San José State University |
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The RelativisticComputation of the Spin of a Down Quark |
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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.

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.

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 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

and

e(½

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.

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

and

μ

Dividing the second equation by 2 gives

Subtracting this equation from the first gives

and hence

μ

The magnetic moment of the Down quark is then

Note the ratio μ_{D}/μ_{U}=0.7899744=1/1.2658638≅4/5.

The magnetic moment of a particle is of the form

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

and

(−1/3)k

Equivalently these are

and

k

Note that the ratio of the RHS of these equations is

If k_{U}=k_{D} and ω_{U}=ω_{D} then

and hence

R

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

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

and

ω

Thus, taking k_{U}=k_{D}=(2/3)

and

ω

Each of these are far greater than the speed of light, 3x10^{8} 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 m_{0} spining at ω radians per second is given by

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

where σ=(m_{0}cR/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 = [0.5355]

σ = 0.6595

The solution for λ is approximately

and thus

β

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

Relative Tangential Velocities

The moment of inertia J can be computed two different ways

and hence

R

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

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.,

Therefore for the Down quark

Thus the rate of rotation is given by

ω = (0.834)(2.9979x10

ω = 2.2457x10

ω = 3.574x10

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 3.574x10^{22} times per second.

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