Interactions between Nucleons: Fusion and Dissociation of Nuclei in Stellar Evolution
O. Manuel, C. Bolon, A. Katragada, M. Insall, and G. Bertrand
University of Missouri-Rolla
(Report to the Foundation for Chemical Research, Inc., 06/08/2001)
Abstract: The quadratic relationship between Z/A and M/A in the Cradle of the Nuclides  is interpreted in terms of interactions between nucleons, assuming that each nucleon interacts with every other nucleon except itself. Aside from Coulomb effects, the n-n and p-p interactions are equivalent, repulsive, and weaker than the attractive n-p interactions. For most values of A, forces of attraction and repulsion cancel at Z/A approximatly equaling 0.15 and 0.85. Repulsive interactions between like nucleons increase their potential energy in the interior of nuclei by about 10 MeV per nucleon; attractive interactions between unlike nucleons decrease their potential energy by about 20 MeV per nucleon. These forces are weaker in lighter nuclei and may not be observed in interactions between single nucleons. Nuclear stability is produced mostly by the interplay of these forces plus Coulomb repulsion.
First generation stars may derive their energy from fusion, and the collapsed supernova core produced in the terminal, "pile-driven" fusion reactions may serve as the accretion site for second generation stars. The repulsion between neutrons may be the driving force for neutron emission from the collapsed core of the supernova that produced our elements [2-6].
Figure 1 is a plot of mass per nucleon, M/A, vs. charge per nucleon, Z/A, for all nuclides in the sixth edition of Nuclear Wallet Cards . Radioactive nuclides are represented by open symbols; stable and long-lived nuclides are represented by filled symbols.
Most nuclides lie within a triangle bounded by three nuclides with extreme values of M/A and Z/A. The neutron (on the left) has the largest value of M/A and the lowest value of Z/A of any nuclide. The Fe-56 nuclide (near the center) has the lowest value of M/A. The H-1 atom (on the right) has the highest value of Z/A. It also has the highest value of M/A among stable and long-lived nuclides.
The data in Figure 1 can be sorted by mass number, A, or atomic number, Z, or neutron number, N, to show the relationship between energy and charge density for isobars, isotopes, or isotones. This was shown in our last report .
Figure 2 is the family of isobaric mass parabolas defined by the data after sorting into a third dimension represented by the mass number, A. The more stable nuclides lie along the valley of this trough that we called the "Cradle of the Nuclides" in the earlier report to FCR . Nuclides that are radioactive or are readily destroyed by fusion or fission occupy higher positions in the cradle.
A slice through this cradle at any given value of A yields the familiar isobaric mass parabola. This is illustrated in Figure 3 for A = 27. This is a typical, odd-A mass parabola, and we will use it to illustrate the interactions between nucleons. For comparison, the values of M/A are also shown in Figure 3 for unbound nucleons, 1n and 1H, at Z/A = 0 and Z/A = 1, respectively.
The Coulomb energy, Ec, associated with the repulsion between the Z positive nuclear charges increases from left to right in Figure 3 as the number of protons increases. Owing to a large contribution from Ec, the value predicted for M/A on the right side of the figure is off-scale, for example, at Z/A = 0 where the nucleus consists of 27 protons in 27Co. This Coulombic repulsion between positive nuclear charges is probably the best understood of all the interactions between nucleons.
To elucidate the other interactions between nucleons, we therefore first subtract the contribution of Ec from each of the known nuclide masses at A = 27: 27F, 27Ne, 27Na, 27Mg, 27Al, 27Si, 27P, 27S. For this correction, we used the recently determined value  of Ec = 0.702 MeV (Z2/A1/3).
Figure 4 shows the "best fit" mass parabola after the subtraction of Coulomb energy, Ec, from the rest mass of each of the eight known isobars at A = 27. Note that this subtraction also reduced the rest mass of the 1H nuclide, shown on the right side of Figure 4, from 1.0078250 amu to 1.0070714 amu. After correcting for Coulomb energy, the parabola is much more symmetric. Its shape suggests that the n-n and p-p interactions are repulsive, increasing the average energy per nucleon above that of free nucleons, while the n-p interactions are attractive, decreasing the average energy per nucleon below that of free nucleons.
