# Fine Structure Constant

The fine structure constant, symbolized as α (alpha), is a fundamental physical constant that indicates the strength of the interaction between charged particles. It plays a crucial role in the realms of quantum mechanics and electromagnetism. Other names for the constant are the coupling constant or the Sommerfeld constant in honor of Arnold Sommerfeld who introduced it.

### Definition and Value

The fine structure constant is dimensionless, meaning it has no units and is a pure number. It approximately equals 1/137, with a more precise value of approximately 0.0072973525693, or 1/137.036.

### Historical Overview

The term “fine structure constant” originates from its role in explaining the fine structure of atomic spectra, particularly the spectral lines of hydrogen.

In atomic physics, spectral lines result from electrons in an atom transitioning between different energy levels, emitting or absorbing light at specific wavelengths. When closely examined, some of these spectral lines are not single lines but actually consist of several closely spaced lines. This splitting of spectral lines is what is referred to as the “fine structure” of the spectrum.

Arnold Sommerfeld introduced the concept of the fine structure constant in 1916, during his extension of the Bohr model of the atom to include the relativistic relationship between mass and velocity and elliptical orbits. This led to the prediction of slight shifts in the energy levels of electrons, especially those moving at speeds close to the speed of light. These shifts in energy levels result in the splitting of spectral lines, explaining the observed fine structure in atomic spectra.

### Significance of the Fine Structure Constant

This constant characterizes the strength of the electromagnetic interaction between elementary charged particles, like electrons and protons. It explains the fine structure splitting of hydrogen atom energy levels, which is how it got its name. The fact that α is dimensionless and close to 1/137 intrigues physicists, leading to deep questions about the fundamental principles of the universe.

### Expression in Terms of Other Constants

The fine structure constant is defined in terms of other fundamental physical constants:

$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c}$

where:

• α represents the fine structure constant
• e is the elementary charge
• π is pi
• ϵ0​ is the vacuum permittivity
• ℏ is the reduced Planck constant
• c is the speed of light in vacuum

### Methods of Measurement

Scientists measure the value of the fine structure constant using various methods, including:

1. Observing the spectral lines of atoms.
2. Quantum Hall effect.
3. Anomalous magnetic dipole moment of the electron.

### The Question of Being the Reciprocal of 137

While α is approximately 1/137, it is not exactly this value. The discrepancy is crucial in physics, as precision measurements and theoretical calculations rely on its exact value. The slight difference from 1/137 continues to be a subject of study and speculation.

### Is the Fine Structure Constant Changing?

One of the ongoing debates in modern physics is whether the fine structure constant is indeed constant or varies over time and space. Some theoretical models and astronomical observations suggest a possible variation, but this is not definitively proven. The idea that α might change is tantalizing because it implies new physics beyond our current understanding of the universe.

### Applications in Physics and Chemistry

The fine structure constant finds its way into several key calculations in physics and chemistry, particularly those involving quantum mechanics and electromagnetism:

Atomic Energy Level Calculations: In atomic physics, α occurs in the calculation of the energy levels of electrons in atoms, especially in the hydrogen atom.

$E_n = -\frac{\alpha^2 m_e c^2}{2n^2}$

Where:

• me​ is the electron mass
• e is the elementary charge
• ϵ0​ is the vacuum permittivity
• ℏ is the reduced Planck constant
• n is the principal quantum number

Quantum Electrodynamics (QED): In QED, α quantifies the strength of the electromagnetic interaction between charged particles. It is in the calculation for the probability of photon emission and absorption by electrons, and in the Feynman diagrams that represent these interactions.

Fine Structure Splitting: The fine structure constant is key to calculating the fine structure splitting in atomic spectra, showing that the splitting of energy levels is proportional to the square of α.:

$\Delta E_{\text{fine}} \approx \alpha^2 E_n$

Sommerfeld Expansion: In solid-state physics, the Sommerfeld expansion, which explains the behavior of electrons in metals at low temperatures, uses α. It determines the electronic heat capacity and density of states.

Anomalous Magnetic Dipole Moment: In particle physics, α is in the formula for the anomalous magnetic dipole moment of the electron. This deviation from the Dirac theory’s prediction is a quantum mechanical effect explained by the interaction of the electron with a cloud of virtual particles. The calculation involves higher-order Feynman diagrams where α plays a crucial role.

Chemistry and Molecular Physics: In chemistry, especially in the study of molecular orbitals and chemical bonding, the fine structure constant helps determine the relative strengths of various types of bonds and interactions between molecules.

### References

• Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-08388-9.
• Sommerfeld, Arnold (1916). “Zur Quantentheorie der Spektrallinien“. Annalen der Physik. 4 (51): 51–52.
• Yu, C.; Zhong, W.; et al. (2019). “Atom‐interferometry measurement of the fine structure constant”. Annalen der Physik. 531 (5): 1800346. doi:10.1002/andp.201800346