Quantum Defect is NOT a Mathematical Defect- It Can Be Calculated

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!

This presentation was first placed on the Internet in July 2007.

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 inverse-square law of: F = G * M * m / r², 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.

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 * v², 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 inverse-square 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 single-electron atoms up to Ge XXXII (Germanium 31-times-ionized). 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 sub-shell, 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 single-electron atoms, based on this simple equation. Since N = 0, the entire denominator (for these specific atoms) is the so-called Quantum Defect number. Therefore we have:

k is the Ionization Potential of neutral Hydrogen or 13.5984340 eV.

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 curve-fit 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 * Z² 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 One-electron 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 electron-Volt 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:

This data has an r² statistical value of 0.9999999981, indicating an extremely accurate curve-fit (for the one-electron 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:

Most of these predicted values are within around one one-millionth of an electron-Volt 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 so-called 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 inverse-square nature of Coulomb's Law. However, note that THIS FACTOR, the so-called 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:

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 so-called Quantum Defects which are due to other effects, such as spin of the electron and the different energies of different sub-shells. 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.

Advanced Physics-related presentations in this Domain:

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 1997-2004, 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 Clock-Time 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 1996-2003, 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 1996-2003, 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 - Carbon-14 Age Determination. Carbon-14, C-14 (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 Clock-Time Correction Factor (Jan 2009)

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Carl W. Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago