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Abstract and Figures

We discuss the nuclear force hypothesis and propose a modified Yukawa potential for it. We also discuss a key issue with Yukawa-like proposals: the non-conservative nature of the force. We suggest two solutions for this issue: (1) a spatially asymmetric potential or, alternatively, (2) the introduction of dynamics: if there is a scalar potential, we should find a vector potential too. The latter requires the elaboration of an equivalent of Maxwell’s equations for nuclear force fields, which we believe is not justified. We believe the nuclear potential is effectively spatially asymmetric because it is nothing but an electromagnetic dipole field. The nuclear binding energy between the proton and neutron in a deuteron nucleus, for example, is of the order of 2.2 MeV, which can effectively be explained by the dipole field from the neutronic combination within the nucleus. The nuclear potential, therefore, appears as an electromagnetic dipole potential, combining an electric dipole and the magnetic fields of the neutral current from the motion of the positive and negative charges, and the charged current from the motion of the two positive charges. Such potential is, typically, spherically non-symmetric but conservative, and the order of magnitude of the presumed nuclear range parameter is the same as that of the distance which separates the charges (femtometer scale).
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Do we need the nuclear force hypothesis?
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
Email: jeanlouisvanbelle@outlook.com
Abstract
We discuss the nuclear force hypothesis and propose a modified Yukawa potential for it. We also discuss
a key issue with Yukawa-like proposals: the non-conservative nature of the force. We suggest two
solutions for this issue: (1) a spatially asymmetric potential or, alternatively, (2) the introduction of
dynamics: if there is a scalar potential, we should find a vector potential too. The latter requires the
elaboration of an equivalent of Maxwell’s equations for nuclear force fields, which we believe is not
justified. We believe the nuclear potential is effectively spatially asymmetric because it is nothing but an
electromagnetic dipole field. The nuclear binding energy between the proton and neutron in a deuteron
nucleus, for example, is of the order of 2.2 MeV, which can effectively be explained by the dipole field
from the neutronic combination within the nucleus. The nuclear potential, therefore, appears as an
electromagnetic dipole potential, combining an electric dipole and the magnetic fields of the neutral
current from the motion of the positive and negative charges, and the charged current from the motion
of the two positive charges. Such potential is, typically, spherically non-symmetric but conservative, and
the order of magnitude of the presumed nuclear range parameter is the same as that of the distance
which separates the charges (femtometer scale).
Contents
Introduction .................................................................................................................................................. 1
Binding energies ............................................................................................................................................ 3
The nuclear force and potential (1) .............................................................................................................. 4
The nuclear force and potential (2) .............................................................................................................. 6
The nuclear range parameter ....................................................................................................................... 7
A theoretical value for the proton and neutron radius ................................................................................ 9
The nuclear potential, energy conservation, gauges, and wave equations ............................................... 11
Do we need the nuclear force hypothesis? ................................................................................................ 13
Power expansions of energies and potentials ............................................................................................ 15
Power expansion of (potential and kinetic) energy ................................................................................ 15
Power expansion of the dipole potential ................................................................................................ 16
Tentative conclusion(s) ............................................................................................................................... 17
References .................................................................................................................................................. 19
1
Introduction
We will use the neutron model to focus our thoughts. The neutron appears as a stable particle in a
nucleus only. We, therefore, do not think of it as an elementary particle.
1
We consider the neutron to
combine positive and negative charge. Thinking of the neutron as a combination of a proton and a
(nuclear) electron its decay products might be helpful too, but charge is the more fundamental
concept when discussing forces and fields (a force acts on a charge, and mass is the inertia to a change
in the state of motion of a charged particle). Imagining the neutron as a dynamic system of a positive
and negative charge, allows us to think of a steady electrically neutral current.
2
Maxwell’s equations
3
then reduce to the equation(s) of magnetostatics:
E = /0 = 0
no (net) charge inside the volume and, hence, zero flux of E through the (closed) surface
E = −B/t = 0
the (electrically neutral) ring current generates a static magnetic field
B = 0
no magnetic charges (no flux of B)
c2B = j/0 + E/t = j/0
no electric field (E/t = 0/t = 0), steady (neutral) current (j), steady magnetic field
The only relevant equation is, therefore, the c2B = j/0 equation. We may add that we can write the
magnetic field as the curl of a vector potential(B = A) and that the magnetic field can be calculated
from the current by using the Biot-Savart law.
We think of the charges inside of the neutron as pointlike but not infinitesimally small particles which,
for the time being, we consider to be spinless. Of course, the neutron itself is not pointlike, and does
have spin: its charge radius is of the order of a femtometer (1015 m), and we think of it as being
determined by the orbital loop(s) of the positive and negative charges. The anomaly in the magnetic
1
We consider the non-zero magnetic moment of the neutron to tentatively confirm this hypothesis. From a more
philosophical perspective, we might add that we associate the idea of an elementary particle with something that
is stable and cannot be further reduced. The neutron is not stable outside of the nucleus (inside of the nucleus,
Schrödinger’s Platzwechsel model might apply and it may, therefore, also not be stable inside of the nucleus). We,
therefore, think of it as a composite particle.
2
The reader should have no difficulty appreciating the difference between a neutral current and a charged
current. A conducting electron in a conductor leaves a positive ion and creates a negative one as it moves through
the lattice: the current is, therefore, neutral (the reader may want to review Feynman’s analysis of the relativity of
electric and magnetic fields here). The positive and negative charge have the same role as the positive and
negative ion in a conductor. The difference between a neutral conductor and the neutron is the drift velocity
(velocity of displacement of the charge), which is of the order of 105 or even 106 m/s (micrometer) in a copper
wire. In contrast, in a mass-without-mass model of elementary particles, we will assume the velocity of the charge
to be equal to lightspeed (c). The current in an electron, for example, can then be calculated as:


This is a rather humongous value, but it is consistent with the observed magnetic moment of an electron, which
the reader can calculate as follows:





3
We should write Maxwell’s equations in integral rather than differential form, but we want to keep the notation
rather light here.
2
moment of an electron suggests charge might have a fractal structure, which is why we think the idea of
finite pointlike charges makes sense.
4
We should say a few words about deuteron models too, perhaps. Erwin Schrödinger originally
considered a Platzwechsel model for the deuteron nucleus (D = 2H+). In classical theory, the deuteron
nucleus (an 2H atom without the orbital electron) consists of a proton and a neutron. The Platzwechsel
model models the deuteron nucleus model as consisting of two protons sharing an electron, but it may
also be more profitable here to think in terms of charge(s) rather in terms of particles here.
5
The
negative charge would then act as a sort of glue holding the two protons together.
A complementary and, possibly, alternative point of view may be offered by considering the
electrically neutral combination of the positive and negative charge to act as an electric dipole
generating a 1/r2 potential which then traps the third (positive) charge.
An even more radical approach based on the dipole idea, is Di Sia’s conception of nucleons as
(electrically neutral) magnetic dipoles
6
, but protons (and, we believe, neutrons) do carry charge, and the
repulsive electrostatic force between them can, therefore, not be wished away. The model does stir
some thinking, however, because energy difference between the deuteron nucleus (about 1875.613
MeV) and its two constituents (neutron and proton) in their unbound state (939.565 MeV + 938.272
MeV = 1,877.837 MeV) is negative and equal to about 2.22 MeV
7
, and Di Sia does get energy values of
the same order of magnitude, which we will discuss below.
4
Such fractal structure may be finite or infinite. Because there is no experimental evidence that suggests an
infinite structure (although one might think of macroscopic superconducting ring currents as a generalization of
atomic electron orbitals), we prefer to think of it as finite. Because the electric constant appears in both the
electromagnetic as well as the nuclear force theory, we assume the dimension of the pointlike charge might be
given by the same fine-structure constant ( which relates the classical (Thomson), Compton and Bohr radii of the
electron. However, we admit that we may never know the true nature of charge: charge comes with a very
variable geometry, it seems. We also note the electron comes in a more massive but unstable variant: the muon-
electron. The proton does not. This suggests a fundamental asymmetry between the structure of the positive and
negative charge: matter and antimatter are each other’s opposite, but positive and negative charge are
fundamentally different (our ontology and physics paper further elaborates this). As a final informative remark, the
reader should note that, since the 2019 revision of SI units), the fine-structure constant is defined as = keqe2/ħc. It
is, therefore, not a fundamental but a derived constant in Nature.
5
We say this because of the scale issue: the classical electron radius (about 2.818 fm) is about 3.35 times larger
than the radius of a neutron. Hence, the pointlike charge which explains the anomaly in the magnetic moment of a
free electron does not fit into a neutron.
6
Paolo Di Sia, A solution to the 80-year-old problem of the nuclear force, 2018. We refer to Di Sia’s nucleons as
zero-charge nucleons because he considers neutral currents only: the electrostatic potential (and the electric field)
does not come into play.
7
Conversely, the energy of a neutron (939.565 MeV) is larger than the sum of energies of a free proton (938.272
MeV) and a free electron (0.511 MeV). Such positive binding energy (about 0.782 MeV) explains why the (free)
neutron (outside of the nucleus) is not stable: it goes into a lower energy state by decaying, which is why the 0.782
MeV is usually referred to as the decay energy. To be precise, the energy difference between a proton and a
neutron, which is of the order of about 1.3 MeV, which is about 2.5 times the energy of a free electron. Hence, the
energy of the electron would explain only about 40% of the mass difference: the rest (about 60%) is to be
explained by some kind of binding energy as well.
3
Binding energies
The numerical example which Di Sia (2018) provides is for nucleons with an approximate size of 0.5 fm
a rather reasonable ballpark number for the radius of the current loop which are separated by the
typical interproton distance (about 2 fm: this corresponds to the usual value for the range parameter in
Yukawa’s formula for the nuclear potential
8
).
Interestingly, Di Sia also considers the phase of the currents, which may effectively be in or out of phase,
and then calculates energy levels for the magnetic binding using the Biot-Savart law
9
, which we can
immediately compare with nuclear binding energies. For the mentioned values (0.5 and 2 fm) he gets an
energy range between 3.97 KeV and a more respectable 0.127 MeV (the latter value assumes in-phase
currents). While this is, without any doubt, significant, it is only 5% of the 2.2 MeV energy difference
between the deuteron nucleus (about 1875.613 MeV) and its two constituents (neutron and proton) in
their unbound state (939.565 MeV + 938.272 MeV = 1,877.837 MeV). The values get (much) better
when changing the parameters (nucleon size and internucleon distance) significantly (2-3 MeV) and,
better still, considering paired nucleons creating dipoles acting on other paired nucleons (values up to 5
MeV). The latter point is interesting because, as mentioned above, we may effectively think of a neutron
as a paired positive and negative charge: in fact, the neutron is the only nucleon which matches Di Sia’s
concept of electrically neutral nucleons.
As we are presenting some energy values here, we may note that the energy of a neutron (939.565
MeV) is larger than the sum of energies of a free proton (938.272 MeV) and a free electron (0.511 MeV).
Such positive binding energy (about 0.782 MeV) explains why the (free) neutron (outside of the nucleus)
is not stable: it goes into a lower energy state by decaying, which is why the 0.782 MeV value is usually
referred to as the decay energy.
10
This 0.782 MeV binding (or decay) energy and the 0.511 MeV energy
of the (free) electron add up to the 1.293 MeV energy difference, and so that should be it, right? So did
Di Sia solve all problems? Do we have a full-blown model of the nucleus based on electromagnetic
theory only here?
Maybe. Maybe not. We request the reader to bear with us and just kindly note the nuclear binding
energies are of the same order of magnitude of the electron energy and effectively correspond to dipole
field energies at nuclear distance scales, which is why, as Di Sia pointed out in his rather provocative
paper, a dipole model makes a lot of intuitive sense. We also ask our reader to note that an added
advantage of the dipole model is that it reduces the three-body problem that is inherent to modeling
the deuteron nucleus as a combination of three charges.
Unlike Di Sia, however, we would rather think in terms of a combined electromagnetic dipole model,
rather than reducing all to electric or magnetic dipoles, but that might sound like a minor correction to
the reader at this point.
The more important question to consider for the reader is this: if the neutron consists of a positive and a
negative charge, why does the negative charge (electron) not go sit right on top of the positive charge
(proton)? Consider this: perhaps it does, but binding energy is binding energy, and the Planck-Einstein
relation (E = hf) tells us the charges might well go and sit right on top of each other so we would have
8
See, for example, Aitchison and Hey, Gauge Theories in Particle Physics, 2013 (4th edition), Vol. 1, p. 16.
9
For a discussion of Di Sia’s model, see our paper on an electromagnetic deuteron model.
10
See, for example, the Wikipedia article on free neutron decay.
4
some kind of n = 0 orbital, which is not an orbital at all
11
but that we should still have local motion: a
truly local Zitterbewegung
12
, and the order of magnitude of the frequency is that of the electron (fe =
ωe/2π = E/h ≈ 0.123×1021 Hz)but on a femto- rather than a picometer scale.
The femtometer scale needs to be explained, however, and so we do not think Di Sia solved all
problems. We will, in this paper, effectively want to think of nuclear electron orbitalsif only because
the assumption of a nuclear force explains the proton (and neutron) radius and, therefore, their rather
humongous masses (which suggest humongous forces too!), exceedingly well, as we will show in one of
the next sections of this paper.
13
Hence, we are reluctant to drop the idea of a nuclear oscillation
altogether, but we readily admit that we should probably revisit Schrödinger’s Platzwechsel model and
think of some linear oscillationsomething like the maser
14
, perhaps, but with a different scale
parameter (a nuclear range parameter, that is).
The nuclear force and potential (1)
The assumption of a combined nuclear and electromagnetic should provide an answer to two basic
questions:
1. What keeps the positive and negative charge inside of a neutron?
2. What keeps the positive charges together inside of nuclei?
The nuclear force hypothesis is a logical answer because the nuclear force (i) keeps positive charges
together (the corollary, of course, is that it keeps opposite charges apart
15
), and (ii) is much stronger
than the electromagnetic force at short range only (the latter conveniently explains why we only see the
electromagnetic force at work at larger distances).
However, while the introduction of a nuclear field with an a/r2 potential seems to be easy enough, it
raises another rather serious problem: the associated force follows an inverse-cube law in spaceas
opposed to the usual inverse-square law. This violates the energy conservation principle. Of course, one
might immediately counter that the force in a dipole field follows an inverse-cube law as well but, in this
11
This is a reference to the principal quantum number, which gives us a gross energy structure. The reader should
note we have not brought spin angular momentum into the analysis yet: we, therefore, are still analyzing all
angular momentum as orbital angular momentum, which comes in two states: up or down (fine structure within
the gross structure). Also, if the two charges would be pointlike, spin coupling might give rise to a hyperfine
structure.
12
The term was coined by Erwin Schrödinger and, apparently, describes the most trivial solution to Dirac’s wave
equation. Zitter is German for trembling or shaking. Think of the English word ‘jittery’. Dirac made a prominent
reference to it in his Nobel Prize Lecture (1933), but we think he did not quite understand its significance because
he stuck to the idea of linear rather than orbital motion, just like Louis de Broglie. See our paper on de Broglie’s
matter-wave idea.
13
The model(s) we use combines Wheeler’s mass-without-mass idea with what is usually referred to as a ring
current, Zitterbewegung, magneton or toroidal ring model (Parson, 1905; Breit, 1928, Schrödinger, 1930; Dirac,
1933; Hestenes, 2008, 2019).
14
See our presentation of the maser in our discussion of Feynman’s view of two-state systems, in which a nitrogen
nucleus constant swaps places in the NH3 (ammonia) molecule, and whose motion is also based on dipole
moments.
15
While this might solve problems in the deuteron model (two positive and one negative charge), it may raise
questions in regard to the neutron model (one positive and one negative charge). However, we think these
questions can be answered.
5
case (dipoles), energy is conserved because the field is not spherically symmetric, and the asymmetry is
such that energy is conserved. This can easily be understood from the equation for the electric field E,
which can be written in terms of a component along the dipole axis (Ez) and a transverse component
(E)
16
: 
Of course, one might suggest fixing the problem of our non-conservative nuclear potential by adding a
unit vector n and assuming the nuclear range parameter is a vector too, whose direction is fixed in
space. We could then write something like this
17
:

The vector dot product na = nacosθ = acosθ (the cosθ factor should be positive so n must be suitable
defined so as to ensure /2 < θ < −/2
18
) introduces a spatial asymmetry (think of an oblate spheroid
instead of a sphere here), which should then ensure energy is conserved in the absence of an inverse-
square law. Alternatively, we could use a vector cross-product na = nnasinθ = nansinθ, but this
trick amounts to the same. At first, such ad hoc solution might not appeal: how can one possibly fix the
direction of the a vector? The answer here is rather straightforward: a would be directed along the axis
connecting the two charges. Are there any other solutions? Of course. We see at least two.
1. One idea might be to try to think of some new curvature of spacesomething along the lines of
how general relativity models gravitation: not as a force, but as a geometric feature of space.
However, this suggestion is totally unappealing because not only would this require the definition
of an entirely new spacetime metric
19
, but it also triggers a very obvious question: why would this
curved space not apply to the electromagnetic force?
16
See, for example, Feynman’s Lectures, II-6-2.
17
We add a vector arrow to the usual notation for vectors (boldface) in the formula to emphasize that its
direction, unlike that of F, n, and r, is fixed in space.
18
Defining a such that it broadly points in the same direction of the line along which we want to measure the force
F should take care of this. Of course, a simple sine or cosine factor does not necessarily ensure energy
conservation. Perhaps we should introduce a cosθ or cos2θ factor. The point is this: we need to integrate over a
volume and ensure that the nuclear potential respects the energy conservation law. We will come back to this at
the end of our paper, and argue the dipole model may, effectively, provide the correct equations.
19
We may refer to electron models using Dirac-Kerr-Newman geometries or, more generally, integrating gravity as
a force (e.g. Burinskii, 2021). However, we think gravity is not relevant to the picture here. We can compare force
magnitudes by defining a standard parameter. In practice, this means using the same mass and charge! in the
equations (we take the electron in the equation below) and, when considering the nuclear force, equating r to a:




Hence, the force of gravity if considered a force is about 1042 weaker than the (electrostatic) Coulomb force. To
demonstrate gravitation is not very relevant, one can also calculate the radius of a black hole with the proton
energy using the Schwarzschild formula:



