First placed on the Internet in June 2007 Many amazing statistical patterns exist in the highly respected NIST data regarding the Ionization Potentials for the elements! It is really obvious, and unavoidable, but it seems to contradict all conventional explanations! It even appears to contradict Coulomb's Law! Consider this: Coulomb's Law says that the electrical attraction between two oppositely charged particles is DIRECTLY proportional to the charge of each of the two (as well as a dependence on the inverse square of the distance between them.) But in 1913, a scientist named Moseley discovered that the radiation created by different elements is proportional to the SQUARE of the electrical charge of the nucleus. Consider the simplest possible situation inside an atom, a single negativellycharged electron orbiting a (very tiny) nucleus which has a single positive charge in it. This is a standard hydrogen atom. There is obviously some energy of attraction between the opposite charges, and we will call that 13.59 electronVolts. Now, let's consider the same situation, but where the nucleus contains TWO positive charges, but there is still one electron orbiting it. This is an atom of helium which has been once ionized. Coulomb's Law indicates that the attraction between the electron and nucleus should be TWICE as great, right? But when this is actually measured, it turns out it is actually FOUR times as great, extremely accurately (54.42 electronvolts). We might think that was just a coincidence, because we KNOW that Coulomb's Law is true! So we now examine a twiceionized Lithium atom, where a single electron orbits a nucleus with THREE charges in it. Is the attraction three times? No. You probably already guess that it is experimentally shown to be NINE times, again, extremely accurately (122.45 eV). And a triplyionized Beryllium atom (one electron circling a nucleus with four charges in it) is experimentally shown to have 16 times the binding energy (217.72 eV). And a quadruplyionized Boron atom has 25 times (340.23 eV). These are all remarkably accurate multiples of the hydrogen binding energy. This is NOT an accident! Researchers have managed to check 32 different elements where there is a single electron orbiting the nucleus, and they all very accurately follow this same pattern, where even a 31timesionized Germanium atom has 32^{2} or 1024 times as great a binding energy of that single electron (14,119.4 eV). This all absolutely contradicts the basic claim of Coulomb's Law! And not just in random ways but in very precise ways. These findings also seem to provide an 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 in any ionized state! They also conclusively prove that the (alleged) Quantum Defect, a number that has always believed to be a random correction, is actually a specific and precise number, accurately specified by equations! No one else has ever noticed that before!


This analysis shows that the concepts can also be extended to describing atoms with multiple electrons, with all the equations still accurately applying. And further, the entire structure of the atomic shells and subshells, are all clearly predicted by surprisingly simple mathematical equations.
It certainly appears that Coulomb was wrong regarding the electrical attractions inside atoms!
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The second distinct way is where the electron is sent to an infinite distance from the nucleus, in other words, entirely removed from the atom, and which is 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 187 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). This also enables a hydrogen atom to emit many thousands of different wavelengths of light, as the electron falls from one of those states to another of them. 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 suborbitals, 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! It turns out that researchers have discovered and experimentally analyzed 32 different atoms which each have a single electron orbiting the nucleus.
The hydrogen atom has one electron. NEUTRAL helium has two, but if the helium is singlyionized, it also has one electron. NEUTRAL lithium has three electrons, but doublyionized lithium has one. NEUTRAL beryllium has four electrons but triplyionized beryllium has one. And so on. The discussion here will stop at element 32, Germanium, which is 31timesionized, 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 subshell) orbiting around various nuclei which have (attractive) (integer) positive charges of from one to thirtytwo.
If we look at Coulomb's Law regarding these 32 atoms, which are essentially identical except for the charge of the nucleus, we might then assume that the attractive force acting should be in the ratio of 1:2:3:...:32. That is NOT the case!
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 electronVolts, 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 electronvolts) and the nucleuschargesquared 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 electronVolts 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 wellknown inversesquare 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 / (1^{2}) 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 * q_{1} * q_{2} / r^{2}, 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 * m_{1} * m_{2} / r^{2}, 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 = m_{1} * v^{2} / r. This can also be written F = I * omega^{2} or I * v^{2} / r^{2}.
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.
nuclear charge  NIST ionization value  Second NIST value 

2  24.5873876  
3  5.3917191  
4  18.211153  136.80273 
5  37.930620  205.99 
6  64.49390  
7  97.89013  
8  138.1196  
9  185.1868  
10  239.0969  
11  299.864  1407.705 
12  367.497  1698.61 
13  441.999  2017.0 
14  523.4203  2362.8411 
15  611.74  2736.31 
16  707.01  3137.35 
17  809.2129  
18  918.3861  
19  1034.5  
20  1158  
21  1288.0  
22  1425.4  
23  1569.6399  
24  1721.1  
25  1879.8  
26  2045.7391  
27  2218.9 
nuclear charge  NIST ionization value 

