LOG#169. A two level problem.

This post is the solution of the following problem: pick an atom with two energy levels. They have a transition wavelength of 580nm. At room temperature $5\cdot 10^{20}atoms$ are in the lower state.

1) How many atoms are in the upper state, if we have thermal equilibrium?

2) Suppose instead we had $9\cdot 10^{20}$ atoms pumped into the upper state, with $5\cdot 10^{20}$ atoms in the lower state. This is a non-equilibrium state. How much energy (in Joules) could be released in a single pulse of light if we restored the equilibrium?

Data: Boltzmann factor is $\exp\left(-E/k_BT \right)$ and $k_B=1\mbox{.}38\cdot 10^{-23}J/K$

Solution to 1). If $N_b$ is the higher energy state and $N_a$ is the lower energy state, then the energy difference between these two atomic states can be calculated using the transition wavelength (580nm), i.e.,

$E_b-E_a=hf=\dfrac{hc}{\lambda}=\Delta E$

$\Delta E=3\mbox{.}43\cdot 10^{-19}J$

But

$k_BT=1\mbox{.}38\cdot 10^{-23}J/K\cdot 300K=4\mbox{.}10^{-21}J$

Now, according to the Boltzmann distribution, the population in any state is given by

$N_i=\exp\left(E_i/k_BT\right)$

Therefore, the ration of $N_a$ and $N_b$ in the thermal equilibrium should be

$\dfrac{N_b(eq)}{N_a(eq)}=e^{-\frac{\Delta E}{k_BT}}=1\mbox{.}1\cdot 10^{-36}$

and thus

$N_b(eq)=N_a(eq)\cdot 1\mbox{.}1\cdot 10^{-36}$

Using the giving value of $N_a$ , we get

$N_b(eq)=5\mbox{.}5\cdot 10^{-16}$

and that is a very tiny number of atoms. This is showing to us that the energy gap between the given atomic states at room temperature is so large that almost all the electrons “choose” to stay in the lower state and hardly we will find electrons in the upper state.

Solution to 2). When we pump atoms into the upper state, we create a non-equilibrium atomic state. Energy will be released up until the equilibrium is restored (in fact, this is the working principle of the laser, in a simplified fashion!). We obtain now

$N_a(non-eq)=5\cdot 10^{20}$