So you tell me how the Bb compares.
Come on you doubting thomas', what is hte predicted value of H at this moment of time by the BB theory?
It is not a lot to ask is it?
So what value of H does the BB predict?
Cheers
Lyndon.
I wait with baited breath.

So you still maintain that H and hr/m can be meaningfully compared, and their similarity gives important information about the universe, even though their numerical values are only similar if you happen to choose the right set of units?

There really is a simple technique called dimensional analysis, and it really is used at about high school level to identify errors in formulae. The formula H=hr/m fails this simple test.

Saying it holds "per cubic meter" does not change a thing. If this worked, you get get any formula of the hook by saying "per mismatched units".

What Lyndon needed was an actual physical constant with dimensions of m^-3, and with magnitude 1 in SI units. He found it in electron number density (or twice the electron number density). The parameter is not known to any accuracy, but the value Lyndon needed was within the range of possibilities.

Electron number density is definitely known to vary considerably from one region of space to another, but the mean value is only guessed to within a couple of orders of magnitude.

This also makes nonsense of the claim to "calculate" H, since Lyndon is tuning an unknown value to obtain the match. Strictly speaking, he should call this a "prediction" of the value for n, not a calculation of the value for H, since H is known to much greater accuracy than n.

Cheers -- Sylas

That sounds like an excellent and constructive suggestion. I am content to point out the places where I disagree, and why; and if we have any readers who are uncertain, they can compare the two and try working out the matter for themselves. For reference and context, I am quoting from this post by Lyndon.

The Compton constant shows up in the Compton scattering formula, which gives the increase in wavelength as (1-cos(a))h/mc, where a is the scattering angle. The Lyndon effect therefore claims to have the same redshift as Compton scattering with an angle of 90 degrees.

Compton scattering can be derived quite easily by using conservation of energy and momentum. When the scattering angle is zero, the increase in wavelength is also zero.

The Lyndon effect has the same redshift as a Compton effect with right angle scattering, and yet the scattering angle in the Lyndon effect is zero. This inevitably violates conservation of energy and momentum. Lyndon has yet to address this problem using a properly quantified calculation of energy and momentum, and the various proposals he has given for extra energy terms have all been considered, quantified, and shown unable to balance the conservation requirements.

This is simply not true. If we look at Lyndon's paper, this is discussed on page 4, and the references are 17, 18 and 19. The actual formula is 2.r.λ.f_2. Lyndon subsequently omits the f_2, which is not done in any published result, except for one special unique value of λ to be identified below.

[[ Added in edit. And not even then. I don't think 2rλ is ever published as the photo-absorption cross section. On checking, I find that the special value of λ gets the f_2 in question up to around 0.96 before dropping back to zero. Where I speak of f_2 being 1 below, I should have said close to 1. ]]

Following the references tells the rest of the story, which I can summarize here.

• Lyndon is using collision cross sections for atoms, not electrons.
  
• The reference is to the Henke Atomic Scattering Factors, which are widely used particularly for X-rays. The Henke ASF is a complex number, of form f_1+i.f_2. The values for f_1 and f_2 vary widely across different atoms and for different wavelengths. They are derived by a combination of theory and empirical observation, and large databases are available tabulating f_1 and f_2 for different light frequencies and different atoms.
  
• Lyndon has focussed upon a Hydrogen atom, presumably because Hydrogen is the simplest atom, and it has a single electron. However, the behaviour of an electron bound to a proton is not a good model for an electron in ionized plasma.
  
• Lyndon needs to cancel the f_2 term to get his formula. His justification for this is given at the bottom of page 4 of his paper, as follows:
  For 10 keV to 30 keV X-rays interacting with Hydrogen, f_2 has values approximately between 0 and 1. ‘One’ meaning that the photon has been absorbed and the electron remaining in an excited state and ‘zero’ meaning that the photon was absorbed and an identical photon reemitted.
  This encapsulates a massive amount of error and confusion, and it is quite illuminating to dig through and see what is really going on.

