Atomic Nucleus: Coulomb without Strong Force

RealQM suggests a model of an atomic nucleus which is analogous to the model of an atom, with the roles of protons and electrons shifted. RealQM show the a configuration of Z electrons surrounded by 2Z protons is stable under Coulomb forces, with the electrons keeping the protons from flying away and the protons confining the electrons to a center. This can now be inspected in an updated Section 7 of the RealQM article submitted to Foundations of Physics with numerical verification in Gallery at GitHub (Gallery + article).  

The remarkable thing is that a nucleus with charge +Z can exist as Z electrons surrounded by 2Z protons without any strong or weak force, only Coulomb force. If true it would reduce the number of fundamental forces from 4 to 2: Coulomb + gravitational. 

The Neutron as Key to a Periodic Table for Nuclei

In recent posts I have tested an idea to view a system comprised of 1 proton + 1 electron in two different ways held together by Coulomb attraction:

• Hydrogen atom H of size $10^{-10] m with point-like proton kernel surrounded by electron density.   (H)
• Neutron N of size $10^{-15}$ m (inside atomic nucleus) as point-like electron kernel surrounded by proton density with a change of spatial scale of $10^5$. (N) 

The observed spatial scale between H and N is thus $10^5$. A transition from H to N would correspond to "shrinking" by a factor $10^{10}$ of the electron density around a proton into forming a kernel of a proton density, thus a a very strong shrinking. 

The observed binding energy of 13.6 eV for H and 0.8 MeV for N correspond to a spatial scale $D=0.6\times 10^5\approx 10^5$, in accordance with the $\frac{1}{r}$ spatial scaling of a Coulomb potential. 

Both systems can be described by a RealQM Schrödinger equation in non-overlapping wave functions $\psi_e(x)$ and $\psi_p(x)$ for electron and proton densities, as minimisers of total energy $E$ given by: 

• $E(\psi_e,\psi_p, m_e, m_p)=\frac{1}{2m_e}\int\vert\nabla\psi_e(x)\vert^2dx+\frac{1}{2m_p}\int\vert\nabla\psi_p(x)\vert^2dx-\int\int\frac{\psi_e^2(x)\psi_p^2(y)}{\vert x-y\vert} dxdy$   (S)

as the sum of separate kinetic energies for electron and proton and common Coulomb potential energy, where $\frac{1}{m_e}$ and $\frac{1}{m_p}$ set spatial scales of electron and proton.    

The standard case H is represented by minimisation of E without proton kinetic energy (formally $m_p=\infty$) and central point-like proton into a binding energy of $13.6$ eV.    

The non-standard case N is represented by minimisation of E without electron kinetic energy (formally $m_e=\infty$) and central point-like electron, which agrees with observation with $\frac{m_p}{m_e}=D$. 

We understand that since the above Schrödinger model does not involve gravitation, only Coulomb attraction between charges of different sign, the physical meaning of the factors $m_e$ and $m_p$ in the kinetic energies, do not connect to mass but rather to (inverse) spatial scale. What determines the roles of protons and electrons is their spatial scale. 

The conception that the mass of proton is about 2000 times that of an electron is thus not in conflict with $D\approx 10^5$ in the above Schrödinger model.   

The basic idea is to view the formation of a neutron inside a nucleus as a form of "capturing" by a proton density of an electron into the center of the proton density in a process at high temperature/pressure driven by Coulomb attraction under release of 1 MeV. The idea of electron capturing by a nucleus was an important element of nuclear physics even before the advent of the Standard Model in the 1960s.

Further capturing of electrons can create nuclei as a negative kernel surrounded by non-overlapping positive proton densities organised into shells, as a direct analog to an atom with a positive kernel surrounded by non-overlapping negative electron densities organised into shells. 

