ENERGY SCIENCE ESSAY NO. 15
THE CHAIN STRUCTURE OF THE NUCLEUS
Copyright, Harold Aspden, 1974
This Essay was first published in 1974 by Sabberton Publications,
P.O. Box 35, Southampton SO16 7RB, England.
ABSTRACT
The atomic nucleus is shown to have a form determined by the
quantum structure of a Dirac-style vacuum. Nucleons occupy a series of holes
in the structured vacuum forming a shell about a core region of unoccupied
holes. These nucleons are linked by electron-positron chains. The lattice
spacing can be related to the binding energy of the nucleus in precise
quantitative terms. The special position of Fe 56 in the nuclear packing
fraction curve is explained in terms of the cubic symmetry of the lattice
system, the optimization of interaction energy with the core charge and the
energy minimization of the chains.
INTRODUCTION
This paper has been prompted by recent
developments in elementary particle research having bearing upon a theory
published in 1969. Chapter 7 of the author's work 'Physics without
Einstein' incorporated some new ideas about nuclear structure. It was argued
that nucleons are located at fixed lattice positions in a cubic structure and
are linked by chains of electron-positron pairs. Each chain had association with
what are now called partons. The mass deficit due to the negative interaction
between a proton-sized parton and a pion-sized parton was deemed to balance the
mass of the chain of electrons and positrons. Indeed, it was the energy of
combination of these two heavy particles to form a nucleon at a nuclear lattice
position which was the source of energy creating the electron-positron
chain.
These ideas have progressed over the past five years and it is
appropriate now to publish some of these developments. The author is indebted to
Dr D M Eagles of the National Standards Laboratory, Sydney, Australia for
helpful communications and encouragement. Dr Eagles recently drew to the
author's attention a paper entitled 'Parton Chains in the Nucleus' by Wojciech
Krolikowski, at p. 2922 of Physical Review D of 1 November 1973. It is this
which has stimulated the publication here of some interesting advances of the
chain nucleus theory at this stage of its development. The theory proposed
offers scope for very detailed computational analysis of the structure of
individual atomic nuclei.
A preliminary note about quark theory is
appropriate before the structure of the atomic nucleus is analysed. This is
important because it is the author's contention that the proton does, indeed,
comprise three particles as demanded by quark theory. Such a structure of the
proton was presented in 'Physics without Einstein' but in the form of a
positive particle having the charge of the positron and associated with an
electron-positron pair.
THE QUARKS
From a study of electron and neutrino scattering
from protons Feynmann, writing in Science at p. 601 of the 15 February 1974
issue, has been able to show that protons have structure as if they comprise a
plurality of particles of more fundamental nature, the so-called quarks. His
paper entitled 'Structure of the Proton' has the introduction:
Protons are not fundamental particles but seem to be made of
simpler elements called quarks. The evidence for this is given. But separated
quarks have never been seen. A struggle to explain this seeming paradox may be
leading us to a clearer view of the precise laws of the proton's structure and
other phenomena of high energy physics.
Feynman explains how, on
quark theory, there are three kinds of quark denoted u, d and s. The s and d
quarks have charge -(1/3) and the u quark charge +(2/3) that of the positron.
The s quark has higher mass than the d and u quarks which have the same mass.
From this he presents a diagram showing how three quarks can combine to produce
ten different particles:
Quark Theory
|
| |
|
| Strangeness |
|
| -3 |
sss |
|
| -2 |
ssd |
ssu |
|
| -1 |
sdd |
sdu |
suu |
|
| 0 |
ddd |
ddu |
duu |
uuu |
|
| Charge |
-1 |
0 |
+1 |
+2 |
Now the unsatisfactory feature of
quark theory is this concept that charge can be quantified in units which are
one third or two thirds that of the electron or positron. It would be so much
more satisfactory if Nature gave us a system of basic particles based
exclusively upon charges which are measured in terms of the unit charge of the
electron or positron. A little speculation shows how this is possible, provided
we pay attention to some of the ideas presented to us by Dirac. It is well known
that Dirac has proposed that the vacuum state is an aether permeated by quantum
states filled by negative mass electrons. This implies that the vacuum has
states with which particles can be associated and in which a negative charge of
-1 electron units will pass undetected, being somehow neutralized by the vacuum
medium. In these states the vacuum appears to add the charge +1. A particle can
exist independently and not occupy such a state. Then we need add no charge to
its own charge. On this basis, consider the following diagram:
Quark Theory
|
| |
|
| True Charge |
-1 |
-1 |
+1 |
+1 |
|
| State |
0 |
+1 |
0 |
+1 |
|
| Effective Charge |
-1 |
0 |
+1 |
+2 |
Given a combination of three
charges, each of which can be -1 or +1, and recognizing that stability criteria
forbid three negative charges and three positive charges from combining, we must
have a net charge of +1 or -1. Also, if we can have a free particle or one
occupying a vacuum-polarized position, effectively adding +1, we see scope for
four different charge entities. It follows that if the s, d and u quarks have
charges +1 or -1, but masses as assumed on normal quark theory, we can have ten
particles satisfying the observed charge system, but without recourse to the
fractional charge features of quark theory.
It is therefore submitted
that, since no experimental evidence exists supporting the fractionally-charged
quarks but since experimental evidence does support other features of quark
theory, then the alternative is to accept that some features of Dirac's aether
theory need scrutiny.
ATOMIC MASS
Bernstein writing in Annals of Physics, 69,
1972, p. 19 has recently pointed out the need to incorporate 'holes' as
constituents of an atomic nucleus. His reason is coupled with the explanation of
energy levels and the inadequacies of the existing shell models. The approach we
will take here is to examine the possibility of substituting nucleons for
electrons in the Dirac continuum. We will presume a hole structure forms around
the charge core of the nucleus and that the holes are occupied by negatively
charged nucleons. This imparts mass to the nucleus but the charge of these
nucleons is merged into the continuum. Interesting quantitative verification of
this principle is available.
It is generally believed that an isolated
electric charge will attract an equal charge of opposite polarity and so one
imagines that two equal and opposite charges will pair together and form a
neutral aggregation. Yet Earnshaw's theorem denies that two equal and opposite
electric charges can rest adjacent one another in stable equilibrium unless they
are immersed in an enveloping electrical medium. Dirac's continuum would, in
effect, be such a medium. The observed vacuum polarization adjacent an atomic
nucleus supports the exception also. Therefore charge neutralization should
occur. Why then is the atomic nucleus itself not a neutral entity?
The
answer is found from classical electrostatic theory. Laplace proved that the
outward forces due to mutual interaction of a surface charge on a conductor are
only half the forces exerted by the field on similar free charge just outside
the surface. Thus, when an electron is added to the surface of a conductor to
charge it, a free electron migrates from the atomic lattice system of the
conductor and joins the added electron. Together the electrons form a surface
charge just outside an inner charge of opposite polarity and half the magnitude.
This latter is the residual charge left by the ionized lattice. This is a
displacement phenomenon. The field on each electron is zero because the
displaced electron has created positive and negative influences which cancel.
The field away from the conductor is that due to the single added electron. In
our atomic case, however, we have no displacement. Instead, a spherical shell of
charge can centre upon a core of opposite polarity of half its strength and be
held stable. A core of Ze charge can and will form a stable aggregation with a
surrounding shell of -2Ze charge. If these added charges are not electrons but
are negative nucleons then the atomic mass number A should be 2Z. If the
nucleons are uniformly distributed over the volume of a sphere because they form
in a structure of some kind then the same principles of Laplace apply except
that a charge of -2.5Ze can be aggregated and held stable. This tells us to
expect the ratio A/Z to increase from 2.0 to 2.5 as an atomic nucleus formed in
shells increases in size.
In line with Bernstein's ideas we need to
recognize that 'holes' are part of the nucleus. These cancel the effects of the
nucleon charge. From another viewpoint we might say that space is pervaded by an
electrically-neutral continuum which nevertheless contains discrete negative
charges (electrons or the like) in a positively charged background continuum.
