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The Incremental Binding Energies
Due to the Interactions of Alpha
Modules and Neutron Spin Pairs

The Fundamental Structure of Matter

The first record we have of human cogitation about the fundamental structure of matter is from the ancient Greeks. They thought about the process of taking a bit of material and cutting it in half, then cutting that half in half and so on. But on a philosophical basis they could not accept that process going on forever in an infinite regress. The bit at which the process ended was given the name a - tom for a=not and tom=cut; i.e., not cuttable. No significant progress in atomic theory was made until John Dalton in the 1700's when he deduced that the regularities of chemistry could be explained by substances being made up of atoms. That led to the notion of molecules made up of combinations of atoms.

In the late 19th century the discovery of particles of electric charge led to the revelation that atoms and molecules contained electrical charge. Since it was known that there are two types of electric charge but that matter is generally electrically neutral there had to be equal amounts of both types in matter. Then a structure was needed that kept the two types of charge separated. For a while investigators favored a model in which one type of charge existed as nuggets in a sea of the other charge. This was known as the plum pudding model of atomic structure. Rutherford established that the positive charge was contained in a small part of the atom, which he called its nucleus, a kernel.

Chemists during this time had identified the elements and their properties. Mendeleev then found that the elements could be organized into a periodic arrangement. The masses (atomic weights) for the elements had been established at that time and periodic table generally adhered to an ordering by atomic weight but not always. When the neutron was discovered in the 1930's, the puzzle was resolved. The chemical properties of an element are determined by the number of electrons, particularly the electrons in the outer shell. The number of electrons of an atom is equal to the number of protons it contained. But the atomic weight for an element includes the mass of is neutrons.

Although the fundamental constituents of nuclei might be protons and neutrons (nucleons), or on an even more fundamental level up quarks and down quarks, the essential structural constituents of nuclei are substructures of nucleons. Whenever possible nucleons form spin pairs, but spin pairing is exclusive. One neutron can form a spin pair with one and only one neutron and with only one proton. This means nucleons are linked together in chains and these chains may close to form rings. Thus the most important substructures of nuclei are these rings made up of what can be called alpha modules. An alpha module consists of a sequence of protons and neutrons linked together by spin pairing. They can be represented by -N-P-P-N- or, equivalently, -P-N-N-P-, where N and P represent a neutron and a proton, respectively. The other possible substructural constituents of nuclei are neutron-neutron spin pairs, neutron-proton or proton-proton spin pairs and possibly a singleton neutron or proton.

The analysis here is limited to those nuclides which contain only alpha modules and neutron spin pairs. This is to simplify the analysis and focus on the major deteriminants of binding energy.

In the display below the relationship between the incremental binding energy of an alpha module and the number of neutron spin pairs appears to be nearly linear.

However visual perception is not very precise. There is an about doubling of the slope of the relationship at about 14 neutron spin pairs from 0.33714 MeV per spin pair to 0.58925 MeV. Fourteen spin pairs corresponds to 28 neutrons, a magic number. But the total number of neutrons is the nuclide is 108=2*40+28, not a magic number. It would be very interesting if the neutron shell magic numbers apply to the ones in the extra spin pairs.

The data for the case of 50 alpha modules displays a sharp .after 26 neutron spin pairs. Usually such a drop is associated with the filling of a shell.

Twenty six neutron pairs contain 52 neutrons, a number near the magic number of 50. But the total number of neutrons is 152=2*50+2*26.

The data for the case of 30 alpha modules displays a sharp increase after 11 neutron spin pairs. The total number of neutrons at that point is 82=2*30+2*11, a magic number.

The data for 20 alpha modules confirms that the drop in the incremental binding energy of an alpha module is associated the total number of neutrons.

The sharp drop comes at 5 neutrons spin pairs. That corresponds with 50=2*20+2*5 neutrons, a magic number.

The data for 10 alpha modules also confirms that the drop in the incremental binding energy of an alpha module is associated the total number of neutrons.

The sharp drop comes at 4 neutron spin pairs which correspond with 28=2*10+2*4 total neutrons, a magic number. There also appears to be a drop after 0 neutron pairs which corresponds with 20 total neutrons, a conventional magic number.

Relationships with the Number of Alpha Modules

The effect of the number of alpha modules is more complex than is the case for the effect of the number of neutron spin pairs shown previously. Here is the case for 12 neutron spin pairs.

The relationship appears to be piecewise linear. The slope of the relationship is related to the binding energy due to the interaction of one alpha module with the preceding one. The linearity represents that interaction being constant within a shell or subshell. Some linear sections are flat, indicating no interaction. In other sections the slope is negative indicating a repulsion between alpha modules.

The drop after 29 alpha modules represents the filling of a shell with 82 neutrons since 2*29+2*12 is equal to 82. The drop after 25 alpha modules does not correspond to a shell since the number of neutrons is 74=2*25+2*12. But the number of neutrons just in the alpha modules is 50, which is a magic number.

The sharp rise at 18 alpha modules does correspond to a magic number; i.e., 50=2*18+2*12. After a flat section after 29 alpha module the slope goes downward at 33 alpha modules. The number of neutrons at that point is 90 which is not a conventional magic number but it perhaps represents the filling of a subshell.

The graph for the case of nuclides with 18 neutron spin pairs is qualitatively different from the preceding ones. There are places where the slope changes but it does not change abruptly at one point. Instead the transition occurs over several alpha modules. As can be seen in the graph, at 30 and 40 alpha modules the incremental binding energy first increases and then decreases producing the appearance of a lump. The only magic number in this range is 126 which corresponds to 45 alpha modules.

