Understanding Energy States and Population Distribution in Thermal Equilibrium"

[amph1bius]

The Question

Consider two energy states, E2 = 2.0 eV and E1 = 1.0 eV. Assume that there are 1.0 x 10^16 electrons/cm^3 in E2 and 1.0 x 10^15 electrons/cm^3 in E1. What temperature is required to create this population distribution in thermal equilibrium?

How do you define the population distribution?

So far, I have a formula for equating number of electrons to the density of states * the fermi distribution.

How do I put the given information into a value for the "number of electrons"?

Thanks in advance

 

[DrClaude]

The probability of being in a given state is given by
$$
P(s) = \frac{e^{-E(s) / k T}}{Z}
$$
therefore one finds that the ratio of probabilities of the two states is
$$
\frac{P(2)}{P(1)} = \frac{e^{-E2 / k T}}{e^{-E1 / k T}}
$$
which one can then solve for $T$ using the given numbers.

Note: the result obtained in this particular case is quite surprising!