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A number of models have been proposed by researchers, starting from time of Newton to Sommerfeld. However, for the current discussion, Bohr’s model of the hydrogen atom and its modification by Sommerfeld will be worth mentioning.

Neils Bohr, proposed his atomic model in 1913 which could successfully explain the observed hydrogen and hydrogen like spectra. Followings are the postulates given by Bohr.

i.            The electron in an atom revolves in a fixed circular orbit about the nucleus, with the essential centripetal force supplied by the Coulomb interaction between the electron and the nucleus

Here, r is the radius of allowed orbit, v is the velocity of electron, Z is the atom
number, m is the mass of electron and ω is the corresponding angular velocity.

i i.            Out of a large number of possible orbits the electron will exist only in those for which the angular momentum of electron is integral multiple of h/2π or h, where h is Plank’s constant.

mvr = mωr2 = nh                                                               … (2)

iii.            The electron, while moving in an allowed orbit does not radiate electromagnetic energy. The electromagnetic energy will be either emitted or absorbed only when electron makes a transition from one allowed stationary state to another frequency of emitted/ absorbed radiation is given by

where v is the frequency of emitted or absorbed radiation, Ei and Ef are the energies of initial and final orbits.

One can obtain expression for radius of allowed orbits by solving equations (1) and (2),

The equation (4) states that the radii of successive orbits are proportional to n2 and have discrete values, which means that they are quantized.

The energy of electron in nth Bohr orbit is given by

Figure 10.1 Few spectral lines of Hydrogen (Z-1) atom arising from transition between the levels

As such the electron bound to nucleus can have any one of discrete set of allowed energy levels.

The Bohr theory could explain the experimental results, of hydrogen and hydrogen like atoms. However, this

  i.            Bohr atomic model can predict only the transition frequencies, but tells nothing about the polarization and intensity of spectral lines.

 ii.            According to the Bohr’s theory, the frequency of the spectral lines in hydrogen are well defined. However, it has been observed that many of the lines in the Balmer and other series are not single lines at all. The two components in Hα  lines with separation is ~ 0.13 A have been reported. This is known as fine structure of spectral lines. The Bohr model could not explain the fine structure.

Extension of the Bohr’s Theory: Sommerfeld Model: In 1916, Sommerfeld attempted to explain the fine structure. He made following assumptions for his model:

(i)      In general, the path of an electron about the nucleus is ellipse. The nucleus is situated at one of the foci of the elliptical path.

(ii)   In an elliptical orbit, the velocity of electron varies substantially at different points of the path, which causes relativistic variation in the mass of the moving electron.

Hence, this model is sometimes, also referred as relativistic atomic model. 

Shortcomings of Bohr Sommerfeld Atomic Models: The above discussed models were quite successful for Hydrogen and Hydrogen like atoms. However, these theories could not explain the following points:

(i)                           The sommerfeld theory could not predict the correct no. of lines in fine structure.

(ii)                        These theories could not explain the complex spectra of alkali metals.

(iii)                      The models were silent about the distribution of electrons in ‘orbits around the nucleus.

(iv)                      Without giving the proper logic, the quantum numbers were introduced.

(v)                         The effect of magnetic and electric fields on spectral lines could not be explained.

In order to explain the complex spectra of atoms and their relation to the atomic structure, a new atomic model, commonly known as Vector Atom Model has been introduced.