For the case of 24 neutron spin pairs there is apparently an abrupt drop after 41 alpha modules and 41 alpha modules have 82 neutrons, a magic number, but the total neutrons at that point is 130=82+2*24.

The incremental binding energy rises after that abrupt drop for four or five modules and then thereafter following. That rise might be indicative of an attraction of additional alpha modules for the ones already existing in the nuclide.

The Constancy of the Interaction Binding Energies within Shells

Consider the plots of the incremental binding energies of alpha modules as functions of the number of neutron spin pairs for 23, 24 and 25 alpha modules.

In the display the incremental binding energies of alpha modules are nearly linear functions of the number of neutron spin pairs over the range of 4 to 14 neutron spin pairs. The values are nearly equal for 23, 24 and 25 alpha modules. In effect this constancy the slope defines a shell. But 14 spin pairs corresponds to 28 neutrons, a magic number. Three neutron pairs as the number preceding the beginning the filling of the new shell corresponds to 6 neutrons, which is arguably also a magic number. However the total number of neutrons for 23 alpha modules and 14 neutron spin pairs is 74, not generally consider a magic number. And likewise for 24 and 25 alpha modules and 14 neutron spin pairs the total numbers of neutrons are not magic numbers.

Visual appearances in graphs are sometimes misleading. Here are the computed slopes of the lines from their minima to their maxima.

The Slopes of the Relationships
between the Incremental Energies
of Alpha Modules and the Number
of Neutron Spin Pairs
Alpha
Modules
Slope:
(MeV per
Neutron Pair)
Range
23 0.63864 4 to 15
24 0.73758 3 to 15
25 0.93925 4 to 16

Those figures represent the binding energies due to the interaction of the alpha module with neutron spin pairs in the 0 to 7 spin pair range. Specifically, the interaction between the 23rd alpha module and each of the neutron spin pairs from the 5th to the 15th is 0.63864 MeV. The interaction of the 25th alpha module with each of the neutron spin pairs from the 5th to the 15th is 0.93925 MeV.

The display for 15, 16, 17 and 18 alpha is similar to the preceding case.

There is near linearity and near equality over a range, in this case 0 to 7 alpha modules. The seven neutron spin pairs for 18 alpha modules corresponds to 14 neutrons, a magic number. But the total number of neutrons for this case is 50=2*18+14, which is a magic number. The 14 for the end of the shell being a magic number is just a coincidence. The 50 for the total number neutrons is not a coincidence. The points of deviation from the pattern for 15, 16 and 17 occur at a total number of neutrons equal to 50.

Again note that visual appearances in graphs can be misleading. Here are the computed slopes of the lines from their minima to their maxima.

The Slopes of the Relationships
between the Incremental Energies
of Alpha Modules and the Number
of Neutron Spin Pairs
Alpha
Modules
Slope:
(MeV per
Neutron Pair)
Range
15 0.99060 to 10
16 0.90800 to 9
17 0.846630 to 8
18 0.813300 to 7

Those figures represent the binding energies due to the interaction of the alpha module with neutron spin pairs in the 0 to 7 spin pair range. Specifically, the interaction between the 15th alpha module and each of the neutron spin pairs from the 1st to the 10th is 0.9906 MeV. The interaction of the 18th alpha module with each of the neutron spin pairs from the 1st to the 7th is 0.8133 MeV. These are the cross differences in binding energies with respect to the number of alpha modules and the number of neutron spin pairs.

A similar sort of analysis can be carried out with respect to the relationship of the incremental binding energy of alpha modules to the number of alpha modules in nuclides with a fixed number of neutron spin pairs.

The near coincidence of the relationships indicate the slopes and curves are definite patterns resulting from the underlying physics. The sharp drops are evidence of shells being filled and in most places correspond to the total number of neutrons being equal to a magic number. Here are the values of the slopes for the right-most linear section of the relationships.

The Slopes of the Relationships of
Incremental Binding Energies of Alpha
Modules and the Number of Alpha Modules
in Nuclides with 4, 5 and 6 Neutron Spin Pairs
Neutron
Spin Pairs
Slope:
(MeV per
Alpha
Module)
4 -0.43400
5 -0.30000
6 -0.36957

The slopes represent an upper limit on the interaction of one alpha module with the preceding one. In other words, the interaction of two alpha modules may be more negative than the value shown. The negativity of the interaction indicates that the force between two alpha modules is a repulsion.

Implication of the Numerical Results

If the nucleonic charge of a proton is taken as 1 and that of a neutron is denoted as q then the nucleonic charge of an alpha module is equal to 2(1+q) and that of a neutron spin pair is 2q. The interaction of an alpha module and neutron spin pair is then proportional to 4q(1+q) and the interaction of two alpha modules is proportional to 4(1+q)², The ratio R of the interaction of an alpha module with a neutron spin pair to the interaction of two alpha is then given by

R = 4q(1+q)/[4(1+q)²] = q/(1+q)

The average of the interactions for an alpha module with a neutron spin pair for the 15th through 18th alpha module is 0.88963 MeV. On the other hand the average of the interaction of two alpha modules is −036786 MeV. Their ratio is −2.418415. >Thus

q/(1+q) = −2.418415
and hence
q = −2.418415 −2.418415q
and
3.418415 = −2.418415
and thus
q =− 0.70747

This is a reasonably close approximation of the value of −2/3 found elsewhere

Conclusions

Alpha modules and neuron spin pairs are attracted to each other, but repelled by those substructures of their own kind.

The incremental binding energies of alpha modules are constant over a range of neutron spin pairs and roughly the same for different numbers of alpha modules.


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