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The Relativistic Angular Momentum,
Magnetic Moment and
Spin of a Proton

Background

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 this so-called spin is not literally particle spin. It is often referred to as intrinsic spin whatever that might mean. However here in the material that follows it is accepted that the magnet moment of any particle is due to its actual spinning and the spin rate can be computed from its measured magnetic moment.

Magnetic Moments

The magnetic moment of a proton, measured in magneton units, is 2.79285. The magneton is defined as

½he/mp in the SI system
and
½he/(mpc) in the cgs 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. Thus the magneton has different dimensions in the different systems of units. In the SI system it has the dimensions of energy per unit time (Joules per second).

For a proton

L/(½h) = 2.79285

`

Aage Bohr and Ben Mottleson found nuclei rotations satisfy the h√I(I+1) rule, where I is an integer representing the number of degrees of freedom of the rotating object. The number of degrees of freedom for a sphere is a bit uncertain. It could be three for rotations about three orthogonal axes. It could be just one for a charged sphere.

The angular momentum according the Bohr-Mottelson Rule is

L = h(I(I+1))½

Thus

h(I(I+1))½ = 2.79285((½h)
and therefore
( I(I+1))½ = ½(2.79285) = 1.396425
which means
(I(I+1)) = 1.95 ≅ 1(2)

Thus, this indicatess that the number of degrees of freedom of the charged spherical proton is 1.

Relativistic Angular Momentum

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 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 proton is the evaluation of the parameter σ. The radius of a proton; i.e. 0.84 fermi.

Thus

σ = [(1.6726x10−27)(3x108)(0.84x10−15) /(2.79285)(0.527x10−34)]2/3
= [2.8637]2/3
= 2.0166

The solution for λ is approximately λ= 0.45065 and thus βm=0.3025

This is the mean relative tangential velocity. The relationship between the mean and maximum tangentential veocities for a spherical ball at velocities far below the speed of light is

βm = (2/5) βmax
hence
βmax = (5/2)βm

Therefore the proton

βmax = (5/2)(0.3025) = 0.7563

Rotation Rate

This means a proton is rotating at a rate of

ω = βmaxc/R
= (0.7563)(2.9979x108/(0.84x10−15)
= 2.6992x1023 radians per second
= 4.296x1022 times per second

This is an increditably high rate but it is what it would have to be to generate its measured magnetic moment. It is comparable to the high rates found for nuclei in general; i.e., 4.74x1021 rotations per second. See Nuclear Rotation.

The rate of rotation found using classical physics and quantum theory was 4.962x1022 times per second with βmax=0.873. The relativistic value of βmax found above was 0.7563.

The relativistic method of computing the rotation rate handled the problem that it might imply the matter of the material of the proton could be traveling faster than the speed of light. The general problem of the determination of rotation rates taking into account Special Relativity is dealt with in Relativistic Angular Momentum.

Conclusion

Taking into account the relativistic nature of angular momentum the measured magnetic moment of a electron is consistent with it deriving from it being a rotating spherical electrostatic charge. Its computed rate of rotation is about 5.663x1022 times per second.

For material on the spin of a neutron see Neutron Spin. For an electron see Electron Spin and Relativistic Electron Spin.


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