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The binding energy of the deuteron is calculated electromagnetically with the Schrödinger equation. In mainstream nuclear physics, the only known Coulomb force is the repulsion between protons, inexistent in the deuteron. It is ignored that a proton attracts a neutron containing electric charges with no net charge and that the magnetic moments of the nucleons interact together significantly. A static equilibrium exists in the deuteron between the electrostatic attraction and the magnetic repulsion. The Heitler equation of the hydrogen atom has been adapted to its nucleus where the centrifugal force is replaced by the magnetic repulsive force, solved graphically, by trial and error, without fit to experiment. As by chance, one obtains, at the lowest horizontal inflection point, with a few percent precision, the experimental value of the deuteron binding energy. This success, never obtained elsewhere, proves the purely static and electromagnetic nature of the nuclear energy.

The purpose of this paper is to calculate the binding energy of the simplest bound nucleus, the deuteron ^{2}H, with only fundamental laws (electromagnetics with Schrödinger equation) and associated constants. Up to now, mainstream nuclear physics is unable to obtain a single nuclear binding energy by applying fundamental laws and constants. The so-called “LQCD fundamental approaches”, with “ab initio predictions of observables”, have no quantitative fundamental laws, only phenomenological models, thus qualitative.

After one century of nuclear physics, “it is an open secret that the underlying force remains a puzzle” [

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

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

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” [

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.

We shall apply, in the Schrödinger equation, the Coulomb [

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 [

The physical constants used are: elementary electric charge

The total electromagnetic potential energy

and the magnetostatic interaction energy

The electrostatic interaction energy

where

where the tensor operator is [

The Schrödinger equation writes [

where

Heitler [

Replacing these expressions in the Schrödinger equation we obtain:

Simplifying by

which is positive only 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 [

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

where

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 [

where

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

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

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

The potential energy having three variables

The lowest saddle point coincides with the deuteron measured binding energy, ^{4}He however with a lower precision due to the approximations used [

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

The energy needed to separate an electron from a proton is given by the Rydberg constant

where

where

The experimental binding energies per nucleon vary from ^{7}H to almost

The Schrödinger equation of the deuteron has been solved using only electric and magnetic interactions. The results obtained confirm the validity of the static approach, simplified with an analytical formula [

In the deuteron, the magnetic moments of the proton and the neutron are opposite and collinear (not antiparallel^{4}He than in ^{2}H, due to the inclination of the magnetic moments at^{4}He binding energy to give

The electromagnetic potential has a real minimum when the positive charge of the neutron is neglected [

term

kinetic energy in the nucleus, contradicting the mainstream belief that the nucleons orbit like the electrons in an atom. With centrifugal force, the result would be incorrect, differing from the experimental value. The centrifugal force in the atom is replaced in the nucleus by the magnetic repulsion between nucleons.

The nuclear interaction is phenomenologically called “strong force”, “LQCD” or other denominations. Unfortunately, after one century of nuclear physics, the fundamental laws of the nuclear interaction remain unknown, still needing empirical formulas fitted to experiment. The Schrödinger equation with the Coulomb and Poisson formulas alone is able to provide the nuclear binding energy of the simplest bound nucleus, the deuteron ^{2}H and also of ^{4}He [

・ Light velocity:

・ Proton-electron mass ratio:

・ Fine structure or coupling constant:

・ Proton mass:

・ Proton Compton radius:

・ Nuclear magneton:

・ Landé factors of the neutron and the proton:

Neutron:

Proton:

Magnetic moments:

・ Relation between vacuum dielectric permittivity and magnetic permeability:

・ Electrostatic energy constant:

This fundamental constant, 4% weaker than the

・ Magnetic energy constant: