After one century of nuclear physics, “it is an open secret that the underlying force remains a puzzle” [1] . The parameters of the potential are still determined by fitting to experimental data [2] . Up to now no fundamental law of the nuclear interaction has been discovered. The electric and magnetic forces are known qualitatively since two millenaries, when the Greeks discovered the properties of amber (elektron) and magnetite (from mount Magnetos). The electromagnetic laws were discovered by Coulomb [3] and Poisson [4] two centuries ago, unified by Maxwell [5] . Born [6] noticed that “From Newton’s law, one can derive that the binding energy of two massive bodies is inversely proportional to the distance between them.” Unfortunately he believed that the neutron was an uncharged particle, thus needing “forces of a different type (...) restricted to a very short range”. The “strong force” hypothesis originated from the idea that the protons would repel one another and the nucleus should therefore fly apart. The attraction between a proton and a neutron seems still to be ignored although the discovery by Bloch [7] of the magnetic moment of the neutron showed its electric charges with no net charge.

The main phenomenological assumptions in nuclear physics are:

a) The forces between nucleons are almost the same according to the assumption of charge independence: NN ≈ pp ≈ nn ≈ np. The so-called “Coulomb force”, repulsive between protons, is the only recognized electromagnetic interaction in a nucleus. The attraction between a proton and a neutron as well as the magnetic moments of the nucleons is ignored.

b) The nuclides have an approximate spherical shape, as for the deuteron (see Figure 1).

c) “The Standard Model has a disturbingly large number of parameters whose numerical values are not explained; many aspects of the model seem unnatural” [8] .

d) “In contrast with the situation with atoms, the nucleus contains no massive central body which can act as a force center. This deficiency is circumvented by the bold assumption that each nucleon experiences a central attractive force” [9] . Moreover, fundamental laws cannot be obtained from magic numbers.

e) Innumerable nuclear forces have been imagined from “Strong force” to LQCD: none has fundamental laws. The “Strong force” is assumed to have a coupling constant of 1 (how come?), thus 137 times the electromagnetic interaction. In fact the fundamental laws of the nuclear interaction are unknown. The “strong force” will disappear as the phlogiston, thanks to Lavoisier, and the Aether, thanks to Einstein.

A completely different approach based on known fundamental laws is necessary.

Every child knows that a rubbed plastic pen attracts small neutral pieces of paper. The same attraction arises between the electric charge of the proton and a nearby “not so neutral neutron”. This attraction, able to create a deuteron, is equilibrated by the repulsion between the collinear and opposite magnetic moments of the proton and the neutron in the deuteron. The dipole and polarizability formulas being invalid in a non-uniform electric field, the exact induced dipole formula has to be used here [10] . No need of a binomial expansion, the exact dipole formula is simple and precise. The following calculations will show that the magnetic repulsion equilibrates statically the electric attraction (Figure 1), giving the binding energy of the deuteron.

The physical constants used are: elementary electric charge, neutron and proton magnetic moments and, magnetic permeability, vacuum electric permittivity, light speed c or, equivalent nuclear fundamental constants, fine structure constant, proton mass, neutron and proton Landé factors, and the proton Compton radius. The formulas in the appendix show the conversion from classical electromagnetic formulas to nuclear physics formulas. The usual fundamental constants of the Coulomb electromagnetic potential are replaced by the rigorously equivalent nuclear fundamental constants shown in the appendix. The formula (3) is not “an arbitrary manipulation of the fine structure constant together with the proton mass”: it shows an interesting similarity with the Hartree constant (twice the Rydberg constant), , characterizing the chemical energy.