Recall that in the Standard Model the strong force appears as an ad hoc invention of remarkable fanciness.  If you ask a professional physicist what keeps a nucleus together thus overpowering Coulombic repulsion between protons, you get the answer that it is a form of "glue" of unknown physical nature named "strong force" transmitted by "gluons" of 8 different "colors" serving as "force carriers" between 6 different "quarks", where a proton is turned into a neutron when one of its two "up-quarks" turns into a "down-quark". If you ask how this can be you get the help that since very much energy is released when H fuses to Helium in the Sun a very strong force must be involved and this is the ”strong force” thus proven to exist. But gravitation is missing in the Standard Model because no “graviton” as force carrier is believed to exist, which is a trauma of modern physics since 50 years without hope.

In RealQM a nucleus has a negative kernel surrounded by positive proton densities held together by Coulomb attraction, as an analog to an atom with a positive kernel surrounded by negative electron densities. The observed "periodic table for nuclei" starting with 2, 8, 20,...appears as an analog to the periodic table for atoms starting 2, 8, 18... 

In the Standard Model a nucleus consists of a collection of protons and neutrons, with each proton and neutron consisting of three quarks held together by gluons, without explanation of the observed periodic table for nuclei.  

The great triumph of modern physics was to model the atom in terms of Coulombic attraction/repulsion between + and - charges using a basic element of classical deterministic physics in a new setting of statistics. RealQM shows that the new setting is not needed. Both atom and atomic nucleus can be modeled within classical deterministic mathematical continuum physics. This should be met with relief by students of physics struggling with weird concepts of modern physics.

The next step is to understand the formation of the nucleus of Deuterium D consisting of 1 proton and 1 neutron, or in RealQM 2 proton densities surrounding 1 electron kernel. In the Standard Model D is held together by a residual strong force as a left-over of the strong force holding proton and neutron together, like a molecule held together by residuals of Coulomb forces holding atoms together.  RealQM makes this analog real for nuclei: Both atoms and nuclei are held together by Coulomb forces. 

New View on Strong Nuclear Force 2

RealQM model of Helium nucleus consisting of 4 protons held together by electron cloud of  2 electrons.

This is a continuation of the previous post. We consider a RealQM model of the nucleus of an atom consisting of Z protons (each with charge +1) and N neutrons with in the normal case N=Z. Since each neutron can be viewed as a proton + an electron (with charge -1) the RealQM nucleus model consists of 2Z protons and Z electrons. 

We compare with a model of an atom ion with kernel charge +2Z surrounded by Z electrons, thus of charge +Z. With the electrons (for simplicity) homogenised into a single charge density $\phi (x)$ around the kernel (then without internal repulsion), the corresponding Hamiltonian takes the following form (in atomic units of length) 

• $-\frac{1}{2}\Delta_x - \frac{2Z}{\vert x\vert}$     (1)

where $\Delta_x$ is the Laplacian differential operator acting on a 3d space variable $x$. Assuming spherical symmetry it follows from the corresponding 1d Schrödinger equation that the bulk of the charge density has a width scaling with $\frac{1}{Z}$. In other words, the charge $Z$ sets the physical scale of the electron charge in terms of atomic units of length from a balance of kinetic energy and potential energy. The total energy then scales with $Z^2$.

Let us now try a model for the nucleus with Hamiltonian 

• $H=-\frac{1}{2}\Delta_x - K(x)$                           (2)

where $K(x)$ is the sum of potentials of the form $ \frac{1}{\vert x -x_i\vert}$, where the $x_i$ for $i=1,..., 2Z$ represent the positions of the 2Z protons of the kernel, with again $H$ acting on a single electron charge density $\phi (x)$ (without internal repulsion). We can alternatively view (2) as model of a molecule ion consisting of 2Z protons held together by an electron cloud of charge Z acting like glue connecting the protons. We compare with (1) with an electron cloud of charge Z surrounding a single nucleon of charge 2Z. 

As a model of a nucleus (2) represents a physical scale which is a factor of at least 1000 smaller than when viewed as a model of a molecule ion, in which the former case the electron appears as being compressed "inside" the nucleus instead of being "outside" as in (1). 

