On the Unification of the Constants of Nature

INTRODUCTION

The magnetic flux quantum Φ₀ is equivalent to^{[1], [2], [3]}

where h is Planck's constant^[4] and Z₀ is the charge of an alpha particle (2e). Planck's reduced constant ћ is

which can also be defined with Bohr's radius r_B as

where α is the fine structure constant, m_e is the mass of a Beta particle, and c is the speed of light in a vacuum. Combining Eqs. (1) through (3) yields

Bohr, however, did not deduce his radius r_B from an Alpha particle (Z₀ = 2e = a Helium nucleus and not a Hydrogen nucleus). The adjusted radius r₀ for a Helium atom can be deduced from

Eq. (4) as r₀ ≈ 0.529177211 × 10⁻¹⁰ m.

WAVE−PARTICLE DUALITY AND A NEW DEFINITION FOR THE FINE STRUCTURE CONSTANT

A particle's wavelength λ can be determined with de Broglie's matter wave equation

where p is the particle's momentum and v is its speed^[6]. With the mass quantized in units of m_e, a Beta particle's ground state wavelength relative to the electric and magnetic flux quanta of a Helium nucleus can be deduced from Eqs. (4) & (5) as

The Beta particle's ground state frequency f₀ is then

and a wave mechanical definition for the fine structure constant can be given as

Eq. (8) suggests v₀ ≈ αc. The energy of electromagnetic radiation (E = hf) is simply the product of the electric, magnetic, and frequency quanta;

CONCLUSION

The Gaussian gravitational constant k₀ (not to be confused with the Coulomb constant k_e)^[7] is

where G is Newton's gravitational constant, T is the orbital period, and M₁ and M₂ are the masses of the system. Setting the total mass of the system to the mass of an Alpha particle M_A (M_A = 2M_P + 2M_N where M_P and M_N are the proton and neutron masses), the quantized relationship between the Gaussian gravitational constant k₀ and the electromagnetic frequency of an Alpha particle can be given as

The standard gravitational parameter µ for the Alpha particle is therefore

and the energy of its electromagnetic frequency is

Could Eq. (13) help explain dark energy? The nuclear frequency f_n of an atom would then be

where n is the nuclear mass number relative to a Helium nucleus. For Hydrogen, n = ½, Helium, n = 1, Lithium, n = 1½, Beryllium, n = 2, etcetera.

REFERENCES

[1] "Magnetic flux quantum Φ0". 2010 CODATA recommended values.

[2] Deaver, Bascom; Fairbank, William (1961). "Experimental Evidence for Quantized Flux in Superconducting Cylinders". Physical Review Letters 7 (2): 43−46.Bibcode:1961PhRvL...7...43D. doi:10.1103/PhysRevLett.7.43.

[3] Doll, R.; Näbauer, M. (1961). "Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring". Physical Review Letters 7 (2):

51–52. Bibcode:1961PhRvL...7...51D.doi:10.1103/PhysRevLett.7.51.

[4] Barrow, John D. (2002) “The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe”, Pantheon Books, ISBN 0-375-42221-8

[5] Eisberg, Robert (1974). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles. New York: John Wiley & Sons Inc. pp. 114–117. ISBN 978-0-471-23464-7.

[6] de Broglie, L. (1923). Waves and quanta, Nature 112: 540.

[7] Gauss, Carl Friedrich; Davis, Charles Henry (1857). Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. Little, Brown and Company, Boston., atGoogle books