San José State University
Thayer Watkins
Silicon Valley,
Tornado Alley
& the Gateway
to the Rockies

Comparison of Alternative
Quarkic Structures of Nucleons

The conventional model of the quarks in nucleons (protons and neutrons) is as point particles linked together in a planar arrangement. At close separation, in this conventional model, these quarks are subject to asymptotic freedom, but at large separations subject to an increasing attraction that prevents any quark from being found isolated.

The conventional model of the quarkic structure of nucleons was created solely to explain the single fact: That an isolated quark has never been found. To explain that fact theorists conjured up a force between quarks that increased with the separation distance between quarks. This is like no other force field known and the nature of the particles which could distribute such a force field is impossible to imagine. (Well not quite unimaginable; one can imagine tiny rubber bands attached to the quarks. But nothing physically feasible would account for such a force field.)

The electrostatic force diminishes with distance squared because the density of the force-carrying photons diminishes with distance squared. If the force-carrying particles decay over time the force formula is of the form α·exp(−βr)/r², where r is separation distance and α and β are parameters. But there is no process that justifies a force increasing with separation distance. Giving the supposed force-carrying particle the catchy name of gluon does not justify its supposed existence. As will be shown below the explanation of the absence of an isolated quark does not require the conjecture of a force of such an impossible nature.

A proton is composed of two up quarks and one down quark while a neutron is composed of two down quarks and one up quark. The up quark has an electrostatic charge of +2/3 and the down quark an electrostatic charge of −1/3.

The conventional model has all quarks attracting each other equally through a nucleonic force usually called the nuclear strong force which, as mentioned above, that increases with distance.

In this diagram depicting a neutron the Down quarks (shown as blue circles) may be rotating about the Up quark (shown as a red circle) thus creating a dipole moment.

It is not clear how the conventional model of quarkic structure would establish the boundaries of nucleons and they do have something in the nature of boundaries. The perceived radius of a proton is 0.8 fermion and that of a neutron is 1.1 fermion.

Each quark has another attribute that is called color although it has nothing to do with visual color. A nucleon is supposed to have quarks of each color so it is said to be color neutral white.

In the alternate model a quark is a spherical shell of charges, electrostatic and possibly a nucleonic force which is not necessarily the so-called nuclear strong force.

A Nucleon as Three Concentric Shells

A nucleon in this alternative model consists of three concentric rotating quarkic shells. This alternate model could appropriately be called the concentric shells model of the quarkic structure of nucleons. The attribute corresponding to color is the radius of the shell. It is obvious in this alternative model why there must be quarks of three different attributes in each nucleon.

The force of attraction is zero between shells of equal or opposite charge if one is located within another but becomes large negative or large positive if they are not concentric. If separated the force of repulsion or attraction decreases with separation distance.

It is impossible to separate them because any action taken against the outer quark equally affects the other quarks in a nucleon.

The conventional model can be compared with an alternate model in terms of measured nucleonic attributes such as magnetic moments.

Magnetic Moments

A magnetic moment is generated by spinning charged particles or charged particles in shells if flowing in a circular path. For some of the details of the technicalities of magnetic moments see Studies.

A magnetic moment of a system composed of charged particles rotating about a center can arise from that rotation of charges, usually called dipole moments, but also from the intrinsic magnetic moments of the particles. This latter phenomenon is usually deemed as being due to the spin of the particles. In 1922 the physicists Otto Stern and Walther Gerlach ejected a beam of silver atoms into a sharply varying magnetic field. The beam separated into two parts. This separation could be explained by the unpaired valence electrons having a spin that is oriented in either of two directions. It has been long questioned whet,her or not there is evidence that this so-called spin is literally particle spin.

The Evidence

The magnetic moments of the proton and the neutron derive from the intrinsic moments of their quarks and any dipole moment of the quarks within the nucleon. The magnetic moment of a proton, measured in magneton units, is +2.79285. That of a neutron is −1.9130. The ratio of these two numbers is −0.685, intriguingly close to −2/3. There is only a 2.7 percent difference. This suggests that the ratio of the intrinsic magnetic moments of the neutron and proton is precisely −2/3.

