How did Arnold Sommerfeld modify the Bohr model?

German physicist Arnold Sommerfeld modified the Bohr model to better account for the fine structure observed in atomic spectral lines.12

His primary modifications included:

• Elliptical Orbits: While Bohr originally proposed only circular orbits, Sommerfeld suggested that electrons also move in elliptical orbits.23
• New Quantum Numbers: To describe these orbits, he introduced the azimuthal quantum number to specify the elongation of the ellipse. He also introduced the magnetic quantum number to account for the orientation of the orbits.23
• Relativistic Effects: The model was further augmented by incorporating the effects of relativity on the motion of the electrons.3

By quantizing the shapes and orientations of these orbits, Sommerfeld successfully introduced additional energy levels that corresponded to the closely spaced fine lines seen in high-resolution spectroscopy.12