The Electromagnetic Inter-Nucleon Quark-to-Quark Bond and Its Effect on the Nuclear Force

Abstract

This chapter discusses the electromagnetic forces inside a nucleus. Previous nuclear theories have ignored the electromagnetic force or treated it as a minor component, considering only Coulomb forces between protons. Since quarks are the centers for both the electric charge and the magnetic dipole moments within a nucleon, such assumptions are not valid. Since the electromagnetic interaction between inter-nucleon quarks may be formidable, electromagnetism can, indeed, be the force holding the nucleons together in a nucleus. Thus, the electromagnetic forces within a nucleus should not be ignored, but rather given the foremost of consideration—specifically with regard to nuclear behavior. New understandings are gained by applying the laws of electromagnetism to the nuclear structure inside an atomic nucleus. In this chapter, historic misunderstandings are debunked and clarified—including the supposed limitations of the electromagnetic force, the miscalculated violations of uncertainty principles, and the misconceived lowest-energy shape of the nuclides.

1. Introduction

The development of a proper theory of the Nuclear Force has occupied nuclear physicists for over eight decades as one of the main topics of physics research in the twentieth and twenty-first centuries. Currently, there is no singular model of the Nuclear Force that can explain all nuclear behaviors [1, 2]. A successful model of the Nuclear Force should explain the more salient nuclear behaviors: the shape of binding energy curve, particle decay, large quadrupole moments, and excited energy states.

The emphasis of this chapter is electromagnetics; however, the concepts of quantum physics are not ignored or contradicted in any way, and they are properly included in this examination of nuclear behavior.

1.1 Definition and clarification of terms

The Nuclear Force binds the nucleons inside the nucleus. Historically, the Nuclear Force was called the “Strong Nuclear Force,” one of four forces of nature, along with the Gravitational Force, the Electromagnetic Force, and the Weak Nuclear Force. After the discovery of quarks, the “Chromodynamic Force” was identified as the force holding the quarks together inside a nucleon. Later, the Chromodynamic Force was renamed the “Strong Nuclear Force,” considered to be the force responsible for binding the sub-nucleon particles together inside a nucleon. The force holding the nucleons together in a nucleus was renamed the “Nuclear Force.”

To add to this confusion, there is a model of the Nuclear Force called the “Residual Chromodynamic Force,” also known as the “Residual Strong Force.” Because of this model, and its association with chromodynamics, the Nuclear Force was thought to be a subset of the Chromodynamic Force. The Strong Nuclear Force is now considered to have two parts: the Chromodynamic Force, holding the quarks inside a nucleon, and the Nuclear Force, holding the nucleons inside a nucleus.

To avoid confusion regarding these forces, the term “Strong Nuclear Force” is not used in this chapter. Rather the term “Chromodynamic Force” is used to describe the force holding the quarks together in a nucleon, and the term “Nuclear Force” is used to describe the force holding the nucleons together in a nucleus.

2. Brief review of other models of the nuclear force

There are a number of models of the Nuclear Force, each of which can explain a limited range of nuclear behavior. The current state of the theory for the Nuclear Force is simply a study of the various models, with no one single model being able to explain the majority of the observed behaviors in an atomic nucleus.

The Liquid Drop Model, developed in 1929, claims that the nucleons bind to their closest neighbors, similar to a drop of liquid. This concept is based on empirical data for the binding energy curve, wherein the binding energy per nucleon is relatively constant. Used in conjunction with the Liquid Drop Model is the Weizsäcker formula [3]. This formula is a mathematical equation that is simply a curve-fitting estimate for the experimental binding energy. It uses five parameters and conditional logic to obtain the best fit to the experimental binding curve, duplicating the experimental binding energies to within a few percent. The Liquid Drop Model assumes that the nucleus is spherical in shape, without any consideration for other shapes. It does not explain excited states, large quadrupole moments, or nuclear particle decay.

The Nuclear Shell Model was developed in 1949 [4], and it mimics the Electronic Shell Model for electrons around an atom. The Nuclear Shell Model attempts to explain the minor deviations that were observed, in the 1940s, between the experimental data and the Weizsäcker formula. At that time, with only a limited amount of experimental data, there were notable deviations that were observed at “magic” numbers. However, when utilizing the current data, these minor deviations are revealed to be either non-existent, relatively minor, or related to obscure behavior [5]. The Shell Model uses the Pauli Exclusion Principle to predict the spins of the various atomic nuclides; however, these predictions are accurate only when the Nilsson terms are included [6]. The Nilsson terms, often referred to as “spaghetti plots” due to their complicated natures, provide two or three different empirical terms per nuclide in order to match the experimental spin data to the theory [7]. The Shell Model does not explain particle decay or large quadrupole moments.

There are several Independent Particle Models [8] of the Nuclear Force. These models hypothesize that nucleons are confined inside a 3-D potential energy well, with no interaction among the nucleons. Although such a concept is unrealistic, it is employed to make the mathematics of the Schrödinger equation easier to solve. Depending on the atomic nuclide being studied, the Independent Particle Models alter the shape of the 3-D potential energy well within the Schrödinger equation, using numerous empirically selected variables to allow the models to duplicate the excited states of the nuclide. These models replicate large quadrupole moments only if the potential energy well is modeled as a 3-D ellipsoid, rather than a sphere. These models do not attempt to explain particle decay or the binding energy curve.

The Alpha-Cluster Model of 1938 [9] explained the stronger binding energy, observed for the nuclei with an integer number of alpha segments, claiming that alpha particles were formed as clusters inside a nucleus. This model was recently extended beyond the alpha nuclei and renamed the Clustering Model [10], with much recent research in the field of a nuclear clustering. According to this model, clusters of nucleons link together to form a chain-like structure. There is much experimental and theoretical evidence that atomic nuclides are made of clusters of alpha particles, as well as other types of segments. Recent research regarding the Clustering Model has indicated that clustering structures do indeed exist within nuclei [11, 12].

