Bohr was able to calculate the radii as well energies of the stationary orbit around the nucleus in an atom and those calculated values were found to be in a good agreement with the experimental values. He also gave the Hydrogen ion spectrum. For these reasons, his theory was widely accepted throughout the world. But a few years later, the use of high resolving power spectroscopes revealed some very fine spectral lines which Bohr was not able to explain. So from this point only, Sommerfeld extended Bohr Theory and gave his postulates.
According to him, the stationary orbits in which electrons are revolving around the nucleus in the atom are not circular but elliptical in shape. It is due to the influence of the centrally located nucleus. The electron revolves in elliptical path with nucleus at one of its foci. So there will be a major and a minor axis of the path. He said that with the broadening of the orbit, the lengths of the two axis approach to equal value and ultimately become equal i.e. the path become circular. So we can say the circular path is just one special case elliptical path.
As electrons travel in elliptical path, it will have an angular momentum and this angular momentum must be quantized according to the quantum theory of radiations. Bohr gave that angular momentum as m=nh/2Ω but Sommerfeld used another integer k instead of n. k is an integer known as azimuthal quantum number. n used by Bohr and k used by Sommerfeld are related as: -
n/k = length of major axis/length of minor axis
With increase in value of k, the path becomes more and more elliptical and eccentric. When k=n, the path becomes circular.
Bohr was not able to explain the reason for the fine spectral lines visible by high revolving power spectroscopes but Sommerfeld explained the reason for the same. He said that the energy of the stationary orbit depends not only on n but on k to some extent as well. So when a transition of electron from a higher level to a lower level occurs, it would be different from what proposed by Bohr as there may be more than one values of k. In this way Sommerfeld was able to explain the reason behind those fine spectral lines. Even the frequencies of some of those fine spectral lines came to be in well agreement with the frequencies by Sommerfeld.
Sommerfeld theory is also sometimes referred to as Bohr-Sommerfeld theory.