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The Relativistic
Computation
of the Spin of
a Down Quark

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)/(mPc) in the cgs system
and
e(½h)/(mP) in the SI system

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 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 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 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.

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

RP/RN = 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,

ωURU = 3.749355/(kURU)
and
ωDRD = 6.61885/(kDRD

Thus, taking kU=kD=(2/3)

ωURU = 3.749355/((2/3)(0.84x10−15) = 6.6953x1015 m/s
and
ωDRD = 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)βm/(1 − βm²)3/2

where βm is average tangential relative velocity on the sphere.

The solution can be found in terms of λ=βm2/3 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 Down quark is the evaluation of the parameter σ. The radius of a large Down quark is 1.1133 fermi.

Thus

σ = [(0.559708x10−27)(2.9979x108)(1.1133x10−15) /(6.61885)(0.527x10−34)]2/3
σ = [0.5355]2/3
σ = 0.6595

The solution for λ is approximately

λ= 0.785
and thus
βm = 0.6955

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 = m0Rm² = k m0
and hence
Rmax = (1/√k)Rm

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

Since βm=ωRm βmax = (1/√(2/3))(βm)
βmax = 1.2247βm

At tangential velocities near the speed of light the relationship is

βmax = βm

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 βmaxm.

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

βmax = (βm)½

Therefore for the Down quark

βmax = √0.6955 = 0.8340

Thus the rate of rotation is given by

ω = βmaxc/R
ω = (0.834)(2.9979x108)/(1.1133x10−15)
ω = 2.2457x1023 radians per second
ω = 3.574x1022 turns per second

Conclusion

Taking into account the relativistic nature of angular momentum the magnetic moment of a Down quark derived from the measured magnetic moments of a proton and a neutron is consistent with it being derived from a rotating spherical electrostatic charge. Its computed rate of rotation is about 3.574x1022 times per second.


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