For reference, people may refer to National Institute of Standards and Technology (NIST). They request the following acknowledgement:

• Chantler, C.T., Olsen, K., Dragoset, R.A., Kishore, A.R., Kotochigova, S.A., and Zucker, D.S. (2005), X-Ray Form Factor, Attenuation and Scattering Tables (version 2.1). [Online] Available: http://physics.nist.gov/ffast [Accessed May 24 2005]. National Institute of Standards and Technology, Gaithersburg, MD. Originally published as Chantler, C.T., J. Phys. Chem. Ref. Data 29(4), 597-1048 (2000); and Chantler, C.T., J. Phys. Chem. Ref. Data 24, 71-643 (1995).

For the range of photon energies Lyndon mentions, f_2 varies from about 6e-7 (at 10 keV) to 4.3e-8 (at 30 keV). The link above allows this to be confirmed by a simple query. The log-log graph is close to linear, and over this range we have a close fit to f_2 = 1.56e-4*E^-2.39, where E is the photon energy in keV. So in this range, f_2 is near enough to zero.

However, f_2 does approach 1 at lower energies. In fact, it hits 1 at energy of 13.595 eV; and you can check this with the above form as well. This corresponds to a wavelength of 912 Angstroms. Nowhere is f_2 ever a simple constant value, and it only hits 1 at this very special frequency where absorption is particularly efficient.

This is the energy required to ionize a neutral Hydrogen atom. Data is not tabulated below this value, unfortunately, but we know for sure that f_2 will drop suddenly for photons with energy below 13.595 eV, and that it must drop precipitously to nearly zero again once the energy drops below about 10.2 eV (1216 Angstroms). This is the requirement to lift an electron one energy level in an unexcited Hydrogen atom. Any residual f_2 value for lower energy photons will involve excitation of an already excited atom, and since Hydrogen atoms tend not to remain excited, such interactions are very rare.

[[ Added in edit. Actually, there is a tabulation of values for f_2 below the cutoff energy of 13.595 eV. To see this, use the database query form with "Display All Tabular Data for Element/Compound" as the "Type of Data", and request the lowest energy range. Request a table of the data. Here is a link for the query I used. The values for f_2 get up to 0.95197, at E=13.668eV, and then become identically zero at smaller energies, which are from 13.5864 down to 10.690 eV. I had thought there might be a small f_2 value for partial excitations, but this does not appear in the data. ]]

Using the WWW X-ray server at http://xray.uu.se/scripts/show_henke?H, we can obtain less accurate plots, showing this sudden drop in f_2 values down to about 1e-16. That there has to be such a drop is definite, and a simple consequence of how Hydrogen atoms absorb photons. The values and the cutoff point suggested in the Swedish server don't quite seem correct, however. I wonder if anyone would care to comment.


Down at the frequencies where Lyndon is really interested, which is visible light with energy of around 2.5 eV, the value of f_2 for Hydrogen is effectively zero.

The really breathtaking error is using a Hydrogen atom at all. The electrons in plasma are not bound to Hydrogen atoms. Ionized electrons don't have excitation levels at all. This means that they cannot absorb photons, and that the value of f_2 is identically zero.

I have not commented on the remainder of Lyndon's post, since it is all founded on the basic errors identified above. Nowhere in the post does Lyndon address the difficulty that his alleged absorption reaction violates conservation of energy and momentum.

Cheers -- Sylas

I agree, and Lyndon also acknowledges this in his website.

Yes, but Lyndon uses the "per cubic metre", so I think that the real issue here should be that can Lyndon justify his use of volume there, not the dimensional analysis of H=hr/m which everyone agrees doesn't work.

__________________
"Stupidity gets denser in a crowd" - Old Finnish saying. [My website and My BLOG] [Nimblebrain forums]

Having worked in the calibration services fields, I totally agree with this statement. If a formula is correct then we can basically use it to determine unknows to a higher accuracy by using other values of greater accuracy.

If I were to calibrate a voltmeter using this same principle, then my "standard voltmeter" would be the tounge testing of voltage in question.

__________________
MrObvious

Ari, that is what I asked him on page 14, and this was his answer:

So it appears that he cannot answer this question, but still he maintains that H = hr/m is significant. Lyndon would be much wiser if he dropped this argument completely and only focused on the H = 2nhr/m, that way he has one fundamental error less to defend.

