Which model extended Bohr's atomic model by allowing elliptical electron orbits?

The Bohr–Sommerfeld model extended Bohr’s atomic model by allowing elliptical electron orbits.

Niels Bohr’s 1913 model treated electrons as occupying quantized circular orbits around the nucleus. Arnold Sommerfeld expanded that old quantum theory by permitting elliptical orbits, much as planets can follow ellipses under classical mechanics, while retaining quantum restrictions on the allowed motion.

The extension introduced an additional radial quantization condition and helped explain details that Bohr’s simpler model could not, including aspects of hydrogen’s fine structure. Sommerfeld also incorporated relativistic corrections and helped develop the idea that orbital orientation could be quantized.

The model was historically important but ultimately incomplete. It struggled with more complicated atoms, could not properly incorporate electron spin, and was superseded by modern quantum mechanics in the 1920s. The electron-cloud model is not a later version of elliptical paths: it describes orbitals as probability distributions rather than literal tracks.