Nuclear Forces

The nuclear forces determine the interaction between two or more nucleons in an atomic nucleus. They are responsible for binding of protons and neutrons. The nuclear force is nearly independent of whether the nucleons are neutrons or protons. This property is called charge independence. To a large extent, this force can be understood in terms of the exchange of virtual light mesons, such as the pions. Sometimes the nuclear force is called the residual strong force, in contrast to the strong interactions that are now understood to arise from quantum chromodynamics (QCD). Since nucleons have no colour charge available in the interaction between quarks, the nuclear force does not directly involve the force carriers of QCD, i.e. gluons.


At the same time, the understanding of how QCD works remains one of the great puzzles of many-body physics. Indeed, the degrees of freedom observed in low energy phenomenology are totally different from those appearing in the QCD Lagrangian. In the case of many-nucleon systems, the question of the origin of the nuclear energy scale is immediately aroused: the typical energy scale of QCD is in the order of 1 GeV, even though the nuclear binding energy per particle is very small, in the order of 10 MeV. Is there some deeper insight from which this scale naturally arises? Or should the reason be looked for in complicated details of near cancellations of strongly attractive and repulsive terms in the nuclear interaction?

Recent high precision measurements [1] of the deuteron electromagnetic structure functions (A, B and T20) extracted from high-energy elastic ed scattering and the cross sections and asymmetries extracted from high-energy photodisintegration $\gamma + d \rightarrow n + p$ have been reviewed and compared with theory. The rigorous results [1] demonstrate that QCD and the meson theory seem to disagree. Hence the origin of nuclear forces in an unperturbed nucleus is still unclear.

Regarding a possibility of deriving nuclear forces from the quark-quark interaction, Santilli [2] reasonably remarks that quarks can only be defined in a mathematical unitary space that has no direct connection to an actual physical reality. Then he continues: “It is known by experts that, because of the impossibility of being defined in our space-time, quarks cannot have any scientifically meaningful gravitation, and their masses are pure mathematical parameters in the mathematical space of SU(3) with no known connection to our space-time.” Indeed, quarks cannot be defined through special relativity and its fundamental Poincaré symmetry. Therefore, this means that mass cannot be introduced as the second order Casimir invariant. However, this is the only necessity for mass to exist in our space-time. In other words, the basic challenge of deducing the gravitational interaction from quarks seems completely unfeasible.

Taking into account the mentioned fundamental difficulties, Santilli, developing hadronic mechanics, started from the hypothesis that in a system of strongly interacting particles the total Hamiltonian cannot be subdivided into kinetic and potential parts. Hadronic mechanics is constructed via a non-unitary transform of orthodox quantum mechanics, namely the unitary character appears only at a distance larger than the radius of nuclear forces $10^{-15}$ m.

Such a transform allows the introduction of a modernized Lie product, Pauli matrices, Dirac equation, etc. So, the conventional Hamiltonian of relativistic quantum mechanics and the appropriate spinors are transformed to new Santilli isomathematical presentations. The mathematics developed enabled the calculation of basic parameters of nuclear systems, which exactly agreed with the experimental data [2].

Unlike conventional quantum mechanics that operates with point particles and their appropriate wave packets, or wave functions, hadronic mechanics deals with the extended particles that feature peculiar shapes in our space-time. Hadronic mechanics predicts a strongly coupled structure for a proton and electron i.e. in hadronic mechanics the neutron is treated as a strongly coupled proton-electron pair. Hence the nuclear forces are associated with the pure Coulomb interaction between protons and neutrons. Thus in hadronic mechanics the origin of nuclear forces is completely plain: this is the usual Coulomb interaction between nucleons.

Sub-microscopic standpoint

None of the quantum theories being developed pays any attention to the background of the systems studied, i.e. the structure and peculiarities of the real physical space. The theories are developed in abstract spaces: energy, momentum, phase, Hilbert and so on. Instead of the background space they use such completely undetermined notions as a “physical vacuum” or/and an ether providing them with every possible and imaginary properties. It seems the aforementioned quotation from Santilli regarding quarks as objects that are not determined in the space-time is the apt turn of phrase, which emphasizes the validity of our criticism.

As was hypothesized in paper [3], we can consider the interaction of two nucleons that closely approach each other to start their interaction through their respective deformation coats that touch each other, because any canonical particle in the tessel-lattice is specified by the deformation coat. For nucleons in a nucleus the radius of such coat coincides with their Compton wavelength, i.e. $\lambda_{\rm Com} = h/(c M_0) \approx 1.32 \times 10^{-15}$ m, which exactly coincides with the experimentally detected radius of nuclear forces (here we set approximately $m_{\rm proton} \approx m_{\rm neutron} = M_0$). The behaviour of the deformation coat can be treated in terms of vibrating particles of a crystallite, see submicroscopic mechanics.


In the crystallite, vibrations of all particles (i.e. superparticles of the tessel-lattice, which have distortions) co-operate and the total energy of superparticles, which is equal to the total energy of the particle, $M_0 {\kern 1pt} c^2$, is quantized,

\begin{align} \hbar \omega_{k} = M_0 c^2 \end{align}

where $k = 2 \pi / \lambda_{\rm Com}$ is the wave number and $\omega = c k$ is the cyclic frequency of an oscillator in the k-space (the quantity $\lambda_{\rm Com}$ is the amplitude of the oscillator, which is given by the crystallite size, i.e. Compton wavelength. Below we designate radius of the crystallite as $R_0$, which in the undisturbed state can be set equal to $\lambda_{\rm Com}$.

Vibrating superparticles of one coat begin to interact with those of the other coat. It is a fact that the interaction between two oscillators reduces the total energy of the oscillators.

Since the interaction of nucleons can reduce their energy by 47 MeV (in agreement with the Fermi gas model, see e.g. Ref. [4]), we can write the equality

\begin{align} \hbar \omega_1 \frac{1}{1+\Delta R/R_0} \approx (938.26 - 47) \ \ {\rm MeV} \end{align}

that makes it possible to estimate an effective range $\Delta R \approx 0.053 R_0$ of the overlapping of deformation coats of two nucleons.

Such an overlapping can virtually draw two nucleons together, but only a little. A deeper penetration into the core can be achieved only in the case of weighty nuclei when the collective motion of a great number of nucleons is allowed for.

The total mass of the nucleon and the superparticles in its deformation coat increases too: $M_0 \rightarrow M_0 + \Delta M$ where $\Delta M = M_0 \cdot \Delta E /(M_0 c^2) \approx (0.037 \ \ {\rm to} \ \ 0.05) \cdot M_0$.

Hence the notion of a “potential well” implies that in the range of space covered by the well, spatial blocks, i.e. superparticles, are found in a more contracted state than in the space beyond the potential well.

Thus the consideration above shows that the coupling of nucleons through their deformation coats is a beneficial process.

One more source of nuclear forces is associated with the overlapping of inerton clouds of moving nucleons. Such an approach allows the study of the deuteron problem from a deeper viewpoint. In particular, the study [3] shows that the radius of the deuteron $R_{\rm d} = 2.85 \times 10^{-15}$ m, which exceeds the double radius $R_0 = \lambda_{\rm Com} = 1.32 \times 10^{-15}$ m. Therefore, in the case of the deuteron the proton-neutron coupling occurs rather through overlapping of their inerton clouds than through the compression of their deformation coats, as is the case for heavy nuclei.

In heavy nuclei nucleons try to gather in clusters such that the number of nucleons $A$ in a cluster is linked with the elasticity constant $\gamma$ of the inerton field in this nucleus, the nuclear density $\rho$ and the elementary charge $e$ of the proton. $A$ is inversely proportional to $\gamma$: an increase in $A$ requires a decrease in $\gamma$ approaching to the critical value of $\gamma_{\rm c}$, such that at $\gamma < \gamma_{\rm c}$ the inerton field is incapable of holding nucleons in the cluster and hence this particular nucleus has to be unstable.


[1] R. Gilman, and F. Gross, Electromagnetic structure of the deuteron, Journal of Physics G: Nuclear and Particle Physics 28, R37-R116 (2002).

[2] R. M. Santilli The physics of new clean energies and fuels according to hadronic mechanics, Journal of New Energies 4, no. 1, 5-314 (1999); Foundations of Hadronic Chemistry with Applications to New Clean Energies and Fuels (Kluwer Academic Publisher, Boston-Dordrecht-London, 2001).

[3] V. Krasnoholovets, Resons for nuclear forces in light of the constitution of the real space, Scientific Inquiry 7, No. 1, 25 –50 (2006); also arXiv:1104.2484

[4] O. H. Sytenko and V. K. Tartakovsky, The Theory of Nucleus (Lybid, Kyiv, 2002), p. 175 (in Ukrainian).

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