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Hofstader's Shells Revisited
Hofstadter's Shells Revisited
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
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
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
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
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
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,
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
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