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The Statistical Explanation of the Binding Energy
of Nuclei in Terms of the Quarks They Contain

Background

A model for explaining the binding energies of nuclides in terms of the number of their nucleon spin pairs and interactions performs very well statistically. It explains 99.98 percent of the variation in binding energy. This webpage is for applying the same methodology in terms of the numbers of up quarks and down quarks contained in the nuclides.

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.

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 neutron\ 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 triplet of down quarks 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. (MeV stands for millions of electron volts.) The negativity of the value indicates that it is an interaction involving repulsion.

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

The interaction of a triplet of up quarks with the previous triplet is -8.8283 MeV. 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.

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.

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.

The Regression Results

The regression equation is just the binding energy as a linear function of the numbers of quark pairs of the three type and of the numbers of quark interactions of the three types.

<---->
Regression Results
(MeV)
VariableCoefficientt-Ratio
uu pairs0.704663.4
dd pairs-1.11370-13.6
ud pairs7.0458561.5
#uu-0.41354-113.9
#dd -0.2936-107.5
#ud0.34400109.6
Const.-53.14840-144.2

The coefficient of determination (R²)for the regression equation is 0.99990 and the standard error of the estimate is 5.0 MeV. This 5.0 MeV is in comparison with an average binding enery of 1072 MeV so the coefficient of variation is 0.467 of 1 percent. This high performance is no surprise because the variables for this equation are linear functions of the numbers of protons and neutrons and the model based upon those nucleons was known to perform There are some surprises. The effect of the pairing of up and down quarks has an extraordinarily high value compared to the other two types of quark pairs. The negative value for the effect of down-down pairs indicates that down-down pairs are not stable. The negative values for the number of uu and dd interactions indicates that like quarks repel each other. The positive value for the number of ud interactions indicate that up and down quarks attrack each other.

If the nucleonic charge of the up quark is taken to be 1.0 and that of the down quark denoted as q then the interaction effects should be proportional to (1: q², q) respectively. Thus

(-0.29365)/(0.34460) = 0.85215 = q
and
(-0.29365) /(-0.41354) = 0.71001 = q²
and hence
q = -0.84267

Therefore the nucleonic (strong force) charges of the up and down quark are opposite in sign and that of the down quark is about 85 percent of the magnitude of that of the up quark. This implies that the nucleonic charges of the proton and neutron should proportional to

proton: 2 − 0.85 = 1.15
neutron: 2(−0,85) + 1.0 = −0.7
and thus in the ratio of
= −0.7/1.15 = −0.60870

The value of −0.60870 for the nucleonic charge of a neutron relative to a proton nucleonic charge of +1.0 is not too far different from the value of −2/3 found previously. The empirical results mix up the effect of the electrostatic charges of quarks with their nucleonic charges. The up quark has an electrostatic charge of +(2/3)e and the down quark one of −(1/3)e and thus their ratio is −1/2. Therefore one could consider the value of −0.60870 as arising from the combined effects involving ratios of −2/3 and −1/2.

Conclusion

The binding energies of nuclides can be overwhelmingly explained by the formation of quark pairs and the interactions of the quarks they contain. Slightly over 99.99 percent of the variation in the binding energies of 2931 nuclides can be explained in this way. The nucleonic (strong force) charges of up and down quards are opposite in sign and different in magnitude. The nucleonic charge of a down quark is roughly 85 percent of that of an up quark.


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