This material stems from a model of quarks being spherical shells of charge and mass. Hadrons are concentric spheres of quarks. A previous study established estimates of densities, volumes and masses for Up and Down quarks in their three different varieties of small, medium and large. These estimates are based upon the characteristics of nucleons and pi mesons such as magnetic moments. It is presumed that these characteristics are intrinsic properties of the quarks and carry over to quarks in other particles.

This material is to investigate the characteristics of quarks in relation to those of hadrons which include a Strange quark along with an Up or Down quark. These are called hyperons. There are also mesons containing Strange quarks. These are called kaons or k-mesons. They include a Strange quark and an anti-version of an Up quark or a Down quark.

A positive sigma particle is composed of two up quarks and one strange quark. Another sigma particle is composed of two down quarks and one Strange quark. The neutralS sigma particle is composed of one up quark, one down quark and a strange quark. The Up quark has an electrostatic charge of +2/3 and a Strange quark an electrostatic charge of −1/3. Therefore a positive sigma particle has an electrostatic charge of +1. A neutral sigma particle has a charge of zero. A sigma particle with two Down quarks and a Strange quark has a charge of −1. But a particle consisting of a Down quark and two Strange quarks would also have a charge of −1.

As mentioned above there are also mesons which contain a Strange quark. The negative kaon consists of a Strange quark and an anti-Up quark. An anti-Up quark has the same characteristics as an Up quark exept its charge is the opposite. Thus a negative kaon has a charge of −1.

A meson consisting of a Strange quark and an anti-Down quark would have a charge of zero. A meson consisting of an Up quark and an anti-Strange quark would have a charge of +1. It would be properly called a positive kaon.

The Radial Distributions of
Charge in the Nucleons

The radial distributions of electrostatic charge are found by sending electrons as probes against collections of positi 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.

In the concentric shells model of the quarkic structure of hadons a quark is a spherical shell of electrostatic charge and mass.

A hyperon in this model consists of three concentric rotating quarkic shells. There is thus three versions of each quark: The small, medium and large versions. It is impossible to separate them because any action taken againt the outer quark equally affects the other quarks in a nucleon.

Conventionally each quark has another attribute that is 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 outer 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.

In the conventional model of hadron structure 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 and mass their effects outside their boundaries is the same as if their charges and mass were concentrated at their centers.

An actual charged point particle would have infinite energy. There is not enough energy in the entire Universe to create even one charged point quark. That is to say, in attempting to create one point particle quark the the effort would fail even after all of the energy of the billions of stars in every one the billions of galaxies is used up. And there would be nothing left over for creating a second quark or any of the zillions upon zillions of other quarks in the Universe.

The Masses of the Strange Quarks

The mass of a positive or negative kaon is 966.1 electron masses. It is uncertain as to whether the Strange quark is the center quark or whether the anti-Up quark is. From the previous study the mass of a small Up quark is 50.2382 electron masses. This means the mass of a medium sized Strange quark would be 915.8616 electron masses. With the mass density of a Strange quark being the same as that of a Down quark the volume occupied by the medium Strange quark would be 2.9264 cubic fermi (f³). This added to the volume occupied by a small Up quark gives the volume occupied by the two inner quarks in a sigma hyperon; i.e., 2.9264+0.0654=2.9918 f³. Dividing this figure by (4/3)π gives the cube of radius of the medium Strange quark; i.e., 0.7142 fermi. This would mean the radius of a Strange quark would be 0.8939 fermi.

If the Strange quark is at the center of the hyperon its mass would be 966.1 electron masses less the mass the mass of a medium anti-Up quark; i.e., 966.1 − 224.7953 = 741.55027 electron masses. Dividing this mass by the density of a Down quark gives the volume of a Strange quark; i.e., 2.3693 f³. Dividing this volume by (4/3)π gives the cube of the radius of the Strange quark; i.e., 0.5656 f³. This means the radius of a Strange quark would be 0.8270 fermi. This seems to be the more plausible alternative especially since the arrangement SUU obey the rule of no particle being linked to more thatn one of its own kind and no more than one of its opposite kind,

The mass of a positive sigma hyperon is 2327.53 electron masses. Subtractin the mass of a charged kaon from this gives the mass of a large anti-Up quark; i.e., a large Up quark as 1361.4342 electron masses, whereas the value from the previous study is 1565.5525 electron masses. These are the same order of magnitude and close enough to provise some confirmation of the concentric spheres model of haron structure..

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 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 outer unpaired electrons of these atoms having a spin that is oriented in either of two directions. It has been long asserted that this so-called spin is not necessarily literally physial particle spin. However there is no evidence that it is not. Here it is accepted that the magnet moment of any particle is due to its actual spinning.

The Evidence

The magnetic moments of the sigma hyperons derive from the intrinsic moments of their quarks and any dipole moment of the quarks within them. The magnetic moment of a positive sigma particle, measured in nuclear magneton units, is +2.458. 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.

No measured magnetic moment for the neutral sigma particle is available at present but it is likely to be nonzero just as is that of the neutron.

The magnetic moment of a negative sigma particle is −1.160. The ratio of these two magnetic moments is −0.47193, intriguingly close to −1/2. This suggests that the ratio of the intrinsic magnetic moments of the negative and positive sigma particles is precisely −1/2.

If the ratio of the intrinsic magnetic moments of the negative and positive sigma particles is −1/2 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 spheresare 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 positive sigma particle is composed of two Up quarks and one Strange quark. For a negative sigma particle its composition is two Strange quarks and one Up quark. Let μ_U and μ_S be the magnetic moments of the Up and Strange quarks, respectively. Then

2μ_U + μ_S = 2.458
and
μ_U + 2μ_S = −1.160

Dividing the second equation by 2 gives

½μ_U + μ_S = −0.580

Subtracting this equation from the first gives

(3/2)μ_U = 3.0380
and hence
μ_U = 2.0253 magnetons

The magnetic moment of the Strange quark is then

μ_S = 2.458 − 2μ_U = −1.5927 magnetons.

Note the ratio |μ_S|/μ_U = −0.7864 ≅ (3/4) This is the ratio of magnetic moments but not necessarily the ratio of quark sizes.

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 Strange quark is −1/3. Let the average charge radii of the Up and Strange quarks be denoted by R_U and R_S, repectively . Likewise let ω_U and ω_S 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 = 2.458
and
(−1/3)k_SR_S²ω_S = −1.160

Equivalently these are

k_UR_U²ω_U = 3.687
and
k_SR_S²ω_S = 3.480

Note that the ratio of the RHS of these equations is

3.687/3.480=1.05948

If k_U=k_S and ω_U=ω_S then

(R_U/R_S)² = 1.05948
and hence
R_U/R_D = 1.0293 ≅ 1

Notably the analysis indicates that the Strange quark is essentially same size as the Up quark.

What this means is that an Up quark and a Strange quark 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. Thus any evidence for a point quark is not evidence for an impossible infinite energy charged point particle, but instead evidence for spherically distributed charged 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 = j_SMR²

where j_S 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 = k_SQR²

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 t. 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 explains 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.)