Heavy negatively charged nucleons can occupy holes from which the negative
charges are displaced. However, these nucleons tend to nucleate, if only by
stronger gravitational effects, in regions immediately surrounding the atomic
core charges Ze. Thus the atomic nucleus is formed, and it may have structural
form characteristic of the properties of this pervading medium.
The
analysis relating A and Z just presented has bearing upon nuclear stability. Z
sets a limit upon the value of A, but one may expect the exact relationships to
depend upon the structural links between the nucleons.
This concept has
already been presented in the author's 1972 book 'Modern Aether Science'.
The relevant part of chapter 14 of this work is reproduced below.
THE NUCLEAR AETHER
The physics of the aether is to many
minds the physics of the nineteenth century. The twentieth century has so far
been concerned with the physics of the atom and its quantum behaviour. Physics
has assumed importance in industry primarily because electrical technology in
the semiconductor field has become the province of the physicist rather than
the electrical engineer. Also, physics has now an undeniable place of
importance because everyone is all too aware of the energy hidden inside the
atomic nucleus. For this reason the minds of many research physicists are
technology-orientated. Theoretical physics is complicated, the aether is dead
and who has the time anyway to be concerned with such an antiquated topic! The
more open minded may say that if the aether has a place it is in cosmology; it
is certainly not in the field of the nucleus. But let us see if we can dispel
this belief.
Is there anything about the atomic nucleus we cannot
explain? The atomic mass does not increment in proportion to the atomic
charge. It seems that over a range of atoms of low atomic mass the number of
nucleons is approximately twice that of the number of proton charge units in
the nucleus. The nucleons comprise the protons and neutrons believed to form
the nucleus. At high mass numbers the ratio of two increases roughly to about
two and a half. An explanation of this would help our understanding of nuclear
physics. Does the reader already have such an explanation? If not, perhaps the
following analysis will have some appeal.
Consider an electric charge
surrounded by a concentric uniform spherical distribution of discrete charges
of opposite polarity. Now calculate the electrostatic interaction energy of
such a system. This quantity will be found to be negative until the spherical
charge distribution has a charge exactly double the magnitude of the central
charge. Thereafter we would have positive interaction energy signifying
instability, because the 'binding' energy associated with the negative
polarity has ceased to 'bind'. We may expect, therefore, an entity to form as
a stable aggregation in which the central charge acquires an enveloping double
charge of opposite polarity, assuming the spherical distribution. If we
consider instead a central charge with a uniform spatial charge distribution
surrounding it, bounded by a sphere, then instability sets in when the
surrounding charge is two and a half times that of the core. Between these two
limiting examples, we could have, say, charge distributed in two concentric
shells of unit and double unit radius, the charge content being proportional
to the area of the spherical shell form. This gives a ratio of 2.166 for
stability.
It needs little imagination to recognize the relevance of
this to our nuclear problem. The atomic mass number is a measure of the number
of negative nucleons clustered around a central core of charge. This charge
has negligible mass compared with the nucleon mass contribution but the charge
is the positive charge we regularly associate with the atomic nucleus. We need
not speak of a combination of neutrons and protons to explain qualitatively
the numerical difference between atomic number and atomic mass number. Somehow
the charges of the nucleons are not detected, because we well know that the
atomic electrons only react to the central charge. They ignore the nucleon
charges just as they ignore charges in the aether medium. Indeed, the
electrons may see these nucleon charges as they see the aether. In fact, the
nucleons may be deemed to be arrayed in a structure and to have displaced
negative aether charge so as to substitute themselves in the structured form
of the aether itself. Their charge is undetected just as the mass of a buoyant
body goes undetected in a fluid of equal mass density.