The Residual Color Model (also known as the Residual Chromodynamic Model or Residual Strong Force Model) is a theoretical model of the Nuclear Force, postulating the exchange of virtual pi-mesons as the binding force between nucleons [13]. In this model, the Nuclear Force is related to the Chromodynamic Force, but with four notable exceptions: it acts outside rather than inside the nucleons, it is a short-range force, it has different mediating particles, and it has a significantly lower strength. The Residual Color Force bonds a quark in one nucleon to a quark in another nucleon, thereby binding the two nucleons together. The Residual Color Model asserts that each nucleon is bonded to its nearest neighbors, similar to the Liquid Drop Model. The derivation of this residual force is mathematically difficult, even for only two nucleons. Only the smallest atomic nuclides have been simulated; however, these simulations resulted in significantly large errors. Thus, this model is unable to predict nuclear behavior. It should be noted that the computer simulations for the Residual Color Model used inflated quark masses, much larger than the theoretical mass of a “naked” quark. This mass inflation of the quarks was done to aid in the convergence of the simulations, since the simulations with small quark masses would not converge. Also, the use of inflated quark masses avoids a violation of Copenhagen interpretation of the Heisenberg uncertainty principle. Fortunately, there is some validity to this mass inflation of the quarks, related with the Constituent Quark Model.

In the Constituent Quark Model [14], the nucleons consist of three constituent quarks that are “dressed” with gluons and quark-antiquark pairs, thereby significantly increasing the effective mass of the quarks. If the quarks could be isolated from each other, then their chromodynamic binding energy could be experimentally measured. The quark rest masses would be the sum of the chromodynamic binding energy plus the rest mass of the nucleon. Unfortunately, the chromodynamic binding energy of the quarks is unknown. However, it is known to be at least as large as the amount of energy required to make a nucleon spontaneously emit a meson. (The actual binding energy could be much larger than this amount.) From this minimum estimate of the chromodynamic binding energy, the minimum value for the quark mass can be determined: the minimum mass of each dressed quark is between 300 and 400 MeV/c². The inflated quark masses used in the Residual Color Model are consistent with the minimum values for a dressed quark.

3. A brief review of quarks

In 1964, the concept of quarks was introduced by Gell-Mann and Zweig [15, 16], stating that the proton and neutron are non-elementary particles. A proton is made of two up quarks and one down quark. A neutron is made of two down quarks and one up quark. The terms “up” and “down” are simply quantum attributes, unrelated to vertical orientation. Up-quarks have an electrical charge that is +2/3 of an elementary charge, and down-quarks have an electric charge that is −1/3 of an elementary charge. The electrical charge and magnetic moments of a nucleon are confined to the quarks, rather than being homogeneously distributed throughout the nucleon. Quarks have the attributes of mass, spin, electric charge, magnetic moments, and a quantum attribute called “color,” which is unrelated to visual color.

Illustrated in Figure 1 are shorthand symbolic representations of the proton and neutron, showing the up- and down-quarks. In this representation, the up-quarks have a “++” charge, and the down-quarks have a “−” charge (2/3 and −1/3 of an elementary charge, respectively). The colors in these illustrations do not relate to the Chromodynamic Force, rather the colors are used to distinguish between the up- and down-quarks. The dots for the quarks do not represent the relative quark size, but rather are sized for easy viewing. The magnetic dipole moments of the quarks, not shown, are perpendicular to the page.

Figure 1.

Shorthand simplified representations of a proton and neutron.

Gell-Mann stated that quark charges are localized, and Richard Feynman asserted that high-energy experiments proved the reality of quarks. Feynman surmised that the quarks have a distribution of position and momentum, like any other particle, and he correctly believed that the diffusion of quark momentum explained diffractive scattering. James Bjorken proposed that point-like partons would imply certain relations in deep inelastic scattering of electrons and protons, relations that were verified in experiments at Stanford Linear Accelerator in 1969 [17].

The deep inelastic scattering experiments indicate that the quarks have a confined three-dimensional spatial distribution for their position. Specifically, the quarks are not moving with a simple random motion throughout the entire volume of the nucleon, rather there is a physical separation of the quarks, as illustrated in Figure 2a.

Figure 2.

In 2a, (left) the quark probability distributions are distinct and separate. In 2b, (right) is an illustration of an incorrect probability distribution of the three quarks, superimposed on one another and coincident with the nucleon boundary, causing a homogeneous charge and a neutral color.

The quarks exist as individual and definitive particles, as in Figure 2a, with each quark having a relative charge distribution in three-dimensional space and each quark being separate and distinct from the other quarks. Nucleons do not have a homogeneous charge or neutral gray color throughout their volumes. This incorrect hypothetical situation is illustrated in Figure 2b, shown as a comparison with Figure 2a.

A quote about the Laureates, who won the 1990 Nobel Prize for their deep elastic experiments at the Stanford Linear Accelerator, describes this concept.

“Earlier investigations of the proton at low energies had shown that this ought to be ‘soft’ with a relatively even internal distribution of its electrical charge. This year’s Laureates therefore had reason to believe there would be a decline in the probability of photon absorption (low number of events). But they found instead a high probability level (many events), i.e. there seemed to be something small and “hard” inside the proton.”

“Thus the new investigations gave the surprising result that the electrical charge within the proton is concentrated to smaller components of negligible size.”

“This unexpected discovery by the 1990 Laureates was noted immediately by certain skilled theoreticians, chiefly R.P. Feynman and J.D. Bjorken. The result was first interpreted within the framework of what is termed the parton model, which, however, soon came to be identified with the quark model.”

Thus, the quark probability distributions are not merged together, but rather they are individual and distinct distributions, as shown in Figure 2a.

3.1 The chromodynamic, nuclear, and electromagnetic forces on quarks

The Chromodynamic Force is much stronger than the Nuclear Force, by several orders of magnitude. However, an actual definitive measurement of its strength has not been made, because particle physicists are unable to separate the quarks from a nucleon. The Chromodynamic Force is thought to be similar to the force of a spring—the further it is stretched from its neutral point, the stronger it pulls back. Presently, there are numerous models of what comprises the mass of a nucleon. These models of the nucleon describe: the number of quarks inside a nucleon, what constitutes the mass of a nucleon, and what gives the nucleon its spin. Some models claim there are several hundred non-valence quarks and gluons inside a nucleon. For the Electromagnetic Model of the Nuclear Force, only the three valence quarks are considered. The Chromodynamic Force is the force inside of the protons and neutrons, and it confines and contains the quarks therein. The Chromodynamic Force abruptly neutralizes itself, falling to zero near the edge of the nucleon. Therefore, the quantum color of a quark strongly affects the other quarks inside that same nucleon; however, the quantum color of a quark does not affect quarks within other nucleons.

Conversely, the Electromagnetic Force does not abruptly neutralize itself at the edge of the nucleon. Rather, the Electromagnetic Force is completely unaffected by the nucleon boundary. These Electromagnetic Forces have an influence on all of the other quarks within the entire atomic nucleus, attracting or repulsing the other quarks within an atomic nucleus, depending on the polarity of the associated electric charges and magnetic dipoles. As a result, a quark inside of one nucleon feels the electromagnetic influences of a quark from another nucleon. Those same two quarks are contained and confined within their respective nucleons by the Chromodynamic Force; however, outside of their respective nucleons, the quarks are able to influence one another due to their electromagnetic forces.

3.2 Quarks and the electromagnetic bond

For the Electromagnetic Model, the nuclear bond is between two quarks forming the bond—a positively charged up quark from one nucleon and a negatively charged down quark from another nucleon. The concept of a nuclear bond is similar to the electronic bonds among the atoms of a molecule. In electronic molecular bonding of atoms, one atom cannot bond to an indefinite number of other atoms. Similarly, one nucleon cannot bond to an indefinite number of other nucleons. The number of bonds is limited by the number of quarks available for bonding. Thus with only three valence quarks, each nucleon can bond only to three other nucleons.

Prior to the 1960s, the proton was incorrectly thought to be homogeneously charged, and to be an elementary particle, with no sub-particles inside of it. As a result, the strongest electrical energy between two such protons was thought to be about 10⁻¹³ joules. The experimental energy required to free a single nucleon from an atomic nucleus is much larger than this. For this reason, the Nuclear Force was believed to be much stronger than the Electromagnetic Force. As a result, this incorrect concept of a homogenously charged elementary proton created the erroneous limitation for the maximum strength of the Electromagnetic Force. Unfortunately, this incorrect concept is still often perpetuated.

Mathematically, in the limit as the distance goes to zero, the electromagnetic energy goes to infinity. Hence, the electromagnetic energy can be extremely large if the quarks are close enough to each other. Since the electric charge of the nucleons resides only within the quarks, a quark from one nucleon can bond electromagnetically with a quark in another nucleon, and the resulting force between two such quarks can be large enough to be the Nuclear Force.

3.3 Slight conceptual change to the inter-nucleon quark-to-quark bonding

If the Nuclear Force is dependent on the up and down flavor of quarks, rather than on the color of the quarks, then many problems are resolved, and many questions can be answered [18]. This one simple conceptual change to the Residual Color Model makes an extremely important difference in the understanding of the Nuclear Force.

Specifically, the concept is that a down-quark is attracted to an up-quark, but not to another down-quark. Similarly, an up-quark is attracted to down-quark, but not to another up-quark. The Nuclear Force is simply the attraction between an up-quark in one nucleon and a down-quark in another nucleon. When a bond is made between these two quarks in two different nucleons, the nucleons themselves become bonded. This bond lowers the overall energy of the nuclide as compared with its constituent parts, thereby giving the nuclide a higher binding energy.

This one simple conceptual change, that the Nuclear Force is dependent upon the up/down polarity rather than the quantum color of the quark, explains why a system of six protons and six neutrons is at a lower energy (and at a higher binding energy) than five protons and seven neutrons. It is because six protons and six neutrons form one bond for every pair of up-down quarks. There are 18 up-quarks and 18 down-quarks in the system of six protons and six neutrons; thus, there are 18 pairs of up-down quarks and 18 bonds. However, for five protons and seven neutrons, there are 19 down-quarks and 17 up-quarks. Thus, only 17 bonds are formed, and two quarks remain unbonded. The nucleus with five protons and seven neutrons is at a higher overall energy (and at a lower binding energy) than the system of six protons and six neutrons. With this simple change, it is now understood why a nucleon only bonds to its nearest neighbors—a nucleon can only bond three times because it only has three quarks available for bonding.

This concept also explains the first term of the Weizsäcker formula; it is because of the limited number of possible bonds for each nucleon. Also, this simple change of concept explains the asymmetry force of the Weizsäcker formula—the greatest number of bonds occurs when there are equal numbers of up-quarks and down-quarks, which means an equal number of protons and neutrons. The Coulomb energy term of the Weizsäcker formula is also easily explained as being related to the electrical energy of the net positive charges within a nucleus.

