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The Analysis of the Binding Energy of Nuclei
in Terms of the Quarks They Contain

Quarks in Nuclides

A proton consists of two up quarks and one down quark. A neutron, on the other hand, consists of one up quark and two down quarks. If n and p are the numbers of neutrons and protons, respectively, then the numbers of up quarks, #u, and down quarks, #d, are given by

#u = 2p + n
#d = p + 2n

The 2931 nuclides can be tabulated in terms of their values of #u and #d. Incremental binding energies can be computed but this means something different for quarks than for nucleons. A single down quark cannot be added to a nuclide.

The Set of Nuclides and their Characteristics which have 80 up Quarks
Nuclide Protons Neutrons Binding
Energy
up Quarks down
Quarks
52Ni 28 24 420.46 80 76
53Co 27 26 449.296 80 79
54Fe 26 28 471.7587 80 82
55Mn 25 30 482.0703 80 85
56Cr 24 32 488.506 80 88
57V 23 34 486.45 80 91
58Ti 22 36 482.5 80 94
59Sc 21 38 470.9 80 97

In order to increase #d but keep #u constant it is necessary decrease the number of protons by one and increase the number of neutrons by two. The decrease in the number of protons by one reduces the number of up quarks by two; the increase in the number of neutrons brings the number of up quarks back up by two. The decrease in the number of protons by one decreases the number of down quarks by one but the increase of two neutrons adds four down quarks for a net increase of three.

Incremental Binding Energies

An incremental binding energy of a triplet of down quarks can be computed. Here is a graph of the incremental binding energy of a triplet of down quarks.

It is downward sloping indicating the interaction of one thriplet of down quards with the previous one is a repulsion. The second differences in binding energy with respect to the number of down quarks are shown below.

The interaction of a triplet of down quarks with the previous triplet is -6.7393 MeV or, since the interaction of triplets involves 9 interactions, -0.7488 MeV per down quark. The negativity of the value indicates that it is an interaction involving repelling.

The data and graphs for the incremental binding energies of up quarks are shown below.

<<<<<<<<<
The Set of Nuclides and their Characteristics which have 80 down Quarks
Nuclide Protons Neutrons Binding
Energy
up Quarks down Quarks Incremental
Binding
Energy
Second
Difference
48S 16 32 362.8 64 80
49Ar 18 31 398 67 80 35.2
50Ca 20 30 427.491 70 80 29.491 -5.709
51Ti 22 29 444.1525 73 80 16.6615 -12.8295
52Cr 24 28 456.3451 76 80 12.1926 -4.4689
53Fe 26 27 458.3802 79 80 2.0351 -10.1575
54Ni 28 26 453.15 82 80 -5.2302 -7.2653
55Zn 30 25 435.38 85 80 -17.77 -12.5398

The interaction of a triplet of up quarks with the previous triplet is -8.8283 MeV or, since the interaction of tiplets involves 9 interactions, -0.9809 MeV per up quark. Again the negativity of the value indicates that it is an interaction involving repelling.

Cross Differences

The increment with respect to one particle of the increments with respect to the other particle is called the cross difference. The binding energy to the interaction of the last particle of one type with the last particle of the other type is equal to the cross difference with respect to the numbers of the two particles. The cross difference is also the slope of the relationship between the incremental binding energy with respect to the numbers of one particle plotted versus the number of particles of the other type. Here are the data for establishing some of the cross differences.

The Incremental Binding Energies of Triplets of Down Quarks as
a Function of the Numbers of Up and Down Quarks in the Nuclide
Number of Up Quarks
Number of
Down Quarks
74 77 80 83 86
73 28.357
76 21.4971 31.874
79 10.7967 19.3371 28.836
82 6.1202 13.4332 22.4627 32.487
85 -2.661 3.4597 10.3116 19.5585 30.382
88 -5.5 0.776 6.4357 12.9944 21.7718
91 -14.2 -7.52 -2.056 4.5801 10.8543
94 -9.5 -3.95 0.8 8.0369
97 -11.6 -5.88 -0.845
100 -7.5 -1.7
103 -10.6

Here is a graph of such a relationship for nuclides containing 85 down quarks.

The positive slope indicates that a triplet of up quarks are attracted to a triplet of down quarks. The amount for the graph shown is 8.26075 MeV. Since there are nine interactions of quarks involved in the interactions of triplets the amount per quark interaction is 0.9179 MeV.

Implication of the Interaction Binding Energies of the Quarks
for the Interaction Binding Energies of the Nucleons

First the above estimates of the interaction binding energies can be put together:

The Binding Energies Involved in the Various
Interactions of Up and Down Quarks
Interaction Binding
Energies
MeV
uu −0.9809
dd −0.7488
ud +0.9179

Now consider the binding energy for the interaction of two protons. This is the interaction of the two up quarks and one down quark in one proton with the two up quarks and one down quark in the other proton. First consider the interaction of one up quark with the quarks in the other proton. This 2uu+ud which equals −1.0439 MeV. This doubled for the two up quarks of the first proton is −2.0878 MeV. The interaction of the down quard of the first proton with the quarks of the second proton is 2ud+dd or +1.087 MeV. Combined the interaction total −1.0008 MeV.

For the binding energy of the interaction of two neutrons the separate components are as follows. The up quark of the first neutron with the quarks of the second neutron has an interaction of 2ud+uu, which totals +0.8549 MeV. One down quark in the first neutron has interactions with the quarks of the second neutron of ud+2dd, which amounts to −0.5797 MeV. Doubled this is −1.1594 MeV and this combined with the interaction of the up quark is −0.3045 MeV.

The interaction of a proton with a neutron involves the interactions of one up quark with the quarks of neutron. These are 2ud+uu, or +0.8549 MeV, which doubled is +1.7098 MeV. The interaction of the down quark of the proton with the quarks of the neutron is −0.5797 MeV. The total is then +1.1301 MeV. Notably the interaction of the unlike nucleons is opposite in sign to those of the interaction of like nucleons.

If the nucleonic charge of a proton is taken to be unity and that of a neutron is denoted as q then the interaction of two neutrons is proportional to q². Then the ratio of the interaction of neutrons to the interaction of protons should be equal to q². The square root of the ratio (-0.3045)/(-1.0008) is ±0.5516. The sign is established as (-) by the difference in the signs of the interaction of the unlike particles to those of the like particle. This differs significantly from the value of −2/3 found by other methods but it is definitely of the right order of magnitude .

Changes in the Character of the
Relationships for Nuclides with Greater Numbers of Quarks

There was an alternation in the previous displays of the incremental binding energies between negative slopes and less negative slopes. For nuclides with more quarks the less negative slope could be a small positive slope, as shown in the case below.

Such anomalies do not alter the fact that generally the relationships are downward sloping, indicating that the force between two like quarks is repulsion.

Conclusion

The data on incremental binding energies confirm that like quarks repel each other and unlike quarks are attracted to each other.


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