Hence, we need
to invoke our aether. Also, we see support for the cubic lattice distribution
of aether charge. An oxygen nucleus can be adequately populated by a single
shell of discrete charges. There are 26 charges disposed in a regular cubic
system about a central charge and 16 of these are presumably replaced by
negative nucleons. The two to one ratio applies, because the oxygen atom has a
atomic number of 8. Now take chromium, for example, which has an atomic number
of 24. Here, we might expect charge to be distributed over another shell as
well. The stability condition, calculated for idealized spherical
distributions, requires 2.166 times as many nucleons as units of central
charge. Hence an atomic mass number of 52, as is found. Similarly, for heavier
atoms we find an appropriate relation between the two quantities conforming
with this theory.
It has to be accepted from this that the nucleus
consists of a central charge surrounded by a cluster of regularly spaced
nucleons of negative charge. As the author has explained in his book
Physics without Einstein, the nucleons form into a lattice structure
with bonds joining the nucleons and, additionally, pions contributing to the
energy of the bonds also derive their energy from an interaction with the
nucleons. These features of the nucleus modify the mass and add some
complication. Different isotopic forms may depend upon alternative structure
configurations rendered possible by the different bond positions available.
This is a matter for further analysis. When the above-mentioned book was
published the author supposed the nucleons to be formed as a system of
neutrons and protons, as is conventional. The later realization of the stable
charge system introduced in this chapter, however, has led to a revision of
the model. All the nucleons are the same. They are negative particles of mass
approximating that of the proton.
The above text appears at pp. 139-141 of the
book.
Contrary to established theory, the author's proposal is
that the enveloping nucleons are neutralized by the occupancy of vacuum states.
The mass of the atomic nucleus is essentially that of these neutralized nucleons
and their related electron-positron chains.
Some recent experimental
evidence from research at the Brookhaven National Laboratory was reported by
Bugg et al in Physical Review Letters, 31, 1973 at p. 475. This research
indicates an abnormally-high probability that a tenuous halo of neutrons may
surround the central charge of the atomic nucleus. This seems to add support to
the role of the vacuum state in compensating charge effects due to nucleons and
gives strength to the author's ideas concerning a Dirac-style aether. Also
encouraging is the reported activity of Lee and Wick of Columbia University in
studying the effects of the properties of the vacuum upon the atomic nucleus.
This is mentioned in Science at p. 51 of the 5 April 1974 issue.
NUCLEAR RADII
It is interesting to digress to examine a
recent proposal by Ross writing in Il Nuovo Cimento, 9A, May 1972 at p. 254.
Ross interprets the muon as an electron orbited by a massless spin-1 wave and we
will contrast this with a classical electron concept.
Ross has suggested
that a particle might orbit the electron at its classical radius. By regarding
the particle as having zero mass and applying the principles of General
Relativity, Ross then shows that this orbit would be a null geodesic and is able
to calculate the energy involved. Though at pains to show that the massless
particle is not a normal photon, Ross rnust have contemplated this possibility.
He derives the quantitative result that:
Mμ = Me[1+ 3/2α],where alpha is the
fine-structure constant. This gives the muon mass mμ as 206.554 times
the electron mass me, in comparison with the observed ratio of
206.767. It is interesting then to note that had we regarded the electron as a
mere sphere of electric charge of radius b and presumed a disturbance of some
kind to ripple around it at this radius but at velocity c, we would have reason
to derive a disturbance frequency of c/2πb. Multiplied by h this could represent
energy, particularly if we are alive to the possibility that the mechanism of
the photon may be involved in this model. Such energy, in mass terms, when added
to the mass of the electron, gives a total mass of:
me[l + e2/αbmec2], since
alpha is 2πe2/hc.Then one can see by analogy with the Ross
result that the muon mass could be derived with the same quantitative success if
the rest mass energy of the electron were 2e2/3b. It is interesting
then to note that this is exactly the rest mass energy found in classical works
from the study of the electromagnetic properties of the electron.
The
purpose of this is to show that we need not appeal to General Relativity to
derive quantitative results in accord with Ross discovery. On the other hand
Ross has come to his result by careful qualitative analysis and has argued that
his muon should not affect the applicability of quantum electrodynamic theory.