6  11.26030 
7  14.53413 
8  35.12111 
9  62.7084 
10  97.1168 
11  138.40 
12  186.76 
13  241.76 
14  303.5381 
15  372.13 
16  447.46 
17  529.2761 
18  618.73 
19  714.6 
20  817.7 
21  927.5 
22  1043.947 
23  1167.931 
24  1299 
25  1437 
26  1575.5911 
27  1735 
28  1894 
29  2060.6172 
30  2234.1951 
31  2418.9316 
32  2665.6601 
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 curvefit included:
It seems appropriate to show the data for sixelectrons 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.
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 nonstatisticians, the r^{2} 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!
electrons  a  b  c  r^{2} 

one  19.3  13.938338  0.21  0.99999693 
.  
two  6.3  3.5312093  1.94  0.99999095 
three  1.5  3.4442806  2.22  0.99999944 
four  2.3  3.4642755  3.21  0.99999939 
five  4.4  3.4478263  3.84  0.99999941 
six  12.8  3.3879534  4.31  0.99996958 
nine  18.1  3.4165106  6.64  0.99999867 
.  
ten  4.1  1.601446  8.60  0.99999826 
eleven  8.4  1.5728455  8.81  0.99998215 
twelve  10.0  1.5421735  9.86  0.9999791 
seventeen  24.9  1.5293853  12.96  0.99997388 
.  
eighteen  15.7  1.6134294  15.79  0.99955473 
nineteen  9.0  1.7198515  17.20  0.99981609 
twenty  1.6  2.1111569  19.14  0.99991352 
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 xaxis, having a vertical differential equal to the a parameter. The graphs were curvefitted 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 12electrons.
The r^{2} 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 xaxis.
This might suggest that there is some physical importance in the fact that the a parameter is not zero. In fact, in progressing from 2electrons to 9electrons, 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 with 10electrons. 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 14/2^{2} or 3.5 and the third is pretty close to 14/3^{2} or 1.56. It will be interesting to see if the next shell (possibly beginning with 18 electrons) might have a b parameter of 14/4^{2} or 0.875, and the shell after that (beginning with 36 electrons) having 14/5^{2} 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 r^{2} for each graph show the very good curvefit.
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 electronVolts!
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 Fourelectron 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 electrochemical processes.
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 3level 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!
Conservation of Angular Momentum  An Exception or Violation (Sept 2006)
Galaxy Spiral Arms Stability and Dynamics A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)
Twins Paradox of Relativity Is Absolutely Wrong (research 19972004, published Aug 2004)
Perturbation Theory. Gravitational Theory and Resonance (Aug 2001, Dec 2001)
Origin of the Earth. Planetary Gravitational Resonances (Dec 2001)
Rotation of the Sun (Jan 2000)
Origin of the Universe. Cosmogony  Cosmology (more logical than the Big Bang) (devised 1960, internet 1998)
Time Passes Faster Here on Earth than on the Moon (but only a fraction of a second per year!) (Jan 2009)
Globular Clusters. All Globulars Must Regularly Pass Through the cluttered Galaxy Plane, which would be very disruptive to their pristine form. (Nov 1997, Aug 1998)
Existence of Photons. A Hubble Experiment to Confirm the Existence of Individual Photons (experimental proof of quanta) (Feb 2000)
Origin of the Moon  A New Theory (June 2000)
Planetary Rotation of Jupiter, Saturn, and the Earth (Jupiter has a lot of gaseous turbulence which should have slowed down its rapid rotation over billions of years) (March 1998)
Cepheid Variable Stars. Velocity Graph Analysis (Feb 2003)
Compton Effect of Astrophysics. A Possible New Compton Effect (Mar 2003)
Olbers Paradox Regarding Neutrinos (Oct 2004)
Kepler and Newton. Calculations (2006)
Pulsars. Pulsars May Be Quite Different than we have Assumed (June 2008)
Sun and Stars  How the Sun Works  Nuclear Fusion in Creating Light and Heat (Aug 2006)
Stars  How They Work  Nuclear Fusion. Lives of Stars and You (Aug 2004)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
General Relativity  A Moon Experiment to Confirm It. Confirming General Relativity with a simple experiment. (Jan 2009)
General Relativity and Time Dilation. Does Time Dilation Result? (Jan 2009)
Geysers on Io. Source of Driving Energy (June 1998)
Mass Extinction, a New Explanation. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Ocean Tides  The Physics and Logic. Mathematical Explanation of Tides (Jan 2002)
Earth's Spinning  Perfect Energy Source (1990, Dec. 2009)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source From the Earth's Spinning (1990, Nov. 2002)
Nuclear or Atomic Physics Related Subjects:
Nuclear Physics  Statistical Analysis of Isotope Masses Nuclear Structure. (research 19962003, published Nov 2003)
Quantum Defect is NOT a Mathematical Defect It Can Be Calculated The Quantum Defect is a Physical Quantity and not a Fudge Factor(July 2007)
Atomic Physics  NIST Atomic Ionization Data Patterns Surprising Patterns in the NIST Data Regarding Atomic Ionization (June 2007)
Nuclear Physics  Logical Inconsistencies (August 2007)
Neutrinos  Where Did they all Come From? (August 2004)
Neutrinos  Olbers Paradox Means Neutrinos from Everywhere (Oct 2004)
Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)
Quantum Physics  Quantum Dynamics. A Potential Improvement (2006)
Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)
Quantum Dynamics (March 2008)
Ionization Potential  NIST Data Patterns. Surprising patterns among different elements (March 2003)
Mass Defect Chart. (calculation, formula) (research 19962003, published Nov 2003)
Assorted other Physics Subjects:
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source (1990, Nov. 2002)
Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)
Accurate Mass of the Earth. Gravitational Constant  An Important Gravitation Experiment. (Feb 2004)
Tornadoes  The Physics of How They Operate, including How they Form. Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels (Feb 2000, Feb 2006, May 2009)
Radiometric Age Dating  Carbon14 Age Determination. Carbon14, C14 (Dec 1998)
Mass Extinction, an Old Explanation. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)
Hurricanes, the Physics and Analysis A Credible Approach to Hurricane Reduction (Feb 2001)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago