First on the Internet in July 2007 The field of Physics was deeply affected in the 1920s and 1930s when Physicists saw electrons ONLY appearing in the same (energy) orbits. It was interpreted that there are many states that are Excluded, such as with the Pauli Exclusion Principle. And Quantum Physics arose entirely on the basis of these observations! In fact, it was recognized that the spectral lines of radiation emitted were always at specific wavelength or frequency, and by Planck, that means specific energy contents. Quantum Physics said that the energy was defined by exact INTEGER values of the denominator in E = k /(n^{2}). This integer was soon called the Principle Quantum Number. It turns out that that was sometimes close but NEVER EXACT. Therefore, a term was added as a "fudge factor" to make corrections to allow the integer requirement of Quantum Physics to still apply. The equation was then still presented in essentially the same form, but now with an asterisk inserted, E = k /(n*^{2}) or by specifically stating the Quantum Defect as δ and saying E = k /((n + δ)^{2}) Many different interpretations have been made to the actual cause of that Fudge Factor, such as extremely elliptical orbits of the electrons where they sometimes get electrically hidden or unhidden by other electrons. But in any case, all have simply assumed that this WAS a Fudge Factor and no serious consideration of it has ever been made. It has simply been experimentally calculated for each specific element and ion, and then that number has been used in the above equation to allow it to work. It is well known that the Fudge Factor is NOT an integer, and in fact is a decimal. I have discovered that there is a REAL PHYSICAL MEANING of that quantity, and that it can be accurately calculated from theory! But the implications of my findings are broader than that, seemingly conclusively proving that at least one of Newton, Coulomb or Planck was dead wrong about a basic assumption of nuclear physics! The following discussion will show that ALL of the previous speculations have been quite incorrect, and even some basic assumptions have been wrong!
I will first present some basic Physics, upon which all of modern Physics has been built, and then show where there is a tremendous flaw in the logic.


Helmholtz presented us with the Conservation of Energy, which is also called the First Law of Thermodynamics, and the equation that describes gravitational attraction, the familiar inversesquare law of: F = G * M * m / r^{2}, where G is a constant, called the constant of Gravitation, the m and M are the masses of the two objects and the r is the distance between the centers of those masses.
Newton also developed Fluxions, or what we now call Calculus. Using Calculus, we can easily show that a Potential Energy can be defined as being the cumulative effect of that Force of gravitation with one of the objects beginning at a point an infinite distance away from the other and moving inward to a particular average distance. This is simply the Integral of that gravitational equation over the distance involved, which gives a result of E =  G * M * m / r. This value is always negative because the Potential Energy is always less than it was at an infinite distance, due to the object "falling" inward.
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An object is defined to have Momentum if it is moving, and that quantity is defined as the mass times the velocity. Newton's Calculus can again be used to determine the amount of energy needed to change the Momentum of an object from zero when at rest up to a specific velocity v. The calculus result is that the energy, called Kinetic Energy is equal to 1/2 * m * v^{2}, the standard formula used in science and Engineering.
We knew that the second object had zero Potential Energy at an infinite distance, and we also define that there is zero Kinetic Energy then. Since we have Helmholtz's Conservation of Energy, and we are carefully making sure that no external energy is being provided or removed, then we have the situation where the (always negative) Potential Energy must always exactly equal the (always positive) Kinetic Energy.
This results in a specific relationship between the orbital velocity (which is related to the Kinetic Energy) and the orbital radius (which is related to the Potential Energy). In fact, the relationship that Newton derived was the mathematical proof of an experimental relationship that Kepler had observed around a hundred years earlier. (It doesn't have any importance in this discussion, but here is the derivation:
Coulomb found that electrostatic attraction and repulsion is also an inversesquare force, but dependent on the charge of the two objects instead of their masses, and with a different constant as a factor.
We are also going to simplify in one additional way, which seems never to have been done before. We will initially consider ONLY atoms that contain a single electron, that is: H I (neutral Hydrogen) (using standard spectroscopic identification system); He II (singly ionized Helium); Li III (doubly ionized Lithium); Be IV (triply ionized Beryllium, and so on. We will exclusively use data from the highly respected NIST database, which contains such singleelectron atoms up to Ge XXXII (Germanium 31timesionized). We therefore will examine 32 specific atoms here from atomic number Z = 1 through 32. Each of these 32 atoms in our current discussion are therefore identical in every way except for the nuclear charge, as each contains a single electron.
All traditional calculations have assumed that for a single electron in the 1s subshell, the N value is 1, as being the Principle Quantum Number for the first shell of electrons in the atom. We shall show that is actually incorrect and we will use the the actual correct value of the Principle Quantum Number N = ZERO, which will soon clearly be seen as correct.
For the Quantum Defect, generally referred to as δ, we will now DEFINE IT as being the reciprocal of the nuclear charge of the atom (which we will call Z here)! It is NOT simply some random Fudge Factor! (For atoms with additional electrons, this is slightly modified, as discussed below.)
Therefore we have the Quantum Defect δ as being: For H I, 1.00; for He II, 0.50; for Li III, 0.333; for Be IV, 0.25 and so on, ONLY FOR THIS SPECIFIC SELECTION OF ATOMS WITH ONE ELECTRON. This is an entirely different understanding than is generally assumed! We can then calculate the actual Ionization Potentials for all of these singleelectron atoms, based on this simple equation. Since N = 0, the entire denominator (for these specific atoms) is the socalled Quantum Defect number. Therefore we have:
k is the Ionization Potential of neutral Hydrogen or 13.5984340 eV.
nuclear charge Z  Published NIST ionization Potential in electronVolts  Predicted Energy By the simple formula given above for the Quantum Defect 
error of the calculated value from the NIST data in percent  

1  H I  13.5984340  13.59843  0.0000 
2  He II  54.4177630  54.39374  0.0441 
3  Li III  122.454353  122.3859  0.0559 
4  Be IV  217.718572  217.575  0.0660 
5  B V  340.225993  339.9608  0.0779 
6  C VI  489.99312  489.5436  0.0917 
7  N VII  667.04602  666.3233  0.1083 
8  O VIII  871.40969  870.2998  0.1274 
9  F IX  1103.1171  1101.473  0.1490 
10  Ne X  1362.1986  1359.843  0.1729 
11  Na XI  1648.70105  1645.411  0.1996 
12  Mg XII  1962.6642  1958.175  0.2288 
13  Al XIII  2304.1401  2298.135  0.2606 
14  Si XIV  2673.1807  2665.293  0.2951 
15  P XV  3069.84143  3059.648  0.3321 
16  S XVI  3494.1877  3481.199  0.3717 
17  Cl XVII  3946.2907  3929.948  0.4141 
18  Ar XVIII  4426.2226  4405.893  0.4593 
19  K XIX  4934.0439  4909.035  0.5069 
20  Ca XX  5469.8614  5439.374  0.5574 
21  Sc XXI  6033.7551  5996.91  0.6107 
22  Ti XXII  6625.81  6581.642  0.6666 
23  V XXIII  7246.1196  7193.572  0.7252 
24  Cr XXIV  7894.8  7832.698  0.7866 
25  Mn XXV  8571.94  8499.021  0.8507 
26  Fe XXVI  9277.6874  9192.542  0.9177 
27  Co XXVII  10012.1  9913.259  0.9872 
28  Ni XXVIII  10775.4  10661.17  1.0601 
29  Cu XXIX  11567.612  11436.28  1.1353 
30  Zn XXX  12388.928  12238.59  1.2135 
31  Ga XXXI  13239.4881  13068.1  1.2946 
32  Ge XXXII  14119.4287  13924.8  1.3785 
33  As XXXIII  15028.6197  14808.7  1.4634 
34  Se XXXIV  15967.6759  15719.79  1.5524 
36  Kr XXXVI  17936.2076  17623.57  1.7430 
37  Kr XXXVII  18964.9937  18616.26  1.8389 
This is pretty remarkable! For a quantity that has always been discarded as a Fudge Factor for 70 years, we have presented a very simple formula to ACCURATELY CALCULATE ITS VALUE!
Notice that the differences gradually get greater in a consistent manner. Here is the graph of the differences, in percent:
A slight adjustment in the simple equation can therefore account for most of that. (Alternately, it can be considered that the definition of the Quantum Defect could have a factor slightly different than one.) Regression Analysis of the curvefit suggests a better equation:
Each element in this series therefore has its own slight adjustment factor, based entirely on the total charge of the nucleus, but they seem to have this simple relationship, of a simple (a + b * Z)^{2} replacement for the k * Z^{2} discussed above, where we can see that the a parameter represents the quantity which has been referred to as the Quantum Defect, but that we now see a physical meaning behind it. This simple equation seems to hold for all atoms of any number of electrons. Given the values of the curve in the Oneelectron Regression Analysis, we can say this equation is therefore E = (0.73156698 + 3.7370056 * Z)^{2}. The resulting calculated values are then within a small fraction of one electronVolt for all values and noting that the 0.0003% error value in the table below is three parts per million. (This particular curve fit has Residuals that seem to have a simple pattern, so there may be another simple improvement to make this match even more impressive.)
Here is a table with this better equation:
nuclear charge Z  Published NIST ionization Potential in electronVolts  Predicted Energy By the better formula given above for the Quantum Defect 
error of the calculated value from the NIST data in percent  

1  H I  13.59843  13.60196  0.0259 
2  He II  54.41776  54.41  0.0143 
3  Li III  122.4544  122.4306  0.0194 
4  Be IV  217.7186  217.6746  0.0202 
5  B V  340.226  340.1571  0.0202 
6  C VI  489.9931  489.8976  0.0195 
7  N VII  667.046  666.9199  0.0189 
8  O VIII  871.4097  871.2521  0.0181 
9  F IX  1103.117  1102.927  0.0172 
10  Ne X  1362.199  1361.981  0.0160 
11  Na XI  1648.701  1648.456  0.0149 
12  Mg XII  1962.664  1962.397  0.0136 
13  Al XIII  2304.14  2303.855  0.0124 
14  Si XIV  2673.181  2672.884  0.0111 
15  P XV  3069.841  3069.543  0.0097 
16  S XVI  3494.188  3493.896  0.0083 
17  Cl XVII  3946.291  3946.011  0.0071 
18  Ar XVIII  4426.223  4425.96  0.0059 
19  K XIX  4934.044  4933.821  0.0045 
20  Ca XX  5469.861  5469.675  0.0034 
21  Sc XXI  6033.755  6033.609  0.0024 
22  Ti XXII  6625.81  6625.713  0.0015 
23  V XXIII  7246.12  7246.085  0.0005 
24  Cr XXIV  7894.8  7894.824  0.0003 
25  Mn XXV  8571.94  8572.035  0.0011 
26  Fe XXVI  9277.688  9277.829  0.0015 
27  Co XXVII  10012.1  10012.32  0.0022 
28  Ni XXVIII  10775.4  10775.63  0.0021 
29  Cu XXIX  11567.61  11567.88  0.0023 
30  Zn XXX  12388.93  12389.21  0.0023 
31  Ga XXXI  13239.49  13239.74  0.0019 
32  Ge XXXII  14119.43  14119.62  0.0014 
33  As XXXIII  15028.62  15029  0.0025 
34  Se XXXIV  15967.68  15968.02  0.0021 
36  Kr XXXVI  17936.21  17935.62  0.0033 
37  Rb XXXVII  18964.99  18964.52  0.0025 
This data has an r^{2} statistical value of 0.9999999981, indicating an extremely accurate curvefit (for the oneelectron atoms).
