Physics/Essays/Fedosin/Fine structure constant

Fine structure constant (usually denoted α, the Greek letter alpha) is a fundamental physical constant, namely the coupling constant characterizing the strength of 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.

Definition

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

$\alpha =\frac{e^2}{4\pi {\epsilon }_0\hslash c}=\frac{m_S^2}{4\pi {\epsilon }_g\hslash c}=\frac{e^{2}c{\mu }_0}{2h}=\frac{k_{\mathrm{e}}{e}^2}{\hslash c}=\frac{Z_0}{2R_K}={\left(\frac{e}{q_p}\right)}^2={\left(\frac{m_S}{m_P}\right)}^2=\frac{G_s{M}_p{M}_e}{\hslash c}=\frac{1}{4\beta },$

where:

• e is the elementary charge;
• h is the Planck constant;
• ħ = h/2π is the Dirac constant or reduced Planck constant;
• c is the speed of light in vacuum;
• ε₀ is the electric constant;
• m_S is the Stoney mass;
• ${\epsilon }_g=\frac{1}{4\pi G}$ is the gravitoelectric gravitational constant of selfconsistent gravitational constants;
• $G$ is the gravitational constant;
• µ₀ is the vacuum permeability;
• k_e is the Coulomb constant;
• R_K is the von Klitzing constant;
• Z₀ is the impedance of free space;
• q_p is the Planck charge;
• m_P is the Planck mass;
• M_p is the mass of proton;
• M_e is the mass of electron;
• G_s is the strong gravitational constant;
• β is the magnetic coupling constant.

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

$\alpha =\frac{e^2}{\hslash c}$

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

$\alpha =\frac{v_B}{c},$

where $v_B$ is the speed of electron’s matter at the Bohr radius $a_B$.

On the other hand

$\alpha =\frac{{\lambda }_0}{2\pi a_B}=\sqrt{\frac{r_e}{a_B}}=4\pi a_B{R}_{\mathrm{\infty }},$

where ${\lambda }_0$ is the Compton wavelength of the electron, $r_e$ is the classical electron radius, $R_{\mathrm{\infty }}$ is the Rydberg constant for wavelength.

The next equation for α is

$\alpha =\sqrt{\frac{{\mathrm{\Phi }}_e}{{\mathrm{\Phi }}_0}},$

where ${\mathrm{\Phi }}_0=\frac{h}{2e}$ is the magnetic flux quantum, ${\mathrm{\Phi }}_e=B{S}_B=\frac{{\mu }_{0}e}{4\pi a_B}{\sigma }_e$ is the electron magnetic flux for the first energy level, $B$ is the magnetic field in electron disc with flat surface area $S_B=\pi a_B^2$, and ${\sigma }_e=\frac{h}{2M_e}$ is velocity circulation quantum for electron.

Another equation for α is

$\alpha =\sqrt{\frac{{\mathrm{\Phi }}_{eg}}{{\mathrm{\Phi }}_p}},$

where ${\mathrm{\Phi }}_p=\frac{h}{2M_p}=1.98\cdot 10^{-7}$ 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

${\mathrm{\Phi }}_{eg}=\mathrm{\Omega }{S}_B=\frac{G_{s}h}{2c^2{a}_B},$

while $\mathrm{\Omega }$ 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:

$\alpha =\frac{G{M}_{ps}{M}_{\mathrm{\Pi }}}{{\hslash }_s{C}_s},$

where $M_{ps}$ and $M_{\mathrm{\Pi }}$ – mass of the star-analogue of proton and the planet-analogue of electron, respectively, ${\hslash }_s$ – stellar Dirac constant, $C_s$ – characteristic speed of stars matter.

See also

Many of articles in Vikiversity make reference to the fine structure constant, or discuss its relationship to other physical quantities. Examples are:

• Planck scale
• Stoney scale
• Strong gravitational constant
• Selfconsistent gravitational constants
• Selfconsistent electromagnetic constants
• Gravitational characteristic impedance of free space
• Magnetic coupling constant
• Monopoles

External links

• Fine structure constant in Russian
• Quotes About Fine Structure Constant
• Wikipedia article fine-structure constant