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Magnetic Moment and Spin of a Proton |
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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.
The magnetic moment of a proton, measured in magneton units, is 2.79285. The magneton is defined as
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. 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
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
Thus
Thus, this indicatess that the number of degrees of freedom of the charged spherical proton is 1.
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 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 proton is the evaluation of the parameter σ. The radius of a proton; i.e. 0.84 fermi.
Thus
The solution for λ is approximately λ= 0.45065 and thus β_{m}=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
Therefore for the proton
This means a proton is rotating at a rate of
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., 4.74x10^{21} rotations per second. See Nuclear Rotation.
The rate of rotation found using classical physics and quantum theory was 4.962x10^{22} times per second with β_{max}=0.873. The relativistic value of β_{max} 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.
Taking into account the relativistic nature of angular momentum the measured magnetic moment of a proton is consistent with it deriving from it being a rotating spherical electrostatic charge. Its computed rate of rotation is about 4.296x10^{22} times per second.
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