What is the Cross Section of a Hydrogen Atom in Thermal Equilibrium at 6000K?

[JohnSimpson]

Homework Statement

I've been asked to find the cross section of a hydrogen atom in thermal equalibrum at 6000K for a photon which induces a transition from the ground state to the first excited state. The density of states for n=2 is 4x the density of states for n=1

i.e. g(E2) = 4*g(E1)

The lifetime of the n=2 state is 1.6 x 10^-9 s.

Homework Equations

The Attempt at a Solution

The incident photon has an energy E = hf = hc/lambda

Since the system is in thermal equlibrum, it has an average energy = kT. I am not sure what exact this is the average energy of though, the atoms? the photons? Everything?

From what I understand, the cross section is some kind of area of interaction for the process that will somehow depend on the energy of the incident photon. With this in mind, I wrote down the radius of the hydrogen atom for n =1

r1 = (epsilon)h^2 / pi * e^2 * m_e

I'm still not entirely comfortable with what the density of states represents for n=1 and n=2 respectively. I know that it sort of represents a "Price" to put an electron at that level, but I might be incorrect in saything that.

It would be very helpful if someone could clear up my misconceptions and nudge me in the right direction, thank you

 

[JohnSimpson]

Update: I've managed to write down the number of particles in the n=1 state at any time, which is given by

g(E1)f(E1) where f(E1) is the maxwell-boltzmann probability of finding a hydrogen atom with energy E1 corresponding to ground state, and similarly the number of n=2 hydrogen atoms is

4g(E1)f(E2)

 

[JohnSimpson]

Anyone able to point me in the right direction?