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Vector Model of Atom; Coupling of Orbital & Spin Angular Momentum

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…. 

The total angular momentum of an atom results from the combination of angular & spin angular momentum of its electrons. 

Since angular momentum is a vector quantity. We can represent the total angular momentum by a means of a vector obtained by addition of orbitals and spin angular momentum vectors. This leads to Vector Model of Atom. 

Let us consider an atom whose total angular momentum is provided by a single electron. The magnitude of orbital angular momentum is given by :
                        L = √l(l+1) .h/ 2π

and its z-component is :

                         Lz = ml. h/2π

where,  l = orbital quantum number &
            ml = magnetic orbital quantum number.
ml can have (2l+1) values which are -l to +l with 0.

Similarly the magnitude of spin angular momentum :

                  S = √s (s + 1). h/2π

Its z-component is :
                  Sz = ms. h/2π

where,  s = spin quantum number &
ms = spin magnetic quantum number.
ms can have two values ±1/2.

Total angular momentum of one electron :

As J is the sum of angular momentum L and S which are quantised. 
Thus J must be quantised 

                      J = √j (j + 1 ).h /2π

wher is called inner quantum number. Its z-component is :

                     Jz = mj.  h/ 2π

where mj is called magnetic quantum numbers which has values from -j to +j. 
i.e.  mj = -j…….. 0…………+j

Therefore, we have 
        j = l ± s
       mj = ml ± ms
In case of one electron atom, there are only two relative orientations possible corresponding to  
      j = l + s, so that J > L
    j = l – s,  so that J < L

The angular momentum of atomic electron L and S interact magnetically which is known as spin orbital interaction. 

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