The n-n and p-p interactions seem to be nearly identical. This may explain why, before subtraction of Ec, differences between values of M/A at the intercepts where Z/A = 0 and Z/A = 1 varied in a regular manner with A in just the manner expected from differences in Ec . In the case illustrated in Figure 4, the n-n and p-p interactions are indistinguishable after subtraction of Ec. The intercept values from the parabola are (M/A)Z/A = 0 = M(1n) + 9.76 Mev and (M/A)Z/A = 1 = M(1H - Ec) + 9.71 Mev.
The symmetry of the n-n and p-p interactions can also be illustrated by noting that the ratio of the masses of the unbound nucleons is the same as that for the masses on the parabola at the interecpts in Figure 4,
(M/A)Z/A=0/(M/A)Z/A=1 = 1.0000 [M(1n)/M(1H - Ec)]
Thus, the potential energy of neutrons and protons each seem to increase by about 10 MeV in the presence of like nucleons, in addition to any changes caused by Coulombic interactions.
Countering this disruptive force between like nucleons is an attractive interaction between unlike nucleons. In a nuclide consisting of Z protons and N neutrons, if each nucleon interacts with every other nucleon in the nucleus then the number of n-n interactions would be (N)(N-1)/2, the number of p-p interactions would be (Z)(Z-1)/2, and the number of n-p interactions would be (N)(Z). Table 1 shows the number of n-n, p-p, and n-p interactions for each isobar at A =27.
The last column in Table 1 (column 6) gives a weighted net number of interactions, assuming that the n-n and p-p interactions are equal and that the n-p interactions have an opposing contribution that is empirically estimated to be 2.5 times greater than the n-n or p-p interactions.
Figure 5 is a plot of the weighted net interactions vs charge density for the 28 isobars at A = 27. The excellent match between Figure 4 and Figure 5 seems to support the idea that the number of interactions between nucleons governs nuclear stability and that the n-n and p-p interactions are repulsive and less than half as strong as the attractive n-p interactions at A = 27.
In comparing the values of M/A at different values of A, two other terms must be considered. The total number of interactions of a given nucleon with any other nucleon in (A-1) and the net effect per nucleon also introduces the term 1/A. The net effect, 1/A(A-1), is constant at any given value of A, as in Figures 3-5, but the numbers of nucleons interactions divided by A(A-1) should be used in comparing nuclides of different A.
As noted above, interactions between nucleons have a decreased effect on the values of M/A (potential energy per nucleon) for lower values of A. The weak binding of nucleons in 2H illustrates this for the n-p interaction. The six known isotopes of hydrogen show this for the n-n interactions.
Figure 6 shows the values of M/A vs Z/A for the six isotopes of H. Hydrogen isotopes with odd numbers of neutrons (6H, 4H and 2H) define the shallow parabola on the left. Those with even numbers of neutrons (5H, 3H and 1H) define the deeper parabola on the right. At Z/A = 0, the shallow parabola yields an intercept of M/A = M(1n) + 0.7 MeV. The other parabola yields an intercept of M/A = M(1n) + 4.6 MeV. Both of these are substantially less than the intercepts for higher values of Z. Interactions between odd numbered neutrons in H seem especially weak.
The interactions concluded here between nucleons offer new insight into the source of stellar energy. Fusion has been widely believed to be the energy source for the sun and other stars. Burbidge et al.  showed that elemental and isotopic abundances in the solar system can be understood in terms of reasonable nuclear reactions that might occur as a first generation star, consisting initially of hydrogen, underwent normal stages of stellar evolution up to and including its terminal explosion as a supernova (SN). If the Sun, a second generation star, formed on the collapsed SN core [1-5], then repulsion between neutrons may be the driving force for neutron emission from the collapsed core of the supernova that produced our elements, and this may be the first, and the rate-determining, step in the production of solar luminosity and the sun's outward flow of solar-wind protons . Figure 7 illustrates this cycle of fusion and dissociation producing energy in first and second generation stars from the attractive and repulsive interactions of nucleons