utah Physics Department"

This just appears to be a pdf of Wolfram Scienceworld. Is there something I'm missing here? --128.153.144.204 (talk) 01:57, 8 March 2010 (UTC)

The correspondent page at ScienceWorld (http://scienceworld.wolfram.com/physics/SahaEquation.html) is empty. 88.195.248.169 (talk) 20:25, 5 September 2015 (UTC)

3 pages (A4) as of 9/26, 2006

87.211.116.227 (talk) 11:40, 23 April 2025 (UTC)

Removal of page tags

I think this page is "good enough." There is a plot of ionization state vs. temperature, and a few references to modern books. I removed the tag requesting an image in the talk, and asking for more citations in the article itself. Dstrozzi (talk) 00:33, 24 June 2024 (UTC)

Notice large fig. (100 10³ K) since 28 Dec has different curve colors.

87.211.116.227 (talk) 13:26, 23 April 2025 (UTC)

Weakly ionized plasma: isn't the Debye length then large?

If there are only few charge carriers, it takes a longer distance for shielding. I have no expertize in plasma physics, but please correct this if it is wrong Peter.steier (talk) 14:53, 20 July 2024 (UTC)

You are right. In general, most of that text is a confusing mess. Evgeny (talk) 15:02, 21 July 2024 (UTC)

I removed the mention of the Debye length, which is irrelevant. Evgeny (talk) 12:17, 30 January 2025 (UTC)

never mind: Debye length section 2

87.211.116.227 (talk) 14:32, 23 April 2025 (UTC)

Why low T is sufficient to ionize

Currently it's said:

There is substantial ionization even though this

$T$

is much less than the ionization energy (although this depends somewhat on density). This is a common occurrence. Physically, it stems from the fact that at a given temperature, the particles have a distribution of energies, including some with several times

$T$

. These high energy particles are much more effective at ionizing atoms.

It's really an entropic thing though, right? Freed electrons have a much larger space to explore. If you insist on a kinetic explanation then you have to give a full picture and say e.g., that for very low densities, once an atom ionizes it is very hard for an ion and electron to find each other again and recombine. The "depends somewhat on density" is not just an aside but actually a key part of understanding this. --Nanite (talk) 18:58, 27 January 2025 (UTC)

k_BT versus ε, T compared to ε/k_B: factor 8 when H-atom/H⁺-ion

total number density n = 1-100 % of (1-100 ×) n_L as parameter in 2 plots: low (high) densities?

nl: Vrije weglengte

87.211.116.227 (talk) 02:07, 20 April 2025 (UTC)

Saha equation is limited to ideal gases

It is known to all physicists that the Saha ionization equation is derived under ideal gas assumption 37.162.168.30 (talk) 17:18, 27 March 2025 (UTC)

No, it is "known" only to a subset of all physicists; I want to believe this subset if very small :).

Perhaps what you mean is that electrons are treated as an ideal, classical gas, for which the chemical potential is given by the simple expression $kT\ln (n_e{\lambda }_{\mathrm{t}\mathrm{h}}^3)$ (which is easily recognizable in the standard Saha equation). It's trivially generalized to an ideal Fermi-Dirac gas, for example, to account for quantum degeneracy effects. In essence, the Saha equation describes equilibrium of a mix of gases expressed through their chemical potentials. It's a very basic thermodynamic principle, not limited to ideal gases. Evgeny (talk) 18:02, 27 March 2025 (UTC)

Sorry to butt in, but I can't resist. :) Even at lower densities where Fermi gas effects are negligible, I would have expected it to be necessary to contend with correlation effects like Debye–Hückel theory (See Landau Lifshitz #5, section 78). But on further digging it seems to be quite subtle! The article right now talks about partition function cutoffs (uncited) but this seems to be a bandaid on a rather intricate problem. It would definitely be nice to see a well-cited single sentence in lead that describes the limit of Saha equation, and also the actual validity condition ($x\ll y$ or whatever) stated in the article body. But I honestly can't say what that condition is.

Plasma physics sources seem to not care so much about the validity condition of Saha. I suspect what is happening is that they are not so much interested in homogeneous thermally equilibrated plasmas since the real plasmas they study will often violate that (radiation escaping, Debye sheaths, etc.). And that's why they are more interested in the nonequilibrium 'detailed balance of rates' (where Saha appears in certain approximations), versus the equilibrium statistical mechanical one. --Nanite (talk) 20:21, 27 March 2025 (UTC)

Some more digging. The stellar physics people of course do care about the homogeneous thermal plasma. It seems currently they like to describe nonidealities with models like 'continuum lowering' i.e. ionization potential lowering, which is basically a modification of Saha equation. This is related to the partition function cutoff but it seems to be a really messy problem, especially for big atoms like iron!

Anyway the key thing seems to be plasma parameter. As far as this article is concerned we can just say Saha equation applies for 1) thermodynamic equilibrium, 2) weak coupling limit (large plasma parameter) and 3) low density (no "pressure ionization" i.e. Fermi gas stuff). That's more accurate than just saying 'ideal gas'.

It's worth mentioning the whole mess of excited states / partition function / cutoff / continuum lowering / Debye-Huckel / Stewart-Pyatt / Ecker-Kröll, but that belongs entirely in a secondary section. --Nanite (talk) 16:42, 29 March 2025 (UTC)

1) is already said upfront, and 2) is not necessary, IMO. Again, it depends on how you read/interpret the equation. If one light-mindedly plugs in values (the ionization potential, in particular) of an isolated atom - sure, the result will be applicable only for a weakly coupled plasma. And yes, it's a mess with the various models of the ionization potential depression. The fact that no static model can, in principle, be universally valid (recall that all plasma particles move and so the Debye-Huckel potential is correct in the best case only on average...) doesn't help. Evgeny (talk) 16:04, 7 April 2025 (UTC)