# alpha Fine Structure Constant

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### Fine Structure Constant

A fundamental quantity known as alpha, fine structure constant,

is pivotal in atomic physics and is

\[

\alpha \ = \ \frac{e^{2}}{4\pi \epsilon_{0} \hbar c} \ = \ 7.2973525698 \times 10^{-3}

\ \sim \ \frac{1}{137}

\]

and

\[

\frac{1}{\alpha} \ = \ 137.035999074

\]

where

$e = \ $ elementary charge,

$\hbar = \ $ Planck’s constant divided by $2\pi$.

$c = \ $ speed of light.

The value of $\alpha$ above is the 2010 value recommended by the Committee on Data for Science and Technology (CODATA), as reported in Mohr, Taylor, and Newell (NIST, March 2012).

The fine structure constant is dimensionless. An interesting result is that the classical orbital speed of an electron in a hydrogenic atom as a fraction of the speed of light is

\[

\frac{v}{c} \sim \alpha \ Z

\]

where $Z$ is the atomic number of the nucleus. Obviously this is not valid if $v/c$ is relativistic, and it is not based on a proper quantum-mechanical treatment, but it does give an idea of whether relativistic effects are important for a given $Z$. Another result is that the binding energy of an electron in a hydrogenic atom can be expressed in terms of $\alpha$ and the electron rest mass-energy, $m_{e}c^{2}$:

\[

E \ = \ -\frac{1}{2} \alpha^{2} Z^{2} (m_{e} c^{2}).

\]

The binding energy is equal to the ionization energy (or potential), or the energy required to remove the electron from the atom. Of course, $\alpha$ sets the scale for the corrections to the basic atomic energy levels due to relativistic effects and the interaction of the magnetic field due to the electron’s orbital motion and the electron’s own magnetic moment. This is known as the spin-orbit interaction. The resulting energy shifts with respect to the gross energy level structure are known collectively as the fine structure of the energy-level spectrum. The energy corrections are of the order of

\[

\Delta E \sim \frac{1}{2} \alpha^{2} Z^{2} \ E_{\rm gross}

\]

where $E_{\rm gross}$ refers to the gross energy difference between levels having different principal quantum numbers ($n$).