In 1919 sommerfeld presented a modified and elaborated version of Bohr’s atomic model. The fine structure of the hydrogen emission lines of the structure is of fine structure.
To explain this, Sommerfeld suggested that it is not necessary that all the electrons of the same orbit have the same energy, the energies of some electrons of the same orbit may be different from the energies of the other electrons.
In other words, the main energy levels are divided into sub-energy levels. It is not necessary that all the electrons of the same orbit rotate on the same circular path, rather some of them can move on the elliptical paths.
The nucleus of an atom is at one of the two foci of elliptical orbitals.
The value of the angular momentum of an electron moving in a circular orbit is nh/2π where n is a whole number.
n is the main energy level of an electron or the number of orbits of the electron. The angular momentum of an electron moving in an elliptical orbit is kh/2π where k is a whole number.
The value of k can be 1, 2, 3, 4…. etc. The value of k represents the sub energy level of the electron.
The value of k depends on the value of n. For any one value of n, there are n values of k. These values of k are 1 2 3 …… n. If n = 1 then k = 1. n = 2 then k = 1, 2, n = 3 then k = 1, 2, 3. n = 4 then k = 1, 2, 3, 4 etc. When the values of k and n are equal, the orbitals of the electron will be circular i.e. the length of the long axis will be equal to the length of the short axis. Therefore
n/k = length of the long axis/length of the short axis
Sommerfeld’s atomic model successfully explains the finer structure of the microstructure of the lines obtained in the emission spectrum of hydrogen.
But this model, like Bohr’s atomic model, fails to explain the emission spectrum of atoms or ions with more than one electron and the Zeeman Effect.
Additionally, electrons have been considered as particles in Sommerfeld’s atomic model. And their speed and place are completely fixed. After the discovery of electron’s dual nature of electrons and the recognition of Heisenberg’s principle of uncertainty, it becomes clear that Somerfield’s atomic model is faulty.