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Demonstration that like nucleons repel each other through the strong nuclear force and unlike nucleons attract through the strong force |
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Nuleons (protons and neutrons) are subject to two types of interaction; Those due to the formation of spin pairs and those due to another force which is usually called the nuclear strong force. Nuclear strong force is not a good name for this force because it is not all that strong. The force associated with the formation of spin pairs is effectively an attractive force. Spin pair formation is however exclusive; i.e., one neutron can form a spin pair with only one other neutron and with one proton. The nuclear strong force is not exclusive and it is distance dependent.
The purpose of this webpage is to demonstrate succinctly that the nuclear force between two neutrons and between to protons is repulsion, but between a neutron and a proton is attraction. This has been demonstrated before but here the database is limited to nuclei in which all of the neutrons and all of the protons are spin paired. The units of analysis are then the number of neutron spin pairs and the number of proton spin pairs.
The subject of analysis is the binding energies of nuclei. Binding energy is the mass deficit of a nucleus expressed in energy units via the Einstein formula E=mc². Binding energy is effectively a potential energy.
The binding energy associated with the formation a spin pair is in the range of 2 to 3 million electron volts (MeV). The binding energy associated with the interaction of two nucleon through the nuclear strong force is a fraction of a MeV. It is easy to see how arose the conventional belief that nucleon attract each other in equal measure regardless of the type. Any interaction of small numbers of nucleons, say two as in scattering experiments, would be dominated by the energies involved in spin pair formation. The nuclear strong force is not dominate except in larger nuclides such that the large number of interactions makes up for the small amount involved in each interaction.
Consider a nucleus with N neutron pairs and P proton pairs. The binding energy of such a nucleus is the sum of the binding energies due to the interactions of the nucleon pairs. This involves a neutron pair with other neutron pairs, a proton pair with other proton pairs and a neutron pair with a proton pair. Let n and m be indices for the neutron pairs and F_{nm} be the binding energy due to the interaction of the n-th neutron pair with the m-th neutron pair, G_{pq} the binding energy due to the interaction of the p-th proton pair with the q-th proton pair and H_{np} the binding energy due to the interaction of the n-th neutron pair with the p-th proton pair. To avoid double counting m is always greater than n. Likewise let p and q be indices for the protons with q greater than p. The binding energy of a nucleus with N neutron pairs and P proton pairs is then
The summation for n is from 1 to N and for m from n+1 to N. Likewise the summation for p is from 1 to P and for q from p+1 to P.
The arrangement is depicted visually below. Both the white and colored squares represent binding energies due to nucleon interactions.
Now consider subtracting BE(N-1, P) from BE(N, P). BE(N-1, P) is represented by the colored squares in the display above. All of the interactions of protons with other protons are obviously eliminated. Also the interactions of the protons with the first N-1 neutrons are eliminated. And the interactions among the first N-1 neutrons are eliminated. The result is Δ_{N}(N, P) = ΣF_{nN} + ΣH_{Np}
where the summations above are from 1 to N-1 for n and from 1 to P for p. This is the first difference or increamental binding energy.
The arrangement is depicted visually above as the squares in white.
Consider now two subtractions from Δ_{N}(N, P). First consider the subtraction of Δ_{N}(N-1, P). This produces
This is the interaction of the last neutron pair with the next-to-last neutron pair. Such a quantity is called a second difference.
Next consider the subtraction of Δ_{N}(N, P-1) from Δ_{N}(N, P). The result of this subtraction is
This is the interaction of the last neutron pair with the last proton pair. This is in the nature of a second difference a;so but usually this quantity is called a cross difference.
The same procedure applies with taking the first differences with respect to the number of protons. An interesting thing is that there are two ways of computing the binding energy due to the interaction of the last neutron pair and the last proton pair.
Rather than computing the second differences it is conventient to observe the slopes of the relationship between first differences and the number of nucleon pairs. The first differences will be referred to as the incremental binding energies of neutron pairs, IBENN. Here is the graph of IBENN for P=25, which is the isotopes of Tin (atomic number 50)
The slope is negative indicating the binding energy associated with the interaction of the 25th spin pair of neutrons with the 24th such spin pair of neutrons is negative. Thus the force associated with that interaction is repulsion. The sharp drop at N=41 is due to the filling of the neutron shell (82 neutrons) and subsequent neutrons go into a higher shell.
On the other hand the incremental binding energy of a neutron pair plotted versus the number of proton pairs has a positive slope, thus indicating that the force between the 25th neutron pair and the proton pairs is an attraction.
This correspondence of a positive slope for IBENN plotted versus P and a negative slope for IBENN plotted versus N prevails for all cases except for the isotopes of Helium. (Helium with its small number of nucleons is a special case.) For an exhaustive display of the correspondence see Neutron Pairs and Proton Pairs.
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