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This material develops an alternative to the conventional model of hadron structure. In this model nucleons and pi mesons are concentric shells of quarks. It provides an explanation of why the charged pi mesons composed of only two quarks have masses of only about 273 electron masses but the nucleons with three quarks have masses of almost 2000 electron masses.
Hadron is a term coined to cover nucleons (protons and neutrons} and mesons such as pi particles. The conventional theory hadron structure has quarks being charged point particles. A charged point particle would require an infinite amount of energy to create. There is not enough energy in the entire Universe to create even one charged point particle.
Instead quarks can be spherical shells of charge and mass. Outside of their shells they have the same effect as if their charge and mass were concentrated at their centers. A nucleon or meson is thus made up of concentric quarkic spherical shells.
For details on this theory of concentric quarkic spheres see Quarkic Structures.
This means that quarks come in three radius sizes: small, medium and large. Conventional theory talks about there being three attributes for point particle quarks which it labels as color. This socalled color attribute could be radius size.
The radial distribution of the charges of nucleons has been determined experimentally; i.e.,
According to the quark theory of nucleonic structure a neutron is composed of two Down quarks and one Up quark. A proton on the other hand has two Up quarks and a Down quark. An Up quark has an electrostatic charge of +2/3 whereas a Down quark has a charge of −1/3.
There are only three possible radial arrangements of the quarks in a neutron: UDD, DUD and DDU, where the left represents the center of the nucleon. The DUD arrangement violates the apparent rule for particle linkages; i.e., that a particle link to no more than one particle of the same kind and no more than one of the opposite kind. The DDU arrangement would result in a positive magnetic moment for the neutron contrary to observation. Thus the arrangement must be UDD for the neutron. Likewise it must be DUU for the proton.
The above empirical charge distribution for a neutron is entirely consistent with the UDD for the concentric shell model. The distribution for a proton should have a radial range of negative charge. Experimental probing of protons with electrons at SLAC found that some electrons bounced back. This was interpreted as evidence of electrons impinging upon hard cores of quarks. It could also be interpreted as evidence of electrons impinging upon a range of negative charge.
The above distribution of charge in a neutron indicates that the positive Up quark is located between 0 and 0.25 fermi in radius. The two negatively charged Down quarks are located between radii of 0.25 fermi and 1.1133 fermi, the radius of a neutron.
This means a small Up quark occupies a volume of
The volume of a neutron is
Thus the volume occupied by the medium and large Down quarks is (5.7800−0.0654)=5.7146 f³.
Let σ_{U} and σ_{D} be the volume mass densities of the Up and Down quark materials, respectively. The units for these densities are electron masses per cubic fermi.
The mass of a neutron is 1838.684 electron masses. Therefore
As noted previously a proton is composed of two Up quarks and one Down quark. For a neutron its composition is two Down quarks and one Up quark. Let μ_{U} and μ_{D} be the magnetic moments of the Up and Down quarks, respectively. Then
Dividing the second equation by 2 gives
Subtracting this equation from the first gives
The magnetic moment of the Down quark is then
Note the ratio μ_{D}/μ_{U}=0.7899744=1/1.2658638≅4/5.
The magnetic moment of a particle is of the form
where Q is charge and k is a constant determined by the spatial distribution of the charge. For a spherical suface k=2/3. For a spherical ball of charge k=2/5. For a spherical charge distributed over a spherical shell of some thickness 2/5<k<2/3. R is the average charge radius and ω is the rate of rotation.
As noted previously the charge of the Up quark is +2/3 and that of the Down quark is −1/3. Let the average charge radii of the Up and Down quarks be denoted by R_{U} and R_{D}, repectively . Likewise let ω_{U} and ω_{D} be their spin rates and k_{U} and k_{D} are the coefficients for the nature of their charge distributions.
Then
Equivalently these are
Note that the ratio of the RHS of these equations is
If k_{U}=k_{D} and ω_{U}=ω_{D} then
This establishes the proposition that the size of an Up quark should be three quarters of the size of the corresponding Down quark. Now this proposition can be tested.
Since in the above it is established on the basis of magnetic moments that the scale of an Up quark is (3/4) the scale of the corresponding Down quark. That means that a proton should have a negatively charged small Down quark occupying the space between its center and 0.3333 fermi. This volume occupied is
The volume of a proton is
Thus the volume occupied by the medium and large Up quarks is (2.4827−0.1551)=2.3276 f³.
