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Guys I hope you like my previous Post i.e. 
Space Quantisation of an Atom
& today I am going to discuss another important topic Electron spin. Without wasting your time let us start… 

The Bohr’s-Sommerfeld quantum theory of elliptical orbit with relativistic corrections explain the fine structure of H-like atom. It suffered from two major drawbacks:-

i) It could not explain fine structure observed in the spectral lines of atom other than Hydrogen. e.g. the spectral lines of alkali atoms are doublets having two close fine structure components. 

ii) The simple quantum theory failed to explain anomalous Zeeman effect, i.e. the splitting of atomic spectral lines into 4, 6 or more components, when atom is placed in external magnetic field. 
To overcome these two drawbacks of the theory, Goudsmit & Uhlenbeck proposed in 1925 that :
“an electron is a charged sphere spinning about its own axes having an intrinsic angular momentum & intrinsic magnetic moment “

These are called spin angular momentum (S) and spin magnetic moment (μs)

We donot take s = -1/2 , otherwise S is imaginary which is not possible.

The component of along magnetic field which is along z-axis. 

where,  ms is spin-magnetic quantum number which takes 2s + 1 = 2× 1/2 +1 = 2 values.

i.e. two values are +1/2 and -1/2.

The gyromagnetic ratio for electron spin is twice the corresponding ratio for electron orbital motion which is e/2m.

Here,  “-ve” sign indicates that μs  & are opposite in direction. 

In magnitude :
I hope you understand it very well, still any query you can always write to us in comment section.