Blackbody radiation spectrum

[David newman]

Homework Statement

I'm trying to calculate the optimum band gap for a solar cell for an essay I'm writing, but I am missing a step somewhere along the way.

A solar cell has an energy transition E.
Any photons with incident energy hv<E are not absorbed.
Any photon with energy hv>=E are absorbed but only energy E is retained*
Therefore a balance must be struck between absorbing many photons but gaining little energy from them (high current) and absorbing few photons but gaining a lot of energy from each (high voltage). (where Power=voltage*current maximum power at asypmtote of this graph)

The energy comes from the sun - which can be considered a black body radiator at ~6000K

It should therefore be relatively easy to calculate the peak energy absorbance (which turns out to be 1.4eV or ~900nm just outside the visible spectrum). My plan to calculate this was to find the blackbody radiation - energy as a function of population (i.e. the energy on the x-axis and the normalised popultation on the y-axis) I was pretty sure i'd seen this before and it was Boltzmann like in distribution. From this you could integrate that between E and oo and multiply the resulting function by E to get a function of power output with gap energy, then find a maximum.

All i have been able to find on BBR is wavelength as a function of power density by Plancks law. Linky
There must be:
1) an alternative formulation to give population
2) a simple way to convert this into the information i want
It's been eluding me for hours now though - is anybody able to help?



*the rest is lost, my explanation for this is that a photon>E promotes the electron to a high vibrational state within the conduction band. The electron then falls down the energy levels in the conduction band as you would expect in florescence. If this is wrong please tell me!

TL;DR
I want a function of BBR energy compared with population density, but can only find BBR wavelength compared with power density and the convesrion between the two has got me running in circles.

 

[NateD]

Homework EquationsPlancks law for BBR:P(λ,T)=2πhc^2/λ^5*(exp(hc/λkT)-1)^-1The Attempt at a SolutionI'm not sure how to proceed. I have tried to calculate the population density from the Plancks law equation but I have been unsuccessful.