Our object in this paper is not to treat the problem of the muon, but rather to
take the classical model of the electron and, guided by the quantitative result
emerging from this analogy with the Ross speculations, examine how the classical
model can be tailored to suit larger particle structures, particularly the
atomic nucleus. We can be encouraged also by a statement made by Dirac in
Scientific American in May 1963. He wrote:
I might mention a third picture with which I have been dealing
lately. It involves departing from the picture of the electron as a point and
thinking of it as a kind of sphere with a finite size.... the muon should be
looked on as an excited electron. If the electron is a point, picturing how it
can be excited becomes quite awkward.
The method of reverting to
a physical model of the electron also takes strength from observations made by
Grandy on the classical Lorentz-Dirac theory of electrodynamics. Grandy was
writing at p. 738 of the February 1970 issue of Il Nuovo Cimento, v. LXV.
Referring to the problem of Schott energy*, he said that an insight into its
nature was outside the scope of classical electrodynamics and also that 'no
relief is to be found in quantum electrodynamics, either, which is totally
unable to account for the structure of the electron'. However, Grandy's comments
about the impossibility of quantum electrodynamics helping an understanding of
electron structure prevail, though this does not preclude the photon-electron
interaction or combination to account for elementary particles or atomic
nuclei.
Footnote:
* The problem of Schott energy has been discussed by the author at
p. 97 of his book Modern Aether Science.
The muon can behave as an atomic nucleus. In muonium a positive muon
replaces the proton in an ordinary hydrogen atom. Also, the muon can replace the
electron in normal atoms. A study of such so-called exotic atoms is reported at
p. 148 in the March 1972 issue of Physics Bulletin by Kim who refers to evidence
of vacuum polarization effects and data showing that the charge radii of nuclei
are given by R = roA1/3, where ro is
approximately 1.2x10-13cm and A is mass number. It is standard to
relate the radius with the mass number, but since we are referring to charge
radii it is very interesting to examine more detailed data and perform a
conversion putting R proportional Z1/3, where Z is the charge number.
Such data is available from Condon and Odishaw's Handbook of Physics, 2nd Ed. at
pp. 9-13. According to these data, the core appears spherical and the charge has
a root mean square radius R given by the formula in ro, where
ro ranges between 0.91 and 1.05 in units of 10-13 cm as A
varies between 12 and 209. We may instead express R as
soZ-13 cm to find that so would vary between
1.22 and 1.32x10-13 cm, a variation of less than 4% about the mean,
in contrast with ro, which varies more than 7% about the
mean.
These data show that it is better empirically to look for
dependence upon Z rather than A. This may well be the outcome as better
measurement data are forthcoming.
Dimensions of Atomic Nucleus
|
| |
|
| A |
Z |
A/Z |
ro |
so |
|
| 12 |
6 |
2.00 |
1.05 |
1.32 |
|
| 40 |
20 |
2.00 |
1.02 |
1.28 |
|
| 51 |
23 |
2.22 |
0.97 |
1.26 |
|
| 115 |
49 |
2.35 |
0.92 |
1.22 |
|
| 122 |
51 |
2.39 |
0.93 |
1.24 |
|
| 197 |
79 |
2.49 |
0.91 |
1.27 |
|
| 209 |
83 |
2.51 |
0.93 |
1.27 |
Numerous writers* have formulated
the energy of the electron of charge e and radius b as 2e2/3b. In the
author's book 'Physics without Einstein' it is shown at p. 209 that this
indicates a uniform field within the radius b and corresponds to a charge
density in cgs. units of e/2(pi)xb2 at radius x. The root mean square
radius of such a charge distribution is b/(2)1/2. The value of b
calculable from the rest mass energy of the electron 8.2x10-7 ergs,
and the value of e of 4.8x10-10 esu, is 1.87x10-13 cm and
its root mean square is 1.32x10-13 cm. There is a remarkable
comparability between this electron radius and so particularly for
smaller Z values.
Footnote:
* Larmor, Phil. Mag. , xliv (1897) p. 503 is but one
example.
It seems obvious from this that if we take the classical formula given above
for the size of the electron and then apply this also to the positron we have
only to conceive the charged core of an atomic nucleus as an aggregation of Z
positrons occupying the same volume as Z separate positrons and the
root-mean-square radius of the resulting core is
1.32xZ1/310-13 cm. This fits the experimental data quite
well.
One is led to suspect that the hydrogen nucleus will be the same
size as a positron, which makes the Ross observations about the nature of the
muon all the more intriguing. However, accepting the empirical implications just
presented, there is need for caution in interpretation. One may wonder how the
inner electrons screening the atomic nucleus really escape involvement with the
measurement of the core radius.
Collectively the majority of the
electrons associated with the atomic nucleus happen to exhibit an aggregate
volume of just the right order to conform with the measurements of core
size.
The interesting feature of the analysis is the applicability of the
classical formula for the size of an electric charge. Also, the table above
indicates a relationship between A and Z such that as Z increases A/Z varies
from 2 to a value close to 2.5. This satisfies the theoretical proposal already
made.
NUCLEAR CHAINS
It is appropriate to reproduce next an
extract from the author's 'Physics without Einstein', noting that some of
the views expressed are subject to modification below. The text preceding this
material involved a rigorous analysis of the structure of the vacuum and the
computation of a lattice dimension d, which was found to be
6.37x10-10 cm. It is also noted that since that work was published,
Dr D M Eagles and Dr C H Burton have made careful calculations using the
computation facilities of CSIRO in Australia and the results reported in Physics
Letters at p. 423 of the 23 October 1972 issue support the value just given for
the lattice dimension d of the likely aether structure.
Nuclear Bonds
Fig. 7.8What is the form of the nuclear bonds? Each of the
six nucleons in Fig. 7.8, three protons, say, and three neutrons, identified
by the full bodied circles, has a bond of its own providing one of the links.
These bonds are the real mystery of the atomic nucleus. We will assume that
their most logical form is merely a chain of electrons and positrons arranged
alternately in a straight line. The reason for the assumption is that
electron-positron pairs are readily formed in conjunction with matter, and we
have seen how an in-line configuration of alternate positive and negative
particles has proved so helpful in understanding the deuteron. Stability has
to be explained. Firstly, the chain is held together by the mutually
attractive forces between touching electrons and positrons. Secondly, it will
be stable if the ends of the chain are held in fixed relationship. This is
assured by the location of the nucleons which these bonds interconnect. In
Fig. 7.9 it is shown how the bonds connect with the basic particles. In the
examples shown, the nucleons are positioned with a chain on either side and
are deemed to be spinning about the axis of the chain. Intrinsic spin of the
chain elements will not be considered. It cancels as far as observation is
concerned because each electron in the chain is balanced by a positron. In
Fig. 7.10 it is shown how, for the neutron, for example, the spin can be in a
direction different from that of the chain. Also, it is shown how another
chain may couple at right angles with this one including the neutron. Note,
that the end electron or positron of the chain does not need to link exactly
with the nucleon. Therefore, it need not interfere with the spin.
Fig. 7.9We will now calculate the energy of a chain of
electrons and positrons. For the purpose of the analysis we will define a
standard energy unit as e2/2a. This is the conventional
electrostatic energy of interaction between two electric charges e of radius a
and in contact. Since 2e2/3a is mc2, as applied to the
electron, this energy unit is 0.75 mc2. On this basis a chain of
two particles has a binding energy of -1 unit. If there are three particles
the binding energy is the sum of -1, 1/2 and -1, since the two outermost
particles are of opposite polarity and their centres are at a spacing of 4a
and not 2a.
Fig. 7.10For N particles, with N even, the total interaction
energy is:
-(N-1) + (N-2)/2 - (N-3)/3 + .... 2/(N-2) - 1/(N-1)which is
-Nlog 2, if N is large. If N is odd, the last term in the above series is
positive and the summation, for N large, is I - Nlog 2. To find N we need to
know how many particles are needed for the chain to span a distance d. This
distance d can be related to m by eliminating r from equation (4.1) (in
'Physics without Einstein'), namely:
r = h/4πmccombined with equation (6.60), namely:
hc/2πe2 = 144π(r/d)Then d/2a is found using
2e2/3a = mc2. It is 54π, so N may be, say, 169, 170 or
possibly 168, particularly if N has to be even and there has to be space for
any nucleons. For our analysis we will calculate the binding energy of the
chain and the actual total energy of the chain for all three of these values
of N. The data are summarized in the following table.
