The Limits of Nuclear Stability
San José State University

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The Limits of Nuclear Stability

Nuclei are held together by the formation of spin pairs of nucleons (neutrons and protons) and the mutual attraction between neutrons and protons. The conventional theory conflates these two disparate forces refers to it as the nuclear strong force. The force involved in the spin pairing of nucleons is strong and an attraction but it is exclusive in the sense that one neutron can pair with one proton and one other neutron and no more. There is another force involved and it is nonexclusive. It is the force of nucleonic interaction and will be referred to as the interactive force. The evidence is that neutrons are repelled from each other through the this interactive force. Protons are also repelled from each other not only through the electrostatic force but also through this interactive force. But a neutron and a proton are attracted to each other through the interactive force. Therefore there has to be some balance between the number of neutrons and the number of protons for a nucleus to hold together. If there are too many protons compared to the number of neutrons the repulsion between the protons overwhelms the attraction between neutrons and protons. Likewise if there are too few protons the repulsion between the neutrons overwhelms the neutron-proton attraction.

There is an asymmetry between the numbers of neutrons and protons in stable nuclei that indicates the strength of the repulsion between protons due to the interactive force is greater than that between neutrons. The interactive force may drop off faster with distance than the electrostatic force so the electrostatic repulsion between protons becomes relatively stronger in larger nuclides where the average distance between protons becomes greater.

The situation is made more complicated by the fact that neutrons form spin pairs with each other despite their mutual repulsion through the interactive force and protons do likewise. Also a neutron where possible will form a spin pair with a proton. But spin pair formation is exclusive in the sense noted above. The interaction force between nucleons is not exclusive so a large number of interactions of smaller magnitude may match or exceed the effect of the spin pair formations. Thus spin pair formation is relatively more important for the smaller nuclides.

There are 2931 nuclides stable enough to have had their masses measured and their binding energies computed. For each number of neutrons the minimum number and the maximum number of protons were compiled. The results are displayed in the following graph.

In the graph there is some piecewise linearity displayed.

A Previous study developed evidence that the nucleonic interactive charge of a neutron is of the opposite sign and smaller in magnitude from that of a proton. Let ν denote the ratio of the nucleonic charge of a neutron to that of a proton. The actual value of ν is undoubtedly a simple fraction. Previous work indicated that the relative magnitude of the nucleonic charge of a neutron relative to that of a proton is − 2/3. Furthermore such a difference in charge of the nucleons can account for the limits to the values of the proton numbers of the known nuclides, shown above.

Another study demonstrated that the binding energy increments experienced by additional nucleons to a nuclide is a function of two components. One is simply the difference in the number of protons and neutrons in the nuclide. This component has to do with the formation of a neutron-proton spin pair. The other component has to do with the interaction force between nucleons and it is a function of the net nucleonic charge of the nuclide. If p and n are the numbers of protons and neutrons, respectively, of the nuclide then the net nucleonic charge ζ is

ζ = p − νn

where ν is the magnitude of the nucleonic charge of the neutron relative to that of a proton.

The binding energy associated with the interaction force between nucleons is a nonlinear function of ζ, but for small values of ζ to a reasonable approximation it is kζ, where k is a constant. Nucleons as noted above also form spin pairs. For example, the addition of another neutron to a nuclide with an odd number of neutrons would result in the formation of a neutron-neutron spin pair. Let Enn be the binding energy associated with the formation of a neutron-neutron spin pair. If there are unpaired protons in the nuclide the addition of another neutron would result in the formation of a neutron-proton spin pair with a binding energy of Enp. The binding energies associated with the formation of spin pairs are not really constants independent of the levels of n and p but for the present they are assumed to be constants.

An Additional Neutron

Let IBEn(n, p) be the incremental binding energy of a neutron in a nuclide with n neutrons and p protons; i.e.,

IBEn(n, p) = BEn(n, p) − BEn(n−1, p)

The energy change associated with the addition of another neutron to a nuclide with p protons and n neutrons in which n is odd and less than p is

IBEn = kζ + Enn + Enp

The minimum number of protons for a nuclide with p protons is reached when IBEn≤0. This means that

k(pmin − νn) + Enn + Enp = 0
and hence
pmin = νn + Enn/k + Enp/k

Thus the slope of the relation between pmin and n will give the value of ν.