__________________
Knowledge is a curse, but ignorance is worse

I agree. This H=hr/m business just distracts people, these threads wouldn't be tens of pages long if there wouldn't be the unit confusion. But maybe it can be seen as good PR-work, after all any publicity is good publicity, right? :wink:

__________________
"Stupidity gets denser in a crowd" - Old Finnish saying. [My website and My BLOG] [Nimblebrain forums]

I still can't get him to commit one way or the other on whether he thinks the value of hr/m is significant by itself... :-?

Well, when you look at pages on his website like this one, then it is very clear that he thinks that H=hr/m is very significant on its own.
This includes very bizarre things like
hr/m m/s per metre? What's that supposed to be? Now we have hr/m per second? Where does the second come from, and where have the cubic metres gone?
Lyndon seems to take H = hr/m very seriously, but doesn't seem to be able to use it very consistently.

__________________
Knowledge is a curse, but ignorance is worse

A bit of history: lyndonashmore posts for the first time in the Tired Light Theory thread on 2004 April 4 19:47 UT. He consistently uses H=hr/m from that point on until he starts the original Ashmore's Paradox thread on 2004 September 17. His first mention of H=2nhr/m is in his post of 2004 September 17 9:10 UT.

A quote from a later post on the same day at 13:07 UT might be of interest in this context:
Of course saying he prefers the "full version" and actually using it are two different things. His claim that he had the full version since 1995 is disingenuous at the very least. If he had actually known this before the 2004 preprint posted in 2004 September, one would think that he would have mentioned it in 2004 April when the first objections to his misuse of units were made.

Of course, he still maintains the importance of hr/m per cubic meter against all objections on dimensional grounds ... :roll:

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

Sylas with a mathematics book or an internet search engine is a bit like a two year old with a machine gun – it doesn’t know what it is doing and it is highly dangerous.
Sylas wrote:
Lyndon has addressed this, and shown beyond doubt that Sylas got his sums wrong – he forgot the vector nature of momentum and ignored that the electron recoils one way on absorption and the other way on emission. Sylas lumped them both together and was then surprised when the final velocity of the electron came out as zero.

Sylas has problems with his sums but when it comes to reading, things become even worse!
The collision cross section for an atom is the sum of the collision cross sections of the electrons within it. However one can’t just ‘add them up’ Sylas’ usual mistake (since he clearly uses his fingers as a calculator) as one gets ‘shielding.’ That is, as one gets more and more electrons they get in the way of each other therefore Hydrogen is the correct one to use.
What Sylas fails to understand is the a collision cross section is a ‘probability’. f_2 =1 means that the photon is absorbed and not re-emitted and f_2 = 0 means that there is no chance of the electron being absorbed a new photon is re-emitted. That is we are ecstatic that this is the case for low photon energies as this is the one we want.


Then why use this information to try to discredit a theory on the board?

Regardless of the value of f_2, the cross section is 2rλ for each and every electron and so this is the one to use. F_2 is the probability function that tells you the probability of the photon being absorbed or not. Since, in plasma the energy of the incoming photon is well away from the resonant frequency at which the electron causes the plasma to resonate, then the photon is always re-emitted.
In fact I will change my original comment,
Sylas with a mathematics book or an internet search engine is not only a bit like a two year old with a machine gun – it doesn’t know what it is doing and it is highly dangerous, but also nothing good will come of it!
Cheers,
Lyndon

Sorry but..
I owe a few of you replies, been buisy today, catch up tomorrow!
Cheers,
Lyndon

You are not in the position of making condescending remarks about other posters.

Irrelevant.
The basic mechanism of your "theory" has been proved to be unphysical.


So, how much time passes between absorption and re-emission?
How much momentum and/or energy does the electron lose during that time?
You never provided any quantitative estimates for this.

And let's not forget that you claimed that the electron is constrained to move along one direction during the scattering with the photon, hence the vector nature of the momentum does not affect Sylas's conclusion.

Electrons that are bopund to the nucleus.
Different from free electrons in a plasma.

So, due to shielding, the Coulomb interaction between electron is even lower.
What were you saying about "restoring forces"?