The mass of a proton is 1836.1529 electron masses. Therefore
The conditions to be satisfied are
2.3276σ_{U} + 0.1551σ_{D} = 1836.1529
0.0654σ_{U} + 5.7146σ_{D} = 1838.684
The solutions for densities in units of electron masses per cubic fermi are
The mass of the small Down quark is then 312.9608*0.1551=48.5402 electron masses. The mass of the small Up quark is 768.1684*0.0654=50.2382 electron masses.
The masses of the medium and large quarks have to be estimated by a procedure that will be given later.
The positive pi meson is composed of a medium Up quark and a small Down antiquark. A small Down antiquark has the same volume and mass as a small Down quark. The mass of a small Down quark has a mass of 48.5402 electron masses. The mass of a positive pi meson is 273.1315 electron masses. Therefore the mass of a medium Up quark is 224.5973 electron masses. The volume occupied by a medium Up quark is then its mass divided by the density; i.e., (224.5973)/(768.1684) = 0.2924 f³
This number divided by (4/3)π gives 0.06980 which is the difference in the cube of the outer radius of the medium Up quarck and the cube of its inner radius. Its inner radius is the same as the outer radius of the small Down quark; i.e., 0.3333 fermi. Its cube is 0.03780. Adding this to 0.06980 give the cube of the radius of the medium Up quark; i.e., 0.1076. The cube root of this number, 0.4756 fermi, is the outer radius of the medium Up quark.
The negative pi meson is composed of a medium Down quark and a small Up antiquark. A small Up antiquark has the same volume and mass as a small Up quark. A small Up quark has a mass of 50.2382 electron masses. The mass of a negative pi meson is 273.1315 electron masses. Therefore the mass of a medium Down quark is 223.0033 electron masses. This divided by the density of Down quark material gives the volume of medium Down quark as 0.7126 cubic fermi.
This number divided by (4/3)π gives 0.1701 which is the difference in the cube of the outer radius of the medium Down quark and the cube of its inner radius. Its inner radius is the same as the outer radius of the small Up quark; i.e., 0.25 fermi. Its cube is 0.0156. Adding this to 0.1701 give the cube of the radius of the medium Down quark; i.e., 0.1857. The cube root of this number, 0.5705 fermi, is an estimate of the outer radius of the medium Down quark.
Since the above analysis made use of the result based upon the magnetic moments of the nucleons that the size of an Up quark should be three quarters of the size of the corresponding Down quark. The ratio of the radii found for the medium quarks is
This is not 0.75 but it is reasonably close enough to represent a notable confirmation of the theory of the concentric shell model of hadrons. To satisfy the three quarters ratio the outer radius of the medium Down quark would have to be (4/3)(0.4756)=(0.6341) fermi.
The simple way to determine the mass of a large quark is the mass of a nucleon less the mass of a corresponding pi meson. For the mass of the large Down that is (1838.6840−273.1315)=1565.5525 electron masses. For the large Up quark it is (1836.1529−273.1315)=1563.0214 electron masses.
There is another method for the calculation of the masses of the large quarks. The volume of the large Down quark is that between the radius of the medium Down quark of 0.5705 fermi and the radius of a neutron of 1.1133 fermi. That volume is
Its mass is then
The volume of the large Up quark is that between the radius of the medium Up quark of 0.4756 fermi and the radius of a proton of 0.84 fermi. That volume is
Its mass is then
The model gives the charge of a type of quark independent of its size and mass. That is to say, the charge of any Up quark is +2/3 the magnitude of the charge of an electron (e) regardless of whether it is small, medium or large in size. Likewise the charge of any size Down quark is −(1/3)e.
Given that the masses of the different sizes of Up quarks are about 50, 225 and 1566 times the mass pf an electron this is somewhat surprising. But consider the masses of the leptons. The masses of an electron, muon and tauon are 1, 207 and 3477, respectively, times that of an electron yet they all have a charge of −1e.
This independence of particle charge on size suggests that charge appears in a particle as a subparticle.
Quark Mass Volume Densities (electron masses per cubic fermi)  

Quark  Up  Down 
Density  768.1684  312.9608 
Radii of Up and Down Quarks (fermi} 


Size  Up  Down 
small  0.25  0.3333 
medium  0.4756  0.5705 
large  0.84  1.1133 
Masses of Up and Down Quarks (electron mass) 


Size  Up  Down 
small  50.2382  48.5402 
medium  224.5973  223.0033 
large  1565.5525  1563.0214 
The mass of a proton should be
That of a neutron should be
The mass of the positive pi meson should be
That of the negative pi meson should be
Thus the discrepancy between the masses of the charged pi mesons and those of the nucleons does not require the existence of hypothetical gluons flitting in and out of existence within the hadrons.
(To be continued.)
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