Ionization Energies from Classical Force Balance: Calcium-like Uranium and Below

We examine a proposed modification (Mod 1 for short) to the Grand Unified Theory of Classical Physics (GUTCP) in which instead of orbitals holding two indistinguishable electrons, each electron has a distinct radius. The technique of creating a force balance equation for each electron in turn, from the inner to the outermost, is extended from Helium (and 2-electron ions such as Li+) and Lithium (and 3-electron ions such as Be+) up to Calcium and Ca-like ions. We release a public domain program to solve the force balance equations which are populated with force terms from GUTCP. We also compare our computations to NIST data in a provided spreadsheet.

Introduction

The Grand Unified Theory of Classical Physics (GUTCP), created by Randell Mills over the last about 30 years, has shown remarkable agreement with observations such as atomic and molecular energies compared to quantum theory. More than this, GUTCP is based on physical laws—Newton's Laws and Maxwell's Equations—and explains the so-called “spooky” quantum behavior as arising from an underlying classical physics [1]. By contrast, quantum mechanics cannot solve for the electron energies for any atom beyond Hydrogen without resorting to “approximations” which are, actually, entirely different algorithms from what they claim to approximate [2]. By contrast, a recent Hydrogen Revolution substack article gathered data from GUTCP for several ions of Iron showing that GUTCP can work even on transition metals without resorting to fitting (not to say fudge) factors or dubious “approximations” [2].

The further observation of the GUTCP-predicted Hydrino form of Hydrogen [3], [4] has been decried by the scientific community as impossible due to its violating quantum mechanical assumptions. However, the existence of Hydrino is now a case of measured data [5] versus 100-year old orthodox postulation. The scientific establishment only discredits itself by failing to replicate GUTCP experiments and join in this groundbreaking work.

The ionization energies of 1 electron atoms are shown in Fig. 1, from Hydrogen to Oxygen. The Schrödinger equation can reproduce this result, but not Helium and above.

Fig. 1. Calculated vs experimental ionization energy in eV. “Calculated” is from GUTCP Mod 1 including the python program [6] and spreadsheet [7] while “Experimental” is from NIST [8].

Method

We created a Python program to solve the force balance equation of the orbit of the outer electron of each atom and ion up to neutral Calcium (electron number) and Uranium (proton number) for the radius of the orbit, referring where necessary to inner electrons already calculated. The centrifugal force term is the same expression for each species, the electric force is a simple function of the electron and proton number, and the magnetic terms contain most of the variation that determines the radius. These force terms have up to quartic powers of r (radius) in their denominators. However, when multiplying through by the largest power of r in order to solve for r, the highest power encountered in the result was no more than quadratic. GUTCP carries the symbolic form through the quadratic equation [1]. Instead, we used Python's “eval” command to get a numeric value for the coefficients, and ultimately, for the radius of the outer electron. Our program creates a database of radii and energies since “earlier” electron radii in the sequence appear in the expressions for “later” ones.

The Python program is available in [6]. It is open source and released to the public domain. A Microsoft Excel spreadsheet containing all the data output by the program is also available in [7]. It contains all the graphs for one through twenty electron atoms and their supporting data as well as relative errors (compared to NIST).

Results

Appendix A shows plots comparing the GUTCP and NIST values for ionization energies of 3 through 20 electron atoms and atomic ions, displaying the first 8 items in each isoelectronic sequence. Many of the sequences remain within a percent or two relative error during this range. It is important to note that Mod 1 does not use all of the correction terms, such as magnetic energies (distinct from magnetic force terms), that GUTCP does.

Equally important is to note that the NIST values are mostly interpolations or extrapolations from sparse measured data. For instance, the Sulfur isoelectronic sequence, up to Uranium, contains two measured values and 74 extrapolated (or interpolated) ones. These can be identified by the parentheses or square brackets surrounding them in the NIST database results. This explains why the NIST data fit low degree polynomial regressions so well: the data themselves come from a curve fit to sparse measurements. For instance, as shown in Fig. 2, the Hydrogen isoelectronic sequence up to H-like Uranium is fit by a third-degree polynomial with R² = 1. Thus, even the NIST reported values may deviate from hypothetical precise measurements.

Fig. 2. One-electron atoms’ ionization energies in eV (H – U91+)—from NIST. NIST H-like data fit by a third-degree polynomial (perfectly; R² = 1).

Some of the higher Z data points, such as H-like Uranium, differ by about 16%, while others, such as N-like Uranium, remain well within a percent relative error even at the highest proton numbers. These are all shown in the Excel file. Comparing the Mod 1 numbers to the GUTCP in relative error, it will be noticed that GUTCP is much closer to the NIST measurements (and interpolations). However, Mod 1 uses fewer terms (it lacks the “magnetic energies”).

