﻿ The Mass and Binding Energies of Nuclides and their Quarkic Composition
San José State University

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The Mass and Binding Energies of
Nuclides and their Quarkic Composition

The masses of the proton and neutron expressed in energy units are 938.272046 and 939.5654133 million electron volts (MeV), respectively. A proton is composed of two up quarks and one down quark whereas a neutron is composed of two down quarks and one up quark. Those compositions and masses imply that the masses of an up quark and a down quark are 312.3262262 and 313.6195935 MeV, respectively.

Another way to obtain estimates of the masses of the up and down quarks is statistically relate the masses of nuclides to their quarkic composition. The regression equation from such analysis is

#### M = 317.3438345U + 314.1191012D         

where U is the number of up quarks in the nuclide and D the number of down quarks. The numbers in square brackets [ ] are the t-ratios for the regression coefficients, the ratios of the coefficients to their standard deviations. The coefficient of determination (R²) for the equation is 0.999999376. The statistical fit of the regression can be improved slightly to 0.999999558 by including a constant term in the regression but it is better for the equation to say that if there are zero quarks the mass is zero.

Another item of interest is the binding energies of the nuclides as a function of their quarkic composition. The regression equation brings quite a surprise; i.e.,

#### BE = 5.017201143U + 0.500367865D [41.2] [4.6]

The coefficient of determination (R²) for the equation is 0.98994. The surprise is that the effect of an up quark on binding energy is ten times the effect of a down quark.

The binding energies of nuclides can be successfully explained by the number of the various spin pairs of nucleons and the interactions of those nucleons. This model applied to the quarkic composition of nuclides gives:

#### BE = −5.533966393DD + 1.868353934UD + 7.96752224DD −0.344234646uu + 0.292778027ud -0.254182109dd [−9.7] [8.3] [24.5] [ −33.6] [32.9 ] [−32.8]

where UU, UD and DD are, respectively, the number of the three possible types of spin pairs. The number of the three types of interactions are given by uu=½U(U-1), ud=U*D, and dd=½D(D-1).

The coefficient of determination (R²) for this equation is 0.9992.

The coefficients for the interactions can be nicely explained by the quarks having a charge. Suppose this type of charge is taken to be 1 for an up quark and q for a down quark. Then the interactions of two down quarks would be proportional to q² and the interaction of an up quark and a down quark would be proportional to q. The interactions of two up quarks are proportional to 1. The ratio of the coefficient for dd to the one for ud should be proportional to q. Likewise the ratio of the coefficient for ud to the one for uu should be equal to q. The ratio of the coefficient for dd to the one for uu should be equal to q². Here are the ratios.

#### Cdd/Cud = q = -0.254182109/0.292778027 = -0.868 Cud/Cuu = q = 0.292778027/ -0.344234646 = -0.851 Cdd/Cuu = q² = -0.254182109/ -0.344234646 = 0.73839781 q = √0.73839781 = 0.859

The results agree in sign and magnitude. The common value is roughly −7/8. The effect of the electrostatic charges of the quarks is not taken into account. The signs of the coefficients indicate the nature of the force between the quarks. The interaction between an up quark and a down quark is an attraction but the other two interactions involve a repulsion. It is another case of an attraction between particles of opposite charge and repulsion between those of the same charge.