Background

Nuclei are held together largely by the spin pairing of the nucleons (protons and neutrons) Such spin pairins are exclusive; meaning one neutron can pair with a proton and with one other neutron and no more. The same applies for protons.

There are other nonexclusivee interactions between nucleons but each each such interaction is an order of magnitude smaller than that of a spin pairing.

pin It is worthwhile to estimate the increments in binding energy due to the formation of spin pairs and explain their variation in terms of the makeup of the nuclei.

The binding energy due to the formation of a spin pair can be computed as the difference in the incremental binding energy at one point and the average of the value at the two adjacent points, as shown below.

This procedure is not valid near where the incremental binding energy makes a big drop.

The results from applying the procedure to the incremental binding energies of neutrons are given below. They include 2368 cases with an average effect of 2.089 million electron volts (MeV) and a standard deviation of 0.777 MeV.

The frquency distribution of the effect of the formation of a neutron-neutron spin pair is shown below.

The database compiled can be used to test the dependence of the increment of binding energy due to the formation of a neutron-neutron spin pair (BEnnSP)

BEnnSP = 3.06881 + 0.02343p −0.02845n
[103.3] [11.4] [-21.2]

The coeffient of determination (R²) for this equation is an unimpressive 0.479 but the t-ratios, shown in square brackets [ ] below the the coeffient, show that BEnnSP and BEppSP are definitely a function of n and p. However the dependence is likely a complicated nonlinear binary function. A comparison of cases in which the values of n and p are interchanged shows that the binary function has a degree of symmetry with respect to n and p. See Comparisons for examples.

A bit more sophisticated model has the BEnnSP being a function of the shell number of the proton creating the spin pair, This involves creating a sequence of variables having the value 1 if the proton is in the shell and 0 if it is not. The shell ranges are shown below:

These are not the conventional shell ranges which have shells filling at 8 and 20 nucleons, respectively. These values, 8 and 20, are taken to be the endpoints of subshells. See Subshells for more on this topic. There is a simple algorithm, analogous to the one explaining the shell ranges for electrons in atoms.