## Introduction:

## To explain the structure of an atom several theories have been proposed. Those are

## 1. J.J.Thomson’s Plum Pudding model

## 2. Rutherford’s Nuclear model

## 3. Bohr’s model

## 4. Sommerfeld’s relativistic model

## 5. Vector atom model.

## Drawbacks of Bohr’s Theory:

## Bohr Theory was able to explain successfully the spectral lines of the neutral hydrogen atom and singly ionized helium atom, etc. by introducing only principal quantum number n. However

## 1. This theory fails to explain the spectra of atoms more complex than hydrogen.

## 2. The theory does not give any information about the distribution and arrangement of electrons in the atom.

## 3. It does not explain the variation in the intensity of the spectral lines of an atom.

## 4. This theory fails to explain the fine structure of spectral lines i.e. one line is composed of several lines.

## 5. This theory cannot be used for the explanation of chemical bonding.

## 6. This theory fails to explain the Zeeman Effect. i.e. splitting of spectral lines in the presence of a magnetic field.

## 7. This theory fails to explain the Stark effect i.e. splitting of spectral lines in the presence of an electric field.

## Sommerfeld Relativistic model:

## In order to overcome the shortcomings of Bohr’s atomic model, Sommerfeld introducing the idea of the motion of electron in elliptical orbits and taking into consideration the variation of mass with velocity. The improved model is known as Sommerfeld’s relativistic model.

## Elliptical Orbits:

## According to Sommerfeld the electron moving around the nucleus in elliptical orbits in addition to circular orbits. In the elliptical orbit the position of the electron at any time may be fixed by two coordinates r and Φ where r is the radius vector and Φ is the angle which the radius vector makes with the major axis of the ellipse, i.e azimuthal angle.

## In order to quantize the elliptical orbits, Sommerfeld introduced the two quantum numbers nr and nΦ where nr is radial quantum number and nΦ is angular or azimuthal quantum number. And Sommerfeld introduced an equation

## nr + nΦ = n.nr

## Here n is the principal quantum number which shows the length of the semi-major axis and nΦ shows the length of the semi-minor axis.

## For n = 1 state, there are two possibilities i.e. nΦ = 1 and nΦ = 0

## If nΦ = 0 then semi minor axis is zero, the ellipse converted into a straight line and the electrons have to pass through the nucleus twice during every period. This type of motion is not possible; hence nΦ may not be zero. Hence lowest possible value of nΦ is one. Thus corresponding to n = 1, the first orbit is a circle identical to Bohr orbit.

## For n = 2 state there are two possibilities

## those are nΦ = 2, a circle and nΦ = 1, an ellipse.

## Similarly for n = 3 there are three possibilities,

## those are nΦ = 3, a circle and

## nΦ = 2, an ellipse

## nΦ = 1, an ellipse.

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