Furthermore, due to the inter-nucleon up-to-down quark bonding with three possible bonds per nucleon, there is a predilection for the formation of an alpha segment, made of two neutrons and two protons. This predilection explains the pairing term in the Weizsäcker formula, giving a higher binding energy (and a lower overall energy) when N and Z are even. The highest binding energy occurs when there are an even numbers of protons and an even number of neutrons, allowing for the formation of alpha segments. The “surface” term of the Weizsäcker formula is related to the two unbonded up quarks on the ends of the chain-like configuration and the two unbonded down quarks in the middle open-alpha segment. (The reason for these unbonded quarks will be explained later.)

Thus, for this one simple change—that the inter-nucleon quark-to-quark bond is related to up/down flavor rather than to color—the five terms of the Weizsäcker formula can be understood as a direct consequence of the inter-nucleon up-to-down quark bonding. Simple calculations show that the concept of inter-nucleon up-to-down quark bonding reproduces the binding energy curve surprisingly well [18]. If the force between the inter-nucleon quarks is assumed to be the Electromagnetic Force, then more rigorous calculations can be made, as done in Section 5.

4. Quarks and the Copenhagen interpretation of uncertainty principles

It is shown here that there is no violation of the Copenhagen interpretation of the Heisenberg uncertainty principle. An example calculation is done to illustrate this. However, some definitions, explanations, and equations are discussed first.

The original derivation of the Heisenberg uncertainty principle states that if the two variables, position and momentum, are measured simultaneously, then the uncertainty of the measurement must be greater than $\mathrm{\hslash }$/2, shown in Eq. (1).

$${\sigma }_x{\eta }_p\ge \raisebox{1ex}{$\hslash $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$E1

Here, σ_x is the noise in the error of the measurement of position, and η_p is the resultant disturbance in the momentum, as a result of the measurement [19]. This equation and these definitions of the variables represent the original form for the Heisenberg uncertainty principle. The Copenhagen interpretation, however, does not involve laboratory measurements. The Copenhagen interpretation claims that the existence of a quantum system—whether observed or not—cannot exist in a state that is more precise than the Heisenberg limit. This changes the original Heisenberg uncertainty principle from that of Eq. (1) to that of Eq. (2),

$${\sigma }_x{\sigma }_p\ge \raisebox{1ex}{$\hslash $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$E2

where σ_x is the standard deviation of the position probability distribution of the particle, and σ_p is the standard deviation of the momentum probability distribution of the particle.

There are probability distributions for position and momentum for each quark. Associated with these probability distributions are the quantum Expectation Values for position and momentum, which are essentially the mean position and momentum of a quark. The mean value for position is a summation of all the possible values that the position can have, with each value being weighted according to the probability P(x) of that value occurring. To normalize these probability distributions, this is divided by the total number of events, which is the summation of P(x). For a smooth distribution, the summation is replaced by an integral, as shown in Eq. (3):

$$\mathit{\text{Mean of}}x=\frac{\underset{-\infty }{\overset{\infty }{\int }}\mathit{xP}\left(x\right)\mathit{dx}}{\underset{-\infty }{\overset{\infty }{\int }}P\left(x\right)\mathit{dx}}$$E3

In a quantum mechanical calculation, the probability function P(x) of the particle is the complex conjugate of the wave function times the original wave function. The wave function $\mathrm{\Psi }\left(x,t\right)$ of the particle is found by solving the Schrödinger equation with the appropriate boundary conditions. For the Expectation Value of position, the probability distribution is multiplied by x, then integrated. Since the overall probability of the particle’s position within infinity is unity, the integral in the denominator of Eq. (3) is defined to be one. Thus, for quantum mechanics, the Expectation Value of the position x is as shown in Eq. (4):

$$\mathit{\text{Expectation Value of}}x=\underset{-\infty }{\overset{\infty }{\int }}{\mathrm{\Psi }}^{\ast }\left(x,t\right)x\mathrm{\Psi }\left(x,t\right)\mathit{dx}$$E4

Since the denominator is defined as one, it is no longer explicitly necessary for the equation. A similar operation is done to find for the Expectation Value of momentum.

As mentioned previously, in the computer simulations of the Residual Color Force Model, the mass of the bare “naked” quarks was inflated to avoid non-convergence. This inflated mass also avoided a violation of the Copenhagen interpretation of the Heisenberg uncertainty principle—since for “dressed” quarks, the quantum uncertainty principles are not inherently violated. This larger mass is thought to be due to either relativistic speeds or to a quantum energy associated with the gluons. If relativistic speeds are involved, then the dressed quarks would also have a significant uncertainty in their momentum probability distribution. Regardless, the Chromo-dynamic Force is considered to be strong enough to confine the quarks inside the nucleus, despite relativistic speeds.

Due to their vibrations and quantum fluctuations, the quarks have a quantum probability distribution in three dimensions for both space and momentum. Thus, the quarks are not in a fixed or static position relative to each other. Rather there are probability distributions for their relative positions and momentums. When the positions are averaged over spacetime, the Expectation Values of the three quarks appear to be in point-like spatial positions, relative to each other.

This is an important concept: the Expectation Value of the three-dimensional spatial positions of the three quarks inside a nucleon can be modeled as point-like positions with respect to each other. At any given instant in time, there is an uncertainty in the position and momentum of the quarks. Although the spatial and momentum probability distributions are not point-like, the positional Expectation Values do appear as point-like positions.

Several experimental studies have been conducted to measure the radial charge distribution inside a proton and neutron [20, 21, 22]. The resulting radial charge densities are shown in Figure 3.

Figure 3.

Experimental data for the radial charge symmetry of the proton and neutron. Note the difference in the vertical axis.

As seen in the Figure 3, these experiments indicate that there is zero charge density at the center of both the proton and neutron—consistent with the quarks having a probability distribution situated more toward the periphery. Using these experimentally determined charge distributions, the quark probability distributions can be found by solving for the up and down probability distributions necessary to achieve these same charge distributions [23]. These quark probability distributions are shown in Figure 4.

