Module 7 : Laser- I
Lecture   : Introduction - Basics of stimulated emission
   
2.2 Boltzmann Statistics:
 

Consider a two level system. Let there be $ N_1$number of atoms per unit volume in the energy level $ E_1$and $ N_2$per unit volume in a higher energy level $ E_2$. Let $ E_2-E_1= h\nu$. Total population density in the system is
                                      $\displaystyle N = N_1 + N_2$
 

If the atoms are in thermal equilibrium with the surrounding at a temperature $ T$, the relative population in the two levels are given by Boltzmann distribution

$\displaystyle \frac{N_2}{N_1} = e^{-h\nu/kT}$

This equation shows that as the temperature increases, the population of excited states increase. However, the population of an excited state always lies lower than the population of the ground state, under equilibrium condition. For large energy gaps such that $ h\nu\gg kT$, the ratio above is close to zero so that very few of the atoms are in the upper energy state.

  Exercise :
  The ground state and the first excited state of Ruby are separated by 1.8 eV. Calculate the ratio of the number of atoms in the excited state to that in the ground state. (Ans. $ e^{-70}\simeq 10^{-30}$)

When two or more states have the same energy, the states are said to be degenerate . The number of states at the same energy level is called the multiplicity of the energy level. As all states having the same energy have the same population, we have
                                             $\displaystyle \frac{N_2}{N_1} = \frac{g_2}{g_1}^{-h\nu/kT}$
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