Hofstadter's Shells Revisited
by
Vernon E. Brown



Dr. Robert Hofstadter of Stanford University was 
awarded the 1961 Nobel prize in physics for discovering the 
structure of atomic nuclei. Using a linear accelerator, he 
bombarded protons and neutrons with electrons at energies 
of 100-600 MeV, and found them to be composed of 
positively charged cores surrounded by shells of alternating 
negative and positive charge. 

There is a newly discovered and very simple 
mathematical relationship that adds great insight to the 
structure that Dr. Hofstadter observed. It's called "the 
square-of-the-shells rule." This paper explores the rule and 
the added impact that Hofstadter's shells may have upon the 
development of twenty-first-century science. 

According to Hofstadter's observations, the charge 
sequence of a proton's shells is positive, negative, and 
positive, going from the innermost to the outermost shell. A 
neutron is the same except that it has an extra negatively 
charged outer shell. 

This outer neutron shell's mass is the difference 
between proton and neutron mass. Measurements show this 
to be about 2.5 electron masses.  When the decimal of this 
difference is extended to 2.54992206745 electron masses 
and this number and the results are repeatedly squared, a 
very curious relationship emerges.

In order to see this, first square the number 
2.54992206745, and square the result, then square the 
result of that, to obtain, 6.50210, 42.27734, and 
1787.37327. These three results added together total 
1836.1527, which is the exact mass of a proton in electron 
masses.  Add the starting number and the result is 
1838.7026, which is very close to the measured mass of a 
neutron. (Measured neutron mass reported in June 1992, 
Physical Review D, was 1838.6839 electron masses.)

Many great scientists of the past, including James 
Clerk Maxwell, H. A. Lorentz, H. Hertz, Erwin 
Schrodinger, and Albert Einstein thought that scientists 
would eventually show mass to be reducible to smaller and 
smaller constituents until they found a smallest-possible 
piece. Einstein said, "...most people [scientists] gradually 
came to believe that the final irreducible constituent of 
physical reality would be the electromagnetic field." 

There were very compelling reasons why most 
scientists believed this and these reasons have never been 
explained by other ideas. The most compelling reason was 
the phenomenon of relativity. H. Ziegler pointed out in 1909 
in a discussion with Einstein and Planck that relativity 
would be a natural result if the most basic components of 
mass moved at the constant speed of light. 

If these scientists were right, Hofstadter's shells 
would be made of photons, and each shell would exist in its 
most simple state as just one sine-wave cycle of 
electromagnetic energy. Since such photon shells must 
complete their loops at the speed of light in one wavelength, 
and we know the mass of each shell, and their wavelength is 
determined by their energy content in accord with Einstein's 
mass-energy equation, we can calculate the exact diameter 
of each shell.  Shell diameters are then as follows:

The diameter of shell(1), the neutron's outer shell, is 
3.3170 x 10-11 centimeters. The next shell in, shell(2) the 
proton's outer shell, is 1.1889 x 10-11 centimeters. The next 
shell, shell(3), the middle shell of the proton, is 1.82854 x 
10-12 centimeters. Shell(4), the final and most inward shell 
of both the proton and neutron, is 4.32511 x 10-14 
centimeters.

These theoretical diameters are consistent with the 
shell diameters that Dr. Hofstadter observed, and the 
structure agrees with the baryon spectrograph of atomic 
nuclei when composite spin states are assumed for each 
shell. The electric and the magnetic fields each must change 
with time so that each completes its cycle in one trip around 
the loop. Both of these have a rate of change that operates 
with a sine function. This combination provides a rich 
diversity of spin states and accounts for the diversity 
observed in hadronic spectra. There is a spin-up, and spin-
down possibility, (clockwise, or counterclockwise), and a 
flatwise spin left, or spin right. In combination, there is also 
the possibility that adjacent shells may spin alike, or 
different, in the spin-up,-spin-down state, and likewise in the 
spin-left,-spin-right state.

Since each shell is charged, the inside shell and the 
next-to-inside shell form a doublet that taken together is 
charge neutral. The next shell out provides the proton's 
positive charge, and is associated with another outer shell in 
the neutron. The two outside shells of the neutron thus form 
another doublet associated by charge. All these spin states, 
associations, and doublets provide the rich and diverse 
nuclear spectra observed at electron-impact energies below 
1000 MeV.

An electron's mass is .51099906 MeV.  This amount 
of energy is also present in a photon whose frequency is 
1.2344046 x 1020 HZ, and whose wavelength is 2.286399 x 
10-10 centimeters. If this photon were curled into a loop 
whose circumference was equal to its wavelength, it would 
form a circle whose diameter would be equal to its 
wavelength divided by pi, or about 7.7 x 10-11 centimeters.

This is much larger than an electron is supposed to 
be. Dr. Samuel Chao Chung Ting of MIT  failed to find a 
solid electron structure larger than about 10-16 meters. His 
methods could not detect structure smaller than that, so he 
concluded that an electron must exist as a point charge 
smaller than 10-16 meters.

Electrons possess a strange characteristic, however. 
It is not possible to predict the exact location that a single 
electron will impact on a flat screen. They seem to exist in a 
"fuzzy" area most easily described by a probability potential 
that the electron is located at any certain point within a 
sphere of about 7 x 10-11 cm. in diameter.  This 
phenomenon is consistent with the idea that an electron may 
be composed of a gamma-ray photon in a spinning in an 
electric and magnetic spin state forming a sphere about 7.7 
x 10-11 cm. in diameter.

According to the standard photon model prior to 
1980, photons did not interact with the fields of other 
photons and certainly not with their own fields as would be 
necessary if a stable photon loop were possible. After 1980 
this changed. Nicolaas Bloembergen, Arthur Schawlow, and 
AK Siegbahn shared the 1981 Nobel Prize in Physics for 
their contribution to this change. They helped develop laser 
spectroscopy, a tool used to investigate photon-photon 
interaction in the new science of non-linear optics. 

Although this science is very new, it is now certain 
that photons do interact with their own electromagnetic 
fields and with those of other photons. All the ingredients 
necessary to cause photons to form stable loops are present. 
These include resonance, electric charge due to the 
asymmetry caused by the bent path of the photon in the 
loop, and positive feedback resulting from the electric 
charge. Feedback and resonance make the loop stable when 
the loop circumference is one wavelength of an electron-
photon's frequency. Some shorter wavelengths can form 
unstable loops that quickly unwind into photons again, 
giving rise to the multitude of unstable particles observed 
downstream of collisions in particle accelerators.

Electrons have antimatter counterparts called 
positrons that were first observed by Dr. Carl D. Anderson 
at the California Institute of Technology in 1932.  
Scientists soon discovered that when electrons and 
positrons collided at very low energies, they became 
photons of energies equivalent to their mass or, 1.234404 x 
1020 HZ. This is consistent with the idea that electrons and 
positrons are composed of photon shells.

That mass can become energy, and energy can 
become mass is so generally accepted today that people give 
little thought to the process. But there must be a process, 
and it must be a reasonable process for those of us who 
believe that nature must operate by reasonable processes. If 
mass is composed of photon shells, the observed 
transformation of mass into electromagnetic energy is a 
reasonable process; otherwise, this process is not 
reasonable.

At high energies, colliding electrons and positrons 
generate much more than just two photons. Electron-
positron colliders today produce many unstable particles as 
well as more electrons, and even protons, neutrons, and 
their anti-matter counterparts. Even a casual observer must 
see that these particles are created by the process. They can 
not possibly come from the colliding particles which are 
orders of magnitude less massive. 

What then is the significance of short-lived unstable 
particles created in these processes. It seems a very 
dangerous assumption to suppose that because such 
particles may be created, they must exist in stable mass. It 
seems even more dangerous to suppose that they somehow 
form the basis of nuclear structure. The danger is that we 
are building an ever-more complex and wobbly foundation 
for fundamental physics. Such a foundation cannot possibly 
survive. It cannot survive because it was admittedly 
unreasonable from the start.

Einstein warned us of this. He said, "There is no 
doubt that quantum mechanics has seized hold of a beautiful 
element of truth, and that it will be a test stone for any 
future theoretical basis. However, I do not believe it will be 
the starting point in the search for this basis, just as, vice 
versa, one could not go from thermodynamics (resp. 
statistical mechanics) to the foundations of mechanics." 

When in a one-wavelength loop, the electrical field 
of a photon, changing as it does with time, must complete 
its negative-positive swing as it moves around the circle. A 
maximum-negative-amplitude point moves around the circle 
changing with time, but since it must traverse a 
circumference described by the same sine function that 
governs its rate of change, the negative field of the photon 
must remain toward the outside of the circle all the way 
around the loop. This gives the electron an overall negative 
charge originating at its circumference and spreading 
outward in a field that decreases in amplitude as the square 
of distance.

Electron charge is a constant; why so has been a 
mystery to scientists of the twentieth century. Now, we 
know that this constant charge is the result of the constant 
amplitude of the electron's photon from which both the 
electron's charge and Planck's constant derive. 

A photon in a loop smaller than an electron's 
circumference must complete the trip around the loop more 
often than an electron's photon around an electron's loop. It 
therefore must present its constant amplitude at any certain 
point around its circumference more often, and so must 
exhibit a more powerful force at its circumference. We 
could then devise a square-of-distance gauge calibrated to 
one-electron force at the radius of the electron's shell and 
measure the force of charge on the circumference of the 
inner photon shells. The inner-shell charge is greater, but 
since the force originates at a smaller radius and diminishes 
as the square of distance, it is the same as an electron's force 
when gauged at an electron's radius.

Consider two protons, each consisting of three 
shells. Assume that they may merge together until their 
outer shells pass through each other and come into close 
proximity with their next-to-outer shells. Four shells then 
have other shells of opposite charge in close proximity, and 
the electric forces of these shells, as calculated by the 
square-of-the-shells rule are, 6.50210, 42.27734, 6.50210, 
and 42.27734.  Added together, these forces total 
97.55888, electron forces. This is the observed force of the 
strong nuclear interaction between two protons.  Add to 
that the force of the neutron's outer shell, 2.54992, and we 
have 100.1088, the force of the strong nuclear interaction 
between protons and neutrons.

Here and now there are fourteen and more 
circumstances that point to this photon-shell structure as 
being what is real. Numbers match to the extended decimal. 
It may be mere coincidence, but coincidence like this would 
be like watching a thousand buffalo stampeding through 
your lawn leaving buffalo chips and hide and hair torn on 
splintered shrubs and hoof-print tracks from the way they 
came to the way they went. They do it every day, and you 
can invite anyone to watch them do it. Would you believe 
any such guest who told you those buffalo did not exist? 
Have we come so far down the unreasonable path that 
Einstein warned us about that we can no longer even 
consider the correct and reasonable one?

If not, then consider this: If massive objects are 
made only of photons, and these photons emit fields that 
move away from their central points at the speed of light, 
the universe must be full of photon fields. They must 
necessarily be greater in intensity near massive objects, and 
diminish as the square of distance away from the massive 
objects. Since the central point of a photon must exist at a 
constant amplitude, and these fields all contribute toward 
that amplitude, all photon points must reach saturation 
amplitude at an offset and so accelerate toward increasing 
field strength. We call this phenomenon gravity.

Scientists have ignored the implications of 
Hofstadter's findings for over thirty years now, but the 
implications are still there. Some things are possible given 
this makeup of mass, and other things are not possible. Not 
possible, for example, is a one-photon loop with anything 
other than one unit of electric charge. What then, is a 
neutrino? What then, is a quark? What then, is a big-bang 
type of singularity?

This model of nuclear structure is much more 
restrictive and exact than the current standard model. It 
would seem then that it could be readily falsified, but this is 
not so easy. Neutrinos, quarks, and big-bang types of 
singularities, for example, have never been observed.

These observations taken together comprise a new 
hypothesis called photon theory. It's being spread through 
the minds of students of science by word of mouth and 
small science news letters such as Photonics, of Cabot, 
Arkansas. Mainstream periodicals have never published a 
thorough treatment of it. Perhaps it is time they did. 
Twenty-first-century science is in the making and twenty-
first-century scientists need to know about it.
 Raymond J. Seeger, "Hofstadter, Robert," Grolier Electronic Publishing, Inc. , 1993.
 Vernon Brown, "Square-of-the-shells rule," November Photonics, Cabot Ark., 1991.  
 Aitchison, I. J. R. and Hey, A. J. G., Gauge Theories in Particle Physics, Bristol England, 1989.
 Physical Review D, June ,1992.
 The decimal was extended using proton mass as the calibration number so it must necessarily be exact.
 Einstein, Albert, Ideas and Opinions, New York, 1954.
 Albert Einstein, "Development of our Conception of Nature and Constitution of Radiation," 
Physikalische Zeitschrift 22, 1909.
 The equation is: Diameter is equal to Planck's constant divided by the product of pi, shell mass, and the 
speed of light. (D = h/(pi m c)
 June 1,   Physical Review D,  American Institute of Physics, New York, 1992.
 Ting, Samuel Chao Chung, Grolier Electronic Publishing, Inc., 1993.
 This is approximately the area of uncertainty predicted by the uncertainty principle.
 Bahaa E. A. Saleh and Malvin Carl Teich, "Nonlinear Optics," Fundamentals of Photonics, New York, 
1991.
 Frank J. Oliver, "Cloud Chamber," Grolier Electronic Encyclopedia, 1993.
 June 1,   Physical Review D,  American Institute of Physics, New York, 1992.
 Albert Einstein, "Quantum Theory and the Fundamentals of  Physics," Jefferson Hane Weaver, ed. The 
World of Physics, New York, 1987.
 Vernon Brown, "How Come the Quantum," Feb. Photonics, Arkansas, 1994.
 Originally calculated as values of massiveness, these numbers also represent electromagnetic energy in 
accord with Einstein's mass-energy equation, E=mc2, and so must also represent forces.
 Force values are in units of an electron's charge. Different experiments yield different results for the 
exact value of the strong nuclear interactions, but there is general agreement that it is about 100 times 
greater than an electron's force and that the neutron-proton interaction is slightly stronger than the 
proton-proton interaction.
12