Figure 4.

Up-quark and down-quark orbital radial distributions. The mean value of the quarks position is roughly at 0.8 femtometers from the center.

As can be seen, the quarks are not homogeneously distributed throughout the entire volume of the nucleon. Their mean position is approximately 0.8 femtometers from the center, in agreement with the calculations of this paper. (Note, these distributions are radial densities.)

While it is not possible to definitively calculate the combined uncertainty in space and momentum, by using reasonable values for these standard deviations, some estimates can be made to determine if the quarks fit within the confines of quantum uncertainty principles.

If the quarks inside a proton cannot be unbound by the addition of 139.6 MeV of energy, then the binding mass is at least this 139.6 MeV/c². The chromodynamic binding mass is subtracted from the masses of the three isolated quarks. In other words, the total mass of the three quarks is equal to the rest mass of the nucleon plus the chromodynamic binding mass. This information gives us a lower limit to the masses of the three constituent quarks. This calculation is shown in Eq. (5), where the mass of a proton is 938.3 MeV/c².

$$\sum {\mathit{\text{Mass}}}_{\mathit{\text{consituent parts}}}\ge 938.3\frac{\mathit{MeV}}{c^2}+139.6\frac{\mathit{MeV}}{c^2}\ge 1077.9\frac{\mathit{MeV}}{c^2}$$E5

Assuming there are no other massive particles, as yet undiscovered, inside the proton, this gives the sum of the masses of the three quarks to be at least 1077.9 MeV/c². This indicates that each of the effective masses of the three constituent quarks is at least 350 MeV/c² = 6.4 × 10⁻²⁸ kg in mass. (Most likely, they are more massive than this.) Regardless of whether this constituent quark mass is the rest mass of the three “dressed” quarks or a relativistically inflated mass of the “naked” quarks, this value of 6.4 × 10⁻²⁸ kg is a reasonable estimate for the lower limit of the quark mass. Also, this value is in good agreement with the Constituent Quark Model and the computer simulations used in the Residual Quark Model.

A reasonable estimate for the standard deviation of the quark vibrational velocity distribution is about 0.8 times the speed of light. Thus, σ_v = 0.8c = 2.4 × 10⁸ m/sec. This gives, σ_p, the standard deviation for the quark momentum as: σ_p = (2.4 × 10⁸ m/sec)(6.4 × 10⁻²⁸ kg) = 1.54 × 10⁻¹⁹ kg-m/sec. A reasonable estimate for the standard deviation of the spatial distribution, σ_x, is about 1/4 the diameter of the nucleon. Thus, σ_x = (1/4) (1.68 fm) = 0.42 × 10⁻¹⁵ m.

Using these reasonable estimates and the lower limit of the quark mass, the product of the two standard deviations falls within the limits of the Heisenberg uncertainty principle, as shown in Eq. (6).

$${\sigma }_x{\sigma }_p=\left(0.42\times {10}^{-15}m\right)\left(1.54\times {10}^{-19}\mathit{kg}m/\mathit{sec}\right)=6.45\times {10}^{-35}\ge \hslash /2$$E6

Thus, using these reasonable estimates for the uncertainties, the product is above the Heisenberg limit; there is no violation of the Copenhagen interpretation. Furthermore, the binding mass is very likely to be more than 139.6 MeV/c², which means that the product of the uncertainties is even larger. Also, the vibrational speed of the quarks could easily exist within the realm of relativistic speeds, again increasing the uncertainty product. Both considerations make a violation of the Copenhagen interpretation less likely. Hence, the concept of a quark that “appears” to be a point-like electric charge and magnetic moment is fundamentally allowed.

5. Calculations of the energies within the nucleus

For this calculation, the energies that are taken into consideration are the centrifugal energy, electric energy, and magnetic energy. The electric charges and magnetic dipole moments of the nucleons are contained in the quarks. Since the electric charges and magnetic dipole moments reside in the quarks, it can easily be realized that the electromagnetic forces between an up-quark and a down-quark are strong enough to bind the two nucleons together. No other force, different from the electromagnetic force, is needed to account for the strong bond between nucleons. Figure 5 shows this electromagnetic bond. Also present is the magnetic force, perpendicular to the page, but not shown.

Figure 5.

The electromagnetic bond between quarks.

The force binding these two nucleons need not be anything other than the electromagnetic forces between the up- and down-quarks.

5.1 The electric energy

The electric energy [24] between two charged particles is shown in Eq. (7):

$${\mathit{\text{Energy}}}_{12}=\frac{q_1{q}_2}{\left(4{\mathit{\pi \epsilon }}_0\right)\left({\mathit{\text{distance}}}_{12}\right)}$$E7

where distance₁₂ is the distance between particles 1 and 2, and q₁ and q₂ are the electric charges on particles 1 and 2. For additional charges, the total electric energy is the double summation over all pairs of charges, as shown in Eq. (8).

$$E_{\mathit{\text{electric}}\_\mathit{\text{total}}}=\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{q_i{q}_j}{\left(4{\mathit{\pi \epsilon }}_0\right)\left({\mathit{\text{distance}}}_{\mathit{ij}}\right)}$$E8

5.2 The magnetic energy

The magnetic energy [25, 26] between two magnets has vector and positional dependence. Given two magnets, with magnetic moments μ₁ and μ₂, the magnetic field of magnet₁ is determined at the location of the magnet₂. This vector field is symbolized as B₁₂, as shown in Eq. (9).

$$\overrightarrow{B_{12}}=\frac{{\mu }_0}{4\pi }\left\{\frac{3\left(\overrightarrow{{\mu }_2}·\overrightarrow{r_{21}}\right)\overrightarrow{r_{21}}-r_{21}^2\left(\overrightarrow{{\mu }_2}\right)}{r_{21}^5}\right\}$$E9

The resultant energy, U_magnetic12, is the negative dot product of the vector of the magnetic moment μ₂ with the vector of B₁₂, as shown in Eq. (10).

$$U_{\mathit{\text{magnetic}}12}=-\overrightarrow{{\mu }_2}·\overrightarrow{B_{12}}$$E10

For a collection of magnets, the total magnetic energy is the double summation over all pairs of magnets, as shown in Eq. (11).

$$U_{\mathit{\text{magnetic total}}}=\sum _{i-1}^{n-1}\sum _{j=i+1}^n-\overrightarrow{{\mu }_i}·\overrightarrow{B_{\mathit{ij}}}$$E11

where $\overrightarrow{B_{\mathit{ij}}}$ is the vector magnetic field established by the ith magnet at the location of the jth magnet.

Combining Eqs. (9)–(11), the total electromagnetic energy of a distribution of charges and magnets is shown in Eq. (12):

$$U_{\mathit{EM}\mathit{\text{total}}}=\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{q_i{q}_j}{\left(4{\mathit{\pi \epsilon }}_0\right)\left(r_{\mathit{ij}}\right)}+\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\mu }_0}{4\pi }\left\{\frac{3\left(\overrightarrow{{\mu }_j}·\overrightarrow{r_{\mathit{ji}}}\right)\left(\overrightarrow{{\mu }_i}·\overrightarrow{r_{\mathit{ji}}}\right)-r_{\mathit{ji}}^2\left({\mu }_i·{\mu }_j\right)}{r_{\mathit{ji}}^5}\right\}$$E12

Due to the vector properties of this energy, the lowest energy configuration for two magnets is a stacked bond, in which the magnetic moments of the magnets are oriented in the same direction, stacked one atop the other, and as close as physically possible. A side-by-side bond is with anti-parallel magnetic moments, oriented side by side, and as a close as physically possible. An angled bond gives intermediate results. From quantum field theory, it is known that quarks behave as point-like Dirac particles, each having their own inherent magnetic moment [27].

5.3 The centrifugal energy

The kinetic energy of spin must be properly included in the total energy of a nuclide. A rotating rigid object comprises numerous individual particles that orbit about a central axis of rotation. The kinetic energy of this orbital movement, E_orbital, is related to the angular velocity, ω, and to the moment of inertia, I_orbital, shown in Eq. (13):

$$E_{\mathit{\text{orbital}}}=\mathrm{1/2}{\mathit{I\omega }}^2=\mathrm{1/2}{L}^2/I_{\mathit{\text{orbital}}}$$E13

where L is the orbital angular momentum of the object. Note that L = ωI_orbital. Putting these concepts into the quantum realm of nuclear physics [28], the equation for the quantum orbital angular momentum, L, is shown in Eq. (14):

$$L=\mathrm{\hslash }\sqrt{l\left(l+1\right)}$$E14

where l is the value of the quantum number for the orbital angular momentum. Combining Eqs. (13) and (14) yields Eq. (15).

$$E_{\mathit{\text{orbital}}}=\frac{\mathrm{1/2}{L}^2}{I_{\mathit{\text{orbital}}}}=\frac{\mathrm{1/2}\left({\mathrm{\hslash }}^2\left[l\left(l+1\right)\right]\right)}{I_{\mathit{\text{orbital}}}}$$E15

5.4 The total binding energy

The total binding energy can be calculated using these equations. There is only one parameter that must be selected, and that is the minimum distance between the inter-nucleon quarks, as shown previously in Figure 5, between two bonded quarks of two different nucleons. The value of 2.11082 × 10⁻¹⁶ meters is used to match the experimental data. This value gives excellent matching to the experimental binding energy, as seen in Section 7. The calculated binding energy is the difference of the total energies of the configuration and the total energies of the isolated constituent parts, shown in Eq. (16).

$$U_{\mathit{\text{binding energy}}}=U_{\mathit{\text{total of configuration}}}-U_{\mathit{\text{total of constituent parts}}}$$E16

Note that a positive binding energy lowers the total energy and mass of the configuration, as compared with the individual constituent parts.

5.5 Nuclear bonds and the quantum hard-core repulsion

The Pauli Exclusion principle states that nucleons cannot overlap in their physical dimensions; this is also known as the hard-core repulsion. For this reason, a pair of double-bonded nucleons, defined as two nucleons bonded twice to each other, is not allowed. Similarly, a triple-quark bond, defined as three quarks attempting to bond together, is not allowed. These conditions are illustrated in Figure 6.

Figure 6.

Left is an illustration of a double bond between a proton and neutron. Right is a triple-quark bond.

6. The determination of the lowest energy configurations

The lowest energy configurations of the atomic nuclei are determined by using the laws and equations for electromagnetics and angular momentum, as applied to the quarks within a structured nucleus. The position of each quark is defined in a matrix with xyz spatial coordinates and an electric charge value of either −1/3 or +2/3 of an elementary charge. The value of the magnetic moment and the three-dimensional vector direction of each magnetic moment are also included in the matrix. Based on the physical constraints of the configuration, a determination is made as to whether the magnetic bond is stacked, angled, or side by side.

The various possible configurations are tested to determine which configuration is the lowest energy state. As a result, every nucleus in this paper is in its lowest energy state in accordance with the rules for electromagnetic and spin energies. Also, double-bonded nucleons or triple-quark bonds are not allowed.

The atomic nuclides from hydrogen ²H up to Copernicium ²⁸³Cn have been simulated and placed in their lowest energy configuration. For this lowest energy configuration, a pattern emerges from ¹²C upward, as discussed below.

The nucleons bond together to form the various basic clusters or segments; these segments subsequently bond together to form the larger atomic nuclides. For the lowest energy configuration, these basic segments bond together in a chain-like configuration, in strong agreement with the Cluster Model.

For stable atomic nuclides, the alpha segment predominates. When there are only three possible bonds per nucleon, the alpha segment is the lowest energy configuration for two protons and two neutrons.

6.1 Alpha segments predominate

The nuclear bond is an attractive electromagnetic bond between the up-quark in one nucleon and the down-quark in another nucleon. This bond lowers the overall energy of the atomic nucleus, similar to the way that chemical bonds between the atoms lower the overall energy of a molecule. The number of times that an atom can bond to another atom is limited by the number of valence electrons available for bonding. Similarly, the number of bonds for each nucleon is limited by the number of valence quarks available for bonding. Thus, with only three valance quarks, each nucleon can only bond three times. Two quarks, one from each of the two nucleons, are needed to form one bond. If every quark were fully bonded in an atomic nucleus, there would be three bonds for every two nucleons.

The alpha segment is predominant because there are only three bonds for each nucleon. Each nucleon can bond a maximum of three times, causing the nucleons to cluster into alpha segments. If there were only two bonds within each nucleon, they would simply string together as shown in Figure 7a. As can be seen in that illustration, each hypothetical proton and neutron can bond only twice. A structure similar to a string of beads is formed.

Figure 7.

(a) Hypothetical nuclear bonding if there were only two bonds for each proton and neutron. (b) Nuclear bonding with three bonds for each proton and neutron. (c) Hypothetical nuclear bonding if there were five bonds for each proton and neutron.

With three bonds allowed for each nucleon, the protons and neutrons form alphas segments, as shown in Figure 7b. This results in clusters of alpha segments, linked together in a chain-like structure.

If there were more bonds within each proton or neutron, the predominant cluster would be something larger than the alpha segment. For example, with five possible bonds per nucleon, the predominant structure inside such a hypothetical atomic nucleus would be ⁶Li, as seen in Figure 7c.

Experimentally, it is known that segments, such as a ⁶Li segment, are not the predominant structure of atomic nuclei, but rather the alpha particle predominates. This can be seen in the scallop-like pattern of the curve of binding energy and in the prevalence of alpha decay for the larger radioactive nuclides. The predominance of the alpha segment strongly implies that there are only three bonds per nucleon, as is expected for a nucleon having three valance quarks.

An important observation to make can be seen in the representation of the alpha segments, shown in Figure 7b. This nuclide represents oxygen ¹⁶O. The left side of the oxygen ¹⁶O configuration would have an unbonded positively charged up-quark on the end, and the right side would have an unbonded negatively charged down-quark on the end. This charge polarity causes an electric dipole moment in the configuration. However, quantum mechanics states there can be no net electric dipole moment within a nuclide [29].

Thus, to prevent this dipole, a bond is broken in one of the alpha segments. This broken bond creates an alpha segment, called an open-alpha segment, which bonds to the rest of the chain via its positive quarks, causing there to be two unbonded down-quarks in its center. This allows a negative charge to be situated near the center of the configuration and also spreads out the net positive charge of the nuclide. Both ends of the chain terminate with a positive charge, and there is no dipole moment.

The lowest energy configuration is one that maximizes the number of up-to-down quark bonds. To reduce the Coulomb energy, the lowest energy configuration spreads out the net positive charge as much as possible, while still maintaining the bonds. Any negative charge, such as an unbonded down-quark of an extra neutron, is situated in what would otherwise be the highest concentration of positive charge.

6.2 Spherical nuclides and the deformation parameter

Thus, the lowest energy configuration of any atomic nuclide is not a sphere. Rather, the electric force of the net positive charge within a nucleus attempts to spread apart that net positive charge as much as possible, without breaking the bonds. This forms a clustered, chain-like structure. When considering the electric force, a spherical shape for a net positively charged nuclide is exactly the opposite of its lowest energy shape.

There is much experimental evidence to substantiate this non-spherical shape. For example, the deformation parameter is a measure of an atomic nuclide’s non-spherical shape. The experimentally known deformation parameters are shown in Figure 8. If the nuclides were spherical, then all of the deformation parameters seen in Figure 8 should be less than 1, indicated by the blue line. However, they are much higher. (All data for Figure 8 were extracted from reference [30].)

Figure 8.

The ratio of the experimental nuclear deformation parameter divided by the predicted deformation parameter of the Shell Model, indicated by the blue line at the value 1.

The deformation parameters are much larger than the Shell Model can explain. This theoretical discrepancy with the empirical data is also true of electric quadrupole moments. Note that the distortion from a spherical shape does not appear in just a few small isolated regions of the nuclear table; rather it covers the entire range of the table. These large experimentally measured deformation parameters for the vast majority of atomic nuclides invalidate the concept of spherical atomic nuclides, in conflict with any model that pre-assumes a spherical shape, as is commonly done in many previous models [31].

For example, when researchers interpret the scattering data from an experimentally probed atomic nucleus, the shape of the atomic nucleus is pre-assumed to be a spherical shape, prior to interpreting the data. Furthermore, only the electric monopole moment is considered. For this pre-assumed spherical shape, the best value of its radius is forced-fit to the data, without any consideration that a different shape may be possible [32].

6.3 The patterns of the lowest energy configurations

For stable atomic nuclides with A ≥ 12, the lowest-energy configurations follow a pattern, as might be expected. Nuclides below A = 12 cannot follow the pattern, simply because there are not enough nucleons to do so. For stable nuclides, with A ≥ 12, the pattern is as follows.

• Alpha segments predominate within the configuration. For every two protons and two neutrons, an alpha segment is formed.

