San José State University
Thayer Watkins
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& Tornado Alley

The Spin of a Proton and its Relationship
to the Spins of its Component Quarks

Up until 1987 it was presumed that the spin and consequent magnetic moment of a proton were due solely to those of its constituent quarks. In 1987 there were some experimental results which indicated that only a small fraction of the spin of a proton is accounted for by the spins of the quarks it is composed of. The additional spin of the proton apparently comes from the virtual gluons which surround a proton. The whole problem of the puzzle concerning the spin of a proton is covered by an article entitled "Proton Spin Mystery Gains a New Clue" by Clara Moskowitz in the July 21, 2014 issue of Scientific American.

The 1987 experimental results came as a surprise, but perhaps they should not have been too much of a surprise. Consider the situation concerning proton mass and quark masses. The mass of a proton is 1736 times that of an electron; a neutron's mass is 1839 times that of an electron. The mass of a positive pi meson is 273 times that of an electron. A proton consists of two up quarks and a down quark, whereas a neutron consists of one up quarks and one down quark. A proton and a neutron together consist of three up quarks and three down quarks. A positive pi meson consists of one up quark and one anti-down quark. An anti-down quark should have the same mass as a down quark. Three positive pi mesons would have three up quarks and the equivalent of three down quarks.

The combination of a proton and neutron has the equivalent of 3575 electron masses, but three pi mesons, having the same number of quarks, has a mass of only 819 electron masses. This accounts for only 23 percent of the proton-neutron combination. Thus the process of bringing the three quarks together in the nucleons creates mass. Logically the bringing together the two quarks of a pi meson may create mass in excess of the mass the two quarks bring to the meson. Therefore it can be said the mass of the constituent quarks of a proton and a neutron account for less than 23 percent of the masses of the nucleons.

However the analysis below does not take into account the 1987 experimental results. It is just an exploration of spin and magnetic moments of particles using the proton and neutron as examples.

Magnetic Moment

On a microscopic level the magnetism of a particle or nucleus is better represented as an electromagnet which results from an electrical current traveling in a circular orbit.

For a particle with a net charge of Q that is spinning at a rate of ω (radians per second) or ν (turns per second) the effective current is i=Qν=Qω/(2π). The area of the loop which the current surrounds is πr². Thus the magnetic moment μ is given by

μ = iA = Qνπr²
or, equivalently
μ = Q(ω/(2π))πr²
= (Q/2)ωr² = (Q/2)vr

where v is the tangential velocity of the charge. This is analogous to the angular momentum of a particle. The angular momentum involves the mass of the particle rather than the term (Q/2).

It is worthwhile at this point to establish the relationship between the magnetic moment and angular momentum. Let angular momentum be denoted by L. If L is divided by the mass of the particle and the result multiplied by the charge divided by 2 the result is the magnetic moment μ.; i.e.,

μ = QL/(2m)

Magnetic Moments of Nucleons and
the Quarks They are Composed of

The magnetic dipole moment of a proton, measured in magneton units, is +2.79285. That of a neutron is −1.9130. The ratio of these two numbers is −0.685, intriguingly close to −2/3.

A proton is composed of two up-quarks and one down-quark. As a preliminary model explaining the derivation of the magnetic moments of protons and neutrons let μU and μD be the magnetic moments of the up and down quarks, respectively. Then

U + μD = +2.79285
μU + 2μD = −1.9130

Multiplying the first equation by and subtracting the second equation from the result gives

U = 7.4987
and hence
μU = 2.4996 magnetons
and therefore
μD = −2.20628 magnetons

Angular momentum is quantized so magnetic moment is inversely proportional to mass. Thus the magnetic moment of an electron should be 1736 times the magnetic moment of a proton. The magnetic moment of an electron is 9284.76×10−27 J/sec whereas that of a proton is 14.106×10−27 J/sec. The ratio of these two figures is 658.2135. This is 1836/2.6375.

Another way of looking at the problem is that the above relation give the quantum of angular momentum; i.e.

L = 2Mμ/Q

Quarks as Spherical Shells of Charge

The magnetic moment M of a spherical charge Q of average radius r spinning at a rate of ν is πr²Qν. If M is expressed in units of magnetons then there is a coefficient C that is a magneton expressed in standard units. Thus

M = Cπr²Qν

Now consider nucleons being made up of three concentric shells of radii of rI, rM and rO, for Inner, Middle and Outer, respectively. Note the radial charge distributions of a proton and neutron.

As an initial trial It is presumed that down-quark for a proton is the inner shell. Therefore

CπrI²(−1/3)ν + CπrM²(2/3)ν + CπrO²(2/3)ν = 2.79285

For a neutron it is presumed the up-quark is the inner shell. Thus

CπrI²(2/3)ν + CπrM²(−1/3)ν + CπrO²(−1/3)ν = −1.9130

Let Cπν/3 be designated as γ. Dividing through the equations by γ gives

−rI² + 2rM² + 2rO² = 2.79285/γ
2rI² −rM² − rO² = −1.9130/γ

Multiplying the first equation by 2 and adding the result to the second equation gives

3rM² + 3rO² = 2.4996/γ

Multiplying the second equation by 2 and adding the result to the first equation gives

3rI² = −1.03315/γ
and hence
rI² = −0.34438/γ

This is an obvious impossibility.

As a second trial assume the down quark in a proton is the middle but the up quark in a neutron remains the inner shell. The equations are then

2rI² −rM² + 2rO² = 2.79285/γ
2rI² −rM² − rO² = −1.9130/γ

Subtracting the second equation from the first gives

3rO² = 4.70585/γ
and hence
rO² = 1.5686/γ


2rI² −rM² = −0.3444/γ

(To be continued.)

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