The Sommerfeld atomic model is an extension of the Bohr atomic model. It was developed in 1916 by the German physicist Arnold Sommerfeld, in collaboration with Peter Debye.
Sommerfeld incorporated concepts from Albert Einstein's theory of relativity, as he discovered that in certain atoms, electrons reached speeds close to the speed of light.
Main modifications of the Sommerfeld model with respect to the Bohr model
- Elliptical orbits: Unlike the Bohr model, where electrons only describe circular orbits, Sommerfeld showed that they can also move in elliptical orbits around the atomic nucleus.
- Energy sublevels: From the second energy level onwards, there are sublevels within the same level, which explains the fine structure of atomic spectra.
- Relativistic corrections: Sommerfeld included relativistic corrections for electrons traveling at speeds comparable to that of light.
- Introduction of the azimuthal quantum number (ℓ): This new quantum number determines the shape of the orbitals and the angular momentum of the electron.
Limitations of the Bohr atomic model
The Bohr model explained the hydrogen spectrum with great precision, but it had problems when applied to atoms with more than one electron. In these cases, electrons in the same energy level could have different energies, which did not fully match the spectra observed experimentally.
For atoms such as hydrogen and the He+ ion, the energy of the layers was equal, but in atoms with several electrons, additional energy levels appeared that generated a greater number of spectral lines.
Sommerfeld's solution
To solve these problems, Sommerfeld proposed the existence of sublevels within an energy level, which allowed for differences in electron energies within a principal level. In addition, his relativistic calculations showed that some electrons reached speeds close to the speed of light, which required adjustments to quantum theory.
Sommerfeld's model introduced two fundamental modifications:
- Consideration of relativistic velocities in the movement of electrons.
- Inclusion of elliptical orbits along with circular ones, better explaining the fine structure of the spectra.
These improvements led to the introduction of new quantum numbers:
- Principal quantum number (n): Determines the energy level.
- Azimuthal quantum number (ℓ): Describes the shape of the orbit.
- Radial quantum number (n'): Related to angular momentum.
- Lateral quantum number (k): Describes the angular momentum of the electron in hydrogen.
Wilson-Sommerfeld formula
Sommerfeld also introduced the Wilson–Sommerfeld formula , a fundamental mathematical expression for the quantization of atomic orbits:
where:
- p is the momentum of the electron.
- dq represents the differential of the generic coordinate function.
- n is a natural quantum number.
- h is Planck's constant.
This equation placed an additional constraint on the quantization of atomic orbits, refining the Bohr model.
Impact of the Sommerfeld model
Sommerfeld's atomic model represented a major advance in the understanding of atomic structure and the emission spectra of atoms. Although it was later superseded by the quantum mechanical model based on the Schrödinger equation, his concepts laid the foundation for the modern theory of atomic orbitals.
In summary, the Sommerfeld model allowed:
- Explain the fine structure of atomic spectra.
- Introduce relativistic corrections into high-speed electrons.
- Extending quantum theory with new quantum numbers.
- Serve as a bridge between the Bohr model and modern quantum mechanics.
Thanks to these contributions, Sommerfeld's model was a crucial step in the evolution of atomic and quantum physics.