• There is an open-alpha segment near the center of the configuration.

• There is a triton segment, made of two neutrons and one proton, with an unbonded down quark, included in all stable nuclides with odd Z (with the exception of ¹⁴N).

• For stable atomic nuclides with N > Z, the extra neutrons are positioned in the location that would otherwise have the highest positive electric charge density.

• For stable atomic nuclides smaller than bismuth ²⁰⁹Bi, the excess neutrons are positioned between the alpha segments.

For unstable nuclides, the pattern is as follows:

• For unstable nuclides larger than ²⁰⁹Bi, the extra neutrons are forced to double up between the alpha segments. There are no stable nuclides with doubled-up neutrons in their configuration.

• In the lowest energy state for radioactive nuclides above the nuclear drip line (either having excess protons or not having enough neutrons), the net positive charge of the nuclide is spread out as much as possible.

• For radioactive nuclides below the nuclear drip line, the excess neutrons, with their negatively charged unbonded down quarks, are spread out as much as possible.

• These chain-like configurations may also curve or curl depending on the orientation of the constraints on the magnetic bonds between the segments. For example, a helix shape may form.

7. Summary of the electromagnetic force within the nucleon

• The electromagnetic force is valid inside the atomic nucleus.

• The electric charge and the magnetic dipole moment of the nucleons are contained within the quarks.

• Within a single nucleon, the three quarks have a spatial quantum probability distribution relative to each other. (An equilateral triangle is assumed for simplicity.)

• There are three possible bonds per nucleon, one for each quark.

• An up-quark and a down-quark, one from two different nucleons, are needed for one bond.

• The configuration, with the lowest electromagnetic energy plus spin kinetic energy, is assumed for the ground state.

• The kinetic energy of the quantum angular momentum of the nucleus [33, 34] is properly taken into account.

• There is a minimum distance between two quarks of two different nucleons, consistent with the hard core repulsion.

• The hard core repulsion of the nucleons prevents a nucleon from bonding more than once to another nucleon. Double-bonded nucleons and triple-quark bonds are not allowed.

Once the lowest energy configurations are determined, there remains only one parameter to be selected for the best fit to the binding energy data—namely, the “minimum quark-to-quark distance.” This model has only one parameter to determine, whereas the Weizsäcker formula uses five empirically fit parameters and a conditional logic statement to achieve its mathematical curve fitting. The electromagnetic model of the Nuclear Force is able to get comparable results with only one parameter.

Over a thousand different configurations for different atomic nuclides have been computer modeled using this method to calculate the binding energy based on electromagnetics. These include stable and unstable atomic nuclides; large, medium, and small atomic nuclides; and ground and excited states. These detailed calculations have been done for every atomic nuclide shown in Figure 9. Each atomic nuclide is placed in the lowest-energy configuration; then the electromagnetic energy of the configuration is calculated, and the binding energy is determined.

Figure 9.

(a) Binding energy per nucleon versus A, for both calculated and experimental data, from A = 2 to A = 208. (b) Binding energy per nucleon versus A, for both calculated and experimental data, from A = 2 to A = 60.

The value for the minimum distance between quarks is 2.11082 × 10⁻¹⁶ meters. The resulting binding energy curve is shown in Figure 9. The top pane shows all the points from A = 2 to A = 208. The lower pane shows only the points from A = 2 to A = 60, for ease of viewing the details. As can be seen, there is excellent agreement in the reproduction of the experimental data.

Most of the atomic nuclides fall within a 1 or 2% error. For the atomic nuclides with large A, the predicted downward slope is not as severe as is experimentally observed. For these larger atomic nuclides, the worst error is 8.32% for ²⁰⁴Pb. (Experimental data are extracted from reference [35].) Using electromagnetic equations, atomic nuclides as large as copernicium ²⁸³Cn have been easily modeled.

By applying the electromagnetic forces to the quarks and nucleons within a nucleus, new insights about many other behaviors of the Nuclear Force can be achieved. For example, by examining the electromagnetic forces that could possibly cause violations of the hard-core repulsion—such as a nuclide attempting to form either a double-bonded nucleons or triple-quark bonds—this model can potentially explain the nuclear behaviors of neutron ejection, proton ejection, and alpha decay.

Also, by considering configurations that are not the lowest energy state, new insights can be provided about the nuclear behaviors of these excited states and their isomeric transitions. Understanding how the electromagnetic forces within a nuclide might change after the occurrence of beta decay also allows for many interesting and discerning insights.

8. Conclusions

This chapter acts as an introduction to the electromagnetic behaviors inside the atomic nucleus. It also serves to clarify the important role of electromagnetics and to debunk the historical misconceptions regarding the electromagnetic forces inside an atomic nucleus. Rather than disregarding the electromagnetic forces and energies of the quarks, when taken into full account and understanding, the electromagnetic forces inside a nucleus can explain much about nuclear behavior. It is the intent of this chapter to highlight the role of electromagnetics, as a topic of nuclear physics that deserves further analysis and serious consideration. By recognizing the Electromagnetic Forces within the nuclear structure, a better understanding of nuclear behavior can be obtained.

The Electromagnetic Model of the Nuclear Force asserts that the electromagnetic properties of the quarks are the force binding the nucleons together, via an inter-nucleon quark-to-quark bond. The ground state configurations of hundreds of atomic nuclides, from ²H to ²⁸³Cn, have been determined and computer-simulated. The calculated binding energies agree with the experimental binding energies to within a few percent. These computations are done by using only one selected parameter. No previous theoretical model of the Nuclear Force is able to demonstrate such an accurate prediction of binding energy with only one parameter.

With this understanding that the electromagnetic properties of the quarks are what that bind the nucleons together in a nucleus, the Nuclear Force is directly unified to the Electromagnetic Force.