If the ratio of the magnetic moments of the neutron and proton is −2/3 then any dipole moment of the quarks would result in a deviation from that value. The question is which of the two models would tend to have the lowest dipole moment. In the alternative model the concentricity of three spheres forces a closeness of their centers. Also if the spheres are subject to a force that drops off faster than distance squared then concentric sphere will line up their centers exactly. See Quarks for detail.

On the other hand, in the conventional with its asymptotic freedom there is no tendency for the quarks to move toward each other. Thus in the conventional model there is nothing that would tend to eliminate the dipole moments of the quarks, whereas in the alternative model there is.

The Radial Distributions of
Electrostatic Charge in the Nucleons

The radial distribution of electrostatic charge is found by sending electrons as probes against collections of protons and of neutrons and analyzing the deviations from a straight path. Here are the results of such experiments.

According to the concentric shell model there should be such a radial distributions and the measured distribution should appear the same in any radial direction. According to the conventional model there should be no such radial distribution. The perceived charge would depend upon the angle between the radial direction and the plane of point quarks.

The experimental radial charge distribution for a neutron, shown above, could not occur unless there is a radial separation of the Up quark and the Down quarks.

The radial distribution of charge for neutrons is entirely in keeping with the concentric shell model. However according to this alternative model there should be radial range of negative charge for the proton. It may well be that the experimentalists who developed the above distribution for protons overlooked such negative charge density because they were not expecting it. This prediction of a radial range of negative charge density for protons would be worth pursuing experimentally.

In the experiments at the Stanford Linear Accelerator (SLAC) probing the structure of protons it was found that some of the electrons bounced backward. This was interpreted of evidence of the electron probes impinging upon a solid quark. It equally well can be interpreted as evidence of the electron probes impinging upon a negatively charged Down quark.


The conventional model of the quarkic structure of nucleons was created to explain a single fact: That an isolated quark has never been found. To explain that fact theorists conjured up a force between quarks that increased with the separation distance between quarks. This is like no other force and the nature of the particle which distributes such a force field is impossible to imagine. The nonexistence of an isolated quark is much more easily explained by an alternative model in which quarks are spherical shells of charge and a nucleon is three concentric spheres of these quarkic shells.

The conventional theory says the quarks are asymptotically free; i.e., the force between quarks asymptotically approaches zero as their separation distance approaches zero. The alternative theory says this asymptotic freedom comes from mistaking the centers of the spherical shell quarks as being the quarks. The centers are close together when the spherical shell quarks are concentric. A charged spherical shell within another charged spherical feels no force. The so-called colors of quarks are the radii of the spherical shells and this explain why a nucleon needs one of each color.

The experimentally determined radial distribution of charge density is compatible with the alternative model but not with the conventional model.

All in all the alternative concentric shell model better explains the single fact, the absence of evidence of an isolated quark, explained by the conventional model and lends itself to further analysis that the conventional model doesn't.

For more on the quarkic spatial structure of nucleons see Sensible Model of Quarkic Structure of Nucleons.

(To be continued.)


Some Implications of
the Alternative Model

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 μU and μD be the magnetic moments of the Up and Down quarks, respectively. Then

U + μD = 2.79285
μU + 2μD = −1.9130

Dividing the second equation by 2 gives

½μU + μD = −0.9565

Subtracting this equation from the first gives

(3/2)μU = 3.74935
and hence
μU = 2.49957 magnetons

The magnetic moment of the Down quark is then

μD = 2.79285 − 2μU = −2.20628 magnetons.

Note the ratio μDU=0.7899744=1/1.2658638≅4/5.

Now remember that the magnetic moment of a particle is of the form

μ = QkR²ω

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 RU and RD, . Likewise let ωU and ωD be their spin rates and kU and kD the coefficients for the shapes of their charge distributions. Then

(2/3)kURU²ωU = 2.49957
(−1/3)kDRD²ωD = −2.20628

Equivalently these are

kURU²ωU = 3.749355
kDRD²ωD = 6.61885

Note that the ratio is

3.749355/6.61885 ≅ 9/16 = (3/4)²

The radius and spin rate of a quark are related to its angular momentum

Moments of Inertia
and Angular Momenta

The moment of inertia J of a mass M uniformly distributed over a spherical shell of radius R is (2/3)MR². For a spherical ball it is (2/5)MR². The general form is then

J = jSMR²

where jS is a coefficient determined by the shape of the distribution of mass. Angular momentum L is given by

L = Jω

The magnetic moment μ is given by

μ = Kω

where K is a moment of charge analogous to the moment of inertia and having the form

K = kSQR²

Generally the magnetic moment of a charged particle is related to its angular momentum by

μ = αL(Q/M)

where α is a coefficient depending upon any difference in the shapes of the distribution of mass and charge. If the shapes are the same then α=1.

This makes μ directly proportional to Q and inversely proportional to M and explains why the magnetic moment of an electron is so much larger than those of the nucleons. Its mass is roughly one two thousandth of their masses. But Q and M can be expected to be proportional to each other. That means that if L is quantized then μ is quantized. Thus μ should approximately be a constant independent of the scale of the structure. In symbols

μ(M/Q) = L
and hence
μ = (Q/M)L

Usually when angular momentum is said to be quantized this is taken to mean that the minimum angular moment is equal to the reduced Planck's Constant, h. But this is not strictly true. If an object has n modes of vibration or oscillation then each of those modes has a minimum of h so the overall minimum is nh. These modes are also called degrees of freedom.

The Masses of the Up
and Down Quarks

It is well known that the masses of nucleons may not be simply the sum of the masses of its quarks. The mass of the positive pion meson, which consists of an Up quark and its corresponding anti-quark Down quark, is a small fraction of the masses of the nucleons. Nevertheless consider what the masses of the nucleons imply about the masses of the quarks. Let mU and mD be the masses of the Up and Down quarks, respectively. Then

2mU + mD = 1.6726231×10-27 kg
mU + 2mD = 1.6749286×10-27 kg


mU = 0.556777×10-27 kg
mD = 0.559708×10-27 kg

Thus mD/mU=1.00526. So essentially mD =mU.

Thus if the shapes of the charge distributions of the Up and Down quarks are the same then the ratio of the angular momenta of the Up and Down quarks should be

LU/LD = nU/nD

a ratio of integers. Since L=jSMR² then if jS=kS for each quark

= 3.749355/6.61885 = 0.566466

Note that 9/16 = 0.5625 and 0.566466=(1.007)(9/16). Thus

LU/LD ≅ (9/16)

So indeed the ratio of the angular momenta of the Up and Down quarks is a ratio of whole numbers. The fact that those whole numbers are greater than unity indicates that quarks may have complicated shapes with multiple degrees of freedom.

It is quite plausible that an Up and a Down quark have the same shape so kU=kD. Furthermore there is a rationale for ωUD. See Sensible Model of Quarkic Structure of Nucleons.

These equalities would mean


But quark radii is likely to also be quantized; i.e.

R = ρν

where ν is an integer and ρ is a constant.

This would mean

LU/LD = νU²/νD² =(νUD

The implication of the above is that

RU/RD = 3/4

Thus it is significant that (9/16)=(3/4)².

According to the Concentric Shells Model of nucleons the outer radius of the large Up quark is equal to the radius of a proton. Likewise the outer radius of the large Down quark is equal to the radius of a neutron. Thus according to this alternative model the ratio of the radius of a proton to that of a neutron should be 3/4. The radius of a neutron is 1.1133 fermi (10−15 m). The accepted radius of the proton had been 0.86 fermi, but a new experimental method puts it at 0.84 fermi. The ratio of 0.84 to 1.1133 is 0.7545. That is solid confirmation of the model and its analysis.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins,