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

The Quantum Defect is a Physical Quantity and not a Fudge Factor

At least one of Coulomb, Newton or Planck had to be very wrong about an important central fact of nuclear Physics

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 /(n2). 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.

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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.

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The second object winds up orbiting the first as it approaches the first, if a stable relationship is to be established. It can orbit on an elliptical or circular orbit, with the equations for the elliptical being more complex than the circular but otherwise identical, so we will choose to consider only circular orbits here.

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.

The simplest way to demonstrate this new perspective is to initially ONLY consider the Ionization Potentials to an infinite distance, such that the single term as presented above is valid, such that the destination value has zero binding energy.

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:

E = k /((n + δ)2) = k /((0 + δ)2) = k /(δ2)

We DEFINED the Quantum Defect as being the reciprocal of the electrical charge of the nucleus Z, or δ = 1/Z or Z = 1/δ. Therefore we have, for only these specific atoms:

E = k * Z2

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

nuclear charge
Published NIST
ionization Potential
in electron-Volts
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 curve-fit suggests a better equation:

E = (a + b * Z)2

this form is very close to reverting to the e = k * Z2 form we have been discussing.

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:

nuclear charge
Published NIST
ionization Potential
in electron-Volts
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 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:

Principle Quantum
Energy change
or radiation
Description of transitionNIST values
n . . . . . . mEnergy eV
0110.198826from 1 shell to 2 shell10.1988101
0212.087498from 1 shell to 3 shell12.0874944
0312.748532from 1 shell to 4 shell12.7485324
0413.054497from 1 shell to 5 shell13.0544979
0513.220700from 1 shell to 6 shell13.2207010
0613.320915from 1 shell to 7 shell13.3209161
0713.385959from 1 shell to 8 shell13.3859595
0813.430552from 1 shell to 9 shell13.4305530
0913.462450from 1 shell to 10 shell13.4624505
01013.486051from 1 shell to 11 shell13.4860510
0infin13.598434from 1 shell to infinity13.5984340

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:

Number of Electrons in the Atom a parameterb parameterr2

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.

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

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago