Background

The magnetic moment μ of a point charge of q rotating about an axis at distance r away from it at a rate of ω radians per second is qr²ω. For a charge of Q uniformly distributed over a surface or volume the magnetic moment must be computed by integration over that surface or volume in exactly the same way the moment of inertia is computed for a mass M distributed over a surface or volume. This means the formula for the magnetic moment of an object is of the form

μ = QkR²ω

where R is in the nature of an average distance of the charge from the spin axis and k is a coefficient dependent upon the shape of the distribution of charge. For example if the charge is uniformly distributed over a spherical shell then k=(2/3). For a solid sphere it is (2/5).

The structures to be considered are nucleons, both as distributions of charge and mass and as compositions of quarks. The net magnetic dipole moments of a proton and a neutron, measured in magneton units, are 2.79285 and −1.9130. respectively. For an electrically neutral neutron to have a magnetic moment might be something of a surprise, but the radial distribution of charge easily explains it.

The average radius of the negative charge of the neutron is so much greater than the average radius of its positive charge that the net value of its magnetic moment is negative. It is notable that direction of the spin of a neutron is the same as that of a proton.

The Magnetic Moments of
Nucleons in Relation to that
of their Composite Quarks

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

2μ_U + μ_D = 2.79285
and
μ_U + 2μ_D = −1.9130

Dividing the second equation by 2 gives

½μ_U + μ_D = −0.9565

Subtracting this equation from the first gives

(3/2)μ_U = 3.74935
and hence
μ_U = 2.49957 magnetons

The magnetic moment of the Down quark is then

μ_D = 2.79285 − 2μ_U = −2.20628 magnetons.

Note the ratio μ_D/μ_U=0.7899744=1/1.2658638≅4/5.

Now remember that the magnetic moment of a particle is of the form

μ = QkR²ω

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, respectively. Likewise let ω_U and ω_D be their spin rates and k_U and k_D the coefficients for the shapes of their charge distributions. Then

(2/3)k_UR_U²ω_U = 2.49957
and
(−1/3)k_DR_D²ω_D = −2.20628

Equivalently these are

k_UR_U²ω_U = 3.749355
and
k_DR_D²ω_D = 6.61885

Note that the ratio is

3.749355/6.61885 ≅ 9/16 = (3/4)²

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 shape of the distribution of mass. Angular momentum L is given by

L = Jω

The magnetic moment μ is 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 the magnetic moment of a charged particle is related to its angular momentum by

μ = αL(Q/M)

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

This makes μ directly proportional to Q and inversely proportional to M and 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. 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.

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 m_U and m_D be the masses of the Up and Down quarks, respectively. Then

2m_U + m_D = 1.6726231×10^-27 kg
m_U + 2m_D = 1.6749286×10^-27 kg

Thus

m_U = 0.556777×10^-27 kg
and
m_D = 0.559708×10^-27 kg

Thus m_D/m_U=1.00526. So essentially m_D =m_U.

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

L_U/L_D = n_U/n_D

a ratio of integers. Since L=j_SMR² then if j_S=k_S for each quark

L_U/L_D = k_UR_U²ω_U/k_DR_D²ω_D
= 3.749355/6.61885 = 0.566466

Note that 9/16 = 0.5625 and 0.566466=(1.007)(9/16). Thus

L_U/L_D ≅ (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 k_U=k_D. Furthermore there is a rationale for ω_U=ω_D. See Sensible Model of Quarkic Structure of Nucleons.

These equalities would mean

L_U/L_D = R_U²/R_D²

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

L_U/L_D = ν_U²/ν_D² =(ν_U/ν_D)²

The implication of the above is that

R_U/R_D = 3/4

Thus it is significant that (9/16)=(3/4)².

For more on the quarkic spatial structure of nucleons see Sensible Model of Quarkic Structure of Nucleons.

(To be continued.)