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Sommerfeld Extension of Bohr’s Theory

Hello friends welcome once again on our blog.. today I’m going to explain Sommerfeld Extension of Bohr’s Theory ,which is also an important topic of Atomic Physics. So let us start…. 

According to Bohr’s theory, an atom has discrete energy levels which only cam be occupied by electrons.

Each level is characterized by an integer ‘n’ called principal quantum number.

The energy of nth level is given by :

,where ‘R’ is Rydberg constant. 
Whenever an electron jumps from one level to another, electromagnetic radiations of distinict frequencis emitted:

According to this theory, each line of Hydrogen Spectrum must be a single line. i.e. it must have single frequency or wavelength.

It was observed by Michelson & others that the Hydrogen lines are not single but they are in small number.

Each line consists of a small number of close components of slightly different frequency.This is called fine structure. 

Sommerfeld extented Bohr’s theory to explain the fine structure by assuming the existence of elliptical orbit rather than circular for the electron with one focus at the nucleus of the atom.
Now an electron in an elliptical orbit has two degree of freedom.

i) Radial Distance (r)
ii) Azimuthal angle(Φ)
Sommerfeld postulated that

each of these degree of freedom ( r, θ) must be quantised separately.

If Pθ and Pr be the angular & radial momenta of the electron then according to Sommerfeld quantisation rule,  we have :

where K & nr are integers called Azimuthal & radial quantum numbers. 

By integrating over a complete revolution it can be proved that :

where,  ‘e’ is the eccentricityof the ellipse. Let us put :

where n is called total quantum number of electron.

If ‘a’ and ‘b’ be the semi-major and minor axis of the ellipse then,

This is the condition of quantisation for the orbit. 

Thus only these elliptical orbit are permitted for the electron for which the ratio of minor to the major axis is equal to the ratio of Azimuthal ( k)  to the total quantum number(n).

Total Energy of electron :

Now,  the angular momentum is given by :-       

Now, the linear momentum is given by :-      
  Multiply & divide by m2( m square)  we get : 
Now, we’ve find polar equation of ellipse. 

Now,  Subsituting the value of r and dr/dθ with the help of equation (vii)  in equation  (vi) we get:
For an isolated system Pθ is                                 constant. Hence, 

and also we have :-                                                

Subsituting,these values in equation (8) we get:
This relationship is exactly equal to the relation for energy given by Bohr’s in circular orbit. Now for a given value of n (n= k + nr), k can take place different values,(k= 1,2,3,.. ) 
This means that for a given n there are n orbits of different eccentricities. Now according to Bohr’s conditions we have :
This is Sommerfeld’s extension of Bohr’s theory…..

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