This material is to investigate the spins of quarks in relation to those of nucleons (protons and neutrons).

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 perceived radius of a proton is 0.84 fermi and that of a neutron is 1.1133 fermi.

The Radial Distributions of
Charge in the Nucleons

The radial distributions of electrostatic charge are 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.

The conventional model of the quarkic structure of nucleons is of quarks as point particles in a plane rotating about their center of mass. The model being considered here is an alternative to that conventional model. In this model a quark is spherical shell of charge(s). A nucleon is three concentric shells.

According to this concentric shell model there should be such radial distributions and they should appear the same in any radial direction. According to the conventional model there should be no such radial distribution. The peceived 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 shells model. However according to this alternative model there should also 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 concentric shells model of the quarkic structure of nucleons a quark is a spherical shell of charges, electrostatic and possibly nucleonic.

A nucleon in this model consists of three concentric rotating quarkic shells. It is impossible to separate them because any action taken againt the outer quark equally affects the other quarks in a nucleon.

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

The attribute corresponding to color is the radius of the quark shell. It is obvious in this model why there must be quarks of three different attributes in each nucleon.

There must be small, medium and large versions of both the Up quarks and the Down quarks.

The force of attraction is zero between shells of opposite charge if one is located within another but becomes large positive if they are not concentric. However, if separated the force of attraction decreases with separation distance. The same applies to concentric shells of the same charge.

Here is a depiction of a cross-section of a neutron according to the concentric shells model:

where blue represent negative charge and red is positive charge.

If quarks were point particles as in the conventional model of the quarkic structure of nucleons there is no mechanism that would account for the radial distributions of charge and their boundedness. On the other hand if quarks are bounded symmmetrical distributions of charge their effects outside their boundaries is the same as if their charges were concentrated at their centers.

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 in part from that rotation of charges. This is usually called a dipole moment. But it is thought that the magnet moment of a rotating particle structure can also come from some 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. In 1926 Samuel A. Goudsmit and George E. Uhlenbeck showed that this separation could be explained by the valence electrons of the silver atoms having a spin that is oriented in either of two directions. It has been long asserted that there is no evidence that this so-called spin is literally particle spin. However here it is accepted that the magnet moments of any particle is due to its spinning.

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 nuclear magneton units, is +2.79285. The nuclear magneton is defined

e(½h)/(m_Pc) in the cgs system
and
e(½h)/(m_P) in the SI system

where e is the unit of electrical charge, h is the reduced Planck's constant, m_P is the rest mass of a proton and c is the speed of light. It has the dimensions of energy per unit time.

The magnetic moment of a neutron is −1.9130. The ratio of these two magnetic moments 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 intrinsic magnetic moments of the neutron and proton is −2/3 then any dipole moment of the rotating quarks would result in a deviation of the overall magnetic moments from that value. The question is what spatial structure of nucleons would tend to have a negligible dipole moment. In the concentric shells 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 spheres will line up their centers exactly. See Quarks for the details.

The Magnetic Moments
of the Quarks

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

2μ_U + μ_D = 2.79285
and
μ_U + 2μ_D = −1.9130

Dividing the second equation by 2 gives

½μ_U + μ_D = −0.9565

Subtracting this equation from the first gives

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

The magnetic moment of the Down quark is then

μ_D = 2.79285 − 2μ_U = −2.20628 magnetons.

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

The Spin and Size of Quarks

The magnetic moment of a particle is of the form

μ = QkR²ω

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

(2/3)k_UR_U²ω_U = 2.49957
and
(−1/3)k_DR_D²ω_D = −2.20628

Equivalently these are

k_UR_U²ω_U = 3.749355
and
k_DR_D²ω_D = 6.61885

Note that the ratio of the RHS of these equations is

3.749355/6.61885 ≅ 9/16 = (3/4)²

If k_U=k_D and ω_U=ω_D then

(R_U/R_D)² ≅ 9/16
and hence
R_U/R_D ≅ 3/4

Notably the analysis indicates that the Down quark is larger than the Up quark.

What this means is that an Up quark and a proton are roughly the same size and in particular an Up quark is not a point particle, as in the conventional model. As noted before, there is a good reason a spherical shell quark would be mistakenly thought to be a point particle. Outside of the spherical shell its physical effects are the same as if its charge were concentrated at its center. In other words at points outside of its shell the effects of a spherically distributed quark cannot be distinguished from that of a point particle.

The radius R_P of a proton should be the same as the radius of the large Up quark and that of the neutron R_N the same as that of the large Down quark. R_P and R_N are known and Indeed

R_P/R_N = 0.84/1.1133 = 0.7545

The crucial terms are the tangential relative velocities at the equators of the particles.These are, from the above equations,

ω_UR_U = 3.749355/(k_UR_U)
and
ω_DR_D = 6.61885/(k_DR_D

Thus, taking k_U=k_D=(2/3)

ω_UR_U = 3.749355/((2/3)(0.84x10⁻¹⁵) = 6.6953x10¹⁵ m/s
and
ω_DR_D = 6.61885/((2/3)(1.1133x10⁻¹⁵) = 6.9179x10¹⁵ m/s

Each of these are far greater than the speed of light, 3x10⁸ m/s, so the analysis must be carried out in relativistic terms..

In another study it was found that the relativistic angular momentum of a spherical particle of radius R and mass m₀ spining at ω radians per second is given by

L = (m₀cR)β_m/(1 − β_m²)^3/2

where β_m is average tangential relative velocity on the sphere.

The solution can be found in terms of λ=β_m^2/3 where λ is the solution to the equation

(1 − λ³) = σλ

where σ=(m₀cR/L)^2/3.

The first step toward a solution for a Up quark is the evaluation of the parameter σ. The radius of a large Up quark is 0.84 fermi.

Thus

σ = [(0.556777x10⁻²⁷)(2.9979x10⁸)(0.84x10⁻¹⁵) /(3.749355)(0.527x10⁻³⁴)]^2/3
σ = [0.7096]^2/3
σ = 0.7956

The solution for λ is approximately

λ= 0.7425
and thus
β_m = 0.6398

This is the mean relative tangential velocity. What is needed is the maximum relative tangential velocity.

The Relationship of Mean and Maximum
Relative Tangential Velocities

The moment of inertia J can be computed two different ways

J = m₀R_m² = k m₀C²
and hence
R_max = (1/√)R_m

where k=(2/3) for a spherical surface and k=(2/5) for a spherical ball.

Since β_m=ωR_{m βmax = (1/√(2/3))(βm)
βmax = 1.2247βm}

At tangential velocities near the speed of light the relationship is

β_max = β_m

This can be rationalized as the result of relativistic contraction of the latitude circumferences of of a sphere. At high rates of rotation the sphere becomes more like a cylinder. On a cylinder β_max=β_m.

Between the two extremes β_max will be between β_m and 1. A relationship that fits over a range of β_m values is β_max being the geometric mean of 1 and β_m; i.e.,

β_max = (β_m)^½

Therefore for the Up quark

β_max = √0.6398 = 0.7999

The rate of rotation of the Up quark is given by

ω = β_maxc/R
ω = (0.7999)(2.9979x10⁸)/(0.84x10⁻¹⁵)
ω = 2.8548x10²³ radians per second
ω = 4.544x10²² turns per second

Conclusion

Taking into account the relativistic nature of angular momentum the magnetic moment of an Up quark derived from the measured magnetic moments of a proton and a neutron is consistent with it deriving from it being a rotating spherical electrostatic charge. Its computed rate of rotation is about 4.544x10²² times per second.