Chain Binding Energy
|
| |
|
| N |
168 |
169 |
170 |
|
| -N log2 |
-116.45 |
-117.14 |
-117.83 |
|
| Binding Energy (units) |
-116.45 |
-116.14 |
-117.83 |
|
| Binding Energy (mc2) |
-87.34 |
-87.11 |
-88.38 |
|
| Add Self Energy (mc2 |
168 |
169 |
170 |
|
| Total Chain Energy |
80.66 |
81.89 |
81.62 |
|
| Ground State Correction |
0.61 |
0.62 |
0.62 |
|
| Corrected Energy (mc2) |
81.27 |
82.51 |
82.24 |
In the above table the binding
energy has been set against the self energy of the basic particles and a
correction has been applied of αmc2 per pair of particles to adjust
for the fact that mass is not referenced on separation to infinity, as was
discussed earlier in this chapter. The total mass energy of the chain is seen
to be about 81 or 82 electron mass energy units, depending upon its exact
length.
This shows that while the electron-positron chain proposed will
provide a real bond between nucleons linked together to form an atomic
nucleus, it will nevertheless add a mass of some 81m per nucleon. This seems
far too high to apply to the measured binding energies. Furthermore, it is
positive and the nature of binding energy is that it must be negative. This
can be explained by introducing the π-meson or pion, as it is otherwise
termed.
The Pion
When an electron becomes attached to a small but
heavy particle of charge e, the interaction energy is very nearly
-e2/a or 1.5 times the energy unit mc2. This means that
the mass of the heavy particle is effectively reduced when an electron
attaches itself to it and becomes integral with it. If we go further and seek
to find the smallest particle which can attach itself to a heavy nucleon to
provide enough surplus energy to form one of the above-mentioned
electron-positron chains, we can see how this nucleon plus this particle plus
this chain can have an aggregate mass little different from that of the
initial nucleon. This can reconcile our difficulties. The fact that an
electron can release the equivalent of about half its own mass indicates that
to form the chain of mass 81m we will need a meson-sized particle of the order
of mass of the muon or pion. To calculate the exact requirement we restate the
inverse relationship between the mass m of a particle of charge e and its
radius a:
2e2/3a = mc2This applies to the
electron, but it can also be used for other particles such as the meson and
the H particle. It may then be shown that if two particles of opposite
polarity charge e are in contact, their binding energy, e2 divided
by the sum of their radii, is 3c2/2 times the product of their
masses divided by the sum of their masses. Let Mo be the mass of
the meson involved and M be the mass of the H particle. The following table
then shows the value of the surplus energy Es:
3MoMc2/2(Mo+M) -
Moc2in terms of units of mc2, for
different values of Mo/M and a value of M of 1836m.
Meson Energy of H particle
|
| |
|
| Mo/m |
*** |
Es |
|
| 230 |
|
76.4 |
|
| 240 |
|
78.3 |
|
| 250 |
|
80.0 |
|
| 260 |
|
81.5 |
|
| 270 |
|
83.0 |
|
| 280 |
|
84.5 |
The above-quoted text appears at pp. 147-150 of 'Physics without
Einstein'.Starting from this basis, we will now seek to improve
this 1969 account. Firstly, a very important advance emerges if we take the
latter equation and find the solution which gives maximum surplus energy. Thus
we put the expression at a minimum with M set at 1836m and Mo
variable. Simple analysis then shows that for this condition Mo is
M(3/2)1/2 - M or 0.225 M or 413m. This is higher than the pion mass
contemplated above. The energy released is found to be (0.225)2M or
93m. Thus subtracting the chain energy of about 81m we find that each chain
together with the parton pair represented by that equation will contribute mass
some 12 electron units m less than that of the proton.
If our atomic
nucleus comprised simple chain bonds and had one per nucleon we should find that
the mass of a nucleus would be 1824 times the number of nucleons when measured
in terms of electron mass units. In fact this mass varies. As the number of
nucleons increases the unit mass rapidly decreases through a minimum of about
1820 for iron and then rises gradually until it is 1823 for the largest
nucleus.
There is a very interesting explanation for this effect. Note
that the energy of a chain is proportional to its length. Then ask how three
nucleons arranged as below can be linked by chains. Three configurations are
shown in Fig. 1.
Fig. 1(a)
Fig. 1(b)
Fig. 1(c)
We now assume that the configuration adopted will
be that of minimum energy, that is minimum total chain length. Simple analysis
shows that 2x+y can be less than 2d. The minimum value is 1.933d when z is
approximately 0.2d. This means that at the corner of the nuclear lattice the
energy of a normal chain of length d is effectively reduced to 0.967 of its
normal value, that is, from 81m to 78m. There is a decrease of three electron
mass units whenever a chain is able to cut a corner so to speak as in Fig.
1(b).
Now consider a nucleus of iron and let us suppose that the charge
of the nucleus is due to 26 vacancies in the vacuum structure, an absence of 26
electron-sized charges which normally neutralize the vacuum state. This core
will be surrounded by nucleons occupying other lattice sites, 56 in number. Now
note that a 3 by 3 by 3 array of a cubic lattice system comprises 27 sites and
that there are 6 faces to this cubic array each having a 3 by 3 array in
adjacent lattice planes. This is 54 sites. We thus see how iron can be close to
an optimum state of symmetry. Also note how most of these 54 sites are
associated with a chain of minimum energy. This is evident from Fig. 2.
Fig. 2
It seems likely that in the iron nucleus of atomic
mass number 56 there are 6 arrays of 8 nucleons as depicted in Fig. 2 and that
four of these arrays have, as illustrated in Fig. 3., central nucleons linked
both to a nucleon in an outer lattice position and to one of the nucleons at P
in Fig. 2.
Fig. 3
In every respect, therefore, iron with an atomic mass
number of 56 is the nucleus for which every chain is at the low energy. Hence it
is not surprising that it appears to be a most stable nucleus. Also, our theory
has shown the unit mass to be three electron rrass units below the extreme of
having all chains lie on the lattice lines. Such an arrangement can be expected
to be more nearly applicable in very large nuclei where multiple shells of
nucleons exist and we have seen how such large nuclei have a unit mass higher by
three electron masses.
But it is of interest to ask about the Helium 4
nucleus. This appears to have four normal chains in its most natural
configuration. The unit mass of the Helium 4 nucleus is about 12. 5 electron
mass units below that of the proton. This compares with the figure of 12m
deduced on the basis of the chain energy of 81m.
CONCLUSION
From such analysis it is concluded that we are
arriving at results which encourage rigorous calculation of detailed structure.
The fact that the value of 80.5m is indicated from the Helium nucleus as the
mass contribution of a chain of standard lattice length checks very well indeed
with the data given in the reproduced tabulation from the author's 'Physics
without Einstein'. By analysing the atomic nucleus and the dependence of its
mass upon its size we can deduce the lattice dimensions of the structured vacuum
state and check a theory which has independently afforded an exact evaluation of
Planck's constant, as reference to the above-mentioned paper by Eagles will
show, and an exact evaluation of the Constant of Gravitation. For the latter
refer to the full text of 'Physics without Einstein' or a new work
'Gravitation' due to be published by the same author early in
1975.
The author is, of course, interested in any work which may advance
the ideas presented above and invites correspondence.
June 30, 1974 H.
Aspden
This concludes the text of that 1974 paper. It will be of
interest to some readers to examine how that theory has evolved from that
point, especially in respect of the development of the theory of the proton
from 1975 onwards.
Harold Aspden
27 May, 1998