San José State University

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Thayer Watkins
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An Explanation for the
Maximum Number of Neutrons
in the Isotopes of Each Element

Background

A nucleus consists of a core of concentric spherical shells having equal numbers of protons and neutrons. These shells are the result of rings of protons and neutrons rotating at fantastically high rates. See Nucleus for the details. Beyond the core there are usually neutrons, called halo neutrons, rotating about the core in shells.

In a previous study it was found that the limit to the number of neutrons can be explained by the shielding of the outermost neutron from the net nucleonic charge of the core by the neutrons in inner shells and in the same shell. More precisely it is the shielding by neutrons in inner shells and subshells and in the same subshell. The shielding ratio is theoretically approximately unity for neutrons in the inner shells and subshells and approximately one half for neutrons in the same subshell. However, at this point the sizes of the subshells in each shell are not definitely known so only the numbers in the same shell (SSmaxn) and in the inner shell (ISmaxn) were computed for each element.

Physical Characteristics for Isotopes with the Maximum Number of Neutrons
#p max #n net nuclear
charge in core
(#p/3) 		#n in
same shell
(SSmaxn) 		#n in
inner shells
(ISmaxn)
1 5 0.333333333 3 0
2 8 0.666666667 0 0
3 9 1 2 0
4 10 1.333333333 3 0
5 14 1.666666667 7 0
6 16 2 1 6
7 17 2.333333333 2 6
8 18 2.666666667 3 6
9 20 3 5 6
10 22 3.333333333 7 6
11 24 3.666666667 9 6
12 25 4 10 6
13 26 4.333333333 11 6
14 28 4.666666667 13 6
15 31 5 2 14
16 33 5.333333333 4 14
17 34 5.666666667 5 14
18 35 6 6 14
19 36 6.333333333 7 14
20 37 6.666666667 8 14
21 38 7 9 14
22 39 7.333333333 10 14
23 40 7.666666667 11 14
24 41 8 12 14
25 42 8.333333333 13 14
26 43 8.666666667 14 14
27 45 9 16 14
28 50 9.333333333 21 14
29 51 9.666666667 0 28
30 52 10 1 28
31 53 10.33333333 2 28
32 54 10.66666667 3 28
33 55 11 4 28
34 58 11.33333333 7 28
35 59 11.66666667 8 28
36 61 12 10 28
37 65 12.33333333 14 28
38 66 12.66666667 15 28
39 67 13 16 28
40 68 13.33333333 17 28
41 69 13.66666667 18 28
42 71 14 20 28
43 72 14.33333333 21 28
44 74 14.66666667 23 28
45 76 15 25 28
46 77 15.33333333 26 28
47 80 15.66666667 29 28
48 82 16 31 28
49 85 16.33333333 2 50
50 87 16.66666667 4 50
51 88 17 5 50
52 90 17.33333333 7 50
53 91 17.66666667 8 50
54 93 18 10 50
55 96 18.33333333 13 50
56 97 18.66666667 14 50
57 98 19 15 50
58 99 19.33333333 16 50
59 100 19.66666667 17 50
60 101 20 18 50
61 102 20.33333333 19 50
62 103 20.66666667 20 50
63 104 21 21 50
64 105 21.33333333 22 50
65 106 21.66666667 23 50
66 107 22 24 50
67 108 22.33333333 25 50
68 109 22.66666667 26 50
69 110 23 27 50
70 111 23.33333333 28 50
71 113 23.66666667 30 50
72 114 24 31 50
73 115 24.33333333 32 50
74 116 24.66666667 33 50
75 117 25 34 50
76 120 25.33333333 37 50
77 122 25.66666667 39 50
78 124 26 41 50
79 126 26.33333333 43 50
80 128 26.66666667 1 82
81 129 27 2 82
82 132 27.33333333 5 82
83 133 27.66666667 6 82
84 134 28 7 82
85 138 28.33333333 11 82
86 142 28.66666667 15 82
87 145 29 18 82
88 146 29.33333333 19 82
89 147 29.66666667 20 82
90 148 30 21 82
91 149 30.33333333 22 82
92 150 30.66666667 23 82
93 151 31 24 82
94 153 31.33333333 26 82
95 154 31.66666667 27 82
96 156 32 29 82
97 157 32.33333333 30 82
98 158 32.66666667 31 82
99 158 33 31 82
100 159 33.33333333 32 82
101 159 33.66666667 32 82
102 160 34 33 82
103 160 34.33333333 33 82
104 160 34.66666667 33 82
105 160 35 33 82
106 160 35.33333333 33 82
107 160 35.66666667 33 82
108 160 36 33 82
109 162 36.33333333 35 82
110 163 36.66666667 36 82

Empirical Results

The relation to be fulfilled for each element is that

#p/3 −cISISmaxn−cSSSSmaxn ≅ 0
and hence
#p/3 ≅ cISISmaxn + cSSSSmaxn

When the net nucleonic charges of the core (#p/3) was regressed upon ISmaxn and SSmaxn the results were:

#p/3 = 0.31235ISmaxn + 0.25919SSmaxn
[130.6] [43.0]

It is notable that both regression coefficients are po sitive and the one for ISmaxn is larger than the one for SSmaxn.

The coefficient of determination (R²) for this equation is 0.9991. The t-ratios shown below the coefficients indicate that there is only an infinitesimal probability that the relationship between #p/3 and ISmaxn and SSmaxn is due only to chance.

The ratio of the same shell shielding ratio to that for the inner shell neutons is about 0.83 rather than the 0.5 previously supposed. This can be accounted for by many of the neutrons in the same shell being in subshells below the subshell of the highest neutron.

The logic of the relationship is that

#p/3 −cISISmaxn−cSSSSmaxn = ε
for a positive ε
so an isotope
with maxn neutrons
does exist but
one for maxn+1
cannot exist

For maxn+1

#p/3 −cISISmaxn−cSS(SSmaxn+1) = −ε

This implies that −cSS=2ε and hence ε=−cSS/2. Thus from the above regression results ε=0.130.

This suggests that a regression of the form

#p/3 = c0 + cISISmaxn + cSSSSmaxn

would be of interest. The results found were

#p/3 = 0.215340861 + 0.31034ISmaxn + 0.25919SSmaxn
[1.7] [117.1] [39.4] .

The value for c0 is not statistically significantly different at the 95 percent level of confidence from the value found for ε previously.

Conclusions

The existence of the isotopes with a maximum number of neutrons and the nonexistence with one more neutron can be explained by the shielding of the maximum neutron by the halo neutrons in inner shells and those in the same shell.

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