Sommerfeld atomic model is an extension of Bohr's atomic model. The new model was developed by the German physicist Arnold Sommerfeld and his assistant Peter Debye in 1916. The model was made with the help of Albert Einstein's theory of relativity. Sommerfeld discovered that electrons in certain atoms reached speeds close to the speed of light.
Sommerfield introduced some to Bohr model in order to explain the observed fine structure of spectral lines:

Electrons move around the atom’s nucleus in circular or elliptical orbits.

There are one (or more) sublevels at the same level after the second energy level.

The electron is a tiny electric current.
The current model of the atom, known as the atomic orbital model, could not have been formulated without the earlier models derived from Bohr's hypotheses.
What were the limitations of Bohr's atomic model?
Bohr's atomic model was seamless when it came to the hydrogen atom. But, on the other hand, when it came to atoms of other chemical elements, the electrons of the same energy level had a different energy.
It does not affect the spectrum concerning the hydrogen atom and the He + ion because both types of shells are energetically equal. However, for multielectron atoms, the number of possible energy levels increases. Therefore, it manifests itself in a more significant number of spectral lines in the spectrum.
What was Sommerfeld's solution to the limitations of the Bohr model?
Regarding these cracks, Sommerfeld postulated that within the same energy level, there were sublevels with slightly different energies. Furthermore, from theoretical calculations, Sommerfeld had found that in certain atoms, the speeds of the electrons reached an appreciable fraction of the speed of light. Sommerfeld also performed these calculations for relativistic electrons.
Sommerfeld's atomic model introduced two essential modifications:

Relativistic velocities.

In atoms, electrons move in circular and elliptical orbits, unlike Niels Bohr's model, in which electrons only rotate in circular orbits.
The eccentricity of the orbit gave rise to a new quantum number that determines the shape of the orbitals: the azimuthal quantum number.
During elaboration, a principal quantum number n = n '+ k. The secondary quantum number n 'determines the angular (radial) momentum and eccentricity of the ellipse. For n '= 0, circular orbits arise. Finally, the lateral quantum number k describes the hydrogen electron’s angular momentum.
What is the WilsonSommerfeld formula?
The WilsonSommerfeld formula represented an essential element for defining the BohrSommerfeld model.
In this model, electrons were supposed to travel around the nucleus in elliptical orbits instead of Bohr's original model in that they moved in circular orbits.
The BohrSommerfeld model contemplated one more restriction on the quantization of the angular momentum of the electron with an additional quantization restriction of the radius determined through the "WilsonSommerfeld quantization restriction formula":
Where:

“p” is the moment

“dq” represents the differential of the generic coordinate function “q(t)

“n” are natural numbers

“h” is Planck's constant.