You still cannot distinguish an atom from a free electron.


However, this is for an atom, not electrons in a plasma.

He does not need that information to discredit your "theory".
You are applying an inappropriate formula anyway.

Electrons in a plasma are not electron in an atom.
So that formula is misapplied.


Why do you keep projecting your behavior on other people?

__________________
papageno


"Why waste time learning, when ignorance is instantaneous?" - Hobbes (Calvin and Hobbes)

"It's all about context!" - Vince Noir (The Mighty Boosh)

"I've never heard of such a brutal and shocking injustice that I cared so little about!" - Zapp Brannigan (Futurama)

"...because the logic of the lines traced from reality is as poor of aesthetic value as it is strict in consistency. " - Paolo Bozzi (Naive Physics - free translation)

Oh, the irony.

Papageno, why did you resurrect this argument? I thought you had more sense. :wink:

__________________
Do try not to take me too seriously.

Well, it was not for Ashmore, but for akirabakabaka.
I was hoping it would not be too late to turn him away from the dark side.
I thought I could train him to be a Jedi...
Hang on... this is the wrong thread!

__________________
papageno


"Why waste time learning, when ignorance is instantaneous?" - Hobbes (Calvin and Hobbes)

"It's all about context!" - Vince Noir (The Mighty Boosh)

"I've never heard of such a brutal and shocking injustice that I cared so little about!" - Zapp Brannigan (Futurama)

"...because the logic of the lines traced from reality is as poor of aesthetic value as it is strict in consistency. " - Paolo Bozzi (Naive Physics - free translation)

Papageno wrote4:
This is hogwash.
no body at any time has done this. stop halluscinating Papageno the theory works. allways did, always will.
cheers Lyndon
PS thanks again for believing this theory worthy of further discussion - you know it makes sense.

Hey, lyndon, can you tell me whether or not you still maintain that comparing hr/m to H numerically is meaningful?

A good description of your idea.

Sylas has provide the calculations for the upper limits on momentum and your idea isn't within those limits. Where's your calculations showing momentum is conserved? You've been asked repeatedly to provide them and haven't. Either you haven't done those calculations or you've done them and know your idea dosen't work. Either way, all you have been doing is handwaving with nothing to back up your claims.

In your mind it works. You haven't provided any hard quantitative calculations for your "mechanism" to show it conserves momentum or spin as you claim. Why not?

__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues.

Neptune- The original Dark Matter.

The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth

Seeing as Lyndon seems to disagree more strongly with Sylas's figures than his process, could Lyndon not just repeat Sylas's calculations, but subsitute in the values he believes to be correct?

He should be able to, or if he disagrees with the process (since his explanation doesn't conserve spin either), provide his own calculations on how the process works. That he either hasn't or can't do this is a rather telling statement on the mathematical validity of his "process".

__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues.

Neptune- The original Dark Matter.

The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth

f_2 = 0 means that the collision cross section is zero, and probability of photo-absorption collision is zero.

Let's make this very simple, so anyone can follow it. If Lyndon just blows off this information again, it does not bother me and I cordially request that others don't bother leaping to my defence. Stick to the substance. If anyone can make a plain statement of where one or other of us is incorrect, then that is very welcome.

The form factors, or Henke Atomic Scattering Factors, are not exactly elementary physics, but some of the points apparently under dispute are very easy. If Lyndon disputes the background information in this post, then anyone can check out the matter themselves. I have recommended the NIST site, which is very good for finding values of all kinds of physical constants. But any reference which actually quantifies the form factors will suffice. Plain references from Lyndon would be welcome as well, of course.

1. The photo-absorption collision cross section. In this post, Lyndon asserts:
I assert that the published result is actually 2rλ.f_2, where f_2 is also highly dependent on wavelength.

My citation is Lyndon's own paper, page 4, and Chantler's 1995 paper on form factors. See equation 5 in chapter 2. It is expressed in terms of energy rather than wavelength, but it is the same equation. Note that this reference makes explicit that f_2 and mu (cross section) are both functions of the photon energy.

Does Lyndon want to justify his claim that the equation is published somewhere without the f_2 factor being included, or would he like to acknowledge that the equation as he has given it in the thread is not from the published literature at all?

