Question

What is a fine structure constant and what are its units, dimensions and formula?

Answer 1

In advanced physics, the fine structure constant is synonymous to Sommerfeld’s constant. Denoted by Greek letter 𝜶 (alpha). It is a dimensionless physical constant. Fine-Structure Constant characterises the strength of electromagnetic interactions between any two elementary charged particles.

Measurement

Being a dimensionless quantity, Fine-Structure Constant has the same numerical value in all system of units. Which is approximately

$\alpha =\frac{1}{137}=0.0072973525664\left(17\right)$.

Definition

Fine-Structure Constant characterises the strength or magnitude of coupling of any elementary charged particle(e) with the given electromagnetic field. Mathematically 𝜶 can be expressed in terms of other fundamental physical constants as follows-

Where,

$\hslash =\frac{h}{2\pi }$ is the reduced Plank constant.

e is the elementary charge ($e=1.602176634\times {10}^{-19}$ C)

c is the velocity of light in vacuum (c = 299792458 m/s)

${\epsilon }_0$ is the permittivity of free space.

${\mu }_0$ is the Permeability of free space.

$K_e$ is the Coulomb constant.

$R_K$ is the von Klitzing constant.

$Z_0$ is the impedance of free space.

In non-SI units

Value of this dimensionless physical quantity can be expressed in many units.

Electrostatic CGS unit

In electrostatic CGS units, electric charge is measured using statcoulomb and is defined assuming permittivity factor $4\pi {\epsilon }_0$ or Coulomb constant $k_e$ is 1 and is dimensionless. Thus, the value of fine structure constant turns out to be-

$\alpha =\frac{e^e}{\left(\frac{h}{2\pi }\right)c}$

Natural unit

High energy physics commonly use natural units, where ${\epsilon }_0=c=ħ=1$. here, ${\epsilon }_0$ is the permittivity of free space. Thus the value of fine structure constant turns out to be-

$\alpha =\frac{e^e}{4\pi }$

Atomic units

According to atomic units, ${ϵ}_0=\frac{1}{4\pi }$ and $e=m_e=ħ=1$, thus the fine structure constant turns out to be-

$\alpha =\frac{1}{c}$

Physical interpretations

The physical interpretation of Fine Structure Constant is plenty. Some are listed below-

In quantum electrodynamics, Fine Structure Constant 𝜶 is directly related to coupling constant.

In solid state physics and electrical engineering, 𝜶 is one-fourth product of the characteristic impedance of free space.

Within the Bohr model, 𝜶 gives the maximum positive charge of an atomic nucleus that will allow a stable-orbit around it.

𝜶 gives the probability about emission or absorption of an electron by a photon.

It also has got wide applications in low energy physics, particularly in electroweak theory. Also, some properties of subatomic particles exhibit a strong correlation with 𝜶.

Hope you have understood about fine structure constant(𝜶). Including, definition, formula, terms along with units, values in non-SI units and physical interpretation of alpha.