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Spin and Size Characteristics
of Quarks and Nucleons

This material is to investigate the characteristics 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 0.8 fermion and that of a neutron is 1.1 fermion.

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.

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.

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.

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.

There is no mechanism that would account for the radial distributions of charge and their boundedness if quarks were point particles. 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 the magnet moment of a rotating particle structure can also come 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 ions into a sharply varying magnetic field. The beam separated into two parts. This separation could be explained by the charged ions having a spin that is oriented in either of two directions. It has been long recognized 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 in the cgs system as eh/(2mPc), where e is the unit of electrical charge, h is the reduced Planck's constant, mP 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 nuclesr 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 μU and μD be the magnetic moments of the Up and Down quarks, respectively. Then

U + μD = 2.79285
and
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.

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 spherial 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 RU and RD, repectively . Likewise let ωU and ωD be their spin rates and kU and kD are the coefficients for the nature of their charge distributions.

Then

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

Equivalently these are

kURU²ωU = 3.749355
and
kDRD²ωD = 6.61885

Note that the ratio of the RHS of these equations is

3.749355/6.61885 ≅ 9/16 = (3/4)²

If kU=kD and ωUD then

(RU/RD)² ≅ 9/16
and hence
RU/RD ≅ 3/4

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

If the same formula of μ = QkR²ω is applied to a proton one obtains

kURU²ωU = 3.749355
and
kPRP²ωP = 2.79285

If kU≅kP and ωU≅ωP then

(RU/RP)² ≅ 1.3425
and hence
RU/RP ≅ 1.159

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 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 nature of the distribution of mass. Angular momentum L is then given by

L = Jω

The magnetic moment μ is similarly given by

μ = Kω

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

K = kSQR²

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

μ = αQ(L/M) = αL(Q/M)

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

This makes μ directly proportional to Q and inversely proportional to M and thus 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 while their charges are of the same magnitude. 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 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.

Conclusions

The concentric shells model of the quarkic structure of nucleons implies that the radii of the Up and Down quarks are roughly the size of nucleons. Their magnetic moments and radii are quantized.

The conventional model of the quarkic structure of nucleons was created to explain a single fact; that an isolated quark has never been found. The nonexistence of an isolated quark is much more easily explained by the alternative model in which quarks are spherical shells of charge and a nucleon is three concentric spheres of these quarkic shells.

The dimensions of the quarks are the same order of magnitude as the nucleons. In particular quarks are not point particles or anything approaching point particles. The centers of the quarks are close together when the spherical shell quarks are concentric and therefore there is little or dipole moment. 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 composed of three concentric quarkic shells needs one of each color.

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

All in all the 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.)

APPENDIX

Some Implications of
the Concentric Shells Model

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

Thus

mU = 0.556777×10-27 kg
and
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

LU/LD = kURU²ωU/kDRD²ωD
= 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

LU/LD = RU²/RD²

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.


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