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Abstract

This paper revisits the (modified) Yukawa potential, as we use it to explain the neutron and deuteron nucleus, and addresses the point which is often used to discredit the model: any potential function should respect the (spatial) energy conservation principle. We show how the introduction of a simple vector potential introduces spatial asymmetry and, therefore, solves the energy conservation question. We also briefly discuss the Lorenz gauge in the context of static and dynamic potentials.
Nuclear potential and energy conservation
Jean Louis Van Belle
30 March 2021
Abstract
This paper revisits the (modified) Yukawa potential, as we use it to explain the neutron and deuteron
nucleus, and addresses the point which is often used to discredit the model: any potential function
should respect the (spatial) energy conservation principle. We show how the introduction of a simple
vector potential introduces spatial asymmetry and, therefore, solves the energy conservation question.
We also briefly discuss the Lorenz gauge in the context of static and dynamic potentials.
Contents
Introduction: the proton oscillator model .................................................................................................... 1
The nuclear potential and Yukawa’s range parameter ................................................................................. 2
The nuclear potential and energy conservation ........................................................................................... 6
Static and dynamic vs. scalar and vector potentials, and gauges ................................................................. 6
1
Nuclear potential and energy conservation
Introduction: the proton oscillator model
The massive nature of the proton indicates that the positive and negative unit charge are not just each
other’s opposite: matter and anti-matter are each other’s opposite, but positive and negative charge
have an additional asymmetry. The positive charge will partake in nuclear oscillations, rather than
electromagnetic ones, in the matter-world, for example. And we also cannot see the equivalents of the
classical, Compton and Bohr radius (all related through the fine-structure constant) for the positive
charge.
1
Of course, in the anti-matter world, all is reversed, including the mentioned asymmetries. Let us
stick to the matter-world.
We modeled the proton as an oscillation in three dimensions or, to be precise, as a combination of two
planar orbital oscillations. Hence, we think of an oscillation which is driven by two (perpendicular) forces
rather than just one, with the frequency of each of the oscillators being equal to = E/2ħ = mc2/2ħ. The
= E/2ħ formula also incorporates the energy equipartition theorem, according to which each of the
two oscillations packs half of the total energy of the nuclear particle. Each of the two perpendicular
oscillations packs one half-unit of ħ only.
2
Of course, we are talking orbital oscillations: each is driven by
a rotating centripetal force according to what is essentially a ring current model.
3
Figure 1: The ring current model
The ring current model generates the electron’s Compton radius as follows:
1
As a final example of how positive and negative charge are different, we may mention we cannot find any
obvious relation between the radii and masses of the muon and proton, respectively, although we do model the
muon as a nuclear oscillation as well. For a more detailed discussion, see our paper on the Zitterbewegung model
and the scattering matrix.
2
This reminds us of our explanation of one-photon Mach-Zehnder interference, in which we assume a photon is
the superposition of two orthogonal linearly polarized oscillations. However, the oscillations of the proton are not
linearly polarized: they are both circular (or elliptical, perhaps).
3
The British chemist and physicist Alfred Lauck Parson (1915) proposed the ring current or magneton model of an
electron, which combines the idea of a charge and its motion to represent the reality of an electron. The combined
idea effectively accounts for both the particle- as well as the wave-like character of matter-particles. It also
explains the magnetic moment of the electron. See our paper on what we think of as de Broglie’s mistake when
interpretating the nature of the matter-wave.
2





For the proton, we imagine two perpendicular oscillations. Such spherical view of a proton fits nicely
with packing models for nucleons and also yields the experimentally measured radius of a proton:
 

The 4-factor in the radius formula is effectively what distinguishes the formula for the surface area of a
sphere (A = 4r2) from that of the surface of a disc (A = r2).
To represent two perpendicular oscillations, we use two imaginary units i from j and think of them as
representing rotations in mutually perpendicular planes. Hence, we write the proton wavefunction as
4
:




The sign of the argument of the two wavefunctions represents the spin direction.
5
Also note the E/mp
= a22 equation assumes the two oscillations are perfectly synchronized. In other words, we have only
one phase, not two.
This is all rather mysterious but it makes sense to us. The idea is that the ring current creates a dynamic
field which maintains the motion of the pointlike charge. Half of the energy is in the field, and half of the
energy is kinetic, and the wavefunction just models how potential and kinetic energy “slosh back and
forth” (Feynman, 1963) over each cycle. Of course, this immediately triggers the stability question: the
motion must be finely tuned. How comes a small disturbance does not make the charge spiral out of
equilibrium? We do not know. Here, we want to focus on the nature of the force and the related
potentials, which triggers a whole other set of questions.
The nuclear potential and Yukawa’s range parameter
The E = mc2 mass-energy equivalence relation models the oscillation of the pointlike charge, whose
tangential velocity equals c (on average, at least
6
). We can then think of the nuclear and electromagnetic
force as comparable forces (they both act on the electric charge
7
) with different magnitudes and/or a
different structure (2D/3D). Both forces would respect the energy conservation and superposition
4
We use an ordinary plus sign, but the two complex exponentials cannot be added in any obvious way (i j). Note
that t is the proper time of the particle (Ep is the rest energy of the property). The argument of the (elementary)
wavefunction a·ei is invariant. We refer to Annexes II and III of our paper on the ring current model and the S-
matrix for an analysis of the wavefunction in the context of SRT and GRT.
5
This triggers an obvious question: combining the plus/minus sign of the two functions, we can have four possible
spin states, which is not in accordance with experiment. We have no answer to that. The most obvious explanation
of the apparent contradiction would be to assume that the spin of both components aligns with the
electromagnetic field in which it is measured, so it adds up to either (all) up or (all) down.
6
We should distinguish circular from elliptical orbitals, which we will do in a moment.
7
That is not the case for gravity, which acts on mass not on charge. That is why we think Einstein’s geometric
approach to modeling gravitation makes eminently sense: there is no (net) energy in gravitational orbitals.
3
principles, and that would be it. But things are not that simple when we start combining the two forces
to explain, say, a neutron (one positive and negative charge), or the deuteron nucleus (two positive
charges and one negative). Let us start with the neutron, which we effectively think of combining a
positive and a negative charge. We duly note the neutron is only stable inside of a nucleus but its mean
lifetime (almost 15 minutes) is extraordinarily long, so we will not consider the decay factor
8
.
So let us present the neutron model. The electromagnetic (scalar) potential energy
9
between the two
unit charges (one positive, one negative) is given by:
We do not put the usual minus sign because we do not take the U = 0 reference point for energy at infinity
(r = ). Of course, ke = 1/40 is just the usual Coulomb constant, and we should also quickly note the
concept of the (standard) force parameter concept, which is the energy per unit mass, and which we can
use to model orbital energies and compare force strengths: C = UC/m.
10
To describe the nuclear potential energy, we add a range parameter a, and we also square the distance r
in the denominator: this ensures the two potentials cross and also fixes the physical dimensions.
11
the
Note that, while the force acts on the same (electric) charge, the force decreases with the cube of the
distance, so we do not have an inverse-square law here (we will come back to this later):

Note we also do not put any minus sign here, because energies should be positive (we will soon revisit
this when discussing orbital energies) and we should choose the U = 0 reference point accordingly.
So the idea is this: the electromagnetic force holds the positive and negative charge together, while the
nuclear force pushes them apart. So it is just the opposite logic of those who first explored possible nuclear
potential functions (e.g. Erwin Schrödinger
12
), and we may think of this range parameter as a new physical
constant, which has to be determined from experiment. We may now compare the order of magnitude
of the electromagnetic and nuclear force may be compared by using the same numerical values for mC =
mN. Evaluating the potential energy functions above at r = a and taking their ratio, we get:
8
We may write the wavefunction of an unstable or transient particle as:

The factor is the inverse of the mean lifetime = 1/.
9
There is also a vector potential, which is related to the magnetic force. We will stick to a static analysis as for
now, which means we consider the charges not to be moving around (we will present the orbital energy equations
shortly, which do incorporate motion). Also note there is a subtle but important difference between the potential
energy U and the potential V: the energy is given by the potential times the charge it acts on, so we use qe2 instead
of qe in the formula.
10
The electron has electromagnetic mass, so we might write that as mC, while the proton has nuclear mass, so we
would denote that by mN.
11
We are not sure why Yukawa did not consider this. Richard Feynman did not seem to have thought of it either.
12
See, for example, Schrödinger’s Platzwechsel model for the nucleus, as presented in the Wikipedia article on the
Yukawa potential.
4

Hence, we cannot calculate a from our potential energy equations. This should not surprise us: we
define the range parameter here as the distance r = a for which the magnitude of the two forces (whose
direction is opposite) is the same. The form factor and, hence, the nature of the two forces, is very
different, though.
13
So how can we go about it? We can use the orbital energy equations. For the
nuclear force, we write:

The mass factor mN is the equivalent mass of the energy in the oscillation
14
, which is the sum of the
kinetic energy and the 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. For the
electromagnetic force, we write:
The total energy in the oscillation is given by the sum of nuclear and Coulomb energies, so we get this:


13
We may compare this with the standard gravitation parameter G in the energy equation for gravitational
orbitals, which follows from Kepler’s laws for the motion of the planets:


It should be noted that the kinetic and potential energy (per unit mass) add up to zero instead of c2 (nuclear and
electromagnetic orbitals), which is why we repeat that a geometric approach to gravity makes eminently sense:
massive objects simply follow a geodesic in space, and there is no (gravitational) force in such geometric approach.
We can now compare the standard parameters by equating m to mC (in practice, this means using the mass and
charge of the electron in the equation below) and, once more, equating r to a:




Hence, the force of gravity if considered a force is about 1042 weaker than the two forces we know
(electromagnetic and nuclear).
14
We use the subscripts (x)N and (x)C to distinguish nuclear from electromagnetic mass/energy/force but there is,
of course, one velocity: the velocity of one charge vis-á-vis the other.
5

The latter substitution uses the definition of the fine-structure constant.
15
Dividing both sides of the
equation by c2, and substituting mN and mC for m/2 using the energy equipartition theorem, yields:



It is a beautiful formula
16
, and we leave it to the reader to further play with it by, for example, evaluating
potential and kinetic energy at the periapsis, where the distance between the charge and the center of
the radial field is closest. Here, we are only interested in the formula because it gives us an order of
magnitude for the nuclear range parameter a. This order of magnitude may be calculated by equating r
to a in the formula above
17
:










The ħ/mc constant is, obviously, equal to the classical electron radius re 2.818 fm (1015 m)which is
of the order of the deuteron radius (about 2.128 fm) and which is the usual assumed value for the range
parameter of the nuclear force. We think it is a significant result that the lower limit for the range
parameter for the nuclear force must be at least twice at large.
An upper limit for this range parameter must be based on the experimentally measured value for the
radius of atomic nuclei. The scale for these measurements is the picometer (1012 m). The nucleus of the
very stable iron (26Fe), for example, is about 50 pm.
18
The radius of the large (unstable) uranium (92U) is
about 175 pm. The fine-structure constant may be involved again (although we noted it only applies to
structures involving the negative charge): 5.536 fm times 1/ yields a value of about 77 pm. This might
be a sensible value for the (range of) the upper limit for the (nuclear) range parameter, which will, of
course, depend on the shape (eccentricity) of the actual orbitals and other factors, such as the magnetic
coupling between the nucleons and the electrons in the atomic (sub)shells. This should, somehow,
15
One easily obtains the keqe2 = ħc identity from the
 formula. See the rationale of the 2019 revision of SI
units.
16
The ħ/mc factor is the classical electron radius. Needless to say, the a in the formula(s) above is the range
parameter of the nuclear force, which is not to be confused with the coefficient a of the wavefunction!
17
The range parameter is usually defined as the distance at which the nuclear and Coulomb potential (or the
forces) equal each other. See: Ian J.R. Aitchison and Anthony J.G. Hey, Gauge Theories in Particle Physics (2013),
section 1.3.2 (the Yukawa theory of force as virtual quantum exchange).
18
This is Feynman’s calculated radius of a hydrogen atom, but the measured radius of the hydrogen nucleus is
about half of it. To be precise, the empirical value is about 25 pm according to the Wikipedia data article on atomic
radii. We leave it to the reader to think about the 1/2 factor and the fine-structure constant as a scaling parameter.
6
explain the ‘magic numbers’ explaining the (empirical) stability of nuclei, but the exact science behind
this seems to be beyond us.
19
The nuclear potential and energy conservation
There are two problems with the analysis above:
1. We started by saying we think of a nuclear oscillation as a 3D oscillation, but so here we are back
with planar oscillations.
2. The nuclear force decreases with the cube of the distance, so we do not have an inverse-square
law here, which suggests we have a potential that does not respect the energy conservation
principle.
Let us deal with both objections by noting that we should probably re-write the orbital energy equations
using vector notation. We can then think of the nuclear oscillation as either 2D or 3D.
20
We should then
also add a unit vector n (same direction of the force but with magnitude 1) in the nuclear potential
formula:
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
21
) introduces a spatial asymmetry (think of an oblate spheroid
instead of a sphere here), which should ensure energy is conserved in the absence of an inverse-square
law. Is this an ad hoc solution? Yes, but one that does not sound all that unreasonable, does it?
Static and dynamic vs. scalar and vector potentials, and gauges
Electromagnetic theory is usually described using four-vectors and invariants, in classical as well as in
quantum physics. The most convenient mathematical representation of fields makes uses of the
concepts of the scalar and vector potential (φ and A)
22
, which also pop up in Feynman’s wave equation
for a free particle
23
:



19
See the Wikipedia article on magic numbers (nuclei).
20
A 2D nuclear oscillator model works perfectly well for the muon-electron. See our paper on ontology and
physics.
21
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. Or perhaps we should introduce a cosθ or cos2θ factor. The point is this: we need to
integrate over a volume and ensure the nuclear potential respects the energy conservation law.
22
See: Feynman, II-18 (Maxwell’s equations and the Lorenz gauge), II-25 (electrodynamics in relativistic notation),
II-26 (Lorentz transformations of fields) and
23
See: Feynman, III-21, Schrödinger’s equation in a magnetic field and his equation of continuity for probabilities.
We took the liberty of writing 1/i as i. We also multiplied the right-hand side of Feynman’s equation with
(1)(1) = +1, and substituted the dot product of the iħ qA operators for the square of the same operator.
7
This equation incorporates the integrity of Planck’s quantum of (physical) action as the unit of the
angular momentum of the oscillation (cf. the factor). The (scalar) potential φ can be electromagnetic,
nuclear or a combination thereof, acting on the (electric) charge q.
For electromagnetic oscillations, it is simple. Assuming the scalar potential varies with time, the vector
potential A can be easily derived from the Lorenz gauge condition:


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.
24
The magnetic field,
therefore, vanishes.
Now, there should be an equivalent scalar and vector potential for the nuclear force too, of course!
However, in light of the above-mentioned asymmetry, it is probably not so easy to use the usual
theorems (Gauss and Stokes) so as to develop the equivalent of Maxwell’s equations for the nuclear
force field. Of course, you may wonder if Gauss’ and Stokes’ Theorems – or vector integral/differential
calculus in general would actually apply to the nuclear force but that seems to be the case: it works for
all conservative forces (no friction)
25
, so the restriction we put on the potential (energy conservation)
should warrant that. So then we could use the superposition principle again to add the electromagnetic
and nuclear scalar and vector potential to get Dirac’s “equations of motion” for everything ! 
We hope readers who are well-versed in math and vector calculus will 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! 
30 March 2021
24
The Lorenz gauge does not refer to the Dutch physicist H.A. Lorentz but to the Danish physicist Ludvig Valentin
Lorenz. It is often suggested one can choose other gauges. We do not think so. We think the gauge is given by
relativity theory, and that is the same for time-dependent and time-independent fields. It does vanish, however,
time-independent fields (cf. electromagnetostatics). See our remarks on the vector potential and the Lorentz
gauge in our paper on the electromagnetic deuteron model.
25
See Feynman, I-14 (work and potential energy), II-2 (vector differential calculus) and II-3 (vector integral
calculus).
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