The "square-of-the-shells" rule states that the elementary particles are composed of single photons locked in resonant shells. The largest and only stable shell is the electron.  The neutron's outer shell ( s1 ) is about two and a half times more massive than an electron and starts a square-of-the-shells sequence that defines the mass of three additional shells.  s2 = s1 * s1, s3 = s2 * s2, and s4 = s3 * s3. The calculator below shows how close this comes to describing the exact mass of electrons, protons, and neutrons.

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350 Last modified July 06, 2008

Square of the Shells Rule How it came about 1986 - 1991 Vernon Brown June 26, 2002 with later edits During my formative years most scientists believed that the final irreducible constituent of all physical reality was the electromagnetic field.  This idea served me well in my career as an electronics engineer and I never saw or read anything to cause me to doubt that idea. When I retired after a career as an electronics engineer and revisited physics, I noticed that scientists seemed to have abandoned the electromagnetic idea in favor of particle theory and quantum mechanics. I began a study to see if it was really necessary to abandon the old idea that made so much common sense. The more I studied the more convinced I was that it was not necessary to abandon the old idea at all. I found that with a little modification the old idea could easily fit into the new quantum scheme of things. The task came down to describing a photon in such a way that it fit all observations while serving as a building block for atomic particles. I knew of Dr. Robert Hofstadter's work at Stanford where he bombarded hadrons with electrons and observed the hadronic spectra. Hofstadter developed a model based upon a shell structure to explain the spectra. Since this was a simple and reasonable way to explain the spectra, I began with a shell model for protons and neutrons. The first obvious thing to me was that there would be four shells for the neutron structure and three Shell structure of protons and neutrons. The inside shell s4, is too small to be seen at this scale. It is only about 4.3 x 10-14 cm in size compaired to the neutron's outer shell which is about 9.5 x 10-11 cm in size. shells for the proton structure. Then it was obvious that a neutron's extra shell mass would be the difference in the mass of a proton and that of a neutron. I remembered that a neutron was about 2.5 electrons more massive than a proton so I started with 2.5 electron masses as the mass of the neutron's outer shell. There would need to be an exponential increase in shell masses going from outer to inner to get the required total mass. The first thing I checked was to see if each inner shell mass was the square of the mass of the next shell out. I squared 2.5 to get the mass of the proton's outer shell. Then I squared the proton's outer shell mass to get the proton's middle shell mass. Finally I squared the proton's middle shell mass to get the mass of the proton's innermost shell. I then added the mass of the inner three shells to get proton mass and added the mass of all four shells to get neutron mass. I extended the decimal of the neutron's outer shell mass to force the calculation to exactly produce the mass of a proton and then checked to see if all the calculated values fit measured values for neutron, proton, and electron masses. The numbers I had at the time were 1836 electron masses for a proton and 1838 for a neutron. These numbers worked out perfectly. But even if they were only close small differences could exist because of forces internal to the structure like the binding forces that slightly distort the calculation of atomic weights. When "Physical Review D", Volume 45 Number 11, published June 1st 1992 became available I ordered a copy to get the latest published values for particle masses. Electron mass was 0.510 999 06 and proton mass was 938.272 31, both in units of MeV/cc To get the number in units of electron masses I divided proton mass / electron mass and came up with 1836.15292789 as the published value for proton mass in electron masses. Published mass values. Numbers in (  ) are the amount of uncertainty. Numbers are in units of MeV/cc Electron: .510 999 05 ( 15 ) Proton: 938.272 31 ( 28 ) Neutron: 939.565 63 ( 28 ) When I extended the decimal of the N - P mass to force the calculation to produce the nominal measured value for proton mass, the calculated value for neutron mass was .009 MeV/cc more than the nominal measured value for neutron mass. This was slightly more than uncertainty in the measurements could account for. Some unknown force like a binding constant could account for the small difference in calculated and measured values. Strong Nuclear Interaction A year or so later ( after 1991 ) I was surprised to find that the nuclear binding forces matched the  mass values of the outside and next to outside shells of the proton and neutron.When I considered the charge on the proton's outer and middle shell surfaces to be the source of the strong nuclear interaction, the square root of the sums of the shells in contact produced the correct values for both the neutron-neutron force and the proton-neutron force. To rationalize how the attracting force on the surface of the shells could be more than an electron's charge I reasoned that in this shell scheme of things an electron would be the largest in size of all the particles. Its charge would come from its radius, not its center. This is because the charge is due to asymmetry resulting from the bent path of the photon. The same is true for the proton and neutron as well as for their inner shells. So the force of attraction at the surface of the proton's outer shell is much greater than an electron's charge at the electron's outer surface. When seen at a distance equal to an electron's radius, however, proton charge is exactly the same as that of an electron. In fact, this is true for any distance greater than an electron's radius since both the electron's force and the proton's force must diminish identically with distance. An electron's size can't be measured the way scientists measure proton and neutron size. Expecting a very small point, measurements fail to detect the fuzzy charge diameter of the electron. It exists as a spinning photon field. It may interact at any place in the spinning field depending upon the momentary state of the field. The exact point that an electron interacts with a target depends upon this momentary state and the momentary state of the fields in the target. My conclusion is still that it is not necessary to abandon that hundred-year-old idea. It may truly be that the final irreducible constituent of all physical reality is the lowly photon. The Photon Theory of matter sums it up.