This clearly shows that, despite the huge energies and forces on these small scales (pico- or femtometer), we
should not be worried that we are modeling black holes here. Now that we are here, we should note that the
6
2. We might think of the nuclear potential as a dynamically changing potential and, therefore,
associate a vector potential with the scalar potential, which might ensure (field) energy
conservation.
We will come back to the latter possibility. Let us first present the proposed nuclear force and potential
without worrying too much about field energy conservation for the time being.
The nuclear force and potential (2)
In previous papers, we introduced the concepts of orbital energies, both electromagnetic and nuclear.
These were associated with the Coulomb and nuclear potential, respectively. To be precise, we imagined
the neutron as a positive and a negative charge in an oscillation which combines both. Assuming
spherical potentials (and, therefore, substituting r for r in the formulas), we can write these potentials
and the associated forces (the negative derivative of the potential) as follows:










To help the reader correctly interpret these equations, the following notes may be made:
The qe2 product combines the positive and negative charge and we should, therefore, talk of
potential energy rather than potential. The concept of a potential effectively assumes a single
charge causing the potential, while potential energy is the energy between charges and,
therefore, assumes the presence of (at least) two charges.
20
A potential depends on position and may not be static. We should, therefore, write the
electromagnetic scalar potential as φ(r, t), while the vector potential will be denoted as A(r, t) =
(Ax(r, t), Ay(r, t), Az(r, t)). However, for the time being, we will only consider a static potential
(φ/t = 0).
The physical proportionality constant ke = qe2/40 is the same for both the electromagnetic and
nuclear potential, but the nuclear potential has an added range parameter (a), which defines
two ranges: the range where the (electromagnetic) attraction between the positive and negative
nuclear potential that we will be introducing produces a ratio of standard parameters equal to one. Indeed,
equating r to a and importantly using the same mass factor, we get this:



Hence, we do get a rather humongous value for nuclear force and field strengths, but these come from the
introduction of the nuclear range parameter only: there is no separate nuclear constant (ke = kN).
20
The reader will appreciate the distinction when we present the derivation of the dipole potential in the last
section of this paper.
7
charge is larger than the (nuclear) repulsive force, and the range where electromagnetism loses
out, as shown below (Figure 1).
The reasoning here is not as straightforward as it may seem at first. Yukawa’s nuclear potential assumed
two like charges will attract each other, while our neutron model (n = p + e) assumes the opposite.
However, we kindly request the reader to forget Yukawa’s potential functionif only because the 1/r
potential in Yukawa’s function results in an inconsistency: the physical dimensions do not work out,
which is why we replaced it by an a/r2 dependence.
Figure 1: Coulomb and nuclear forces for a = 2re and k = 1 (fm units)
Note that, for the convenience of the reader, we show not only the sum of forces, but also the sum of
potentials in Figure 1. The model, then, gives us a potential energy well, which is nice because it
suggests equilibrium: we want the neutron to be stable, of course! As for the value of a, we used a lower
range based on a calculation in a previous paper. We found this lower range to be equal to twice the
classical electron radius (about 2.818 fm), but we now think this calculation was erroneous (we think we
made a sign mistake with the potentials).
In any case, the reader should note that we equated the electric constant ke = qe2/40 to one, so we
should not add much importance to the graph and values above: the idea is just to illustrate that the
model effectively generates a potential well, and that does not depend on the value of the range
parameter a.
The nuclear range parameter
We can now do a few calculations. We may, for example, apply the usual rule for finding the maximum
of the combined FN + FC function:
 








8
This value reminds us of the approximate neutron radius (0.82 to 0.84 fm) but that is, of course, a mere
coincidence: we can, effectively, not give much meaning to the sum of forces here because we equated
the Coulomb constant to one in the graph above. To get energies and meaningful values, we must
integrate forces over a distance and use the correct numerical value for the Coulomb constant. These
energies are, of course, the potential energies as well as the kinetic energy of the two pointlike charges.
For the nuclear force, we write:

The mass factor mN is the equivalent mass of the energy in the (nuclear) oscillation, which is the sum of
the kinetic energy and the (nuclear) potential energy between the two charges. The velocity v is the
velocity of the two charges (qe+ and qe) as measured in the center-of-mass (barycenter) reference frame
and may be written as a vector v = v(r) = v(x, y, z) = v(r, , ), using either Cartesian or spherical
coordinates. Note that we take the potential energy to be negative here. This is rather tricky: the sign of
the potential energy depends on the U = 0 reference point, which we can choose at either r = 0 or r = ,
and we will choose it at r = 0 here, so the nuclear potential energy is − at r = 0 and 0 at r = +. This is
not the usual reference point for electromagnetic energy, and so we must swap the sign for UC as well.
We write:
Of course, the charges qe+ and qe are the same in each equation, and the Coulomb and nuclear energies
have to add up, and then the whole equation has to respect the mass-energy equivalence relation: E =
EN + EC = (mN + mC)c2 = mc2. To be precise, this mass-energy equivalence relation defines
electromagnetic and nuclear mass, respectively, and we may assume that the energy equipartition
theorem applies: half of the total energy is electromagnetic, and half is nuclear and, therefore, mN = mC
= E/2. We must, therefore, add the UN and UC terms. The kinetic energy is the kinetic energy, of
coursebut now we must wonder: should we add the v2/2 terms too? We have two charges: should
we, therefore, have twice the kinetic energy?
We suggest the relevant addition is, effectively, equal to (mN + mC)v2/2. Furthermore, the ring current
model that we have been using assumes the pointlike charge has no rest mass and, therefore, whizzes
around at lightspeed: we, therefore, equate v to c to obtain the following sum of equations:




What can we say about this statement? What does it mean, really? Nothing much: the laws of physics
give us a radius for the oscillation which must be equal to the nuclear range parameter a. So we have a
truism here: a self-obvious summary of the laws (plain electromagnetic theory) that we think should
apply to elementary particles as well. We must obtain a empirically: in this case, it must be the effective
9
charge radius of the neutron which, as mentioned above, is in the same range as the proton radius, for
which the CODATA value is about 0.841015 m.
21
Let us redraw the illustration above and equate both a and ke to 1 to show the result of two other
evident calculations: (1) the sum of forces reaches a minimum at r = 3a/2 = 1.5a (we calculated that
above already), and (2) the sum of potentials reaches a minimum at r = 2a, so that is the distance at
which the potential well bottoms out.
Figure 2: Coulomb and nuclear forces for a = k = 1
So far, so good. Let us take a step back and explore some other, related, line of reasoning.
A theoretical value for the proton and neutron radius
We mentioned the empirical value for the proton rms charge radius. Can we present a theoretical value?
Sure. We think of the proton as a nuclear oscillation of a pointlike positive charge in a modified ring
current model. However, instead of assuming a 2D oscillation, we imagine the nuclear oscillation to be
driven by two (perpendicular) forces rather than just one, with the frequency of each of the oscillators
being equal to ω = E/2ħ = mc2/2ħ. Each of the two perpendicular oscillations would, therefore, pack
one half-unit of ħ only.
This ω = E/2ħ formula also incorporates the energy equipartition theorem, according to which each of
the two oscillations should pack half of the total energy of the nuclear particle (so that is the proton, in
this case). This spherical view of a proton fits nicely with packing models for nucleons and yields the
experimentally measured radius of a proton:
21
The point estimate is 0.8414 fm, with an uncertainty of 0.0019 fm. Hence, the 2 rules gives a lower value of
0.8376 fm and an upper value of 0.8452 fm. The CODATA value for the proton radius (0.8414 0.0019 fm) takes all
past measurements into account but gives very high weightage to the measurements of Pohl (2010) and Antognini
(2013), which are both based on muonic-hydrogen spectroscopy. In contrast, the PRad experiment which is based
on a proton-electron scattering quite a different technique established the following new value for the proton
radius: rp = 0.831 ± 0.007stat ± 0.012syst fm. Prof. Dr. Randolf Pohl is of the opinion that the PRad measurement
and the muonic-hydrogen spectroscopy measurements are basically in agreement. He replied this to an email on
this: “There is no difference between the values. You have to take uncertainties seriously (sometimes we spend
much more time on determining the uncertainty.” (email from Prof. Dr. Pohl to the author dated 6 Feb 2020)
10
 

You can see that the 4 factor is the same factor 4 as the one appearing in the formula for the surface
area of a sphere (A = 4πr2), as opposed to that for the surface of a disc (A = πr2). This leads us to
represent a proton by a combination of two wavefunctionssomething like this:




But what about the neutron? The proton and neutron mass do not differ much: 939.565 MeV 938.272
MeV 1.3 MeV, so their radii should be more or less the same, and they are.
22
The reader should note
that the mass of an electron is 0.511 MeV/c2, so that is only about 40% of the energy difference, but the
kinetic and binding energy could make up for the remainder. If anything, the effective charge radius of a
neutron might be slightly smaller than that of a proton because these ring current models yield an
inverse proportionality between the energy and the radius of (elementary) particles. To be precise, a 2D
ring current model yields the following:





Hence, what is also referred to as the Zitterbewegung radius (Schrödinger, 1930; Dirac, 1933; Hestenes,
2008, 2019
23
) is nothing but the Compton radius of a particle.
24
Finally, we should note the calculations
above are consistent with the experimentally measured values for the magnetic moment of the proton
and neutron.
25
Having developed the rationale for thinking of a neutron as a proton and a nuclear (or deep) electron,
22
CODATA does not list a rms charge radius. We think this is because standard theory considers the neutron to not
carry any charge, while our model considers it to be a composite particle. It should be noted that the neutron is
not stable outside of a nucleus, which we take to confirm Schrödinger’s Platzwechsel model: it is the nuclear
electron (or, to be precise, the negative charge) which acts as the gluon, so to speak, between protons in the
nucleus. The reader should also note that the mass of a proton and an electron add up to less than the mass of a
neutron, which is why it is only logical that a neutron should decay into a proton and an electron. Binding energies
think of Feynman’s calculations of the radius of the hydrogen atom, for example, are effectively measured as
negative energy.
23
The Zitterbewegung or ring current model actually goes back much further in time. It was established as soon as
it became clear that an electron had a magnetic moment (Parson, 1905; Breit, 1928). It is also a direct application
of Wheeler’s suggested mass-without-mass model of elementary particles, which he pushed as an alternative to
mainstream theory in the 1960s.
24
The reader will be more familiar with the Compton wavelength but, paraphrasing Prof. Dr Patrick LeClair, we
understand the related radius (a = /2) to be the actual “scale above which the electron can be localized in a
particle-like sense. Such interpretation clarifies what Dirac, in his Nobel Prize Lecture (1933), referred to as the
law of (elastic or inelastic) scattering of light by an electron”Compton’s law, in other words. Finally, we should
add mainly for the informed reader that the CODATA value for the Compton wavelength incorporates the same
4 factor which we associated with the spherical model of protons and neutrons. We are not sure about the
CODATA methodology here.
25
See our paper on the mass-without-mass model of protons and neutrons.
11
we must note the n = p + e model tells us that the proton accounts for most of the mass and, therefore,
for most of the energy of the neutron, which seems to contradict the intuition that the nuclear and
electromagnetic mass of the neutron must each account for half (1/2) of the total mass (energy
equipartition theorem). This contradiction may be resolved by considering the scales (femtometer
versus picometer
26
) and, more generally, a model which thinks of the neutron as a combination of two
electric charges rather than as a combination of a proton and an electron. Such approach would also
allow to think of excited energy states and more exotic versions of the neutron.
27
However, let us leave this question aside for the time being, and let us return to the key issue we raised
in our introduction: how do we reconcile an a/r2 or 1/r2 potential with the energy conservation
principle?
The nuclear potential, energy conservation, gauges, and wave equations
We hoped we have been able to demonstrate that the introduction of a nuclear field with an a/r2
potential might solve our neutron or deuteron modeling problem, but we also were quite clear we are
now stuck with an issue which is at least as problematic as the problems we tried to solve: a force that is
associated with a 1/r2 potential suggests an inverse-cube law in spaceas opposed to the usual inverse-
square law. This violates the energy conservation principle. The best way to solve this issue, is to suggest
the nuclear force is not spherically symmetric. We may model this by adding a unit vector n and
assuming the nuclear range parameter is a vector too, whose direction is fixed in space. We wrote
28
:

However, one might also imagine something else. The proposed nuclear potential formula only models a
scalar potential, and it is static: it does not vary with time. This can, of course, not be true: the charges
move constantly, and at lightspeed. The potential must, therefore, vary with time, and we must,
therefore, also have a vector potential.
For electromagnetic oscillations, this corresponds to the distinction between the electric and magnetic
force, respectively, and to the distinction between the scalar and vector potential. Indeed, assuming the
scalar potential varies with time, one can derive the vector potential A from the Lorenz gauge condition:


26
The Compton radius of an electron is about 0.3861012 m (pm), i.e. 386 fm, or about 460 times the proton
radius. We must note another apparent contradiction here. The small anomaly in the magnetic moment of an
electron suggests the pointlike charge is not infinitesimally small: the anomaly may be explained by assuming it has
a radius itself, and the anomaly suggests this radius equals the classical electron radius (Thomson radius), which is
equal to re = rC 2.818 fm. This is the femtometer scale alright, but it is still much larger than the neutron radius.
Electric charge in Nature seems to have a very variable geometry.
27
We might refer to the ongoing research on deep electron orbitals (Meulenberg and Paillet, 2020), more exotic
versions of the hydrogen atom (see, for example, Jerry Va’Vra, 2019) or, more in general, to all of the research on
low-energy nuclear and/or anomalous heat reactions (aka cold fusion).
28
See footnote 17: we add a vector arrow to the usual notation for vectors (boldface) in the formula to emphasize
that its direction, unlike that of F, n, and r, is fixed in space.
12
For a time-independent scalar potential, which is what we have been modeling so far, the Lorentz gauge
is zero (·A = 0) because the time derivative is zero: φ/t = 0 ·A = 0.
29
The magnetic field,
therefore, vanishes. However, as mentioned above, it is pretty obvious that the time derivative cannot
be zero. The question is thus this: can we use the very same Lorenz gauge for the nuclear force fields?
We should, effectively, be able to define an equivalent scalar and vector potential for the nuclear force
too!
Can we use the Lorenz gauge? We are not sure, but we think it should be possible. The Lorenz gauge
incorporates the usual theorems from vector differential and integral calculus (Gauss and Stokes) as well
as special relativity. It, therefore, also incorporates the principles of energy (and, most probably,
momentum conservation).
30
Hence, if the nuclear force is a conservative force which it should be (no
concepts of entropy or friction here!) then we should use the Lorenz gauge to relate the nuclear scalar
potential to the nuclear vector potential, and then we can use the superposition principle again to add
the electromagnetic and nuclear scalar and vector potential to get Dirac’s “equations of motion” for
everything !
What do we need for that? We need something like the equivalent of Maxwell’s equations for the
nuclear fields and that, we do not have. We, therefore, have no ideas on how to calculate the time
derivative of the nuclear scalar potential. Will this question ever be solved? We hope so: readers who
are well-versed in math and vector calculus should try their hand at that! There is a lot of useful material
out there. Just google, for example, for papers on non-paraxial fields. This site (Alonso Research Group,
University of Rochester), for example, offers a fine point of departure! 
A more direct approach might be to substitute the 1/r12 factor in the scalar and vector potential integrals
below
31
by the a/r122 factor but again we should probably use some vector product instead to ensure
field energy conservation:
So as to motivate the reader to go into this direction, we note that, if we could define the nuclear scalar
and vector potentials, we could probably use the combined electromagnetic and nuclear scalar and
29
The Lorenz gauge does not refer to the Dutch physicist H.A. Lorentz but to the Danish physicist Ludvig Valentin
Lorenz.
30
We say so because the derivation involves the consistent use of relativistic four-vectors. See Feynman, I-14
(work and potential energy), II-2 (vector differential calculus), II-3 (vector integral calculus), II-25 (electrodynamics
in relativistic notation), III-26 (Lorentz transformation of the fields) and III-27 (field energy and momentum).
31
We took these from the excellent overview table of electromagnetic theory in Feynman’s Lectures (table 15-1 in
Chapter 15).
13
vector potentials in a more general wave equation, such as the one which Feynman suggests in his last
Lecture on Physics (III-21)
32
:



If this equation would be, effectively, the most general form of a wave equation, then substituting the φ
and A scalar and vector potential, respectively, by the combined electromagnetic and nuclear scalar and
vector potentials should give us the solutions we are all seeking for.
33
However, something inside of us tells us the use of one rotation operator only (i) might not do the trick:
as we show in our previous paper in our previous introductory paper on the nuclear force), we suspect
quaternion algebra may be necessary to take the 3D geometry of the nuclear force into account.
Do we need the nuclear force hypothesis?
Let us briefly review Feynman’s derivation of the (electrostatic) dipole potential. The point of departure
is the (electrostatic) Coulomb potential of zero-spin (electric) charges
34
and the superposition principle
(which drives our concern on the need to ensure any model of a nuclear force if it exists respects the
(field) energy conservation principle). With multiple (zero-spin) charges, the potential at some point x =
(x, y, z)
35
will be equal to:


With two (opposite) charges only (remember we are trying to develop an n = qe+ + qe model here),
separated by the distance d along the chosen z-axis, this gives us the electrostatic dipole potential:


 
 
We now will let Feynman speak. We can expand the term(s) in the denominator above in the (small)
distance d, and simplify by keeping the first-order terms only:
32
We wrote Feynman’s ħ/i factor as iħ, moved minus signs out of the brackets and combined them into one square
factor. We hope we did not make any mistake. Note that Feynman refers to this wave equation as the wave equation
for an electron in any electromagnetic field. After re-reading Feynman’s Lectures several times, we sometimes get
the feeling that Feynman might have a secret drawer with the answers to all questions: he just did not want to tell
us all! 
33
In another paper on the nuclear potential, we suggested a wave equation using nuclear (kinetic and) potential
energy directly (instead of scalar and vector potentials) directly. However, the formally correct approach is really to
think of how the time derivative of the nuclear (static) potential (modified Yukawa potential combined with
Schrödinger’s Platzwechsel idea for modeling a nucleus) could possibly look like.
34
Apart from not considering the incongruency of the physical dimensions of his equations, Yukawa also
introduced a new nucleon (or nuclear) charge (gN), which opened the flood gates to ontologizing mathematical
niceties such as strangeness and, ultimately, led to the even stranger concepts of quarks and gluons. We think of
the Higgs particle as the latest addition (we hope it is the last) to what we refer to as smoking gun physics. We
made our viewpoint on the ontological status of these mathematical concepts clear in our paper on the
Zitterbewegung hypothesis and the S-matrix.
35
Switching to spherical coordinates will usually be more convenient.
14
Putting all in vector form (defining p as a vector with magnitude p and a direction along the axis of the
dipolefrom qe to + qe), then yields the magnitude of the electric dipole field:

So what does this show? We are not sure, but we offer the following reflections:
The magnitude of the electric (electrostatic) and magnetic (dynamic) fields (the E and B in
Lorentz’s F = q(E + vB) = q(E + viE/c) force law) is equal for v = c, and Lorentz’ force law,
therefore, can then be written as F = q(E + ciE/c) = q·(E + 1×iE) = q·(E + j·E) = (1+ j)·q·E. The
magnetic force may be viewed as a relativistic correction to the electric force, but it becomes an
equal component of the electromagnetic force.
Dipole moments comes with 1/r2, 1/r3 and higher-order terms, with complicated directional
factors in the numerator, when expanding the basic electromagnetic equation(s) for the dipole
field(s).
At short (i.e. nuclear) range, dipole potentials explain the 0.782 and 2.224 MeV binding energy
between the charges in a neutron and a deuteron nucleus, respectively.
None of our calculations so far considered the spin angular momentum of chargesi.e. the spin
of the charges themselves, as opposed to the orbital angular momentum of charge orbitals.
We think all of the above, somehow, justifies an intuition that there may be no need to invoke a nuclear
force or potential. Correctly modelling the energy in Schrödinger’s Platzwechsel model along the lines
of any two-state quantum-mechanical system that is based on the concept of (opposite) spin angular
momentum and/or opposite dipole moments might give us a reasonable explanation of the stability,
size, and other intrinsic properties (most importantly, the residual or resultant magnetic moment) of
both the neutron as well as the deuteron nucleus.
15
We are far from proving this, however, but we hope our reader(s) will be able to do so, one day. To
encourage the discussion, we provide some mathematical remarks in the last and final section of this
paper.
Power expansions of energies and potentials
As mentioned above, we are not convinced that there is a need to invent a new potential. If such need
would be there, then it should probably be based on the idea of spin applied to the pointlike charges
themselveswhich is an entirely different concept than the spin of the (elementary or composite)
particles themselves, which we think of orbital angular momentum. When analyzing the math which
would be required for this, the following rather peculiar power expansions probably merit some
attention.
Power expansion of (potential and kinetic) energy
Orbital energy is kinetic and potential, but the total energy must obey Einstein’s mass-energy
equivalence relation. Using the binomial theorem, we can rewrite this E = mc2 relation as follows:
36
:


The relativistically correct formula for kinetic energy defines kinetic energy as the difference between
the total energy and the potential energy: KE = E PE. The potential energy must, therefore, be given by
the m0c2 term. This term is zero for r = 0 but non-zero because of the potential energy in the radial field
at distances r 0. The total energy of a charge in a (static) Coulomb field is given by
37
:

The potential itself is equal to V(r) = U(r)/qe:


36
The total energy is given by E = mc2 = m0c2 which can be expanded into a power series using the binomial
theorem (Feynman’s Lectures, I-15-8 and I-15-9 (relativistic dynamics). He does so by first expanding m0:

This is multiplied with c2 again to obtain the series in the text.
37
U(r) = V(r)·qe = V(r)·qe = (ke·qe/r)·qe = ke·qe2/r with ke 9109 N·m2/C2. Potential energy (U) is, therefore,
expressed in joule (1 J = 1 N·m), while potential (V) is expressed in joule/Coulomb (J/C). Since the 2019 revision of
the SI units, the electric, magnetic, and fine-structure constants have been co-defined as ε0 = 1/μ0c2 = qe2/2αhc.
The CODATA/NIST value for the standard error on the value ε0, μ0, and α is currently set at 1.51010 F/m, 1.51010
H/m, and 1.51010 (no physical dimension here), respectively.
16
We could define the kg (mass) in terms of newton (force) and acceleration (m/s2). Can we do the same
for the coulomb? Rewriting the energy equation as a function of the relative velocity and the radial
distance r does the trick:


We may say this defines the mass of the pointlike charge as electromagnetic mass only, which now
consists of a kinetic and potential piece. The energy in the oscillation, therefore, defines the total mass
m = E/c2 of the neutron electron (n = p + en). The kinetic energy is thus given by
38
:





The first term in the series gives us the non-relativistic kinetic energy

. In line with the usual
convention for measuring potential energy, we will now set the reference point for potential energy at
zero at infinity, and the potential energy will, therefore, be defined as negative, going from 0 for r
to − for r 0. This makes for a negative total energy which is in line with the concept of a negative
ionization energy for an electron in an atomic orbital which, for a one-proton atom (hydrogen), is given
by the Rydberg formula.
Is this power series relevant to the discussion at hand? Should we distinguish a nuclear from the usual
electromagnetic potential? Again, we do not know: all that we want to do here is to trigger the
imagination of the reader and encourage him to think this through for him- or herself.
Power expansion of the dipole potential
Another interesting power series may be obtained, perhaps, from substituting the electromagnetic
potential which gives rise to the electric dipole field by the proposed nuclear potential, and see how it
changes the derivation that follows from it. We write:


With two (opposite) charges only, this becomes:


 
 
We have the same [z (d/2)]2 term in the denominator but no square root function anymore, which
should obviously affect the subsequent derivation. We leave it to the reader to work it all out: if he
or she would obtain something interesting, we would sure hope to get feedback! 
38
In line with the usual convention for measuring potential energy, we will set the reference point for potential
energy at zero at infinity.
17
Tentative conclusion(s)
If reality is that what is the case, what do we believe might be the case?
39
We think it might be this:
1. The (planar) ring current model of an electron, and the (spherical) oscillator of a proton, suggest
(electric) charge comes in two fundamental oscillations. The intriguing thing here is the energy level
of these two elementary particles: we can imagine a lower- or higher-energy electron, or a lower- or
higher-energy proton (and apply the Planck-Einstein and mass-energy relations to them to obtain
their radius and mass), but the energy of an electron is the energy of an electron, and the energy of a
proton is that of a proton. This indicates the negative and positive charge are not just each others
opposite.
40
2. We do not need electromagnetic theory to explain the radius (or mass) of these two elementary
particles. The Zitterbewegung hypothesis, applied to the idea of pointlike charges (with zero rest
mass), combined with the Planck-Einstein and energy-mass equivalence relations, will do. The
negative charge is associated with a 2D planar oscillation, while the proton is associated with a 3D
(spherical association). To refer to the first as an electromagnetic and the second as a nuclear
oscillation may be slightly misleading. We do so because we believe these two different matter-
waves are associated with two different types of lightlike particles: the photon and the neutrino, and
we only observe the neutrino from nuclear reactions, which explains why the term nuclear
oscillation is a convenient shorthand.
41
We also do not need electromagnetic theory to explain
unstable composite particles and particles reactions: the combination of the zbw hypothesis and
standard matrix algebra will do.
42
3. However, we do need electromagnetic theory to explain the magnetic moment of the electron,
proton, neutron, and composite (stable or unstable) particles: we must assume the Zitterbewegung is
regular (or regular enough), and that the Zitterbewegung amounts to a ring current which generates
a magnetic field which keeps the charge in motion.
43
39
This is an obvious reference to Wittgenstein’s first statement in his Tractatus Logico-Philosophicus: Die Welt ist
alles, was der Fall ist.We admire Wittgenstein for his search of a formal language which would encompass each
and everything and would be entirely logical/unambiguous, but we are surprised Wittgenstein was apparently
unaware of the scientific revolution (especially Einstein’s relativity theory) that was taking place. The later
Wittgenstein also acknowledged non-ambiguity in language may be unachievable or, worse, that it is actually the
key to sense-making and understanding.
40
For example, the electron comes in a more massive but unstable variant: the muon-electron. The proton does
not. Unstable particles can be modeled by combining the elementary wavefunction (complex-valued exponential)
and a decay factor (real-valued exponential decay function).
41
Such nuclear reactions may be low- or high-energy (decay versus high-energy particle collisions).
42
See our paper on the Zitterbewegung hypothesis and the scattering matrix.
43
The magnetic field is quantized too. The analogy with a superconducting ring and perpetual currents comes to
mind. There is no heat: no thermal motion of electrons, nuclei or atoms or molecules as a whole and, therefore, no
(heat) radiation. Also, the perpetual currents in a superconductor behave just like electrons in some electron
orbital in an atom: they do not radiate their energy out. That is why superconductivity is said to be a quantum-
mechanical phenomenon which we can effectively observe at the macroscopic level. Hence, we have a magnetic
field but no radiation and, since 1961 (the experiments by Deaver and Fairbank in the US and, independently, by
Doll and Nabauer in Germany), we know this field is, indeed, quantized. To be precise, the product of the charge
(q) and the magnetic flux (Φ), which is the product of the magnetic field B and the area of the loop S, will always
be an integer (n) times h: q·Φ = q·BS = n·h. However, superconducting rings are made of superconductors. The
18
4. A neutron is only stable inside of a nucleus. The smallest nucleus is the deuteron nucleus, which
combines two positive charges and one negative charge. The nuclear binding energy is of the order of
2.2 MeV, which can be explained by the dipole field from the neutronic combination within the
nucleus. The nuclear potential, therefore, appears as an electromagnetic dipole potential, combining
an electric dipole and the magnetic fields of the neutral current from the motion of the positive and
negative charges, and the charged current from the motion of the two positive charges. Such
potential is, typically, spherically non-symmetric but conservative, and the order of magnitude of the
presumed nuclear range parameter is the same as that of the distance which separates the charges
(femtometer scale).
In short, we believe there is no such thing as a nuclear charge (no gN, only qe), nor is there a nuclear
equivalent of the electric constant.
44
There is, therefore, no real nuclear force, but we do think of two
different fundamental oscillations in spacetime, and combinations thereof. Such combinations,
however, can all be explained by standard electromagnetic theory, including electromagnetic dipole
theory. We, therefore, tentatively agree with Di Sias conclusion: the 80-year-old problem of the nuclear
force has been resolved.
We started by quoting the first statement of Wittgensteins Tractatus Logico-Philosophicus, and find it
appropriate to conclude with his last: Wovon man nicht sprechen kann, darüber muss man schweigen.
Brussels, 18 April 2021
question in regard to the ring current model of elementary particles is this: what keeps the charge in place? We
think of this as the fine-tuning problem, but we do not see it as a fundamental problem of the theory: elementary
particles in free space may not sit still either. Their motion in space may probably be modelled by some
combination of the random walk model (in 3D space) and thermal motion. Both the random walk and thermal
motion must respect the Planck-Einstein relation too. Any random walk model should, therefore, be combined
with the quantum-mechanical least action principle: the action associated with any path should be equal to h or ħ
(linear versus orbital path), or an integer multiple thereof.
44
We interpret the 2019 redefinition of SI units as confirming this hypothesis: over the past 100 years, no
specifically nuclear-related new constant in Nature has appeared, and our current knowledge of fundamental
constants incorporate all laws of physics.
19
References
The reference list below is limited to the classics we actively used, and publications of researchers whom
we have been personally in touch with:
Richard Feynman, Robert Leighton, Matthew Sands, The Feynman Lectures on Physics, 1963
Albert Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik, 1905
Paul Dirac, Principles of Quantum Mechanics, 1958 (4th edition)
Conseils Internationaux de Physique Solvay, 1911, 1913, 1921, 1924, 1927, 1930, 1933, 1948
(Digithèque des Bibliothèques de l'ULB)
Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 1970 (2nd edition)
Patrick R. LeClair, Compton Scattering (PH253), February 2019
Herman Batelaan, Controlled double-slit electron diffraction, 2012
Ian J.R. Aitchison, Anthony J.G. Hey, Gauge Theories in Particle Physics, 2013 (4th edition)
Timo A. Lähde and Ulf-G. Meissner, Nuclear Lattice Effective Field Theory, 2019
Giorgio Vassallo and Antonino Oscar Di Tommaso, various papers (ResearchGate)
Diego Bombardelli, Lectures on S-matrices and integrability, 2016
Andrew Meulenberg and Jean-Luc Paillet, Highly relativistic deep electrons, and the Dirac equation,
2020
Ashot Gasparian, Jefferson Lab, PRad Collaboration (proton radius measurement)
Randolf Pohl, Max Planck Institute of Quantum Optics, member of the CODATA Task Group on
Fundamental Physical Constants
David Hestenes, Zitterbewegung interpretation of quantum mechanics and spacetime algebra (STA),
various papers
Alexander Burinskii, Kerr-Newman geometries (electron model), various papers
Ludwig Wittgenstein, Tractatus Logico-Philosophicus (1922) and Philosophical Investigations
(posthumous)
Immanuel Kant, Kritik der reinen Vernunft, 1781
ResearchGate has not been able to resolve any citations for this publication.
ResearchGate has not been able to resolve any references for this publication.