An Additional Proton

The incremental binding energy of a proton, IBEp(n, p), in a nuclide with n neutrons and p protons is given by

IBEp(n, p) = BEp(n, p) − BEp(n, p−1)

The binding energy of an additional proton to a nuclide with p protons and n neutrons in which p is odd and less than n is

IBEp = −kζ + Epp + Enp
and thus
IBEp = kνn - kp + Epp + Enp

For IBEp to be positive requires a maximum p of

pmax = νn − Epp/k − Enp/k

Thus the slope of the relationship between pmax and n should be ν the same as the slope of the relationship between pmin and n. p>The relationships are not linear over the whole range of values of n but over some intervals they are reasonably close to being linear. For the maximums an approximating line goes from the nuclide with 31 neutrons and 34 protons to the nuclide with 88 neutrons and 77 protons. The difference in the numbers for this line is 57 neutrons and 43 protons and. The ratio of these two quantities is 0.75439. This should be ν.

For the minimums an approximating line runs from the nuclide with 49 neutrons and 28 protons to the nuclide with 94 neutrons and 55 protons. The difference in the proton numbers is 27 and in the neutron numbers 45. Their ratio is 0.6. This is an alternate estimate of ν.

Thus the maximum proton line gives an estimate of ν of (3/4) and the minimum proton line an estimate of (3/5). The average of these two values is 27/40=0.675, or about 2/3. This is the magnitude of the ratio of the nucleonic charge of a neutron to that of a proton; the numerical ratio is −2/3.

Another study shows that a given value of (n+p) the nuclide with the minumum energy and hence the one likely to be most stable is

n = (1/nu;)p + ½(1 −1/ν)

For ν=2/3 this evaluates to

n = (3/2)p −1/4

The stablest nuclides typically do have 50 percent more neutrons than protons.

APPENDIX

The data for the previous graph are displayed below.

The Maximum and Minimum
Number of Protons Possible
for Each Level of the
Number of Neutrons
#n min #p max #p
1 1 4
2 1 6
3 1 7
4 1 8
5 1 9
6 2 10
7 2 11
8 2 14
9 3 15
10 4 16
11 5 17
12 5 18
13 5 19
14 5 20
15 6 21
16 6 22
17 7 23
18 8 24
19 9 26
20 9 26
21 10 27
22 10 28
23 11 29
24 11 30
25 12 31
26 13 32
27 14 33
28 14 33
29 15 33
30 15 33
31 15 34
32 16 35
33 16 36
34 17 37
35 18 38
36 19 38
37 20 38
38 21 39
39 22 40
40 23 41
41 24 42
42 25 43
43 26 44
44 27 45
45 27 46
46 28 46
47 28 47
48 28 48
49 28 49
50 28 50
51 29 50
52 30 51
53 31 51
54 32 52
55 33 53
56 33 54
57 34 55
58 34 56
59 35 56
60 36 57
61 36 58
62 37 59
63 37 59
64 37 59
65 37 59
66 38 60
67 39 61
68 40 62
69 41 63
70 42 63
71 42 63
72 43 64
73 44 65
74 44 66
75 45 67
76 45 68
77 46 69
78 47 70
79 47 71
80 47 71
81 48 71
82 48 72
83 49 73
84 49 74
85 49 75
86 50 76
87 50 76
88 51 77
89 52 77
90 52 78
91 53 78
92 54 79
93 54 79
94 55 79
95 55 80
96 55 81
97 56 81
98 57 81
99 58 82
100 59 82
101 60 82
102 61 83
103 62 83
104 63 83
105 64 83
106 65 84
107 66 85
108 68 85
109 68 85
110 69 86
111 70 86
112 71 86
113 71 87
114 72 87
115 73 88
116 74 88
117 75 88
118 76 89
119 76 89
120 76 90
121 77 90
122 77 91
123 78 91
124 78 91
125 79 91
126 79 92
127 80 92
128 80 92
129 81 92
130 82 92
131 82 92
132 82 93
133 83 93
134 84 94
135 85 94
136 85 95
137 85 95
138 85 96
139 86 98
140 86 98
141 86 99
142 86 100
143 87 100
144 87 101
145 87 101
146 88 101
147 89 102
148 90 103
149 91 104
150 92 105
151 93 105
152 94 106
153 94 107
154 95 108
155 96 108
156 96 109
157 97 110
158 98 110
159 100 110
160 102 110
161 108 111
162 109 110
163 110 110

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