# Physics/Essays/Fedosin/Fine structure constant

< Physics‎ | Essays‎ | Fedosin

In physics, the fine structure constant (usually denoted α, the Greek letter alpha) is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units. Arnold Sommerfeld introduced the fine-structure constant in 1916.

The current recommended value of α is 7.29735257×10−3. [1]

## Definition

Some equivalent definitions of α in terms of other fundamental physical constants are:

${\displaystyle \alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}={\frac {m_{S}^{2}}{4\pi \varepsilon _{g}\hbar c}}={\frac {e^{2}c\mu _{0}}{2h}}={\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}={\frac {Z_{0}}{2R_{K}}}=\left({\frac {e}{q_{p}}}\right)^{2}=\left({\frac {m_{S}}{m_{P}}}\right)^{2}={\frac {G_{s}M_{p}M_{e}}{\hbar c}}={\frac {1}{4\beta }},}$

where:

In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, ke, or the permittivity factor, 4πε0, is 1 and dimensionless. Then the expression of the fine structure constant becomes the abbreviated

${\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}}$

an expression commonly appearing in physics literature.

## Bohr model

In the Bohr model of hydrogen atom α is connected to the atom parameters for the first energy level

${\displaystyle ~\alpha ={\frac {v_{B}}{c}},}$

where ${\displaystyle ~v_{B}}$  is the speed of electron’s matter at the Bohr radius ${\displaystyle ~a_{B}}$ .

On the other hand

${\displaystyle ~\alpha ={\frac {\lambda _{0}}{2\pi a_{B}}}={\sqrt {\frac {r_{e}}{a_{B}}}}=4\pi a_{B}R_{\infty },}$

where ${\displaystyle ~\lambda _{0}\ }$  is the Compton wavelength of the electron, ${\displaystyle ~r_{e}\ }$  is the classical electron radius, ${\displaystyle ~R_{\infty }}$  is the Rydberg constant for wavelength.

The next equation for α is

${\displaystyle ~\alpha ={\sqrt {\frac {\Phi _{e}}{\Phi _{0}}}},}$

where ${\displaystyle ~\Phi _{0}={\frac {h}{2e}}}$  is the magnetic flux quantum, ${\displaystyle ~\Phi _{e}=BS_{B}={\frac {\mu _{0}e}{4\pi a_{B}}}\sigma _{e}}$  is the electron magnetic flux for the first energy level, ${\displaystyle ~B}$  is the magnetic field in electron disc with flat surface area ${\displaystyle ~S_{B}=\pi a_{B}^{2}}$ , and ${\displaystyle ~\sigma _{e}={\frac {h}{2M_{e}}}}$  is velocity circulation quantum for electron.

Another equation for α is

${\displaystyle ~\alpha ={\sqrt {\frac {\Phi _{eg}}{\Phi _{p}}}},}$

where ${\displaystyle ~\Phi _{p}={\frac {h}{2M_{p}}}=1.98\cdot 10^{-7}}$  m2/s

is the strong gravitational torsion flux quantum, which is related to proton and to its velocity circulation quantum,

and the strong gravitational electron torsion flux for the first energy level of hydrogen atom is

${\displaystyle ~\Phi _{eg}=\Omega S_{B}={\frac {G_{s}h}{2c^{2}a_{B}}},}$

while ${\displaystyle ~\Omega }$  is the gravitational torsion field of strong gravitation in electron disc.

The Bohr model for hydrogen system at the level of star introduces α in such way:

${\displaystyle ~\alpha ={\frac {GM_{ps}M_{\Pi }}{\hbar _{s}C_{s}}},}$

where ${\displaystyle ~M_{ps}}$  and ${\displaystyle ~M_{\Pi }}$  – mass of the star-analogue of proton and the planet-analogue of electron, respectively, ${\displaystyle ~\hbar _{s}}$ stellar Dirac constant, ${\displaystyle ~C_{s}}$ characteristic speed of stars matter.