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Attraction and Repulsion in Nuclides Composed
Entirely of Alpha Modules and Neutron Spin Pairs

Background

The details of the alpha module model of nuclear structure are given elsewhere. Briefly the basis for it is that whenever possible a nucleon (neutron or proton) forms a spin pair with one other nucleon of the same type and with one nucleon of the opposite type. This leads to chains composed of modules of the form -n-p-p-n- (or equivalently -p-n-n-p-). These chains form rings which rotate in four modes; as a vortex ring, rotation about an axis through the center of the ring and perpendicular to its plane and flipping rotation about two diameters of the ring. These rotations take place so rapidly that the alpha module ring dynamically appears as a sphere with the nucleons smeared uniformly throughout a spherical shell.

The occupancies of the spherical shells are basically what were found by Maria Goeppert Mayer and Hans Jensen and labeled nuclear magic numbers. The Goeppert Mayer and Jensen magic numbers were based upon the numbers of stable isotopes and isotones. Their values were (2, 8, 20, 28, 50, 82, 126). The patterns of incremental binding energies justify a modified sequence of (2, 6, 14, 28, 50, 82, 126). The numbers 8 and 20 instead represent the filling of subshells within shells.

The Forces Involved in a Nucleus

There are three types of forces involved in maintaining nuclear structure. First there are the forces involved the formation of spin pairs. This type of force is exclusive. Second there is the nonexclusive, distance dependent force usually called the nuclear strong force. The name is inappropriate because it is not all that strong compared with the forces involved in spin pair formation. The formation of a spin pair results in an increase in binding energy on the order of two to three million electron volts (MeV). The interaction of two nucleons throught the so-called strong force results in a change in bindinging energy of only a fraction of 1 MeV. A better name for the nuclear strong force is the nucleon force, the force between nucleons. In larger nuclei the force between one nucleon and all rest of the nucleons of the nucleus is larger than that involved in spin pair formation. The third force is the electrostatic (Coulomb) repulsion between protons, which is less powerful than the nucleonic force at small distances but greater at greater distances. In other words, the nucleon force drops off faster than inverse distance squared.

It must be noted that the alpha modules in a ring are held together not by the nucleon forces but by the linkages of spin pairs. The nucleonic force is however important in the internal structure of the alpha modules.

Attraction and Repulsion in Nuclides

When the force F between two particles is of the form

F = −f(s)

where f(s) is a positive function of the separation distance s, then the force is an attraction that tends to reduce the separation distance. If the form is F=+f(s) then it is a repulsion that tends to increase the separation distance.. nuclear terms, there is an increase in binding energy. Likewise under the action of a force of repulsion there is a decrease in binding energy. Therefore if some action results in a decrease in binding energy the force involved is a repulsion and if it results in an increase in binding energy the force is an attraction.

Nuclei are ultimately composed of neutrons and protons but those n.ucleons are whenever possible combined into spin pairset;euutron-neuutron, proton-proton and neuutron-proton. These spin pairs are further organized into alpha modules which are linked together into chains which close to form rings. There can be neutron and proton spin pairs in a nucleus outside of the alpha module rings.

In order to examine the question of attraction and repulsion between alpha modules and neutron spin pairs the binding energies of nuclides composed entirely of these structures were compiled.

The Binding Energies of Nuclides Composed
Entirely of Alpha Modules and Neutron Spin Pairs
Number of Neutron-Neutron Spin Pairs
0 1 2
0 0.0
1 28.29567429.2691 31.408
2 56.49951 64.9772 68.65
3 92.161728105.284508110.753
4 127.619336 139.807 151.3707
5 160.644859 177.76991 191.836
6 198.25689 216.68063 231.6275
7 236.53689 255.61963 271.4103
8 271.78066 291.83916 308.71394
9 306.7157 327.3427 343.81044
10 342.052 361.8953 380.9602
11 375.4747 398.1944 418.6987
12 411.462 435.0441 456.3451
13 447.697 471.7587 492.2539
14 483.988 506.4538 526.8416
15 514.992 538.119 559.0936
16 545.95 569.29 590.792
17 576.4 600.33 622.43
18 607.1 631.28 654.235
19 638.1 663.008 686.281
20 669.8 694.7 718.19
21 700.9 725.8 750.118
22 731.4 757.4 782.5
23 762.1 789.1 815.03
24 793.4 820.9 843.88
25 824.9 848.9 871.85
26 872.9 896.7
27 921.67
28 946.8
29 972.9

The Incremental Binding Energies of Alpha Modules

If the binding energy BE is given as a function of the number of alpha modules #α and the number of neutron spin pairs #nn then the incremental binding energy IBEα of an alpha module is given by

IBEα(#α, #nn) = BEα(#α, #nn) − BEα(#α−1, #nn)

And likewise the incremental binding energy IBEnn of a neutron spin pair is given by

IBEnn(α, #nn) = BEα(#α, #nn) − BEα(#α, #nn−1)

The Incremental Binding Energies
of Alpha Modules

   

There are similarities and differences.

