Absorbtion, Spontaneous Emission & Stimulated Emission of Radiation

An atom has a number of possible quantized energy states defined by principal quantum number “n”.

If it is initially in a lower state “1”, it can rise to higher states ‘2’ by absorbing a quanta of radiation (photon) of frequency “ν” given by:     
      

where, E1 and E2 are energies of atom in the state “1” and “2”. This is absorption of radiation.This is a stimulated process .

The absorbed photon is the stimulating photon.

The probable rate of occurance of this absorption transition 1 –>2 depends upon the property of states 1 and 2 and is proportional  to the energy density  u(ν) of the radiation of frequency ν incident on the atom.

               

          

The proportionality constant B12 is known as Einstein coefficient of absorption of radiation.
SPONTANEOUS EMISSION :
Let us now consider an atom initially in higher excited state “Z”.
Observation shows that its life time in higher state is very small (10-8s).
After this ,it jumps to the lower energy side “1” emitting a photon of frequency ν. This is spontaneous emission of radiation.
If there is an assembly of atoms , the radiation emitted spontaneously by each atom has a random direction and a random phases and is therefore incoherent from atom to atom.






The probability of spontaneous emission 2–>1 is determined by the properties of states 2 and 1.
Einstein denoted this probability by A21 , which is known as Einstein coefficient of spontaneous emission of radiation.
STIMULATED EMISSION (INDUCED EMISSION)
According to Einstein an atom in excited energy state make under the influence of electromagnetic field of a photon of frequency ν incident upon it jump towards lower energy state , emitting an additional photon of same frequency ν.
Thus now two photons , one  original and the other emitted move on. This is stimulated emission of radiation.

  “The direction of propogation ,energy, phases of emitted photon is exactly same as that of incident stimulating photon.”

In other words ,the stimulated radiation is completely coherent with the stimulated radiation.
As a result of this process ,radiation passing through assembly of atom is amplified.

The probability of simulated emission transition 2→1 is proportional to the energy density u(ν)

 of stimulating radiation and is written as:

                           

where, B21 is Einstein coefficient of stimulated emission of radiation.

The total probability for atom is state 2 to drop to the  lower state 1 is
                       


Relation between spontaneous and stimulated Emission Probabilities:
Let us consider an assembly of atom in thermal equilibrium. At temperature T with radiation of frequency v and energy density u(v)dv .
Let N1 and N2 be the number of atoms in states 1 and 2 respectively.
 from eq. (1) we have:

                          



For equilibrium ,the objective and emission must occur equally. Thus, 
              

 

                         



Enstein proved thermodynamically that:
 “Probability of absorption (stimulated) is equal to probability of stimulated emission”
Einstein proved thermodynamically that:


According to Boltzmann law the distribution of atoms among different energy states is given by:
                             
            





where,  No = Number of atoms in ground state.
Er = Energy of rth state.
Nr = Number of atoms in rth state.
Now,




This is the formula for the ratio between spontaneous  and stimulated emission.

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