2. The values of f_2. In the preprint of Lyndon's paper, page 4, Lyndon says:
While this is true, it is very misleading, because for photons in this range, f2 ranges from 4.3e-8 to 6.0e-7.

Would Lyndon like to concede that the values of f2 are consistently less than 0.0000006 for X-ray photons with energy of 10 keV or more?

This may simply have been a typo. If we consider photons in the range 10 to 50 eV, we get this plot of f2 values:

In this range, f2 ranges from 0 to 0.96. There is a sharp cutoff at 13.595 eV, which is the ionization energy of a Hydrogen atom. Does Lyndon agree that f2 is basically zero ( less than 1e-8 ) for energies below 13.595 eV, and less than 0.1 for energies above 50 eV?

[[ By the way, this is a point on which I am uncertain myself. I am not merely regurgitating material from a quick google. I'm studying it, with a combination of internet references, library references, and my own worked problems. I'm dubious about the extent of the drop for f2, because I would have expected significant collision cross sections for partial excitations, with Lyman photons in the range 10.2 eV to 13.6 eV. I'd be very grateful for anyone who could give photoabsorption cross sections for Hydrogen at lower energies, either tabulated or with a formula; and how they relate to the form factors. This is not simply about Lyndon's theory for free electrons. I'm also working on the conventional physics of interactions with neutral Hydrogen, since it crops up frequently in various contexts, and help is appreciated. ]]

3. Excitation energy The dramatic cut-off seen in the above plot is at the energy required to eject an electron from a neutral Hydrogen atom.

That is, when bound to a proton, an electron has a negative potential energy of 13.595 eV, and can absorb this from a photon to become a free electron. Photoabsorption is an inelastic collision where some energy is given to excitation of the particles.

In plasma, the electron is already ionized. There is no further excitation state, and hence no way for an electron in plasma to absorb more energy in an inelastic collision. It can only exchange kinetic energy in an elastic collision.

Cheers -- Sylas

Regarding ‘the paradox’ and the various posters, let me answer them all at once. Yes I do think it worthy of note in that one expects this sort of thing when the equation is of the correct form. Tired Light gives H = 2nhr/m so, when observation shows H = hr/m in each cubic metre of space, it is a result. Not an Earth shattering one but a result. More so in that it shows just how ludicrous the Bb is. Why does each and every length of space increase by hr/m for the electron every second if the expansion is not related to the electron. Ergo, it is the expansion bit that is wrong.
I can just imagine Newton announcing his theory of gravity on the Net!
“F = GMm/r^2” he boldly said on the “Bad Alchemy” site.
“But it is impossible” said Mamageno, “the units are different on each side, F is in Newton and Mm/r^2 in kg^2 m^-2”
“Ah”, said Isaac, “but the constant has units itself so that makes it correct”
But you can make any theory work if you trump up a constant with units chosen to fit the equation” said Ourlas.
“Bet he can’t make it work in imperial units,” said a voice from the side.
“That’s right” said Ourlas, “Plus it contravenes the conservation laws” said Ourlas, If a gravitational force acts on the Earth the we would speed up and the years would get shorter”.
But that’s how gravity works” said Isaac.
“Show me the mechanism” said Mamageno, “the space between the Sun and the earth is too sparse. There is no way that a force could be transmitted through it. Any particle is free and so couldn’t exert forces between them. Go on, show me the mechanism.”
“How big is this constant”
“6.67x10^-11 Nm^2kg^-2” Rreplied Newton.
“6.67x10^-11” said Heavenly blacksmith, that is a fiddle factor if ever I saw one, ‘2’, or ‘3’ I might accept but not “6.67x10^-11”. Your formula is 11 orders of magnitude out without your constant.
And so on, and so on.
See what I mean?
In fact Newton stopped publishing because of the criticisms – well he did wait till after Hooke’s death to prevent the whinging.
Cheers,
Lyndon.
P.S. the names used in this post are fictitious and any similarity between these names and those of real people is purely coincidental.

In this case, Newton would be correct and Mamageno would be the clueless critic.