These graph reveals that there are changes in the level of the incremental bindind energies at critical values of #α. One critical value is #α equal to 14, which corresponds to 28 neutrons and 28 protons. Twenty eight is a magic number corresponding to the filling of a shell. Another critical number is at #α equal to 4, which corresponds to 8 neutrons and 8 protons. Another is for #α equal to 7 which corresponds to 14 neutrons and 14 protons. The changes in slope are not so sharp as is the case with the alpha nuclides.

When the relationships of IBE to #α are plotted together the correspondence is more easily seen.

The Incremental Binding Energies
of Neutron Spin Pairs in Nuclides
Composed Entirely of Alpha
Modules and Neutron Spin Pairs
IBEnn(#α, 1) IBEnn(#α, 2) Second
Difference
1 0.973426 2.1389 1.165474
2 8.47769 3.6728 -4.80489
3 13.12278 5.468492 -7.654288
4 12.187664 11.5637 -0.623964
5 17.125051 14.06609 -3.058961
6 18.42374 14.94687 -3.47687
7 19.08274 15.79067 -3.29207
8 20.0585 16.87478 -3.18372
9 20.627 16.46774 -4.15926
10 19.8433 19.0649 -0.7784
11 22.7197 20.5043 -2.2154
12 23.5821 21.301 -2.2811
13 24.0617 20.4952 -3.5665
14 22.4658 20.3878 -2.078
15 23.127 20.9746 -2.1524
16 23.34 21.502 -1.838
17 23.93 22.1 -1.83
18 24.18 22.955 -1.225
19 24.908 23.273 -1.635
20 24.9 23.49 -1.41
21 24.9 24.318 -0.582
22 26 25.1 -0.9
23 27 25.93 -1.07
24 27.5 22.98 -4.52
25 24 22.95 -1.05

There are theorems demonstrated elsewhere (cross differences and second differences) that the second differences (increments in the increments) give the binding energy due to the interaction of the last two particles added to the nuclide. In this context that means that the cross difference, the increment in the incremental binding energy of a neutron spin pair with respect to the number of alpha modules, gives the interaction binding energy of the last neutron pair with the last alpha module added to the nuclide. The cross difference is also given by the increment in the incremental binding energy of an alpha module with respect to the number of neutron pairs.

The second difference, the increment in the incremental binding energy of one nuclear particle with respect to an iincrease in the number of the same type of nuclear particle gives the binding energy due to the interaction of the last nuclear particle of that type with the next-to-last nuclear particle of the same type.

In the above table the second differences in binding energy due to an increase in the number of neutron spin pairs are shown in the last column. In all except the first the values of the second differences are negative. This means that the interaction between the neutron pairs is a repulsion. It is another case of like particles repeling each other.

The cross difference be perceived when the incremental binding energies of the neutron pairs are plotted versus the number of alpha modules as shown below:

The slope is positive everywhere except where there is a transition to a different nuclear shell. This means that the force between a neutron spin pair and an alpha module is an attraction.

The second differences in the binding energy of alpha modules presents more of a problem. Their values are shown below. The problem is that the alpha modules are not held together or held in the nucleus by the nucleonic force but instead by the linkages of spin pairs.

The Incremental Binding Energies of
Alpha Modules in Nuclides
Composed Entirely of
Alpha Modules and Neutron Spin Pairs
IBEα(#α, 0) IBEα(#α, 1) IBEα(#α, 2)
1 28.295674
2 28.203836 35.7081 37.242
3 35.662218 40.307308 42.103
4 35.457608 34.522492 40.6177
5 33.025523 37.96291 40.4653
6 37.612031 38.91072 39.7915
7 38.28 38.939 39.7828
8 35.24377 36.21953 37.30364
9 34.93504 35.50354 35.0965
10 35.3363 34.5526 37.14976
11 33.4227 36.2991 37.7385
12 35.9873 36.8497 37.6464
13 36.235 36.7146 35.9088
14 36.291 34.6951 34.5877
15 31.004 31.6652 32.252
16 30.958 31.171 31.6984
17 30.45 31.04 31.638
18 30.7 30.95 31.805
19 31 31.728 32.046
20 31.7 31.692 31.909
21 31.1 31.1 31.928
22 30.5 31.6 32.382
23 30.7 31.7 32.53
24 31.3 31.8 28.85
25 31.5 28 27.97
26 24.85
27 24.97
28 25.13
29 26.1

The second difference of binding energy with respect to the number of alpha modules is sometimes positive and sometimes negative. When an alpha module is added to an existing system there are formed two new spin pair linkages. The effect on binding energy due to the interaction of two consecutive alpha modules is overwhelmed by the effect of the two new spin pairs formed.

Conclusions

The set of nuclides considered can be treated as being made up of alpha modules and neutron spin pairs. Alpha modules and neutron spin pairs are attracted to each other. The evidence indicates that neutron spin pairs repel each other. The evidence is not clear on the nature of the force between two alpha modules. The alpha modules are linked together by spin pairing and the interaction through the nucleon force and the electrostatic force is relatively unimportant and the electrostatic repulsion is relatively small.


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