Any of these atoms can be ionized to other energy states other than complete removal to infinity. This same equation gives these remarkably accurate (compared to actual measured NIST data) results!
For example, using this approach in the traditional way to calculate the energy states of Hydrogen, we have:
Start/End Principle Quantum numbers  Calculated Energy change or radiation  Description of transition  NIST values  

n . . . . . . m  Energy eV  
0  1  10.198826  from 1 shell to 2 shell  10.1988101 
0  2  12.087498  from 1 shell to 3 shell  12.0874944 
0  3  12.748532  from 1 shell to 4 shell  12.7485324 
0  4  13.054497  from 1 shell to 5 shell  13.0544979 
0  5  13.220700  from 1 shell to 6 shell  13.2207010 
0  6  13.320915  from 1 shell to 7 shell  13.3209161 
0  7  13.385959  from 1 shell to 8 shell  13.3859595 
0  8  13.430552  from 1 shell to 9 shell  13.4305530 
0  9  13.462450  from 1 shell to 10 shell  13.4624505 
0  10  13.486051  from 1 shell to 11 shell  13.4860510 
0  infin  13.598434  from 1 shell to infinity  13.5984340 
Most of these predicted values are within around one onemillionth of an electronVolt of the NIST data.
These cited NIST values are all for transitions to the s subshell of the target shell, and where the spin has not changed and remains at 1/2. The match is truly impressive, to many decimal points! This is essentially true for all the 32 atoms described above. This is a solid confirmation that this DEFINING of the socalled Fudge Factor Quantum Defect is absolutely valid.
Previous researchers have always assumed that everything in the denominator of the basic equation above MUST BE distance dimensions, in order to comply with the inversesquare nature of Coulomb's Law. However, note that THIS FACTOR, the socalled Fudge Factor or Quantum Defect, is not even a distance at all! It is directly related to the CHARGE of the nucleus! No one had even imagined that before, as it seems to be an outrageous concept, because it is charge rather than distance! Yet, the data above, and the data for many other atoms, confirms the validity of this approach.
For atoms that have more than one electron, this same equation is used, but with different values of a and b.
Further examination of the NIST data provides these a and b parameters:
Number of Electrons in the Atom  a parameter  b parameter  r^{2} 

one  0.00000097324  0.073519785  0.9999999981 
two  0.00000332928  0.079094268  0.99992134 
three  
four  
five  
six  
nine  
ten  
eleven  
twelve  
17  
18  
19  
20 
THIS effect provides even more interesting results! In the Periodic Table, we have Hydrogen and Helium in the first row, then elements 3 through 10 in the second row, etc. This additional variable (which we have called F here) suggests why! As additional electrons are added to the situation, some parameters in the Quantum Defect show that the Principle Quantum Number remains unchanged.
There ARE some rather small contributions to the socalled Quantum Defects which are due to other effects, such as spin of the electron and the different energies of different subshells. That shell difference seems likely to be the reason that neutral Hydrogen and Helium have a slightly different energy than the graph of all the other elements would imply. There seems to be evidence that a similar jump occurs between element nine and ten, but that it is so small relative to the higher Ionization Potentials as to be barely noticeable. However, you might note that in the "better" table above, elements two through nine all have remarkably consistent error values of around 0.003%, which drops to much less at element 10 and above.
This seems to suggest that each shell of electrons has its own generalized F factor, for more complex elements with additional electrons. These patterns seem to get pretty complex when a lot of electrons are present, as electrons do not always seem to be in the lowest possible energy states. This new approach of analysis might better aid in explaining why that is and what is happening.
This approach seems to have another value. When the NIST data is compared to these smooth, simple equation curves, there are sometimes data points that are clearly off the curve. An example here is that the Sodium (11) NIST value seems low by 6 parts in a million! Instead of the published NIST value being 1648.701 eV, this suggests that the correct value might actually be 1648.711 eV. So this approach may aid researchers in checking data they develop.
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)
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