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The Binding Energies Due to the Formation of Neutron-Neutron and Proton-Proton Spin Pairs |
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Dedicated to Chuck
A friend for sixty years
and an always willing reader
A nucleus is held together by the net result of two types of forces. One is the so-called nuclear strong force and the other is the forces associated with the formation of spin pairs; neutron-neutron, proton-proton and neutron-proton. The binding energies involved in spin pair formation are relatively large, on the order of two to three million electron volts (MeV). The strong force binding energy involved in a single interaction of two nucleons is a only fraction of 1 MeV. But the spin pair formation involves an exclusivity in the sense that one neutron can form a spin pair with only one neutron and with only one proton. The strong force interaction is not exclusive so m nucleons will have ½m(m-1) interactions. Thus in nuclei involving only a small number of nucleons the binding energies created in the formation of spin pairs are dominant, but in larger nuclei the large number of interactions results in the binding energies due to the strong force being dominant.
The material below is an attempt to estimate the magnitudes of the binding energies association with the formation of neutron-neutron and proton-proton spin pairs. This anaysis was carried out in two previous studies for nuclides involving 12 protons (magnesium isotopeco s) and 24 protons (chromium isotopes) with satisfactory results. But the values found differed. What is needed is an analysis that takes into account as many different nuclides as possible.
The binding energy due to neutron-neutron spin pair formation is manifested in the variation in the incremental binding energies of neutrons in nuclides with different numbers of neutrons, as shown below.
The enhancement of incremental binding energy when an additional neutron completes a spin pair involves the formation of a neutron-neutron spin pair but something more is probably involved such as a rearrangement of the other nucleons as a result of the formation of the spin pair.
The ampliltude of the odd-even fluctuations in the incremental binding energies of neutrons can be computed by taking the absolute value of the difference between the incremental binding energy of neutrons at n neutrons and the average of the values at n-1 and n+1 neutrons. This is depicted in the diagram below in which the black dots represent the values of incremental binding energies of a neutron. The red dot represents the average binding energy of the adjacent cases and the length of the red line is the estimate of the binding energy due to the formation of the neutron-neutron spin pair.
However this value cannot be computed at lowest and the highest level in a range such as the isotopes of an element. Furthermore there are situations in which the incremental binding energies drop sharply for an additional neutron. Therefore for this method of estimation at n neutrons to be valid neither of the values at n-1 and n+1 may involve a sharp change in values due to other factors. One of those other factors is the associated with the formation of neutron-proton spin pairs. If the number of neutrons n is less than the number of protons p then an additional neutron will form a neutron-proton pair. But once n is greater than p an additional neutron will not form a neutron-proton spin pair. This means that the cases for which n=p and n=p+1 have to be eliminated. Also when a neutron shell is filled the incremental binding energy of an additional neutron drops because the additional neutron goes into a higher shell. The numbers at which neutron shells are filled are called magic numbers. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. It also has been found that 6 and 14 are magic numbers. Thus the cases for which n is equal a magic number m or m+1 have to be eliminated.
However the data for the cases involving n less than p involve the formation of neutron-proton spin pairs as well as neutron-neutron spin pairs.
After those cases have been eliminated the for the incremental binding energies of neutrons the data set is reduced from about 2800 to 2251. The average of the estimates of the binding energy involved in the formation of a neutron-neutron pair is 1.956 million electron volts (MeV) and the standard deviation is 0.648 MeV. The histogram of these estimates is
This distribution is remarkably close to a normal (Gaussian) distribution.
The same analysis can be applied to estimate the binding energy due to the formation of a proton-proton spin pair. However the cases for which the number of protons is greater than the number of neutrons is relatively rare so the data set is reduced to only 57. The average of the values of the binding energies due to the formation of proton-proton spin pairs is 2.856 MeV and the standard deviation is 0.682 MeV. This average is notably higher than the value of 1.956 MeV for neutron-neutron spin pairs, but given the values of the standard deviations the difference is not statistically significant at the 95 percent level of confidence. The histogram for the estimates of the binding energy due to the formation of a proton-proton spin pair is
This is not notably close to a normal distribution but given the smaller size of the data set it is not unreasonable to presume that the population distribution for these estimates is normal.
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