Surprising Patterns in the NIST Data Regarding Atomic Ionization

These findings also seem to provide actual physical meaning behind the equations used for the Balmer series of lines of Hydrogen and the Rydberg equation, as well as suggesting how Rydberg could be extended to essentially any element!

Each neutral element has equal numbers of protons in the nucleus and electrons orbiting it. We are going to look at this situation in the old-fashioned traditional way, where discrete electrons were actually seen as orbiting the nucleus!

Atoms can be ionized in two distinct ways. The most common experimental way is where external energy is provided and an electron is kicked up into a higher energy state orbital (of many possible choices) and then it soon drops back down and in that process radiates away the exact amount of energy difference between the two energy states. The wavelength of that light photon can be used to determine the exact amount of energy involved in such transitions. NIST has data for all neutral elements and many already ionized elements, based on the wavelengths emitted. In some cases, there are as many as 780 different possible energy states in a given element, and so an immense number of different wavelengths of light can be emitted as an electron falls from one of those states to a lower energy one.

The second distinct way is where the electron is sent to an infinite distance from the nucleus, in other words, the greatest possible amount of energy which can be inserted in that electron. Effectively, this is the common situation when chemical reactions occur.

The exact wavelengths of the resulting radiation from all elements have been extremely accurately determined. This means that the exact amount of energy transfer regarding any of these ionizations is also known extremely accurately.

There are an immense number of ways that the NIST data might be studied. The transitions from one energy state to a lower energy state must certainly include many valuable insights, but there are a number of variables involved, including for example a very large number of possible target energy states. The single electron in a hydrogen atom is known to have at least 167 raised energy levels available to it (which is known by the fact that there are at least 187 different wavelengths of radiation emitted when the electron later falls back to its neutral state). For now, we choose to avoid this complexity and only consider electrons that are ionized to an infinite distance.

The electrons in atoms are in many different quantum orbitals and sub-orbitals, and the total effect of many of them on each other figures to be quite complex, so we wish to temporarily eliminate this complexity as well. It turns out that we can do that!

The hydrogen atom has one electron. NEUTRAL helium has two, but if the helium is singly-ionized, it also has one electron. NEUTRAL lithium has three electrons, but doubly-ionized lithium has one. NEUTRAL beryllium has four electrons but triply-ionized beryllium has one. And so on. The discussion here will stop at element 32, Germanium, which is 31-times-ionized, such that it too only has one remaining electron.

Therefore, we have simplified the situation into one where there only is a single electron (we will not now specify any shell or sub-shell) orbiting around various nuclei which have (attractive) (integer) positive charges of from one to thirty-two.

The EXPERIMENTALLY MEASURED NIST data on these atoms has an amazing pattern in it! Consider for a moment only the H I and He II atoms. The NIST ionization potentials, in electron-Volts, are 13.5984340 and 54.4177630, meaning that these values represent the actual binding energy of the single electron for these two cases, as compared to zero, the potential at infinite distance.

This is quite close to being a 1:4 proportion, within 1/20 of one percent! We might them look at the Lithium II value of 122.454353 and see that it is remarkably close to NINE times the hydrogen value, again around a match of around 1/20 of one percent. Could this be some sort of fluke, where the universally accepted experimental data seems to have a remarkable match with the square of the electric charge of the nucleus? No! The match continues, consistently amazingly accurate, throughout the entire sequence of these 32 elements.

Here is the NIST data (in electron-volts) and the nucleus-charge-squared multiple of hydrogen's.

nuclear charge	NIST ionization value	multiple of hydrogen
1	13.5984340	13.5984340
2	54.4177630	54.393736
3	122.454353	122.385906
4	217.718572	217.574944
5	340.225993	339.96085
6	489.99312	489.543624
7	667.04602	666.323266
8	871.40969	870.299776
9	1103.1171	1101.473154
10	1362.1986	1359.843
11	1648.70105	1645.4105
12	1962.6642	1958.1745
13	2304.1401	2298.1353
14	2673.1807	2665.2931
15	3069.84143	3059.648
16	3494.1877	3481.199
17	3946.2907	3929.947
18	4426.2226	4405.8926
19	4934.0439	4909.0347
20	5469.8614	5439.3736
21	6033.7551	5996.9094
22	6625.81	6581.6421
23	7246.1196	7193.5716
24	7894.80	7832.70
25	8571.94	8499.02
26	9277.6874	9192.54
27	10012.1	9913.26
28	10775.4	10661.17
29	11567.612	11436.28
30	12388.928	12238.59
31	13239.4881	13068.10
32	14119.4287	13924.80

This is very troublesome! By doubling the charge in the nucleus, and being careful not to change any other variables, standard understandings in Physics do NOT allow any conclusion where the Binding Energy becomes four times as great!

