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The Statistical Relationships
for the Stable Nuclides

Nuclei are composed of nucleons (protons and neutrons) in variable proportions. There are almost three thousand nuclides that are stable enough to have their mass measured and their binding energies computed. But most are unstable. The following beautiful display from Wikipedia shows the nature of their instabilities.

Only the ones shown in black in the middle of the distribution are stable. The proton numbers p and neutron numbers n for those stable nuclides are given in the Appendix. What is sought here is the statistical relationship between the proton and neutron numbers of stable nuclides. Their neutron numbers could be regressed upon their proton numbers. Or equally well the proton numbers could be regressed upon their neutron numbers. But these two regressions would give somewhat different relationships and neither has any claim to being the right relationship.

Instead a different approach is taken. The stability of each of the stable nuclides is taken to be unity; i.e., fully stable. Then a linear function of their p and n values is sought which best approximates this stability; i.e.,

1 ≅ cpp + cnn

No constant term is allowed because the regression analysis would just give 1=1.

The regression equation found is

1 ≅ 0.145808276p − 0.092720326n
[9.4] [−7.7]

The number in a square brackets below a coefficient is its t-ratio, the ratio of the coefficient to its standard deviation. For a coefficient to be statistically significantly different from zero its t-ratio must be greater than or equal to 2 in magnitude.

The coefficient of determination (Rsup2;for this equation is 0.8615.

Now n can be solved for. The result is

n = 1.572560002p − 10.7851217

The coefficient of determination (R²) for this equation is 0.8615. It can be improved by including quadratic terms in p and n. :

The above equation is familiar. In another study the values of n for p≥26 were regressed on p.

The result was:

n = 1.57054p − 10.83610

Here n represents an average of all n, stable and unstable.

If the regression equation for stable nuclides is used to estimate n and the actual values are plotted versus those values the result is:

These regression estimates do not do very well for the small nulides. The statistical fit is improved by including in the regression quadratic terms for p and n. The coefficient of determination (R²) for that equation is 0.9454

The intriguing aspect of this equation is that it seems to correspond to the equation for minimizing energy and hence maximizing the binding energy due to nucleon inter action. See Energy minimization for the details.

However if the number neutrons in stable nuclides are regressed up a quadratic function of p the coefficient of determination is 0.99017. The equation is

n = 1.089712498p + 0.004997229p² −0.499982822
[22.1] [5.7] [−0.8]

Here n repesents an average number of neutrons for the stable nuclides with proton number p. The graph of the actual number of neutrons in stable nuclides plotted against the regression estimates based upon the above equation is shown below.

Note that

(dn/dp) = 1.089712498 + 0.009994458p
and for p=50
(dn/dp) = 1.589435398

Conclusions

From various approaches for the stable nuclides the relationship between the neutron number n and the proton number p is approximately

n = 1.572560002p − 10.7851217
and
(dn/dp) ≅ 1.57

This corresponds to a neutron having a nucleoni charge of −2/3 relative to a proton having a nucleonic charge of +1. The ratio of the electrostatic force to the nucleonic interaction force between protons is 0.08. rum ,,

Appendix

The Proton and
Neutron Numbers
for the Stable
Nuclides
p n stable
1 0
1 1
2 1
2 2
3 3
3 4
4 5
5 5
5 6
6 6
6 7
7 7
7 8
8 8
8 9
8 10
9 10
10 10
10 11
10 12
11 12
12 12
12 13
12 14
13 14
14 14
14 15
14 16
15 16
16 16
16 17
16 18
16 20
17 18
17 20
18 18
18 20
18 22
19 22
20 20
20 22
20 23
20 24
20 26
21 24
22 24
22 25
22 26
22 27
22 28
23 28
24 26
24 28
24 29
24 30
25 30
26 30
26 31
26 32
27 32
28 30
28 32
28 33
28 34
28 36
29 34
29 36
30 34
30 36
30 37
30 38
30 40
31 38
31 40
32 38
32 40
32 41
32 42
33 42
34 40
34 42
34 43
34 44
34 46
35 44
35 46
36 44
36 46
36 47
36 48
36 50
37 48
38 46
38 48
38 49
38 50
39 50
40 50
40 51
40 52
40 54
41 52
42 50
42 52
42 53
42 54
42 55
42 56
44 52
44 54
44 55
44 56
44 57
44 58
44 60
45 58
46 56
46 58
46 59
46 60
46 62
46 64
47 60
47 62
48 58
48 60
48 61
48 62
48 64
48 66
49 64
50 62
50 64
50 65
50 66
50 67
50 68
50 69
50 70
50 72
50 74

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