We can now play with the model (2) and vary Z as an "effective charge" setting a scale where the size of the electron matches the distance between the protons so that it can act as glue keeping protons together even under proton-proton repulsion. This scale can be found by minimizing the total energy normalised by $Z^2$.  Here you can play case of Helium nucleus (2P+2N) and here with a (hypothetical) Beryllium nucleus (4P+4N). 

This analysis suggests that it may be possible to replace the strong force supposed to keep an atomic nucleus together, by the electromagnetic force between protons and electrons acting on a smaller nuclear scale. Quite a bit of simplification if true...

To sum up, we are led to explore the possibility that protons and electrons can interact on two different scales: 

1. Electrons on atomic/molecule scale surrounding smaller scale nuclei. 
2. Electrons on smaller nuclear scale "compressed" with protons as neutrons and then acting like a glue between protons just like in a molecular ion. (It is not clear if "compressed" electrons inside a nucleus are subject to repulsion).

The scale difference is around a factor 1000 with thus the energy dissociation of one electron increasing from eV for atoms to MeV for nuclei. 

Summary so far as concerns the composition of a nucleus (connecting to Platons ideal solids):

• Hydrogen: 1P
• Deuterium: 1P+1N, shape: 1d linear
• Tritium: 1P+2N, 2d triangle (compare with H3+ tritium cation stable)
• Helium: 2P+2N, 3d tetrahedron
• Lithium: 3P+3N, octahedron
• Beryllium: 4P+4N, cube
• ...
• Coal: 6P+6N, icosahedron 
• ...
• Neon 10P+10N, dodekahedron
• ...  

 

New View on the Strong Nuclear Force 1

Atomic nuclei consist of about the same number of protons and neutrons 

This is a continuation of a previous post on the possibility of viewing a Neutron consisting of a proton and electron as a smaller analog of a Hydrogen atom again consisting of a proton and electron. In particular it suggests that atomic nuclei consisting of nearly the same number of protons and neutrons is held together by a strong force which is an electromagnetic force acting on a smaller scale than the electromagnetic force keeping an atom/molecule together. 

To see the analog, consider the H2+ molecule consisting of two protons and one electron which acts like an glue keeping the two protons together into a molecule from the attractive force between the protons and the electron in between, which is stronger than the repulsive force between the protons and forms the H2+ molecule having a dissociation energy of about 1eV  at a distance of about 1Å. This is what the Schrödinger equation says. In atomic units the Schrödinger equation is parameter-free with unit length scale, and the physical dimension is set through Planck's constant and the mass/charge of the electron. You can view and play with the Schrödinger equation for H2+ here.

We next consider a nucleus consisting of a proton and a neutron, as the nucleus of deuterium as an isotop of hydrogen, thus two protons and one electron just as the H2+ molecule. The difference is the scale with the nucleus being much smaller than the molecule. It now appears to be conceivable that a nucleus is again held together by electromagnetic forces in the same way as the molecule, just on a smaller physical scale scale. In other words also the nucleus would be governed by the Schrödinger equation just on a smaller physical scale.  How much smaller physical scale?

 It follows from Schrödingers equation that energy scales like $d^{-2}$ with the dimension $d$. The decay energy of a neutron is around $10^6$ eV thus around $10^6$ times the dissociation energy of H2+ of about 1 eV. This indicates a scale factor of about 1000 between atomic and nuclear scale. 

The typical size of an atom is 1Å or $10^{-10}$ meter. The size of a nucleus is decided by shooting high energy alpha particles at the nucleus and measuring the point of closest approach to be about $10^{-14}$ meter, thus with a scale difference of 10.000. But it is possible that the effective size is bigger, and so the factor 1000 may capture reality.

We are thus led to the following

Question: Is the strong nuclear force in fact an electromagnetic force acting on small scale?