# Deriving Expression for Population Fraction (Statistical Mechanics)

1. Dec 8, 2012

### Collisionman

1. The problem statement, all variables and given/known data

A quasi-3 level solid-state laser gain medium consists of a ground state manifold containing two energy levels within which a single electron can be promoted, with the second energy 10meV above that of the lowest level.

A. Where the gain medium is not optically pumped, derive the relationship for the population fraction held in the ground energy level.

B. Given the value for the ground state population fraction where the coolant holds the gain medium to a temperature of 25°C.

2. Relevant equations

Population Fraction: $\frac{n_{i}}{N} =$$\frac{g_{i}e^{-\frac{1}{K_{B}T}E_{i}}}{\sum_{i}g_{i}e^{-\frac{1}{K_{B}T}E_{i}}}$

3. The attempt at a solution

For the ground energy level, $E_{0}=0$ and, for the second energy level, $E_{1}=10meV$. So, for the ground energy level, the population fraction is;

$\frac{n_{i}}{N} = \frac{g_{0}e^{0}}{g_{0}e^{0}+g_{1}e^{-\frac{10meV}{K_{B}T}}}$

I'm unsure of what the degeneracies are for $E_{0}$ and $E_{1}$. Would I be right in assuming they're just $g_{0}=g_{1}=1$?

If I knew this I'd have no problem finishing the rest of the question which is just a matter of substitution.

Any help appreciated and thanks in advance.

Regards,

Collisionman

2. Dec 11, 2012

### clamtrox

Yes.

Note that you don't really care about all possible degeneracies. Like for example, you don't care at all that electrons have spin 1/2, as those factors cancel out. All you care about is the relative number of states corresponding to each energy.