Newton's equation explicitly identifies the constant and gives it the correct units; the theory gives a way to measure the value of the constant and determine if it really is constant or not. The theory would be refuted if repeated measurements of G for different masses and distances gave different values. The measurements can be done in any unit system, since G is not merely a number, but can be converted between units by the scale factors determined by its dimensions.

I have not seen anybody on this board engage in the foolishness exhibited here by Mamageno.

Isaac is right on. Ourlas doesn't get it. Since the variables m and r are measurable quantities that can be varied across all kinds of circumstances, there is a real substantive test of the theory, and of whether or not G is truly a constant as Isaac proposes.

I have not seen anybody on this board engage in the foolishness exhibited here by Ourlas.

And so on....

That's good. Anyone who did think there was a similarity between Isaac's critics and people on the board would have to be a total klutz at physics!

Cheers -- Sylas

Sylas, we are talking about photoabsorption cross sections. f -2 = 0 means that the photon was not absorbed but continues. Going by our knowledge of how light travels through matter, this means that it has been absorbed and re-emitted. f_2 =0 means that the electron is not retained but re-emittted. However, the 2r(lambda) term remains constant. f_2 modulates this to show whether the electron is absorbed (never to reappear) or re-emitted.
What, and show you how to remove the safety catch? I give references in my paper.

Values of f_2 are given on the same page!!! Therefore I am correct. It is standard practice to publish a formula with a constant and include the values on the same page.

The 10 to 30 keV came from the title of the paper in which they gave f_2 as having a maximum value of 0.95 (just about one) for Hydrogen. Sylas is right in that there range of results exceeds that given in the title. However, the tables have been extended since my papaer was accepted. Present values give f_2 as 0.95197 when the photon energy is 13.66800 eV(wavelength of photon 8.7x10^-8m)


The elctrons in plasma are not free. This is a common mistake of Sylas'. The point I make with these results is that the collision cross section between photons and electrons for this interaction where the electron absorbs the photon is 2r9lamda) f_2 merely represents the probability of whether the electron retains the photon or 'spits' it out.
Cheers,Lyndon

This part of Sylas' post got lost somewhere.
I believe the sharp edge represents the limit of their data and not some remakable cliff hanger of an absorption edge for Hydrogen. If one looks at other elements the all have the gradual absorption edges that Sylas expects.
Cheers,
Lyndon

First, I'll point out that I'm specifically addressing the subject of comparing H directly hr/m in this message, since that's what I've been asking about. I'll leave the entirely different matter of comparing H to 2nhr/m until we get this sorted out.

There's nothing wrong with introducing a constant of proportionality in a relation, of course, and that constant can indeed have units. But in that case, you should be talking about a realtionship like H = Lhr/m, where L is "Lyndon's constant". As Sylas has already pointed out, in Newton's relation, it's possible to determine G by measurement, and to determine experimentally whether F and Mm/r^2 really are proportional. So, depending on how you look at it, you're either arbitrarily assuming that L has a value of 1 m^-3, or else your actually just determing L from the known value of H. In neither of those case could you be said to be predicting the value of H, and in neither case are you directly comparing the values of H and hr/m.

This implies that a constant of proportionality should be expected to have some simple value, but actually that's almost exactly the opposite of what one would expect. I'd be extremely suspicious of a theory where a constant of proportionality came out to be exactly 1 in whatever my favorite units are. That's because I know that my choice of units is arbitrary, so there shouldn't be any particular reason for a constant to come out to 1 m^-3. The universe shouldn't give special preference to the meter, unless you think God really likes the metric system. Lyndon, can you name me any fundamental physical constants with a nice round value (other than in a case where the units in question were defined specifically to make that constant come out even)?

Let me give two examples of numerical coincidences, one which is a genuine coincidence, and one which led to a deeper understanding of physics, and try to show the difference between them.

First, have you noticed that the major axis of Earth's orbit (measured in kilometers) is exactly equal (well, within 0.2%) to the speed of light, measured in meters per second? Obviously, one might say, this is much too close to be a coincidence, so we must be missing something. Somehow the speed of light must be tied to Earth's orbit, and we should come up with a theory that explains this.

