The table of data on the binding energies of nuclides displays an interesting phenomenon. Let P be the number of protons and N the number of neutrons. As in the sample displayed below, the number of digits after the decimal point for binding energy (BE) goes through a cycle. This occurs for all segments of the nearly three thousand nuclides.

Elem  P  N   BE
24P	 15  9	150
25O	  8 17	168.4
25F	  9 16	183.48
25Ne  10 15	196.02
25Na 11	14	202.5346
25Mg 12	13	205.58756
25Al 13	12	200.5282
25Si 14	11	187.005
25P	 15	10	171.18
26O	  8	18	168.4
26F	  9	17	184.53
26Ne 10	16	201.6

What is being held constant over the cycle is the total number of nucleons, P+N. In the display the number of protons is increased as the number of neutrons is decreased. This sort of phenomenon occurs for multiples of a fraction. For example, consider the multiples of 1/16.

M  M/16
0   0
1   0.0625
2   0.125
3   0.1875
4   0.25
5   0.3125
6   0.375
7   0.4375
8   0.5

One possible explanation of the phenomenon is that the effects of an additional proton and an additional neutron on binding energy differ by an amount that is some rational number, say

BE_P−BE_N=(r/s)

Thus as P increases by m and N decreases by m the binding energy gets incremented by an amount proportional to 2m(r/s). The fractional value of the binding energy would then depend upon the ratio of m to s.

However there are actually two phenomena involved in the apparent cycle. For any value of the mass number A=P+N, the data ranges from the minimum value of P for a a stable nuclide (maximum N) to the maximum value of P (minimum N). The maximum value of the binding energy occurs somewhere in the middle of that range. For low values of A it occurs where P equals N. The maximum binding energy is strongly correlated with stability. Higher stability corresponds to a longer life and thus the opportunity to obtain greater accuracy in the measurement of mass and hence binding energy. Thus there would be more significant digits in the measured binding energy of the nuclide with the greatest binding energy.

Overwhelmingly the binding energy of the nuclide with the maximum binding energy for a given mass number A is expressed to five significant digits beyond the decimal point. For the nuclide with the minimum binding for a given A the number of significant digits beyond the decimal point is overwhelmingly one. For example, the nuclide with minimum binding energy might be 50.3. The means the binding energy is 50.3±0.05. The range of uncertainty for this case is then 0.1. In general if a value is given to k digits beyond the decimal point then the range of uncertainty is 10^-k. Therefore for the nuclear binding energies the ratio of the range of uncertainty for the least certain to the most certain is about 10⁴. This means in any statisical analysis it is very important to take into account this relative uncertainty of the data points.

Regression Analysis for Data of Varying Certainties

Suppose data are generated according to this scheme

y_j = α + βx_j + u_j
where u_j has mean zero and standard deviation σ_j.

The probability of getting a particular sample depends upon

J = Σ(u_j/σ_j)²
which can be represented as J = Σ((y_j−α−βx_j)/σ_j)²
or, equivalently
J = Σ((y_jσ_j−α(1/σ_j)−βx_j/σ_j)²

The probability for a particular sample depends inversely on J. One approach to obtaining estimates of α and β from a particular sample is to find the values that minimize J and hence maximize the likelihood of the the sample. The values of a and b which minimize J are the same as the regression coefficients obtained when (y_j/σ_j) is regressed on (x_j/σ_j) and (1/σ_j) with no constant.

(To be continued.)