Chapter 17. Bohr and Nuclear Magnetons

This chapter examines and compares the formulas for the Bohr and nuclear magnetons. This examination is a prime example of how the use of FUPCONs in equations prompts us to come to false conclusions.

A. Magnetons, Historically

The Bohr magneton, µ_B, deals with the electron. Its historical, SI-based formula is:

µ_B = ½ h_bar q_e·m_e^-1 = 9.3 × 10^-24 A·m² (178

We can express this SI unit of magnetic moment, A·m², also as J·T^-1, which is one joule of energy per one tesla of magnetic induction.

The nuclear magneton, µ_N, deals with the proton. Its historical formula is:

µ_N = ½ h_bar q_p·m_p^-1 = 5.1 × 10^-27 A·m² (179

B. Analyzing the Magneton Formulas

The formulas for both magnetons contain the charge and mass for their respective particles. Note that the formulas do not include the wavelengths of these particles. Because the magnitudes of their respective charges equal each other, we can reasonably assume that the magnitudes of their masses contribute to the values of the magnetons. Right? Also, we can confirm this observation by dividing the value of µ_B by that of µ_N, as follows:

δ = µ_B µ_N^-1 = (9.3 × 10^-24 A·m²) ×

(5.1 × 10^-27 A·m²) = 1836.15 ... (180

which is the ratio of the mass of the proton to that of the electron. Therefore, we can see that the magnitude of each particle's mass is solely responsible for the ratio between the values of two magnetons. Right? Perhaps, we should look at the magnetons more closely before we come to this obvious conclusion.

C. Magnetons, Revisited

Upon a second look at Equations 178 and 179, we notice from the SI units, A·m², that a magneton's dimensions are QT^-1L². The mass dimension does not exist. Why not?

To find out, we convert the FUPCONs in the two equations into their constituent parts.

For the Bohr magneton:

µ_B = ½ h_bar q_e·m_e^-1 = 9.3 × 10^-24 A·m² (181

µ_B = ½ (m_e·λ_e²·t_e^-1)(2 π)^-1 q_e·m_e^-1 =

(4 π)^-1 q_e·t_e^-1·λ_e² (182

Polykarp Kusch received the Nobel prize for determining empirically the magnitude of the magnetic moment of the electron. It is larger than the magnitude of the Bohr magneton by a factor of [(2 π β) + 1][2 π β]^-1. The electron g-factor possesses twice the value of this factor.

For the nuclear magneton:

µ_N = ½ h_bar q_p·m_p^-1 = 5.1 × 10^-27 A·m² (183

We convert each SE component of Planck's constant into its SP counterpart (see Equations 76 through 79). As we know, the SE components of Planck's constant, h, are (m_e·λ_e²·t_e^-1). They convert, as follows:

h = (δ m_e) (δ^-1 λ_e)² (δ^-1 t_e)^-1 = m_p·λ_p²·t_p^-1

Therefore, when using the quantum attributes of the proton, the nuclear-magneton formula is:

µ_N = ½ (m_p·λ_p²·t_p^-1) (2 π)^-1 q_p·m_p^-1 =

(4 π)^-1 q_p·t_p^-1·λ_p² (184

Now, dividing the SE-based value of µ_B by the SP-based value of µ_N gives:

δ = µ_B _N^-1 = [(4 π)^-1 q_e·t_e^-1·λ_e²] [(4 π)^-1 q_p·t_p^-1·λ_p²]^-1 =

[c (4 π)^-1 q_e·λ_e] [c (4 π)^-1 q_p·λ_p]^-1 =

λ_e·λ_p^-1 = (λ_e)(δ^-1 λ_e)^-1 = δ = 1836.15 ... (185

where: c = λ_e·t_e^-1 = λ_p·t_p^-1, and q_e = q_p.

Clearly, the value of each magneton is a function of the wavelength of the elementary particle concerned and not of its mass as the historical formula appears to intimate.