However, this information is in exact accordance with the Balmer:

and Rydberg:

with the special case for Hydrogen Balmer being:

mathematical equations for spectral lines. (These equations are presented in their traditional form, for solving for the wave number nu. By multiplying that value by Planck's constant, the energy is determined, so we could have presented these to solve for energy in electron-Volts by that simple multiplication.)

This investigation seems to give a physical reality to the mathematics of those equations. A critical point is that it has generally been assumed that all the numbers in the denominators were DISTANCES (because of the well-known inverse-square distance rule of electromagnetic phenomena), but this investigation seems to definitely prove that the a and b parameters are rather the charge in the nucleus! The fact that these quantities of charge need to be squared for these equations appears to contradict Coulomb's Law. (The Rydberg constant is 109,677.8/cm or 13.5978 eV equivalent, the ionization potential of neutral hydrogen.).

For clarification, the first term of the Rydberg equation describes the initial energy state of the electron and the second term describes the final energy state. The initial state is therefore described by R / (n + a)². For Hydrogen, we are describing here that a is the reciprocal of the nuclear charge of one (which is still one), and the first n is zero, such that we have the energy become 13.6 eV / (1²) or 13.6 eV. Regarding the final state being at infinite radius, we have m is infinite and the energy is 0 eV.

And with our calculated parabola of EXACT INTEGERS in yellow:

Traditional thought would say that with double the central electrostatic attraction, then the attractive force, that is, the centripetal force must also be exactly doubled. But then all the traditional analysis of energy and angular momentum do NOT result in the energy going up as the square of that centripetal force! But it clearly must, per this parabola!

Here is the general reasoning that has always been applied, for each of planets in a gravitational field and electrons in an electrostatic field.

Coulomb's Law says that the force of attraction between two charged objects is given by F = k * q₁ * q₂ / r², where the q's are the amounts of charge and r is the distance separating them (with k a constant depending on the system of measurements). Newton's Law similarly says that the force of attraction between two massive objects is given by F = G * m₁ * m₂ / r², where the m's are the amounts of mass and r is the distance separating them (with G, the Gravitational constant, depending on the system of measurements).

Newton's Law F = m * a for central force for circular motion is F = m₁ * v² / r. This can also be written F = I * omega² or I * v² / r².

It might also be speculated that the radius of the orbit of the electron would adapt itself to account for this different force. However, calculations show that that is not a viable explanation. Yes, the orbital radius WOULD change to some extent, but it is physically impossible that it could change enough to account for this full effect. An easy proof of that is that the Binding Energy is known to be in the inverse proportion to the orbital radius. We have just been discussing a Germanium XXXII and a Hydrogen I, which each have one electron. The NIST data shows that the Ge atom has a binding energy of over 1,000 times that of the Hydrogen, meaning that its orbital radius would necessarily be 1/1,000th the radius. That electron would be extremely close to being inside the nucleus!

But the standard calculations do not even result in this solution! There is NO conventional way where multiplying the charge of the nucleus by some integer can possibly result in the Binding Energy increasing by the square of that integer! At least, not if Coulomb's Law is assumed to still apply.

Ions that have Two Electrons

Ions that have Six Electrons

Here are the three graphs above repeated, and some other numbers of electrons, but with the statistical data and the parameters now included and the parabolic curve-fit included:

It seems appropriate to show the data for six-electrons again. You probably noted that the Residuals for six- had a large excursion outlier at a nuclear charge of 31. Look at the red dot on the following graph of six- again, to see that even that extreme outlier is actually quite close to the parabolic curve. This demonstrates just how good these parabolas are.

Apparent NIST Errors in the Data

Look at element 26 and element 32.

Look at element 26.

Look at element 20.

These graph analyses present some amazing results! We will collect numbers from these parabolic graphs. For non-statisticians, the r² value is an indication of how well a mathematical curve fits the data, with 1.000 meaning a perfect fit. These are mighty close, meaning that the data is truly extremely close to parabolic!

All of the electron numbers from two through nine have a b parameter very close to 3.45. These match up to the Quantum level 2 electrons, and it is surprising that they share that same parameter. Starting with ten electrons, that b parameter suddenly and consistently drops to around 1.57, which seems to be true for the Quantum level three electrons. The single electron situation with the b parameter of 13.9 matches up to the Quantum level 1 of Hydrogen. Note that TWO electrons seems to belong with the next quantum level, implying that the Periodic Chart may be wrong! It could be that each shell starts out with an EMPTY state as the first entry, meaning the Noble Gases might belong on the LEFT side of the Periodic Table rather than the right where they have always been shown (as being FILLED shells). In the same way, TEN electrons appears to belong with the next shell!

These results show parabolas that are NOT tangent to the x-axis, having a vertical differential equal to the a parameter. The graphs were curve-fitted with a requirement that there was that tangent requirement, but every such curve was substantially worse than the ones presented above. Here is one example, for 12-electrons.

The r² value is good, but not nearly as good as the parabola presented well above which would tangent around the -10 eV line. That seems to suggest that it is not appropriate to force the parabolas to be tangent to the x-axis.

This might suggest that there is some physical importance in the fact that the a parameter is not zero. In fact, in progressing from 2-electrons to 9-electrons, it is clear from the chart above that the a value constantly becomes more negative. The same trending toward negative a seems to exist starting witt 10-electrons. There may be a physical meaning for this.

It is probably silly to note a potential relationship between these b parameters. If we say that the first value is 14, then the second shell parameter is pretty close to 14/2² or 3.5 and the third is pretty close to 14/3² or 1.56. It will be interesting to see if the next shell (possibly beginning with 18 electrons) might have a b parameter of 14/4² or 0.875, and the shell after that (beginning with 36 electrons) having 14/5² or 0.56

There does not seem to be any logical reason for this to be the case, and I am merely noticing a possible pattern in the data numbers.

Certain errors in the NIST data for specific ions probably cause some variation and the consistency may even be better than that. The amazingly high values for r² for each graph show the very good curve-fit.

Therefore, it seems that a simple equation certainly exists which takes the (number of electrons) and (number of nuclear charges) and gives the Ionization Potential in electron-Volts!

For real ions, the number of electrons cannot be greater than the number of nuclear charges, although this equation seems to allow certain limited exceptions. (notice the Five- and Four-electron parabolas which clearly extend to the left of the physically possible numbers. This might aid in explaining how and why some chemical molecules can form. The upward curve of the graphs at the left might be describing the REAL situation when an atom is in a compound. In salt, a neutral Chlorine has 17 electrons, but Chemistry has taught us that the single electron that a Sodium atom has handy can "complete the shell" for the Chlorine atom. That is, the Chlorine atom technically has an extra electron in that compound. These graphs might provide some insight into which compounds are possible and what energy amounts are required for each bonding.

I don't know if this is credible reasoning or not, but it seems an interesting possibility. It might help explain a lot about electro-chemical processes. It might also just be a silly thought.

Keep in mind that most of these atoms have already been multiply ionized, meaning that some electrons are already gone. The point being made here is that it seems possible to know the SEQUENCE of where the electrons are emitted from! A Ca XI ion has already given up ten electrons, suggesting that all the 1s, 2s, and 2p electrons are already gone. Therefore, the only electrons that should still remain would be 3-level electrons, and the graph (10 C) seems to show that.

There are clearly complex changes that occur within the electron movements, which apparently are needed to try to describe how S VI would be able to expel a 1s electron which would normally be expected to have been long gone.

The present analysis has shown the physical reality of the a parameter in the Rydberg Equation, being the nuclear charge. The n and m parameters appear to still be distance/radius, which brings up an interesting question regarding how an equation could mix and match charge and distance! There appears to be a level of understanding still beyond us all!

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C Johnson, Physicist, Physics Degree from Univ of Chicago