Nuclei are composed of neutrons and protons in various proportions. Tin is the element with the largest number of known isotopes. They range from a minimum of 50 to a maximum of 87. The number of protons is 50. Thus the ratio of neutrons to protons ranges from 1.00 to 1.74.

In the conventional theory of nuclear structure all nucleons are attracted equally to each other by the hypothetical so-called strong force. In this theory there is nothing which limits the nucleonic composition of nuclides. There could be nuclides which are 90 or even 100 percent neutrons. Or zero percent neutrons. Such do not exist.

There is evidence that like nucleons repel each other and unlike ones attract. This can be explained by neutrons and protons having nucleonic charges. If the nucleonic charge is taken as 1 then a regression analysis of nuclear binding energies indicates that the nucleonic charge of the neutron is −2/3.

Nuclear binding involves two phenomena. One is spin pairing of nucleons; a proton with a proton, a neutron with a neutron and a neutron with a proton. This spin pairing is exclusive in the sense that a neutron can spin pair with one other neutron and with a proton and no more. The same applies to a proton.

The other phenomenon involved is interactions among the nucleons. This is not exclusive, but the magnitude of an interaction in binding energy is an order of magnitude smaller than that of a spin pairing.

The Limits of the Known Nuclides

The graph of the maximums and minimums of the numbers of neutrons of known nuclides is given below.

One possibility is that the limits have to do with the level of binding energy, say that beyond the limits the binding energy is negative. That is not the case. There are high levels at all extremes. Another possibility is that the stability for a nuclide might depend upon its net nucleonic charge. If n and p are the numbers of the neutrons and protons, respectively, the net nucleonic charge of a nuclide C is given by

C= p + (−2/3)n

The graph of C versus p is given below

This has possibilities. For p<80 it is as though a value of C for a nuclide is less than −5 means there are too many neutrons for a stable nuclide. On the other hand if the value of C is greater than 17 then there are not enough neutrons for a stable nuclide.

The problem is for the low values of p in which the values of C for the extremes are roughly proportional to p. More generally there are piecewise linear segments which establish the limits.

Empirical Determination
of Nuclear Limits

An alternative to a piecewise linear function is a quadratic function; i.e.

L = c₁p + cp² + c₀

When the equation is used to estimate the values of the minimum number of neutrons in nuclides the results

L_min = 0.60337p + 0.00803p² − 0.81144
[20.6] [31.2] [-1.1]

The coefficient of determination (R²) is 0.9974. The number in square brackets below a coefficient is its t-ratio; i.e. the ratio of the coefficient to its standard deviation. the t-ratio must greater than 2.0 for the coefficient to be significant different from zero at the 95 percent level of confidence,

For the maximum number of neutrons they are

L_max = 1.71331p − 0.00183p² + 3.5051
[50.6] [-6.1] [4.3]

The coefficient of determination (R²) is 0.9966.

When the regression equation is used to estimate the minimum numbers of neutrons and those values are plotted along with the actual numbers the lines nearly coincide, as shown below:

Essentially the same thing prevails for the maximum numbers of neutrons:

Conclusions

The limits of maximum and minimum number of neutrons as a function of their proton number for the known nuclides can be expressed as quadratic functions