But on further reflection, if the Earth's rotation period were different, or we'd chosen to divide the day into more or less than 86,400 seconds (so the length of the second was different), these values wouldn't match at all. Similarly, if we had twleve fingers, we'd probably do our math in base twelve, and the metric system we developed would probably have a factor of 1,728 between a kilometer and a meter, so again, these numbers wouldn't match at all. So, realizing that these numbers are only close because of a particular choice of measurement system, I instead conclude that this similarity must be a genuine coincidence. Note that the fact that it's very easy to find coincidences like this should make one wary of assuming that any numerical coincidence must have some reason for it.

For the other, I'll mention something that you alluded to here, though if you were talking about what it sounded like, I think you're not quite right on the details. What Maxwell noticed was not a similarity between the speed of an electric pulse and the speed of light. Rather, there are two constants that show up when analyzing electric and magnetic fields, the permittivity of free space and the permeability of free space. These relate the strength of the fields to the charge and current present, and you can actually measure them directly just by working with static charges and magnets.

What Maxwell noticed was that, since changing electric fields induce magnetic fields and vice versa, it would be possible to have a self sustaining oscillation of both fields, and that it would propogate through space as a wave. Using his theory, he was able to show that this wave would have a velocity that depended on these two constants, and when he worked out that velocity, it turned out to match the speed of light, leading him to predict that light was in fact, just such an electromagnetic wave. Notice that in this case, Maxwell was directly comparing two values that were both velocities,* and if we were to convert the enitre calculation into some other arbitrary set of units, the two values would still match.

As you point out on your webstie, units have to match when comparing values. That doesn't mean, however, that you can just decide to arbitrarily decide to divide by one cubic meter, without explaining why you didn't instead decide to divide by a cubic foot, or a cubic parsec,

* This is actually a slightly tighter constraint than just that the units match. For example, if I multiply the acceleration due to gravity on the Earth's surface by the distance from the Earth to the Moon, and then divide by the speed of light (or just take the square root, if you'd prefer not to introduce an arbitrary constant), I'll get a result that is in units of length over time, but it's not actually a meaningful velocity.

f_2 = 0 means that the cross section for photo-absorption is zero, which means that there is no photo-absorption at all. It has nothing to do with "re-emission"; it is about cross sections. Zero cross section means the interaction doesn't happen at those frequencies.

When f_2 has a value approaching 1, as is the case for neutral hydrogen and photons with energy just over 13.6 eV, absorption is very efficient, and the light is blocked. The re-emitted photons are scattered in all directions, resulting the Hydrogen cloud glowing with Lyman photons, but light from distant stars having a Lyman absorption line in the spectrum. The effect is of great importance in astronomy.

We can consider transparent transmission of light as photons being briefly delayed in an interaction with matter particles, but it is not the same interaction as photo-absorption and re-emission from an excited particle. With conventional photo-absorption, followed by emission, there is a random delay with a Poisson distribution for the excited atom to decay back to a ground state. This delay time reflects the stability of the excited state of an atom after absorption.

Transmission in a transparent medium, on the other hand, is modelled using the complex index of refraction, which depends on form factors and bears no relation to the stability of excited states for matter. You can find the relation between form factors and the complex index of refraction as equation 2 in chapter 1 of Chantler (1995), cited previously. The imaginary part of the complex refraction index is related to f2, and it is called the "extinction co-efficient". In a transparent material, like glass, this is zero. There is no absorption, no loss of energy and no redshift. But when f2 is non-zero, the electromagnetic wave is absorbed as it transfers energy into the medium.

Similar phenomena occur with transmission of electromagnetic waves in plasma. For light with frequency below the plasma frequency, the extinction co-efficient is non-zero and the wave does not pass through the plasma, because its energy is rapidly transferred to plasma oscillations. But for higher frequency light, there is no extinction.

Lyndon and John and others who mention transparent transmission in this context never give a thorough quantified model for the transparent transmission of an electromagnetic wave. It's not the same as photo-absorption.

The answer to my question is apparently no. Lyndon is not able to justify his claim that the formula for photo-absorption cross-section is ever given as 2rλ. The correct formula is 2rλ.f_2, and that is how it is published. People who publish the tables also publish the correct formula.