The Schrödinger and Dirac equations cannot reproduce Fig. 3 nor any of the Figs. 4–21 in Appendix, showing the superiority of GUTCP and Mod 1 over quantum mechanics (QM).

Fig. 3. Calculated vs experimental ionization energies of 2-electron atoms, from Helium to Fluorine.

Fig. 4. The ionization energies of 3-electron atoms, from Lithium to Neon. The energy comes from the radius of the orbit, which in turn comes from the force balance equation. The force balance equation terms come from GUTCP and are hardcoded into the program.

Fig. 5. The ionization energies of 4-electron atoms, from Beryllium to Sodium. This is the worst fit out of all isoelectronic series computed despite having the most terms. There is an apparent typo in one of the GUTCP force terms which probably contributes to the mismatch.

Fig. 6. The ionization energies of 5-electron atoms, from Boron to Magnesium.

Fig. 7. The ionization energies of 6-electron atoms, from Carbon to Aluminum.

Fig. 8. The ionization energies of 7-electron atoms, from Nitrogen to Silicon.

Fig. 9. The ionization energies of 8-electron atoms, from Oxygen to Phosphorus.

Fig. 10. The ionization energies of 9-electron atoms, from Fluorine to Sulfur.

Fig. 11. The ionization energies of 10-electron atoms, from Neon to Chlorine.

Fig. 12. The ionization energies of 11-electron atoms, from Sodium to Argon.

Fig. 13. The ionization energies of 12-electron atoms, from Magnesium to Potassium.

Fig. 14. The ionization energies of 13-electron atoms, from Aluminum to Calcium.

Fig. 15. The ionization energies of 14-electron atoms, from Silicon to Scandium.

Fig. 16. The ionization energies of 15-electron atoms, from Phosphorus to Titanium.

Fig. 17. The ionization energies of 16-electron atoms, from Sulfur to Vanadium.

Fig. 18. The ionization energies of 17-electron atoms, from Chlorine to Chromium.

Fig. 19. The ionization energies of 18-electron atoms, from Argon to Manganese.

Fig. 20. The ionization energies of 19-electron atoms, from Potassium to Iron.

Fig. 21. The ionization energies of 20-electron atoms, from Calcium to Cobalt.

Discussion

The Pauli Exclusion principle states that “no two electrons may simultaneously occupy the same quantum state” in an atom (or atomic ion) [9]. The principal, azimuthal and magnetic quantum numbers, but not the spin quantum number, were initially known. With the discovery of spin, and its promotion to a fourth quantum number, it became possible to assign unique sets of quantum numbers to every electron, with, for instance, filled s orbitals containing one each of a spin-up and spin-down electron (thought to exist in the same physical space). Similarly, filled p orbitals would contain three spin-paired electron pairs. GUTCP, while proposing extended-body electrons quite different from QM, has carried over the concept of paired electrons in an orbital being indistinguishable except for their spin. Thus, both QM and GUTCP obey the Exclusion Principle.

Mod 1 has challenged this one-orbital, two-electron paradigm [10]. While spin pairing still occurs between pairs of electrons in Mod 1, even paired electrons occupy nested spherical shells, distinct in radius. Also criticized in Mod 1 is the QM doctrine (also possessed by GUTCP) called “relaxation,” in which energy levels exist in the atom which have no spectroscopic evidence. The substack article explaining Mod 1 contains energy level diagrams that demonstrate this [11].

Neither here, nor in earlier Mod 1 work, is there a derivation of the magnetic force terms from classical physics. That is done in GUTCP, and Mod 1 only plugs these GUTCP force terms into the force balance equation to solve for each electron radius. The agreement we obtained to NIST data suggests that GUTCP and Mod 1 are on firm theoretical footing (better than the “gold-standard” QM). We also found a minor error in GUTCP for neutral Beryllium, where terms like (Z-3)/(Z-2) or (Z-5)/(Z-4) exist for Li and B, but instead of (Z-4)/(Z-3) which would be expected for neutral Be, the term for Li is given again. The general equation (10.447) given at the end of GUTCP Book I, which summarizes the earlier equations, does not suffer from this error [1].

Conclusion

The next scientific revolution is quietly underway. As always, it is opposed. This time is full of opportunity for young (and established) researchers to make a mark in areas as diverse as clean energy and dark matter, which are both topics of interest in GUTCP. Replicating GUTCP experiments, including over a dozen forms of spectroscopy, is a great place to start [3], [12].