The total electromagnetic potential energy is the sum of the electrostatic interaction energy

and the magnetostatic interaction energy between two point-like electric charges and:

The electrostatic interaction energy between particles and is due to the electric charges and. The magnetostatic interaction energy is due to the magnetic dipoles and. The general formula of the electric and magnetic interactions between two particles and is [5] [7] [11] -[13] :

where (Figure 1) is the distance between the centers of the point-like particles and the corresponding internucleon vector. Formula (2) shows that the Coulomb potential is attractive or repulsive depending on the sign of the product of the electric charges and on the orientation and position of the magnetic moments. An alternative presentation of formula (2) is

where the tensor operator is [14] :

is the Landé factor of the particle. is the mass. and are the Compton radii of the electron and the proton. is positive for magnetic repulsion and negative for magnetic attraction, depending on the relative orientation and position of the magnetic moments of the nucleons. Formulas (3) and (4) are rigorously equivalent to (2), with equivalent fundamental constants.

The Schrödinger equation writes [9] :

where is the wave function, the mass, the reduced Planck constant, E the fundamental state binding energy, V the potential energy. is the kinetic energy, always positive. Without kinetic energy, the Schrödinger equation reduces to the Laplace equation. For a spherically symmetrical potential, one has:

Heitler [15] has calculated the binding energy of the atom with the exponential wave function, where is the range of the wavefunction. For or at the center, the wavefunction is 1. Here we don’t need normalization of the wave function, only the binding energy:

Replacing these expressions in the Schrödinger equation we obtain:

Simplifying by, the kinetic energy becomes:

which is positive only for meaning that there is no kinetic energy for.

Formula (10) will be used for both hydrogen atom and heavy hydrogen nucleus. The atomic potential is electrostatic, equilibrated by the centrifugal force. The nuclear potential is electrostatic, equilibrated by the magnetic repulsion. The concept of eigenfunction is useless for the fundamental state [15] , the only state known in the deuteron.

From formula (10), using the attractive Coulomb potential, the fundamental state potential of the hydrogen atom is:

where is the attractive potential between the proton and the electron. The first term corresponds to

the quantized centrifugal movement, the same as in the Bohr model but obtained with the Schrödinger equation. This expression will be identically null if the constant and variable terms in 1/r are identically nullified, giving two equations:

The well known formula for the hydrogen atom fundamental state energy is thus [15] :

where is the fine structure constant and the electron mass.

In the deuteron, the negative charge of the neutron is attracted by the proton positive charge; its positive charge is repelled farther away. The result is a net attractive electrostatic force.

The magnetic moments being opposite and collinear, the magnetic energy is repulsive (14). The nuclear electromagnetic potential is the sum of the Coulomb electric and Poisson magnetic potentials [10] :

2a is the distance between the positive and negative electric charges of the neutron and r the distance between the centers of the nucleons. Although this potential is not really spherical we may use it because we need only the forces along the neutron-proton axis. The kinetic energy, always positive, represented by the first term of Equation (15) below, is repulsive, needing the condition, as for the H atom but with the proton mass. The binding energy being defined as the minimum of the potential energy, the Schrödinger equation of the deuteron becomes, according to (10) and (14):

Numerically, the energy is given in MeV and the distances in fm:

The potential energy having three variables we have to find their values to obtain the potential minimum. This equation was solved by trial and error until the horizontal inflection point attains a minimum by varying a and b as shown on Figure 2 (not to be confused with an ad hoc adjustment). The lowest horizontal inflection point (saddle point) of the thick curve is only a relative minimum of the potential energy due to the Coulomb singularity in (Figure 2). On the other hand, b cannot decrease towards zero, giving a repulsion, larger than the experimental binding energy, attainable only for b infinite, practically annihilating the wave mechanics term, representing the repulsive kinetic energy:

The lowest saddle point coincides with the deuteron measured binding energy, , with a 5% precision (Figure 2). Only electromagnetic forces, without kinetic energy [10] are needed. The distance is the distance between the centers of the nucleons, not the radius of the deuteron, around 2 fm. This approach is confirmed for ⁴He however with a lower precision due to the approximations used [16] .

To check the graphical result, let us ignore the magnetic repulsion. The distance between the positive charge of the proton and the negative charge of the neutron is. To have equidistance of the positive charges with the negative charge, we have the condition thus. Using the graphical solution, gives, without magnetic repulsion, not to too far from the result of the complete graphical calculation,. The magnetic repulsive potential (in) is small but its force (in) is important for the equilibrium.