I asked Lyndon to justify his claim that "The published result for the photo-absorption collision cross section is 2rλ." He can't do that, because he is wrong. The formula for cross section is 2rλ.f_2, and the f_2 is not omitted in any published source. When f_2 is zero this means the cross section is zero.

Lyndon's own sources refute this. Cross section is proportional to f_2, and therefore f_2 "represents" (is proportional to) the probability that the photo-absorption interaction occurs at all.

As for being free, electrons are "bound" if they are linked to a particular nucleus, and they interact as a pair. This is why for very high energy photons, even bound electrons are considered "effectively free". The binding energy is much smaller than the photon energy, and so the photon interacts with the electron as a free individual, and not with the atom as a whole.

Electrons are "free" if they respond to external forces as individual particles, without any special association to a particular nucleon. In all circumstances they continue to be impacted by electromagnetic fields, of course. Being free is not about being immune to forces, but about whether you respond as an electron, or as an electron/nucleon pairing. This is the standard usage in any reference on plasma you care to mention. For example, Lyndon has previously cited some lecture notes by Kalevi Mursula. See section 1.2, page 5:
This is really elementary. Of course electrons in plasma are free, and free electrons do not have excitation levels. Excitation levels relate to the binding to a particular nucleon.

And finally, on the cutoff of f2 for Hydrogen at 13.6 eV
I don't expect gradual change. The sharp edges to photo-absorption are basic to this phenomenon, and you'll see the same thing for every other element. To see the cutoffs, use the low energy range from 1 eV up. I'm simply querying the value to which f2 falls after the 13.6 eV cutoff for Hydrogen.

The absorption edges for Hydrogen are basic to astronomy. That's why we get absorption lines in spectrum at certain characteristic wavelengths. The wavelengths of 91.2nm (13.6 eV) and 121.6 nm (10.2 eV) will be instantly recognizable to almost any astronomer.

The NIST tables do actually include data for Hydrogen at energies below 13.6, and they incorporate a zero value for f2. Try this query. My guess at this point is that they put a zero in the table because they know it is below some measurement limit. I suspect this, because the www xray server gives a plot below 13.6 eV, showing f2 dropping precipitously to about 1e-16 or so, and then declining slowly thereafter.


Here are some plots for other elements, showing the sharp edges that are characteristic of photo-absorption.


[[ Added in edit. Anyone who checked the data supplied by NIST using my query links should have seen this explicit comment:
1 edge. K Edge energy (keV) = 1.36000E-02
So it's definitely not a limit to the data. Photoabsorption has strong edges at various characteristic absorption frequencies, and these are well known to real astronomers. ]]

Cheers -- Sylas

Sylas wrongly wrote
So if these photons are not absorbed and re-emitted, how is it that the group velocity is reduced and the phase velocity increased?
Sylas has a problem. How can the phase and group velocity change if the photons do not interact?
When he answer, duck!!!!!
cheers,
,lyndon

The answer was already given in my previous post. There are more ways than photo-absorption for light to interact with matter. Photons do interact with matter in the straight-line propagation of an electromagnetic wave through a transparent medium, but the interaction is not photo-absorption.

Lyndon, I don't mind honest disagreement. But it is merely deliberate obfuscation to ignore the bits of my post where I answer your question already. You may disagree with the answer, but just pretending it isn't there is dishonest.

Here is the bit from my post that Lyndon ignored.

Photo-absorption is covered within the complex index of refraction as the extinction factor, and this relates to f2. Extinction means that a signal is absorbed into the medium. Subsequent re-emission is not part of the refraction index, and it relates to the stability of excited atoms.

You can't ignore f1 and the real part of the refraction index. This defines the phase velocity for light, and it does not involve the photo-absorption interaction. It is generally analysed by considering the photon as a wave, though you can of course get right into quantum mechanics to deal with photons as particles. The group velocity can be obtained in terms of the derivative of the index with wavelength.

Cheers -- Sylas

PS. Don't just ignore the problem with photo-absorption cross section. The published formula is 2rλ.f_2. You ought to retract your claim that it is published as 2rλ.